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Lysis dynamics and membrane oligomerizationpathways for Cytolysin A (ClyA) pore-forming toxin
M S Vaidyanathandaggera Pradeep Sathyanarayanabc Prabal K Maitid
Sandhya S Visweswariahbc and K G Ayappaab
Pore-forming toxins are known for their ability to efficiently form transmembrane pores which eventually
leads to cell lysis The dynamics of lysis and underlying self-assembly or oligomerization pathways
leading to pore formation are incompletely understood In this manuscript the pore-forming kinetics and
lysis dynamics of Cytolysin-A (ClyA) toxins on red blood cells (RBCs) are quanti1047297ed and compared with
experimental lysis data Lysis experiments are carried out on a 1047297xed mass of RBCs under isotonic
conditions in phosphate-buffered saline for different initial toxin concentrations ranging from 294ndash147
nM Kinetic models which account for monomer binding conformation and oligomerization to form the
dodecameric ClyA pore complex are developed and lysis is assumed to occur when the number of
pores per RBC (np) exceeds a critical number npc By analysing the model in a sublytic regime ( np lt npc)
the number of pores per RBC to initiate lysis is found to lie between 392 and 768 for the sequential
oligomerization mechanism and between 5300 and 6300 for the non-sequential mechanism Rupture
rates which are 1047297rst order in the number of RBCs are seen to provide the best agreement with the lysis
experiments The time constants for pore formation are estimated to lie between 1 and 20 s and
monomer conformation time scales were found to be 2ndash4 times greater than the oligomerization times
Cell rupture takes places in 100s of seconds and occurs predominantly with a steady number of pores
ranging from 515 to 11 000 on the RBC surface for the sequential mechanism Both the sequential
irreversible and non-sequential kinetics provide similar predictions of the hemoglobin release dynamics
however the hemoglobin released as a function of the toxin concentration was accurately captured only
with the sequential model Each mechanism develops a distinct distribution of mers on the surface
providing a unique experimentally observable 1047297ngerprint to identify the underlying oligomerizationpathways Our study offers a method to quantify the extent and dynamics of lysis which is an important
aspect of developing novel drug and gene delivery strategies based on pore-forming toxins
1 Introduction
Pore-forming toxins (PFTs) are a class of proteins produced by a
wide variety of organisms including bacteria1 and humans2
They have the unique property of generating pores in the
membranes of target cells3ndash5 PFTs are classied on the basis of
the secondary structure (a or b) of the pore-forming region and
usually undergo a monomer to oligomer transition that is a
pre-requisite for pore formation67 Cytolysin A (ClyA HlyE or
SheA) from E coli is a well characterized a-PFT38 and crystal
structures are available for both the water-soluble monomeric9
and membrane-associated oligomeric forms10 The monomer
of ClyA possesses 5 a helices and a hydrophobic b sheet the
ldquob-tonguerdquo that is buried within the helices More recent
crystal structure data10
indicates that the toxin oligomerizes asa dodecamer A pathway for the transition from the monomer
to the membrane-bound protomer and nally to the dodeca-
meric pore complex has been proposed based on a comparison
of the monomeric and oligomeric crystal structures10 Large
conformational changes in the N-terminus and the b sheet
regions need to occur during the monomer to oligomer tran-
sition that appear to involve initial ipping out of the b-
tongue to bind to the lipid membrane followed by larger
translocation of the N-terminus into the lipid membrane ClyA
forms cation-selective pores which have an internal diameter
of 4ndash7 nm10
a Department of Chemical Engineering Indian Institute of Science Bangalore India
E-mail ayappachemengiiscernetin Fax +91 80 23608121 Tel +91 80 22932769
b Bioengineering Programme Indian Institute of Science Bangalore India E-mail
pradeepmrdgiiscernetin Fax +91 80 23600999 Tel +91 80 22932659
c Department of Molecular Reproduction Development and Genetics Indian Institute of
Science Bangalore India E-mail sandhyamrdgiiscernetin Fax +91 80
23600999 Tel +91 80 22932659
d Department of Physics and Center for Condensed Matter Theory Indian Institute of
Science Bangalore India E-mail maitiphysicsiiscernetin Fax +91 80
22932602 Tel +91 80 22932315
dagger Present address Department of Chemical Engineering University of Texas at
Austin Te xas USA
Cite this RSC Adv 2014 4 4930
Received 16th September 2013
Accepted 22nd October 2013
DOI 101039c3ra45159c
wwwrscorgadvances
4930 | RSC Adv 2014 4 4930ndash4942 This journal is copy The Royal Society of Chemistry 2014
RSC Advances
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7252019 c3ra45159c
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A variety of experimental techniques such as lytic experi-
ments gel electrophoresis site-directed mutagenesis and cryo-
electron microscopy have been used to unravel the mechanisms
of pore formation in ClyA 11ndash14 and in other widely studied PFTs
such as cholestrol-dependent toxins (CDCs)15 and S aureus a-
Hemolysin116 Despite the interest in unravelling structural and
mechanistic pathways for the action of PFTs10 the kinetics of
membrane oligomerization rates of pore formation and the
dynamics of ensuing lysis have not been the subject of muchquantitative investigation17 Pore formation kinetics and
dynamics of the release of self-quenching dye molecules from
liposomes have been quantied by Schwarz and co-workers1819
where the marker release dynamics is tted to either a single or
double exponential function with suitably motivated kinetic
models for pore formation Models which quantify the perme-
ation rates across bacterial membranes due to pore-forming
protegrin peptides20 and lysenin-induced permeation in giant
unilamellar vesicles21 have appeared in the recent literature
Recently an investigation of permeation rates due to a-Hemo-
lysin on liposomes using optical contrast microscopy and
micropipette experiments
22
reveal pore densities of about 100pores mm2
Quantifying the phenomenon of pore formation and lysis
is crucial for developing PFT-based druggene delivery thera-
pies and controlling pore formation in vesicle-based biore-
actors23 during the development of articial cells In this
study the lytic activity of ClyA is modeled based on experi-
ments carried out on red blood cells We formulate an
adsorption-kinetic model which incorporates monomer
binding conformational changes and sequential as well as
non-sequential oligomerization pathways to determine the
rate of hemoglobin released as a function of time and ClyA
concentration A rst order rupture model is used to quantify
the lysis dynamics Our model captures the experimentalhemoglobin data as a function of toxin concentration From
this observation we extract the critical number of pores per
RBC above which cell lysis occurs
2 Experimental procedure21 Expression and purication of His-tagged Cytolysin A
(ClyA)
pGS1111 plasmid containing the ClyA gene as a fusion with
glutathione S-transferase was obtained from Dr J Green
University of Sheffield UK The ClyA gene was subcloned from
pGS1111 into pPRO Ex-HTb using EcoRI and SalI to obtainpPROb ClyA containing an N-terminal hexahistidine tag E coli
BL21 endo cells transformed with pPROb ClyA were grown in
terric broth ClyA full length (ClyA FL) proteins were expressed
on induction with 500 mM isopropyl thiogalactopyranoside
Cells were lysed by sonication in buff er containing 100 mM
TrisndashHCl (pH 80) 5 mM b-mercaptoethanol 100 mM NaCl
1 mM benzamidine 2 mM phenylmethylsulfonyl uoride and
10 glycerol Centrifugation was carried out at 30 000 g and the
cell-free extract was interacted with nickelndashnitrilotriacetic acid
beads Beads were washed with buff er containing 100 mM Trisndash
HCl (pH 80) 5 mM b-mercaptoethanol 500 mM NaCl 20 mM
imidazole to remove nonspecic proteins on the beads His6
ClyA was eluted in buff er containing 100 mM TrisndashHCl (pH 80)
5 mM b-mercaptoethanol 100 mM NaCl 300 mM imidazole
10 glycerol Proteins were desalted in buff er (100 mM Trisndash
HCl (pH 80) 5 mM b-mercaptoethanol 100 mM NaCl and 10
glycerol)
His6 ClyA was treated with TEV protease to obtain tagless
protein 1 part of puried hexahistidine TEV protease was
taken per 30 parts by mass of ClyA and incubated overnight at 4 C TEV was separated by interacting further with Ni-
NTA beads Protein quantity was estimated by the Bradford
method24
22 Hemolysis assay
The hemolysis assay was carried out as described previously 12
Rabbit erythrocytes were washed and diluted 1 100 vv in PBS
(phosphate-buff ered saline pH 74) Aliquots of RBC suspension
were transferred to microcentrifuge tubes ClyA was added to
suitable aliquots of RBCs and incubated at 37 C in a shaking
incubator for 1 hour Lysis experiments were carried out for
ClyA concentrations ranging from 294ndash147 nM These corre-spond to 100ndash500 ng ml1 respectively since ClyA is a 34 kDa
monomer Unlysed cells and debris were sedimented by
centrifugation at 5000 rpm for 1 min Released hemoglobin in
the supernatant was quantied by spectrometric detection at
540 nm The numbers of cells remaining a er lysis were
counted in a hemocytometer
23 Turbidity assay
A suspension of rabbit erythrocytes (1 vv in phosphate-buff -
ered saline 1 ml) was treated with varying amounts of Cytolysin
A as indicated To assess turbidity 200 ml of the cell suspension
was transferred to a clear-bottomed 96-well plate and light scattering was measured at 620 nm The cells in the plate were
centrifuged at 3000 rpm for 2 min and the extent of haemolysis
was estimated by measuring the absorbance of the supernatant
at 570 nm Optical density measurements were carried out on a
Tecan Innite F50 microplate reader
3 Modeling31 Membrane binding and bulk toxin concentration
We develop a model to predict the hemoglobin release kinetics
of the RBCs as a function of initial toxin concentration The
series of steps that lead to pore formation are illustrated inFig 1 The model is developed in the mean eld framework
wherein all cells are assumed to be identical Diff usion is
assumed to be fast relative to membrane binding and oligo-
merization Assuming a protein diff usion coefficient of 1013
m2 s1 the diff usion time on the membrane is of the order of
15ndash50 ms for toxin concentrations ranging from 294ndash147 nM
Membrane binding is assumed to be irreversible and of similar
time scale to that of oligomerization The amount of hemo-
globin released due to cell lysis is signicantly larger (108
times) than that released from the pores of unlysed cells both
due to the size of the hemoglobin molecule25 (5ndash6 nm) as well as
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the small eff ective pore diameter available for transport The
inner pore diameter exposed to the cytosol is 4 nm in the crystal
structure and in a fully solvated environment the eff ective
diameter is expected to decrease further Further osmoticprotection assays of ClyA conclude that the eff ective pore sizes
range from 20ndash35 nm26 The solution is assumed to remain
isotonic as lysis proceeds since lysis did not occur in the
absence of toxin when RBCs were incubated in buff er solution
made up of fully lysed (sonicating 1 RBC (vv)) RBCs We have
assumed that the conformational change follows rst order
irreversible kinetics since the conformational step involves a
transition from a water-soluble monomer to a membrane-
inserted protomer via a series of conformational changes in the
regions around the b-tongue region of the monomer and the N-
terminus10 This is succeeded by a fast oligomerization step to
form the pore complex
The rate equation for the membrane-bound monomer
whose surface molar concentration is denoted as m is
dm
dt frac14 k aC m
ms m
Xnl frac141
pl
k dm k cm (1)
where k a is the adsorption rate constant k d is the desorption
rate constant ms is the saturated surface molar concentration
and the last term represents the rate at which the membrane-
bound monomer (m) undergoes a conformational change
to the membrane-bound protomer ( p1) with a rate constant
k c and pl is the surface molar concentration of the oligomer
containing l -mers If the bulk concentration of the toxin
monomer is constant eqn (1) is similar in form to the
LangmuirndashHinshelwood equation traditionally used to
describe the concentration of surface species undergoing
both adsorption and reaction
Since the initial toxin concentration (C in) in the aliquots is in
the range of 294ndash147 nM an additional balance is used to
describe the concentration change of toxins in solution This
yields
V sol
dC m
dt frac14
k dm k aC m
ms m
Xnl frac141
pl
ARBCN RBC (2)
where V sol denotes the volume of solution in the aliquot ARBC is
the area of a single RBC and N RBC is the number of erythrocytes
present in V sol at any instant
32 Oligomerization kinetics
Oligomerization involves the formation of dimers trimers and
higher mers from the protomer until an n-mer complex (pore)
is formed Data obtained from scanning transmission electronmicroscopy (STEM) and single-wavelength anomalous diff rac-
tion (SAD) indicate that the Cytolysin A (ClyA) pore complex
consists of n frac14 12 and 13 mers respectively1014 Oligomerization
can occur in a number of distinct kinetic pathways In Fig 2
the two main mechanisms are illustrated In the sequential
mechanism the nth mer is formed by the addition of a 1 mer to
a (n 1) mer complex In the non-sequential mechanism the
nth mer can be formed by allowed integer combinations of the
smaller mers As an example a 4 mer can be formed by a
combination of 2 + 2 mers as well as a 3 + 1 mers as illustrated
in Fig 2
If oligomerization occurs sequentially and irreversibly thereaction mechanism is
p1 thorn pl k l pl thorn1 l frac14 1 n 1 (3)
where k l is the reaction rate constant for the l th oligomerization
step If the l th oligomer is formed in an irreversible non-
sequential process the reaction mechanism is
pr thorn pl rk l pl l frac14 2 n
r frac14 1 to l =2 l even
1 to ethl 1THORN=2 l odd
for the formation of the l th oligomer In the above non-
sequential mechanism for oligomerization the number of
distinct reaction rate constants for the formation of a 12-mer
pore complex is 66 In what follows we develop the model for
the irreversible sequential mechanism We are unaware of any
Fig 1 Schematic indicating the various steps leading to pore forma-
tion The water-soluble monomer adsorbs onto the cell membrane
and undergoes a conformational change to form the membrane-
bound protomer This is followed by an oligomerization step to form
the dodecameric pore complex
Fig 2 Two possible modes of oligomerization (top) sequential olig-
omerization and (bottom) non-sequential oligomerization are shown
In sequential oligomerization a protomer is necessary for the forma-
tion of a higher oligomer whereas in the non-sequential mechanism a
higher oligomer can be formed from an allowed combination of lower
oligomers
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experiments which shed light on either of these mechanisms
Results for the reversible sequential mechanism and non-
sequential irreversible kinetics are presented later in the text
In the sequential mechanism there are 11 rate constants
Molecular simulation of hydrophobic association of small
solutes in water27 reveal that a sequential aggregation procedure
is favored during cluster formation The sequential aggregation
mechanism is also used for modelling micellar aggregation28
We assume that all the rate constants for the sequentialmechanism are identical This assumption is widely used in
sequential polymerization reactions With the assumption that
all rate constants for oligomerization (k l ) are identical a balance
on the protomer yields
d p1
dt frac14 k cm k l
Xn1
l frac141
p1 pl (4)
where the rst term on the right hand side represents the
formation of the protomer from the monomer and the other
terms represent sequential oligomerization steps wherein the
protomer binds with the other lsquomersrsquo to form the higher lsquomersrsquo
with a rate constant k l From rate considerations for dimerformation (l frac14 2) a prefactor of 12 appears in the term which
corresponds to the formation of the dimer This is a necessary
condition for satisfying the species mass balance The govern-
ing equation for the dimer (l frac14 2) is given by
d p2
dt frac14
1
2k 2 p1 p1 k 2 p1 p2 (5)
where p2 represents the concentration of dimer and k 2 repre-
sents the rate constant for the reaction The equation for the
formation of the l th oligomer (l gt 2) is
d pl
dt frac14 k l p1 pethl 1THORN eth1 dl 12THORNk l p1 pl l frac14 3 12 (6)
where d l 12 represents the Kronecker delta function The corre-
sponding number of pores per RBC is obtained using
np frac14 p12N avARBC (7)
where N av is the Avogadro number
33 Cell lysis
Every dodecamer corresponds to a stable pore in the
membrane If the rate at which lysis occurs is directly propor-
tional to the number of cells that are present at any instant of
time then cell lysis follows a rst order process On physical
grounds we further assume that cell lysis occurs only when the
number of pores exceeds a critical number of pores in each cell
Since our experiments are carried out under isotonic condi-
tions lysis is associated with rupture Lysis can be described
using the following rst order process
dx
dt frac14 klxR
np npc
(8)
where
R frac14
0 npnpc
np npc np $npc
x represents the fraction of unlysed cells at any instant of time
np is number of pores per RBC at any instant and npc is the
critical number of pores per RBC above which cell lysis occurs
The constant kl represents the decay rate constant for cell lysis
In eqn (8) the ramp function R(np npc) incorporates the
increased lysis as a function of the excess pores np npc We
also investigate other functional forms for R such as a unit step
function and a higher power dependence on np npc The
inuence of these on the model predictions are discussed laterin the text As cells lyse the number of RBCs N RBC at any
instant is
N RBC frac14 N inRBCx (9)
N inRBC is the initial number of RBCs
The rate at which hemoglobin is released from the RBCs into
solution is
dH out
dt frac14 V hrhN
inRBCkl xR
np npc
thornDhAp
ms
h H out
N RBCnp
V soll p
(10)
where the rst term represents the contribution due to lysis
(rupture) and the second term is the diff usive ux contribution
from the pores of unlysed cells In the above equation V h is the
volume of hemoglobin present in a single RBC rh is the density
of hemoglobin Dh is the diff usivity of hemoglobin l p is the
diff usion length along the pore Ap is the average area of a pore
msh H out and rh represents the saturated hemoglobin mass in
one RBC amount of hemoglobin present in the solution at any
instant of time and the density of hemoglobin respectively
4 Solution procedure While analyzing the problem it is useful to recast the equations in
suitable dimensionless forms If t frac14 t s C m frac14 C mC in m frac14 mms
and pl frac14 pl ms then eqn (1) (2) and (4)ndash(6) in dimensionless forms
are
dm
dt frac14
s
sa
C m
1 m
Xnl frac141
pl
s
sd
m s
sc
m (11a)
dC m
dt frac14
ARBCN RBCm
s
V solC inb
s
sd
m s
sa
C m
1 m
Xnl frac141
pl
(11b)
d p1
dt frac14
s
sc
m s
sl
Xn1
l frac141
p1 pl (11c)
d p2
dt frac14
s
2s2
p1 p1 s
s2
p1 p2 (11d)
d pl dt frac14
s
sl
p1 pethl 1THORN eth1 dl 12THORNs
sl
p1 pl l frac14 3 12 (11e)
From eqn (11a)ndash(11e) we can extract the following set of time
constants
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sa frac14 1
k aC inis the adsorption time constant
sd frac14 1
k dis the desorption time constant
sc frac14 1
k cis the conformational time constant
sl frac14 1
k l ms is the reaction time constant
s frac14 1
k aC in thorn k c thorn k d
(12)
41 Model parameters
Since we do not have experimental data to independently
determine the various time constants it is more convenient to
dene a ratio between time constants We dene the ratio
between conformational and adsorption times as
l frac14sc
sa
(13)
It has been observed that the conformational times sc arelarger than the time for membrane binding and oligomeriza-
tion1014 suggesting that l gt 1 We can also rewrite the adsorption
time constant and conformational time constant in terms of l
provided we have an estimate of the time required for initiation
of pore formation Since the processes leading to pore forma-
tion occur in series the total time constant snet for pore
formation is the sum of the time constants for the individual
steps
snet frac14 sc + sa + stl + sd (14)
Using eqn (12) and (13) and with the added assumption that
desorption rate is negligible and the adsorption and reaction
time constants are similar (sa st
l frac14 (n 1)sl ) the constants k aand k c can be expressed in terms of l Hence
k a frac14 2 thorn l
snetC in(15a)
and
k c frac142 thorn l
lsnet
(15b)
Under these assumptions for a xed initial concentration of
toxins and cell mass specifying l snet and C in is sufficient to
make predictions for the rate at which pores are formed in thesublytic regime With these assumptions eqn (11) can be
expressed solely in terms of the constant l The values of various
system properties used in the simulation are given in Table 1
and the values of diff erent parameters are given in Table 2 For
snet the model predictions were tested for a range of values as
indicated The parameters related to the pore geometry radius
of the pore r p and length of the pore l p are obtained from the
crystal structure of the ClyA pore10 The initial number of RBCs
are counted using the hemocytometric technique Typical
liquid diff usivities are used for hemoglobin Since the diff u-
sivity only inuences the hemoglobin release in the sublytic
regime obtaining a precise value of the diff usivity is not of
special consequence
42 Simulation details
We used an explicit Euler scheme for discretizing the governing
ordinary diff erential equations and the equations were solved
with a reduced time step of 0005 (0016 s l frac14 4) Calculations
performed with a reduced time step of 0001 did not alter the
reported results We developed an in-house program using
Matlab 70 to solve the discretized equations Calculations werechecked with a mass balance on the monomers
5 Results and discussions51 Lysis experiments
The OD data from lysis experiments at 60 minutes are illus-
trated in Fig 3a as a function of the bulk monomer toxin
concentration C in The data represents an average over 5
independent experiments Based on the time evolution (Fig 3b)
data no further lysis was observed above 30 minutes for all the
toxin concentrations investigated in the study Hence data at 30
minutes is expected to represent the steady state in the systemFrom the cell counts in the hemocytometer we nd 98 lysis at
147 nM and about 10ndash15 lysis at 588 nM The data clearly
reveals that the RBC lysis occurs only above the critical toxin
concentration (npc) which we estimate at 544 nM Below this
critical concentration lies the regime of low hemolytic activity
where the absolute OD values are an order of magnitude below
values obtained with lysis indicating that leakage from pores is
not signicant Although pore formation occurs in this regime
the concentration of pores is not sufficient to initiate lysis To
further support this hypothesis we carried out turbidity assay
experiments (Fig 4) for initial toxin concentrations ranging
Table 1 Various system properties and parameters used in the
simulation In some cases only the range of parameters that were
tested are given
Area of RBC29 ( ARBC) 136 mm2
Volume of RBC29 (V h) 90 fLRadius of pore10 (r p) 35 nmLength of pore10 (l p) 13 nmDiff usivity of hemoglobin ( Dh) 109 m2 s1
Initial number of RBC ( N inRBC) 32 107 cells per ml Volume of lysis assay (V sol) 1 mlSaturated surface concentration (ms) 109 mol m2
Net reaction time constant (snet ) 1ndash25 s
Table 2 Values of constants obtained from sequential and non-
sequential oligomerization
ParameterSequentialoligomerization
Non-sequentialoligomerization
l 2ndash4 2ndash4npc 392ndash768 pores 5300ndash6300 poresDecay rate
constant (kl )
15ndash18 107 s1 125ndash135 107 s1
4934 | RSC Adv 2014 4 4930ndash4942 This journal is copy The Royal Society of Chemistry 2014
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from 294ndash294 nM The decrease in turbidity is seen to occur
simultaneously with an increase in the OD supporting the view that the increase in OD is due to cell lysis The turbidity
decrease is also mirrored with the corresponding OD data Since
the cell mass used in the turbidity experiments is lower than
that used in the lysis experiments sublytic toxin concentrations
lie below 294 nM
52 Model predictions
In this section model predictions for the sequential irreversible
kinetics are compared with the lysis data
521 Sublytic regime
In the sublytic regime (np lt npc) the evolution of np with time is
obtained by solving eqn (11a)ndash(e) The number of pores np is
obtained from eqn (7) From the simulations we observe that the number of pores per RBC for a given initial toxin concen-
tration saturates within 10 s (Fig 5a) This saturation in np is
due to the limiting amount of toxin present in solution In
Fig 5b we plot the variation in np at saturation (20 s) with
l frac14sc
sa for diff erent initial toxin concentration assuming that
np lt npc The curve corresponding to 544 nM is tted to the
form y frac14 axb and the relation npc frac14 20033l097 is obtained
which can be used to x the value of npc for a given value of l In
order to simulate the lysis data a value of snet (eqn (14)) which
is the time constant associated with the time required for
formation of the rst pore has to be specied Initial estimates
are in the range of 1ndash25 s and we use a value of 1 s in all our
simulations unless specied We show later that our results are
Fig 3 (a) The normalized optical density values as a function of the
initial toxin concentration observed after 60 minutes during the lysis
experiments A distinct jump is observed above a toxin concentration
of 544 nM (b) Time evolution data of optical density values during
RBC lysis At 147 nM 98 lysis is observed
Fig 4 Turbidity (left axis) and lysis data (right axis) show that the
decrease in turbidity occurs simultaneously with an increase in the
OD The sublytic initial toxin concentration is less than 294 nM
(100 ng ml1)
Fig 5 Relation between l and npc in the sublytic regime (a) Number
of pores per RBC as a function of time The data plotted are for l frac14 2
and snet frac14 1 s (b) A regression analysis of the curve at 544 nM (dash-
dotted line) yields the relation npc frac14 20033l097
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relatively insensitive to the value of snet in this range The
parameters l npc and snet are obtained in the sublytic regime as
discussed above
522 Lysis regime
In the lysis regime in addition to the kinetic eqn (11andashe) we also
solve the lysis and hemoglobin release equations eqn (8) and
eqn (10) Cell lysis occurs once np gt npc and the OD increases with increasing toxin concentration as shown in Fig 3a Once
values of l npc and snet are xed in the sub-lytic regime the only
unknown parameter in the model is the value of kl (eqn (8)) In
all cases kl is xed by matching the maximum extent of lysis of
98 obtained at 147 nM Subsequent simulations are run with
diff erent values of C in to compare with the experimental data
Fig 6 illustrates the comparison between the model prediction
and the experimental OD data Since the OD varies linearly with
the amount of hemoglobin released during lysis30 we scale both
the experimental and predicted data by their respective
maximum OD to facilitate a meaningful comparison We also
carried out independent lysis experiments to verify the linear
relationship between the OD versus hemoglobin data Hence we
normalize the long time data and dene H max as the ratio of the
mass of the steady state hemoglobin released at a given C in to
the corresponding value at C in frac14 147 nM which is the highest
C in considered in the study This facilitates a comparison of
H max predicted from the model directly with the normalized OD
values For l frac14 2 the corresponding npc frac14 392 and the
comparison of H max at a value of kl frac14 18 107 s1 is
illustrated in Fig 6a The comparison for lfrac14 4 (npcfrac14 768) and kl
frac14 15 107 s1 is illustrated in Fig 6c Comparison of the
hemoglobin release dynamics ( H out vs time) for the corre-
sponding set of parameters are illustrated in Fig 6b and d
respectively
We observe that in this range of l (2 l 4) values and
kl 15ndash18 107 s1 the OD vs C in data is captured quite
accurately Since l is the ratio of the ClyA monomer confor-
mation time to the monomer adsorption time this range of l value is consistent with experiments10 which indicates that
conformation is preceded by fast adsorption followed by rapid
oligomerization The H out dynamics predicted by the model is
seen to capture the experimental data quite well (Fig 6b and d)
Upon increasing l we nd that a lower value of kl is required to
match the OD vs C in data (Fig 6d) A value of snet frac14 1 s captures
the early time release in the H out data at 147 nM quite accu-
rately and increasing snet to 24 s results in a short delay at early
times Since snet represents the time taken to form the rst pore
in situ monitoring of the hemoglobin release dynamics would be
required to determine snet more precisely In our experiments the
time evolution of the lysis data is carried out by intermittently arresting lysis at diff erent times and quenching the aliquots in
ice for a period of 3ndash5 minutes while the OD is determined
Experiments carried out continuously for the diff erent time
points shown in Fig 6b and d did not alter the data obtained
from the intermittent experiments Fluorescence permeation
experiments by Yamazaki and co-workers21 by lysenin (334 kDa)
induced pore formation on single giant unilamellar vesicles show
that pore formation is complete within about 10 s for toxin
concentrations of 200 ng ml1 and 50 s at 40 ng ml1 These are
similar to the time scales deduced in our model
At a toxin concentration of 147 nM a steady distribution of
mers is achieved on the time scale of 10ndash20 s (for a snet frac14 1 s)
Within this time interval the pore density goes through a phaseof rapid increase to exceed the critical pore density (Fig 7)
during which very little lysis is observed (Fig 6a and d) Lysis is
predominantly observed a er a steady number of pores have
formed on the RBC surface This steady number of pores ranges
Fig 6 Comparison of model predictions (open circles) with experi-
mental results (open squares) The amount of hemoglobin released
(Hmax) atthe end of30min for(a) npcfrac14 392 (c) npcfrac14 768 Dynamics of
hemoglobin released (Hout) for (b) npcfrac14 392 (d) npcfrac14 768 A value of l
between 2 and 4 is seen to accurately capture the Hmax versus C in data
[(a) and (c)] The amount of hemoglobin released is scaled with the
maximum amount to facilitate a comparison Simulations corresponding
to snet frac14 24 s are shown in bold lines and snet frac14 1 s in dashed lines
Fig 7 The pore density is plotted as a function of time in thepost-lysis
regime The number of pores (np) per RBC ranges from 515 at 588 nM
to 11 657 at 147 nM l frac14 2 and kl frac14 18 107 s1 npc frac14 392
4936 | RSC Adv 2014 4 4930ndash4942 This journal is copy The Royal Society of Chemistry 2014
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from 515 at 588 nM to 11 657 at 147 nM Given this situation it
is instructive to dene an eff ective lysis time constant kleff frac14
kl (np npc) where np is the steady state value of the number of
pores at a given value of C in (Fig 7) The value of kleff at 147 nM
is 2027 103 s1 which results in an eff ective lysis time
constant of 493 s
523 Oligomer and pore concentration
In both the low (lt544 nM) and high toxin ($544 nM) regimes
the amount of toxin is found to be limiting Even at the highest
toxin concentration C in frac14 147 nM the toxin in bulk solution is
depleted within 10 s Selected oligomer concentrations as a
function of time are plotted in Fig 8a and b for both high and
low toxin concentrations as predicted by the kinetic model (eqn
(4) and (6)) The pore density is illustrated in Fig 7 The gov-
erning equations for the formation of an l -mer are given in eqn
(6) Since pore formation occurs via a sequential oligomeriza-
tion mechanism a protomer (1-mer) is necessary for the
formation of all other l mers Hence a steady monomer
concentration on the RBC is achieved once the 1-mer concen-
tration reduces to zero and lag times are observed for theformation of the l + 1-mer From the distribution of mers on the
membrane we observe that a large fraction of protomers remain
trapped as intermediate mers on the membrane At 147 nM the
number of monomers per ml is 8854 1012 The initial
number N RBC frac14 32 107 the monomers per RBC is 276
105 If all the monomers were converted to pores each RBC
would have 23 105 pores However the number of pores
formed per RBC at 147 nM is 11657 105 pores (Fig 7) indi-
cating that about 50 of the mers remain on the membrane
surface as intermediate n-mers (n frac14 1ndash11) At a sublytic
concentration of 544 nM only about 45 of the mers are
converted to pores resulting in 375 pores per RBC (Fig 5a)
The eff ect of l is more prominent at the higher toxin
concentration where both the life time and the maximum
concentration for 1-mers is found to decrease as l is increased
from 2 to 4 The steady state concentrations of various n-mers aspredicted by the model are illustrated in Fig 8c and d A change
in the value of l results in a shi in the distribution for a
particular initial toxin concentration An increase in l implies an
increase in the conformational time relative to the adsorption
and reaction times Hence as l is increased occurrence of the
lower mers on the surface decreases due to the faster reaction
time scales relative to conformation At low bulk toxin concen-
trations (Fig 8c) the distribution of higher mers and conse-
quently the number of pores (12 mers) is very low due to the
limited supply of monomers in the system However at higher
concentration (Fig 8d) the number of monomers is no longer
the limiting factor and the distribution shi
s towards the highermers thereby increasing the number of pores on the surface
524 Parameter sensitivity
We brie y summarize the results of simulations carried out to
test the inuence of the estimated parameters on the model
predictions In the absence of monomer membrane binding
equilibria the value of saturated surface concentration ( ms) is
Fig 8 Model predictions for oligomer concentration pro1047297les ( pl) for l frac14 2 and kl frac14 18 107 s1 (a) Initial bulk concentration frac14 544 nM (b)
Initial bulk concentration frac14 147 nM Steady state distribution of oligomers for (c) C in frac14 544 nM (d) C in frac14 147 nM
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unknown In order to test the inuence of ms on the model
prediction we carried out a few simulations for ms frac14 1 108
mol m2 and msfrac14 1 1010 mol m2 for various C in values For
ms frac14 1 108 mol m2 a negligible number of pores were
formed and the H max ndashC in data (Fig 3a) is underpredicted At
ms frac14 1 1010 mol m2 pore formation was extremely rapid
and little variation in pore density between C in frac14 882 nM and
147 nM was observed As a consequence H max ndashC in data (Fig 3a)
is grossly overpredicted Hence a value of ms frac14 1 10
9 molm2 was used in the simulations (Fig 6) We further note
that the amount of saturated surface concentration ms
implicitly changes the reaction rate constant k l (eqn (12))
Increasing ms eff ectively decreases the reaction rate constant
(eqn (12))
Once snet is xed npc is related to l through the relation npcfrac14
alb with the constants a and b being xed for a given initial toxin
concentration C in (Fig 5) We have found that 2 l 4 ts the
hemoglobin release data very closely (Fig 6) and although the
hemoglobin released as a function of time is slightly under-
estimated by the model the agreement is reasonable Upon
increasing l
to 7 and keeping m
s
frac14 10
9
mol m
s1
the corre-sponding value of npc is 1349 With a value of kl frac14 13 107 s1
although H max versus C in data is accurately predicted the
hemoglobin versus time data is grossly underpredicted Varying
ms between 108 and 1010 mol m2 further deteriorated the
prediction Finally we point out that other functional forms of the
dependence on np npc in the cell lysis equation eqn (8) such as
the unit step function or a quadratic dependence (np npc)2 only
overestimated the H max versus C in data
53 Sequential oligomerization with reversible kinetics
In the previous discussion we present the results for the oligo-
merization kinetics which was assumed to be irreversible Thesequential oligomerization model with reversible kinetics is
p1 thorn pl ) k f
k b pl thorn1 l frac14 1 n 1 (16)
where k f and k b represent the forward and backward reaction
rate constants The kinetic equations are
d p1
dt frac14 k cm k f
Xn1
l frac141
p1 pl thorn k bXn1
l frac142
pl (17a)
d p2
dt frac14
1
2k f p1 p1 k f p1 p2 thorn k b p3
1
2 p2 (17b)
d pl
dt frac14 k f p1 pethl 1THORN eth1 dl 12THORNk f p1 pl
thorn k beth1 dl 12THORNetheth1 dl 11THORN pl thorn1 pl THORN l frac14 3 12 (17c)
A reversible time constant can be dened from eqn (17) as
sb frac14 1
k b The ratio R is dened as the ratio of forward to back-
ward time constants ( R frac14 sf sb) to study the eff ect of revers-
ibility Upon examining the number of pores as a function of
time we observe that the time taken to reach a steady number of
pores is signicantly larger than the time taken to reach steady
state in the lysis experiments In order to make comparisons
with the irreversible mechanism we evaluated the number of
pores at the threshold concentration of 544 nM It is observed
that the number of pores required for lysis initially increases
and then decreases for increments in R values The distribution
of oligomers at steady state are shown for diff erent R values in
Fig 9a and b for C in frac14 147 nM The steady state concentrations
of lsquomersrsquo change from a predominantly 11-mer concentration to
a predominant 1-mer concentration as R is varied between0 and 1 For R lt 1 we nd that the concentration of pores (12
mers) are signicantly higher than the intermediate lsquomerrsquo
concentrations shown in Fig 9b and range from 0142 nmol
Fig 9 Oligomer distribution as a function of the ratio of forward and
backward time constants R frac14 sfsb for C in frac14 147 nM (a) R varies
between 0 and 1 (b) R varies between 0 and 01 (c) Hemoglobin
release data as a function of toxin concentration is shown forvarious R
R frac14 0 has the closest agreement with the experimental data (open
squares)
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m2 ( R frac14 0) to 028 nmol m2 ( R frac14 001) For R frac14 1 the
concentration of 12 mers is 024 nmol m2 and decreases with a
further increase in R The predictions using reversible sequen-
tial kinetics for the H max vs C in data (Fig 9c) indicate
greater deviation from the experimental data when compared
with R frac14 0 These results indicate that the irreversible mecha-
nism provides the best agreement with the experimental data
We point out that the critical number of pores npc in
the reversible framework is non-monotonic ranging from 392( R frac14 0) to 5128 ( R frac14 1) for l frac14 2
54 Non-sequential oligomerization
In contrast to the 392 pores obtained for the critical number of
pores via the sequential mechanism a substantially larger
critical number of pores are observed via the non-sequential
oligomerization (6000 pores per cell) The mass balance for
the protomer ( p1) concentration remains identical to that of the
sequential oligomerization mechanism (eqn (4)) The governing
equations for pl (l lt l n) oligomer undergoing non-sequential
irreversible oligomerization are
d pl
dt frac14
1
2k l Xl 1
ufrac141
pu pl u eth1 dl 12THORNXn1
ufrac141
k l pl pu for 1l n (18)
where n represents the number of monomers in a pore The
relation between the critical number of pores and npc for non-
sequential oligomerization is npc frac14 5439l015 The critical
number of pores for l frac14 2 is 6035 pores which is about 20 times
greater than that obtained from a sequential oligomerization
mechanism In this scheme the hemoglobin release (Fig 10a)
data is overpredicted at intermediate toxin concentrations when
compared with the sequential oligomerization The predictions
of the H out vs time data at 147 nM (Fig 10b) are similar whencompared with the sequential oligomerization (Fig 6b and d)
A comparison of the tted parameters between the sequential
and non-sequential oligomerization mechanisms are given in
Table 2
Oligomer distributions obtained from the non-sequential
mechanism (Fig 11b) show an entirely diff erent trend when
compared to that obtained from the sequential mechanism
(Fig 8) In the sequential mechanism the higher lsquomersrsquo
attained a steady state once the protomer was depleted In the
non-sequential mechanism the contribution to the dodeca-
mers (12 mer) can be obtained from a large number of combi-
nations of the lower oligomers leading to a larger dodecamer orpore concentration The time scale required to attain the olig-
omer steady state concentration (Fig 11a) is about 20 s for an
initial concentration of 147 nM Similar time scales are
observed in the sequential mechanism as well
Further experiments are required to distinguish between the
various mechanisms Western Blot experiments conducted on
Hemolysin E12 and Clostridium perfringens 3 toxins31 showed the
presence of intermediate oligomers On the other hand single-
molecule uorescence imaging of a-hemolysin on a droplet
interface bilayer showed the presence of only monomers and
heptamers (pores)32
6 Discussion and conclusions
Lysis experiments on RBCs with the ClyA pore-forming toxin
show that a threshold initial toxin concentration is required to
initiate lysis From this observation we analyzed the problem in
two regimes a low toxin concentration regime where rupture of
cells is absent and a high toxin concentration regime where
lysis occurs and hemoglobin is released Kinetic models which
accounts for monomer binding conformation (membrane-
bound monomer to protomer) and oligomerization to form the
dodecameric pore complex are developed Models which
account for sequential and non-sequential oligomerization are
tested Cell rupture is assumed to be
rst order in the number of live cells and directly proportional to the pores in excess of the
critical number of pores npc In the sublytic regime the number
of pores is found to have a power law dependence on l which is
the ratio of conformational time to the reaction time This leads
to the construction of a ldquophase diagramrdquo between the number
of pores np and l for diff erent values of the initial toxin
concentration Comparing simulations with experimental data
the range of npc was 392ndash768 for the sequential mechanism and
5300ndash6300 pores for the non-sequential mechanism for 2 l 4
The range of l values is consistent with available experimental
data on ClyA which indicates that the membrane-bound
Fig 10 (a) Model predictions from the non-sequential mechanism
Simulated Hmax C in (open circles) curves grossly overpredicts the
experimental data (open squares) (b) Simulated hemoglobin release
compares well with the experimental data Dashed line snetfrac14 1 s solid
line snet frac14 24 s
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conformational step is slower than the preceding adsorption
and subsequent oligomerization steps10
From the model we are also able to comment on the time
constants for the various processes The time constant for poreformation is about 1 s indicating that pore formation itself is a
fast process relative to the time taken for the pore population on
a single RBC to reach steady state which is about 20ndash30 s Since
rupture kinetics is dynamic and depends on the fraction of live
cells as well as the number of pores on the cell lysis occurs in
the time scale of 10s of minutes Due to this separation of time
scales lysis is seen to occur once the number of pores has
reached a steady state This steady number of pores ranges from
515 to 11 657 as the toxin concentration ranges from 588ndash
147 nM For the non-sequential mechanism the critical
number of pores required to initiate lysis is about 20 times
higher when compared to that of sequential oligomerizationComparison of three diff erent kinetic models reveals that the
irreversible sequential kinetics provides the closest match with
the hemoglobin released as a function of the initial toxin
concentration Although we observe an overprediction of the
hemoglobin release data with the non-sequential mechanism
the hemoglobin release kinetics are similar to that of the
sequential mechanism The distribution of lower oligomers is
distinctly diff erent in both cases with a negligible numbers of
lower mers observed in the non-sequential oligomerization
These diff erences in the distribution of mers off er a ngerprint
to identify the underlying mechanism for pore formation
Experiments which can determine the number of pores or the
steady state lsquomerrsquo distributions on the membrane surface will
shed light on the pathways for oligomerization and enable a
more denite conclusion of the underlying kinetics Furtherthe rupture kinetics model contains only one adjustable
parameter Lysis experiments conducted with Vibrio cholerae El
Tor cytolysin33 and Monalysin34 show similar lysis times (in the
order of 10s of minutes) as observed in our study suggesting
similarities in the underlying kinetic pathways that lead to pore
formation and rupture The model developed in this manu-
script is generic and could be recast with some variation to
study the dynamics of other PFTs
We brie y discuss some of the limitations of the model in its
present form The model is based on the mean eld approxi-
mation where all cells are assumed to be identical and for the
purpose of binding and oligomerization this is an adequatestarting point The more complicated process is the mechanics
of rupture with the correct functional dependence on the pore
density In general there could exist a distribution of cells with
diff erent densities of pores Preliminary experiments by varying
the number of RBCs at a xed toxin concentration led to an
increase in lysis suggesting that cell heterogeneity could be
playing a role Although a population balance model35 could
include these variations this is at an added cost of complexity A
second aspect inherent to the model is the presence of lysis
beyond the time at which steady state is observed in the
experiments (30 minutes) Once the number of pores has
Fig 11 Oligomerconcentrationsas a function of time areshownfor (a) 544 nM and(b) 147 nMSteady state oligomer concentrations areshown
for (c) 544 nM and (d) 147 nM At steady state 1ndash4 mer concentrations are zero for both 544 and 147 nM The 12-mer concentration at steady
state is quite large compared to the other oligomer concentrations present in the system
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reached a steady state which occurs within 30 s (Fig 7a) cell
lysis continues to occur at a xed number of pores proportional
to np npc Running the simulation to steady state would
eventually lead to lysis of all the cells in the system albeit at an
exceedingly slow rate at the lower and intermediate toxin
concentrations We have also assumed complete and irrevers-
ible binding of the monomer to the membrane and hence the
number of pores predicted represent an upper limit within the
proposed kinetic framework Despite the mean eld approxi-mation the model proposed in this manuscript is able to
capture the inherent time scales in the process of pore forma-
tion and rupture as well as predict the variation of hemoglobin
release as a function of the initial toxin concentration observed
in the experiments We nally point out that the eff ect of
temperature on the kinetics and lysis has not been investigated
in this work Preliminary experiments at 147 nM toxin
concentration carried out to steady state indicated a marked
drop in lytic activity for temperatures below 15 C and lysis was
not observed at 10 C Data on binding isotherms would be
required to understand temperature eff ects in these systems
Our work has implications in optimizing dosage and devel-oping novel drug delivery strategies based on PFTs The obser-
vation that lysis occurs in a given window of toxin concentration
has implications while developing PFT-based drug and gene
therapy protocols In recent studies E coli used in conjunction
with radiation therapy was found to retard the growth of cancer
tumors when compared with only radiation therapy protocols36
Quantifying the dynamics and extent of lysis is an important
aspect of developing appropriate treatment protocols in these
combination therapies In other applications where pores are
used for gene delivery and in the development of articial cells
lysis must be prevented and pore formation restricted to
concentrations below the lysis threshold
Acknowledgements
We acknowledge funding from Department of Science and
Technology (DST) India under the IRHPA grant The authors
thank Sanjeev Kumar Gupta and Jaydeep Basu for several useful
discussions on the development of the model as well as the
reviewers for their critical comments
References
1 S Bhakdi and J Tranum-Jensen Microbiol Rev 1991 55
733ndash
7512 J Thiery D Keefe S Boulant E Boucrot M Walch
D Martinvalet I S Goping R C Bleackley
T Kirchhausen and J Lieberman Nat Immunol 2011 12
770ndash777
3 A Ludwig C von Rhein S Bauer C Huttinger and
W Goebel J Bacteriol 2004 186 5311ndash5320
4 A W Bernheimer and L S Avigad Proc Natl Acad Sci U S
A 1976 73 467ndash471
5 M V Sousa M Richardson W Fontes and L Morhy
J Protein Chem 1994 13 659ndash667
6 G Anderluh and P Macek Toxicon 2002 40 111ndash124
7 I Iacovache F G van der Goot and L Pernot Biochim
Biophys Acta Biomembr 2008 1778 1611ndash1623
8 N R Wyborn A Clark R E Roberts S J Jamieson
S Tzokov P A Bullough T J Stillman P J Artymiuk
J E Galen L Zhao M M Levine and J Green
Microbiology 2004 150 1495ndash1505
9 A J Wallace T J Stillman A Atkins S J Jamieson
P A Bullough J Green and P J Artymiuk Cell 2000 100
265ndash27610 M Mueller U Grauschopf T Maier R Glockshuber and
N Ban Nature 2009 459 726ndash730
11 J Oscarsson Y Mizunoe L Li X-H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
12 A Atkins N R Wyborn A J Wallace T J Stillman
L K Black A B Fielding M Hisakado P J Artymiuk and
J Green J Biol Chem 2000 275 41150ndash41155
13 A Ludwig G Volkerink C von Rhein S Bauer E Maier
B Bergmann W Goebel and R Benz J Bacteriol 2010
192 4001ndash4011
14 N Eier M Vetsch M Gregorini P Ringler M Chami
A Philippsen A Fritz S A Muller R Glockshuber A Engel and U Grauschopf EMBO J 2006 25 2652ndash2661
15 L A Shepard O Shatursky A E Johnson and R K Tweten
Biochemistry 2000 39 10284ndash10293
16 A J Ratner K R Hippe J L Aguilar M H Bender
A L Nelson and J N Weiser J Biol Chem 2006 281
12994ndash12998
17 I Iacovache M Bischoerger and F G van der Goot Curr
Opin Struct Biol 2010 20 241ndash246
18 G Schwarz and C H Robert Biophys Chem 1992 42 291ndash296
19 G Schwarz R-t Zong and T Popescu Biochim Biophys
Acta Biomembr 1992 1110 97ndash104
20 D Bolintineanu E Hazrati H T Davis R I Lehrer and
Y N Kaznessis Peptides 2010 31 1ndash821 J M Alam T Kobayashi and M Yamazaki Biochemistry
2012 51 5160ndash5172
22 J Lemiere K Guevorkian C Campillo C Sykes and T Betz
So Matter 2013 9 3181ndash3187
23 V Noireaux and A Libchaber Proc Natl Acad Sci U S A
2004 101 17669ndash17674
24 M M Bradford Anal Biochem 1976 72 248ndash254
25 H Xu E K Bjerneld M Kall and L Borjesson Phys Rev
Lett 1999 83 4357ndash4360
26 J Oscarsson Y Mizunoe L Li X H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
27 T M Raschke J Tsai and M Levitt Proc Natl Acad SciU S A 2001 98 5965ndash5969
28 L A Rodrıguez-Guadarrama S K Talsania K K Mohanty
and R Rajagopalan Langmuir 1999 15 437ndash446
29 C E McLaren G M Brittenham and V Hasselblad Am J
Physiol 1987 252 3181ndash3187
30 J G Kim M Xia and H Liu IEEE Eng Med Biol Mag 2005
24 118ndash121
31 M Nagahama H Hara M Fernandez-Miyakawa
Y Itohayashi and J Sakurai Biochemistry 2006 45 296ndash302
32 J R Thompson B Cronin H Bayley and M I Wallace
Biophys J 2011 101 2679ndash2683
This journal is copy The Royal Society of Chemistry 2014 RSC Adv 2014 4 4930ndash4942 | 4941
Paper RSC Advances
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1313
33 A Zitzer I Walev M Palmer and S Bhakdi Med Microbiol
Immunol 1995 184 37ndash44
34 O Opota I Vallet-Gely R Vincentelli C Kellenberger
I Iacovache M R Gonzalez A Roussel F G van der Goot
and B Lemaitre PLoS Pathog 2011 7 e1002259
35 S Kumar and D Ramkrishna Chem Eng Sci 1996 51
1311ndash1332
36 S N Jiang T X Phan T K Nam V H Nguyen H S Kim
H S Bom H E Choy Y Hong and J J Min Mol Ther
2010 18 635ndash642
RSC Advances Paper
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 213
A variety of experimental techniques such as lytic experi-
ments gel electrophoresis site-directed mutagenesis and cryo-
electron microscopy have been used to unravel the mechanisms
of pore formation in ClyA 11ndash14 and in other widely studied PFTs
such as cholestrol-dependent toxins (CDCs)15 and S aureus a-
Hemolysin116 Despite the interest in unravelling structural and
mechanistic pathways for the action of PFTs10 the kinetics of
membrane oligomerization rates of pore formation and the
dynamics of ensuing lysis have not been the subject of muchquantitative investigation17 Pore formation kinetics and
dynamics of the release of self-quenching dye molecules from
liposomes have been quantied by Schwarz and co-workers1819
where the marker release dynamics is tted to either a single or
double exponential function with suitably motivated kinetic
models for pore formation Models which quantify the perme-
ation rates across bacterial membranes due to pore-forming
protegrin peptides20 and lysenin-induced permeation in giant
unilamellar vesicles21 have appeared in the recent literature
Recently an investigation of permeation rates due to a-Hemo-
lysin on liposomes using optical contrast microscopy and
micropipette experiments
22
reveal pore densities of about 100pores mm2
Quantifying the phenomenon of pore formation and lysis
is crucial for developing PFT-based druggene delivery thera-
pies and controlling pore formation in vesicle-based biore-
actors23 during the development of articial cells In this
study the lytic activity of ClyA is modeled based on experi-
ments carried out on red blood cells We formulate an
adsorption-kinetic model which incorporates monomer
binding conformational changes and sequential as well as
non-sequential oligomerization pathways to determine the
rate of hemoglobin released as a function of time and ClyA
concentration A rst order rupture model is used to quantify
the lysis dynamics Our model captures the experimentalhemoglobin data as a function of toxin concentration From
this observation we extract the critical number of pores per
RBC above which cell lysis occurs
2 Experimental procedure21 Expression and purication of His-tagged Cytolysin A
(ClyA)
pGS1111 plasmid containing the ClyA gene as a fusion with
glutathione S-transferase was obtained from Dr J Green
University of Sheffield UK The ClyA gene was subcloned from
pGS1111 into pPRO Ex-HTb using EcoRI and SalI to obtainpPROb ClyA containing an N-terminal hexahistidine tag E coli
BL21 endo cells transformed with pPROb ClyA were grown in
terric broth ClyA full length (ClyA FL) proteins were expressed
on induction with 500 mM isopropyl thiogalactopyranoside
Cells were lysed by sonication in buff er containing 100 mM
TrisndashHCl (pH 80) 5 mM b-mercaptoethanol 100 mM NaCl
1 mM benzamidine 2 mM phenylmethylsulfonyl uoride and
10 glycerol Centrifugation was carried out at 30 000 g and the
cell-free extract was interacted with nickelndashnitrilotriacetic acid
beads Beads were washed with buff er containing 100 mM Trisndash
HCl (pH 80) 5 mM b-mercaptoethanol 500 mM NaCl 20 mM
imidazole to remove nonspecic proteins on the beads His6
ClyA was eluted in buff er containing 100 mM TrisndashHCl (pH 80)
5 mM b-mercaptoethanol 100 mM NaCl 300 mM imidazole
10 glycerol Proteins were desalted in buff er (100 mM Trisndash
HCl (pH 80) 5 mM b-mercaptoethanol 100 mM NaCl and 10
glycerol)
His6 ClyA was treated with TEV protease to obtain tagless
protein 1 part of puried hexahistidine TEV protease was
taken per 30 parts by mass of ClyA and incubated overnight at 4 C TEV was separated by interacting further with Ni-
NTA beads Protein quantity was estimated by the Bradford
method24
22 Hemolysis assay
The hemolysis assay was carried out as described previously 12
Rabbit erythrocytes were washed and diluted 1 100 vv in PBS
(phosphate-buff ered saline pH 74) Aliquots of RBC suspension
were transferred to microcentrifuge tubes ClyA was added to
suitable aliquots of RBCs and incubated at 37 C in a shaking
incubator for 1 hour Lysis experiments were carried out for
ClyA concentrations ranging from 294ndash147 nM These corre-spond to 100ndash500 ng ml1 respectively since ClyA is a 34 kDa
monomer Unlysed cells and debris were sedimented by
centrifugation at 5000 rpm for 1 min Released hemoglobin in
the supernatant was quantied by spectrometric detection at
540 nm The numbers of cells remaining a er lysis were
counted in a hemocytometer
23 Turbidity assay
A suspension of rabbit erythrocytes (1 vv in phosphate-buff -
ered saline 1 ml) was treated with varying amounts of Cytolysin
A as indicated To assess turbidity 200 ml of the cell suspension
was transferred to a clear-bottomed 96-well plate and light scattering was measured at 620 nm The cells in the plate were
centrifuged at 3000 rpm for 2 min and the extent of haemolysis
was estimated by measuring the absorbance of the supernatant
at 570 nm Optical density measurements were carried out on a
Tecan Innite F50 microplate reader
3 Modeling31 Membrane binding and bulk toxin concentration
We develop a model to predict the hemoglobin release kinetics
of the RBCs as a function of initial toxin concentration The
series of steps that lead to pore formation are illustrated inFig 1 The model is developed in the mean eld framework
wherein all cells are assumed to be identical Diff usion is
assumed to be fast relative to membrane binding and oligo-
merization Assuming a protein diff usion coefficient of 1013
m2 s1 the diff usion time on the membrane is of the order of
15ndash50 ms for toxin concentrations ranging from 294ndash147 nM
Membrane binding is assumed to be irreversible and of similar
time scale to that of oligomerization The amount of hemo-
globin released due to cell lysis is signicantly larger (108
times) than that released from the pores of unlysed cells both
due to the size of the hemoglobin molecule25 (5ndash6 nm) as well as
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the small eff ective pore diameter available for transport The
inner pore diameter exposed to the cytosol is 4 nm in the crystal
structure and in a fully solvated environment the eff ective
diameter is expected to decrease further Further osmoticprotection assays of ClyA conclude that the eff ective pore sizes
range from 20ndash35 nm26 The solution is assumed to remain
isotonic as lysis proceeds since lysis did not occur in the
absence of toxin when RBCs were incubated in buff er solution
made up of fully lysed (sonicating 1 RBC (vv)) RBCs We have
assumed that the conformational change follows rst order
irreversible kinetics since the conformational step involves a
transition from a water-soluble monomer to a membrane-
inserted protomer via a series of conformational changes in the
regions around the b-tongue region of the monomer and the N-
terminus10 This is succeeded by a fast oligomerization step to
form the pore complex
The rate equation for the membrane-bound monomer
whose surface molar concentration is denoted as m is
dm
dt frac14 k aC m
ms m
Xnl frac141
pl
k dm k cm (1)
where k a is the adsorption rate constant k d is the desorption
rate constant ms is the saturated surface molar concentration
and the last term represents the rate at which the membrane-
bound monomer (m) undergoes a conformational change
to the membrane-bound protomer ( p1) with a rate constant
k c and pl is the surface molar concentration of the oligomer
containing l -mers If the bulk concentration of the toxin
monomer is constant eqn (1) is similar in form to the
LangmuirndashHinshelwood equation traditionally used to
describe the concentration of surface species undergoing
both adsorption and reaction
Since the initial toxin concentration (C in) in the aliquots is in
the range of 294ndash147 nM an additional balance is used to
describe the concentration change of toxins in solution This
yields
V sol
dC m
dt frac14
k dm k aC m
ms m
Xnl frac141
pl
ARBCN RBC (2)
where V sol denotes the volume of solution in the aliquot ARBC is
the area of a single RBC and N RBC is the number of erythrocytes
present in V sol at any instant
32 Oligomerization kinetics
Oligomerization involves the formation of dimers trimers and
higher mers from the protomer until an n-mer complex (pore)
is formed Data obtained from scanning transmission electronmicroscopy (STEM) and single-wavelength anomalous diff rac-
tion (SAD) indicate that the Cytolysin A (ClyA) pore complex
consists of n frac14 12 and 13 mers respectively1014 Oligomerization
can occur in a number of distinct kinetic pathways In Fig 2
the two main mechanisms are illustrated In the sequential
mechanism the nth mer is formed by the addition of a 1 mer to
a (n 1) mer complex In the non-sequential mechanism the
nth mer can be formed by allowed integer combinations of the
smaller mers As an example a 4 mer can be formed by a
combination of 2 + 2 mers as well as a 3 + 1 mers as illustrated
in Fig 2
If oligomerization occurs sequentially and irreversibly thereaction mechanism is
p1 thorn pl k l pl thorn1 l frac14 1 n 1 (3)
where k l is the reaction rate constant for the l th oligomerization
step If the l th oligomer is formed in an irreversible non-
sequential process the reaction mechanism is
pr thorn pl rk l pl l frac14 2 n
r frac14 1 to l =2 l even
1 to ethl 1THORN=2 l odd
for the formation of the l th oligomer In the above non-
sequential mechanism for oligomerization the number of
distinct reaction rate constants for the formation of a 12-mer
pore complex is 66 In what follows we develop the model for
the irreversible sequential mechanism We are unaware of any
Fig 1 Schematic indicating the various steps leading to pore forma-
tion The water-soluble monomer adsorbs onto the cell membrane
and undergoes a conformational change to form the membrane-
bound protomer This is followed by an oligomerization step to form
the dodecameric pore complex
Fig 2 Two possible modes of oligomerization (top) sequential olig-
omerization and (bottom) non-sequential oligomerization are shown
In sequential oligomerization a protomer is necessary for the forma-
tion of a higher oligomer whereas in the non-sequential mechanism a
higher oligomer can be formed from an allowed combination of lower
oligomers
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experiments which shed light on either of these mechanisms
Results for the reversible sequential mechanism and non-
sequential irreversible kinetics are presented later in the text
In the sequential mechanism there are 11 rate constants
Molecular simulation of hydrophobic association of small
solutes in water27 reveal that a sequential aggregation procedure
is favored during cluster formation The sequential aggregation
mechanism is also used for modelling micellar aggregation28
We assume that all the rate constants for the sequentialmechanism are identical This assumption is widely used in
sequential polymerization reactions With the assumption that
all rate constants for oligomerization (k l ) are identical a balance
on the protomer yields
d p1
dt frac14 k cm k l
Xn1
l frac141
p1 pl (4)
where the rst term on the right hand side represents the
formation of the protomer from the monomer and the other
terms represent sequential oligomerization steps wherein the
protomer binds with the other lsquomersrsquo to form the higher lsquomersrsquo
with a rate constant k l From rate considerations for dimerformation (l frac14 2) a prefactor of 12 appears in the term which
corresponds to the formation of the dimer This is a necessary
condition for satisfying the species mass balance The govern-
ing equation for the dimer (l frac14 2) is given by
d p2
dt frac14
1
2k 2 p1 p1 k 2 p1 p2 (5)
where p2 represents the concentration of dimer and k 2 repre-
sents the rate constant for the reaction The equation for the
formation of the l th oligomer (l gt 2) is
d pl
dt frac14 k l p1 pethl 1THORN eth1 dl 12THORNk l p1 pl l frac14 3 12 (6)
where d l 12 represents the Kronecker delta function The corre-
sponding number of pores per RBC is obtained using
np frac14 p12N avARBC (7)
where N av is the Avogadro number
33 Cell lysis
Every dodecamer corresponds to a stable pore in the
membrane If the rate at which lysis occurs is directly propor-
tional to the number of cells that are present at any instant of
time then cell lysis follows a rst order process On physical
grounds we further assume that cell lysis occurs only when the
number of pores exceeds a critical number of pores in each cell
Since our experiments are carried out under isotonic condi-
tions lysis is associated with rupture Lysis can be described
using the following rst order process
dx
dt frac14 klxR
np npc
(8)
where
R frac14
0 npnpc
np npc np $npc
x represents the fraction of unlysed cells at any instant of time
np is number of pores per RBC at any instant and npc is the
critical number of pores per RBC above which cell lysis occurs
The constant kl represents the decay rate constant for cell lysis
In eqn (8) the ramp function R(np npc) incorporates the
increased lysis as a function of the excess pores np npc We
also investigate other functional forms for R such as a unit step
function and a higher power dependence on np npc The
inuence of these on the model predictions are discussed laterin the text As cells lyse the number of RBCs N RBC at any
instant is
N RBC frac14 N inRBCx (9)
N inRBC is the initial number of RBCs
The rate at which hemoglobin is released from the RBCs into
solution is
dH out
dt frac14 V hrhN
inRBCkl xR
np npc
thornDhAp
ms
h H out
N RBCnp
V soll p
(10)
where the rst term represents the contribution due to lysis
(rupture) and the second term is the diff usive ux contribution
from the pores of unlysed cells In the above equation V h is the
volume of hemoglobin present in a single RBC rh is the density
of hemoglobin Dh is the diff usivity of hemoglobin l p is the
diff usion length along the pore Ap is the average area of a pore
msh H out and rh represents the saturated hemoglobin mass in
one RBC amount of hemoglobin present in the solution at any
instant of time and the density of hemoglobin respectively
4 Solution procedure While analyzing the problem it is useful to recast the equations in
suitable dimensionless forms If t frac14 t s C m frac14 C mC in m frac14 mms
and pl frac14 pl ms then eqn (1) (2) and (4)ndash(6) in dimensionless forms
are
dm
dt frac14
s
sa
C m
1 m
Xnl frac141
pl
s
sd
m s
sc
m (11a)
dC m
dt frac14
ARBCN RBCm
s
V solC inb
s
sd
m s
sa
C m
1 m
Xnl frac141
pl
(11b)
d p1
dt frac14
s
sc
m s
sl
Xn1
l frac141
p1 pl (11c)
d p2
dt frac14
s
2s2
p1 p1 s
s2
p1 p2 (11d)
d pl dt frac14
s
sl
p1 pethl 1THORN eth1 dl 12THORNs
sl
p1 pl l frac14 3 12 (11e)
From eqn (11a)ndash(11e) we can extract the following set of time
constants
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sa frac14 1
k aC inis the adsorption time constant
sd frac14 1
k dis the desorption time constant
sc frac14 1
k cis the conformational time constant
sl frac14 1
k l ms is the reaction time constant
s frac14 1
k aC in thorn k c thorn k d
(12)
41 Model parameters
Since we do not have experimental data to independently
determine the various time constants it is more convenient to
dene a ratio between time constants We dene the ratio
between conformational and adsorption times as
l frac14sc
sa
(13)
It has been observed that the conformational times sc arelarger than the time for membrane binding and oligomeriza-
tion1014 suggesting that l gt 1 We can also rewrite the adsorption
time constant and conformational time constant in terms of l
provided we have an estimate of the time required for initiation
of pore formation Since the processes leading to pore forma-
tion occur in series the total time constant snet for pore
formation is the sum of the time constants for the individual
steps
snet frac14 sc + sa + stl + sd (14)
Using eqn (12) and (13) and with the added assumption that
desorption rate is negligible and the adsorption and reaction
time constants are similar (sa st
l frac14 (n 1)sl ) the constants k aand k c can be expressed in terms of l Hence
k a frac14 2 thorn l
snetC in(15a)
and
k c frac142 thorn l
lsnet
(15b)
Under these assumptions for a xed initial concentration of
toxins and cell mass specifying l snet and C in is sufficient to
make predictions for the rate at which pores are formed in thesublytic regime With these assumptions eqn (11) can be
expressed solely in terms of the constant l The values of various
system properties used in the simulation are given in Table 1
and the values of diff erent parameters are given in Table 2 For
snet the model predictions were tested for a range of values as
indicated The parameters related to the pore geometry radius
of the pore r p and length of the pore l p are obtained from the
crystal structure of the ClyA pore10 The initial number of RBCs
are counted using the hemocytometric technique Typical
liquid diff usivities are used for hemoglobin Since the diff u-
sivity only inuences the hemoglobin release in the sublytic
regime obtaining a precise value of the diff usivity is not of
special consequence
42 Simulation details
We used an explicit Euler scheme for discretizing the governing
ordinary diff erential equations and the equations were solved
with a reduced time step of 0005 (0016 s l frac14 4) Calculations
performed with a reduced time step of 0001 did not alter the
reported results We developed an in-house program using
Matlab 70 to solve the discretized equations Calculations werechecked with a mass balance on the monomers
5 Results and discussions51 Lysis experiments
The OD data from lysis experiments at 60 minutes are illus-
trated in Fig 3a as a function of the bulk monomer toxin
concentration C in The data represents an average over 5
independent experiments Based on the time evolution (Fig 3b)
data no further lysis was observed above 30 minutes for all the
toxin concentrations investigated in the study Hence data at 30
minutes is expected to represent the steady state in the systemFrom the cell counts in the hemocytometer we nd 98 lysis at
147 nM and about 10ndash15 lysis at 588 nM The data clearly
reveals that the RBC lysis occurs only above the critical toxin
concentration (npc) which we estimate at 544 nM Below this
critical concentration lies the regime of low hemolytic activity
where the absolute OD values are an order of magnitude below
values obtained with lysis indicating that leakage from pores is
not signicant Although pore formation occurs in this regime
the concentration of pores is not sufficient to initiate lysis To
further support this hypothesis we carried out turbidity assay
experiments (Fig 4) for initial toxin concentrations ranging
Table 1 Various system properties and parameters used in the
simulation In some cases only the range of parameters that were
tested are given
Area of RBC29 ( ARBC) 136 mm2
Volume of RBC29 (V h) 90 fLRadius of pore10 (r p) 35 nmLength of pore10 (l p) 13 nmDiff usivity of hemoglobin ( Dh) 109 m2 s1
Initial number of RBC ( N inRBC) 32 107 cells per ml Volume of lysis assay (V sol) 1 mlSaturated surface concentration (ms) 109 mol m2
Net reaction time constant (snet ) 1ndash25 s
Table 2 Values of constants obtained from sequential and non-
sequential oligomerization
ParameterSequentialoligomerization
Non-sequentialoligomerization
l 2ndash4 2ndash4npc 392ndash768 pores 5300ndash6300 poresDecay rate
constant (kl )
15ndash18 107 s1 125ndash135 107 s1
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from 294ndash294 nM The decrease in turbidity is seen to occur
simultaneously with an increase in the OD supporting the view that the increase in OD is due to cell lysis The turbidity
decrease is also mirrored with the corresponding OD data Since
the cell mass used in the turbidity experiments is lower than
that used in the lysis experiments sublytic toxin concentrations
lie below 294 nM
52 Model predictions
In this section model predictions for the sequential irreversible
kinetics are compared with the lysis data
521 Sublytic regime
In the sublytic regime (np lt npc) the evolution of np with time is
obtained by solving eqn (11a)ndash(e) The number of pores np is
obtained from eqn (7) From the simulations we observe that the number of pores per RBC for a given initial toxin concen-
tration saturates within 10 s (Fig 5a) This saturation in np is
due to the limiting amount of toxin present in solution In
Fig 5b we plot the variation in np at saturation (20 s) with
l frac14sc
sa for diff erent initial toxin concentration assuming that
np lt npc The curve corresponding to 544 nM is tted to the
form y frac14 axb and the relation npc frac14 20033l097 is obtained
which can be used to x the value of npc for a given value of l In
order to simulate the lysis data a value of snet (eqn (14)) which
is the time constant associated with the time required for
formation of the rst pore has to be specied Initial estimates
are in the range of 1ndash25 s and we use a value of 1 s in all our
simulations unless specied We show later that our results are
Fig 3 (a) The normalized optical density values as a function of the
initial toxin concentration observed after 60 minutes during the lysis
experiments A distinct jump is observed above a toxin concentration
of 544 nM (b) Time evolution data of optical density values during
RBC lysis At 147 nM 98 lysis is observed
Fig 4 Turbidity (left axis) and lysis data (right axis) show that the
decrease in turbidity occurs simultaneously with an increase in the
OD The sublytic initial toxin concentration is less than 294 nM
(100 ng ml1)
Fig 5 Relation between l and npc in the sublytic regime (a) Number
of pores per RBC as a function of time The data plotted are for l frac14 2
and snet frac14 1 s (b) A regression analysis of the curve at 544 nM (dash-
dotted line) yields the relation npc frac14 20033l097
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relatively insensitive to the value of snet in this range The
parameters l npc and snet are obtained in the sublytic regime as
discussed above
522 Lysis regime
In the lysis regime in addition to the kinetic eqn (11andashe) we also
solve the lysis and hemoglobin release equations eqn (8) and
eqn (10) Cell lysis occurs once np gt npc and the OD increases with increasing toxin concentration as shown in Fig 3a Once
values of l npc and snet are xed in the sub-lytic regime the only
unknown parameter in the model is the value of kl (eqn (8)) In
all cases kl is xed by matching the maximum extent of lysis of
98 obtained at 147 nM Subsequent simulations are run with
diff erent values of C in to compare with the experimental data
Fig 6 illustrates the comparison between the model prediction
and the experimental OD data Since the OD varies linearly with
the amount of hemoglobin released during lysis30 we scale both
the experimental and predicted data by their respective
maximum OD to facilitate a meaningful comparison We also
carried out independent lysis experiments to verify the linear
relationship between the OD versus hemoglobin data Hence we
normalize the long time data and dene H max as the ratio of the
mass of the steady state hemoglobin released at a given C in to
the corresponding value at C in frac14 147 nM which is the highest
C in considered in the study This facilitates a comparison of
H max predicted from the model directly with the normalized OD
values For l frac14 2 the corresponding npc frac14 392 and the
comparison of H max at a value of kl frac14 18 107 s1 is
illustrated in Fig 6a The comparison for lfrac14 4 (npcfrac14 768) and kl
frac14 15 107 s1 is illustrated in Fig 6c Comparison of the
hemoglobin release dynamics ( H out vs time) for the corre-
sponding set of parameters are illustrated in Fig 6b and d
respectively
We observe that in this range of l (2 l 4) values and
kl 15ndash18 107 s1 the OD vs C in data is captured quite
accurately Since l is the ratio of the ClyA monomer confor-
mation time to the monomer adsorption time this range of l value is consistent with experiments10 which indicates that
conformation is preceded by fast adsorption followed by rapid
oligomerization The H out dynamics predicted by the model is
seen to capture the experimental data quite well (Fig 6b and d)
Upon increasing l we nd that a lower value of kl is required to
match the OD vs C in data (Fig 6d) A value of snet frac14 1 s captures
the early time release in the H out data at 147 nM quite accu-
rately and increasing snet to 24 s results in a short delay at early
times Since snet represents the time taken to form the rst pore
in situ monitoring of the hemoglobin release dynamics would be
required to determine snet more precisely In our experiments the
time evolution of the lysis data is carried out by intermittently arresting lysis at diff erent times and quenching the aliquots in
ice for a period of 3ndash5 minutes while the OD is determined
Experiments carried out continuously for the diff erent time
points shown in Fig 6b and d did not alter the data obtained
from the intermittent experiments Fluorescence permeation
experiments by Yamazaki and co-workers21 by lysenin (334 kDa)
induced pore formation on single giant unilamellar vesicles show
that pore formation is complete within about 10 s for toxin
concentrations of 200 ng ml1 and 50 s at 40 ng ml1 These are
similar to the time scales deduced in our model
At a toxin concentration of 147 nM a steady distribution of
mers is achieved on the time scale of 10ndash20 s (for a snet frac14 1 s)
Within this time interval the pore density goes through a phaseof rapid increase to exceed the critical pore density (Fig 7)
during which very little lysis is observed (Fig 6a and d) Lysis is
predominantly observed a er a steady number of pores have
formed on the RBC surface This steady number of pores ranges
Fig 6 Comparison of model predictions (open circles) with experi-
mental results (open squares) The amount of hemoglobin released
(Hmax) atthe end of30min for(a) npcfrac14 392 (c) npcfrac14 768 Dynamics of
hemoglobin released (Hout) for (b) npcfrac14 392 (d) npcfrac14 768 A value of l
between 2 and 4 is seen to accurately capture the Hmax versus C in data
[(a) and (c)] The amount of hemoglobin released is scaled with the
maximum amount to facilitate a comparison Simulations corresponding
to snet frac14 24 s are shown in bold lines and snet frac14 1 s in dashed lines
Fig 7 The pore density is plotted as a function of time in thepost-lysis
regime The number of pores (np) per RBC ranges from 515 at 588 nM
to 11 657 at 147 nM l frac14 2 and kl frac14 18 107 s1 npc frac14 392
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from 515 at 588 nM to 11 657 at 147 nM Given this situation it
is instructive to dene an eff ective lysis time constant kleff frac14
kl (np npc) where np is the steady state value of the number of
pores at a given value of C in (Fig 7) The value of kleff at 147 nM
is 2027 103 s1 which results in an eff ective lysis time
constant of 493 s
523 Oligomer and pore concentration
In both the low (lt544 nM) and high toxin ($544 nM) regimes
the amount of toxin is found to be limiting Even at the highest
toxin concentration C in frac14 147 nM the toxin in bulk solution is
depleted within 10 s Selected oligomer concentrations as a
function of time are plotted in Fig 8a and b for both high and
low toxin concentrations as predicted by the kinetic model (eqn
(4) and (6)) The pore density is illustrated in Fig 7 The gov-
erning equations for the formation of an l -mer are given in eqn
(6) Since pore formation occurs via a sequential oligomeriza-
tion mechanism a protomer (1-mer) is necessary for the
formation of all other l mers Hence a steady monomer
concentration on the RBC is achieved once the 1-mer concen-
tration reduces to zero and lag times are observed for theformation of the l + 1-mer From the distribution of mers on the
membrane we observe that a large fraction of protomers remain
trapped as intermediate mers on the membrane At 147 nM the
number of monomers per ml is 8854 1012 The initial
number N RBC frac14 32 107 the monomers per RBC is 276
105 If all the monomers were converted to pores each RBC
would have 23 105 pores However the number of pores
formed per RBC at 147 nM is 11657 105 pores (Fig 7) indi-
cating that about 50 of the mers remain on the membrane
surface as intermediate n-mers (n frac14 1ndash11) At a sublytic
concentration of 544 nM only about 45 of the mers are
converted to pores resulting in 375 pores per RBC (Fig 5a)
The eff ect of l is more prominent at the higher toxin
concentration where both the life time and the maximum
concentration for 1-mers is found to decrease as l is increased
from 2 to 4 The steady state concentrations of various n-mers aspredicted by the model are illustrated in Fig 8c and d A change
in the value of l results in a shi in the distribution for a
particular initial toxin concentration An increase in l implies an
increase in the conformational time relative to the adsorption
and reaction times Hence as l is increased occurrence of the
lower mers on the surface decreases due to the faster reaction
time scales relative to conformation At low bulk toxin concen-
trations (Fig 8c) the distribution of higher mers and conse-
quently the number of pores (12 mers) is very low due to the
limited supply of monomers in the system However at higher
concentration (Fig 8d) the number of monomers is no longer
the limiting factor and the distribution shi
s towards the highermers thereby increasing the number of pores on the surface
524 Parameter sensitivity
We brie y summarize the results of simulations carried out to
test the inuence of the estimated parameters on the model
predictions In the absence of monomer membrane binding
equilibria the value of saturated surface concentration ( ms) is
Fig 8 Model predictions for oligomer concentration pro1047297les ( pl) for l frac14 2 and kl frac14 18 107 s1 (a) Initial bulk concentration frac14 544 nM (b)
Initial bulk concentration frac14 147 nM Steady state distribution of oligomers for (c) C in frac14 544 nM (d) C in frac14 147 nM
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unknown In order to test the inuence of ms on the model
prediction we carried out a few simulations for ms frac14 1 108
mol m2 and msfrac14 1 1010 mol m2 for various C in values For
ms frac14 1 108 mol m2 a negligible number of pores were
formed and the H max ndashC in data (Fig 3a) is underpredicted At
ms frac14 1 1010 mol m2 pore formation was extremely rapid
and little variation in pore density between C in frac14 882 nM and
147 nM was observed As a consequence H max ndashC in data (Fig 3a)
is grossly overpredicted Hence a value of ms frac14 1 10
9 molm2 was used in the simulations (Fig 6) We further note
that the amount of saturated surface concentration ms
implicitly changes the reaction rate constant k l (eqn (12))
Increasing ms eff ectively decreases the reaction rate constant
(eqn (12))
Once snet is xed npc is related to l through the relation npcfrac14
alb with the constants a and b being xed for a given initial toxin
concentration C in (Fig 5) We have found that 2 l 4 ts the
hemoglobin release data very closely (Fig 6) and although the
hemoglobin released as a function of time is slightly under-
estimated by the model the agreement is reasonable Upon
increasing l
to 7 and keeping m
s
frac14 10
9
mol m
s1
the corre-sponding value of npc is 1349 With a value of kl frac14 13 107 s1
although H max versus C in data is accurately predicted the
hemoglobin versus time data is grossly underpredicted Varying
ms between 108 and 1010 mol m2 further deteriorated the
prediction Finally we point out that other functional forms of the
dependence on np npc in the cell lysis equation eqn (8) such as
the unit step function or a quadratic dependence (np npc)2 only
overestimated the H max versus C in data
53 Sequential oligomerization with reversible kinetics
In the previous discussion we present the results for the oligo-
merization kinetics which was assumed to be irreversible Thesequential oligomerization model with reversible kinetics is
p1 thorn pl ) k f
k b pl thorn1 l frac14 1 n 1 (16)
where k f and k b represent the forward and backward reaction
rate constants The kinetic equations are
d p1
dt frac14 k cm k f
Xn1
l frac141
p1 pl thorn k bXn1
l frac142
pl (17a)
d p2
dt frac14
1
2k f p1 p1 k f p1 p2 thorn k b p3
1
2 p2 (17b)
d pl
dt frac14 k f p1 pethl 1THORN eth1 dl 12THORNk f p1 pl
thorn k beth1 dl 12THORNetheth1 dl 11THORN pl thorn1 pl THORN l frac14 3 12 (17c)
A reversible time constant can be dened from eqn (17) as
sb frac14 1
k b The ratio R is dened as the ratio of forward to back-
ward time constants ( R frac14 sf sb) to study the eff ect of revers-
ibility Upon examining the number of pores as a function of
time we observe that the time taken to reach a steady number of
pores is signicantly larger than the time taken to reach steady
state in the lysis experiments In order to make comparisons
with the irreversible mechanism we evaluated the number of
pores at the threshold concentration of 544 nM It is observed
that the number of pores required for lysis initially increases
and then decreases for increments in R values The distribution
of oligomers at steady state are shown for diff erent R values in
Fig 9a and b for C in frac14 147 nM The steady state concentrations
of lsquomersrsquo change from a predominantly 11-mer concentration to
a predominant 1-mer concentration as R is varied between0 and 1 For R lt 1 we nd that the concentration of pores (12
mers) are signicantly higher than the intermediate lsquomerrsquo
concentrations shown in Fig 9b and range from 0142 nmol
Fig 9 Oligomer distribution as a function of the ratio of forward and
backward time constants R frac14 sfsb for C in frac14 147 nM (a) R varies
between 0 and 1 (b) R varies between 0 and 01 (c) Hemoglobin
release data as a function of toxin concentration is shown forvarious R
R frac14 0 has the closest agreement with the experimental data (open
squares)
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m2 ( R frac14 0) to 028 nmol m2 ( R frac14 001) For R frac14 1 the
concentration of 12 mers is 024 nmol m2 and decreases with a
further increase in R The predictions using reversible sequen-
tial kinetics for the H max vs C in data (Fig 9c) indicate
greater deviation from the experimental data when compared
with R frac14 0 These results indicate that the irreversible mecha-
nism provides the best agreement with the experimental data
We point out that the critical number of pores npc in
the reversible framework is non-monotonic ranging from 392( R frac14 0) to 5128 ( R frac14 1) for l frac14 2
54 Non-sequential oligomerization
In contrast to the 392 pores obtained for the critical number of
pores via the sequential mechanism a substantially larger
critical number of pores are observed via the non-sequential
oligomerization (6000 pores per cell) The mass balance for
the protomer ( p1) concentration remains identical to that of the
sequential oligomerization mechanism (eqn (4)) The governing
equations for pl (l lt l n) oligomer undergoing non-sequential
irreversible oligomerization are
d pl
dt frac14
1
2k l Xl 1
ufrac141
pu pl u eth1 dl 12THORNXn1
ufrac141
k l pl pu for 1l n (18)
where n represents the number of monomers in a pore The
relation between the critical number of pores and npc for non-
sequential oligomerization is npc frac14 5439l015 The critical
number of pores for l frac14 2 is 6035 pores which is about 20 times
greater than that obtained from a sequential oligomerization
mechanism In this scheme the hemoglobin release (Fig 10a)
data is overpredicted at intermediate toxin concentrations when
compared with the sequential oligomerization The predictions
of the H out vs time data at 147 nM (Fig 10b) are similar whencompared with the sequential oligomerization (Fig 6b and d)
A comparison of the tted parameters between the sequential
and non-sequential oligomerization mechanisms are given in
Table 2
Oligomer distributions obtained from the non-sequential
mechanism (Fig 11b) show an entirely diff erent trend when
compared to that obtained from the sequential mechanism
(Fig 8) In the sequential mechanism the higher lsquomersrsquo
attained a steady state once the protomer was depleted In the
non-sequential mechanism the contribution to the dodeca-
mers (12 mer) can be obtained from a large number of combi-
nations of the lower oligomers leading to a larger dodecamer orpore concentration The time scale required to attain the olig-
omer steady state concentration (Fig 11a) is about 20 s for an
initial concentration of 147 nM Similar time scales are
observed in the sequential mechanism as well
Further experiments are required to distinguish between the
various mechanisms Western Blot experiments conducted on
Hemolysin E12 and Clostridium perfringens 3 toxins31 showed the
presence of intermediate oligomers On the other hand single-
molecule uorescence imaging of a-hemolysin on a droplet
interface bilayer showed the presence of only monomers and
heptamers (pores)32
6 Discussion and conclusions
Lysis experiments on RBCs with the ClyA pore-forming toxin
show that a threshold initial toxin concentration is required to
initiate lysis From this observation we analyzed the problem in
two regimes a low toxin concentration regime where rupture of
cells is absent and a high toxin concentration regime where
lysis occurs and hemoglobin is released Kinetic models which
accounts for monomer binding conformation (membrane-
bound monomer to protomer) and oligomerization to form the
dodecameric pore complex are developed Models which
account for sequential and non-sequential oligomerization are
tested Cell rupture is assumed to be
rst order in the number of live cells and directly proportional to the pores in excess of the
critical number of pores npc In the sublytic regime the number
of pores is found to have a power law dependence on l which is
the ratio of conformational time to the reaction time This leads
to the construction of a ldquophase diagramrdquo between the number
of pores np and l for diff erent values of the initial toxin
concentration Comparing simulations with experimental data
the range of npc was 392ndash768 for the sequential mechanism and
5300ndash6300 pores for the non-sequential mechanism for 2 l 4
The range of l values is consistent with available experimental
data on ClyA which indicates that the membrane-bound
Fig 10 (a) Model predictions from the non-sequential mechanism
Simulated Hmax C in (open circles) curves grossly overpredicts the
experimental data (open squares) (b) Simulated hemoglobin release
compares well with the experimental data Dashed line snetfrac14 1 s solid
line snet frac14 24 s
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conformational step is slower than the preceding adsorption
and subsequent oligomerization steps10
From the model we are also able to comment on the time
constants for the various processes The time constant for poreformation is about 1 s indicating that pore formation itself is a
fast process relative to the time taken for the pore population on
a single RBC to reach steady state which is about 20ndash30 s Since
rupture kinetics is dynamic and depends on the fraction of live
cells as well as the number of pores on the cell lysis occurs in
the time scale of 10s of minutes Due to this separation of time
scales lysis is seen to occur once the number of pores has
reached a steady state This steady number of pores ranges from
515 to 11 657 as the toxin concentration ranges from 588ndash
147 nM For the non-sequential mechanism the critical
number of pores required to initiate lysis is about 20 times
higher when compared to that of sequential oligomerizationComparison of three diff erent kinetic models reveals that the
irreversible sequential kinetics provides the closest match with
the hemoglobin released as a function of the initial toxin
concentration Although we observe an overprediction of the
hemoglobin release data with the non-sequential mechanism
the hemoglobin release kinetics are similar to that of the
sequential mechanism The distribution of lower oligomers is
distinctly diff erent in both cases with a negligible numbers of
lower mers observed in the non-sequential oligomerization
These diff erences in the distribution of mers off er a ngerprint
to identify the underlying mechanism for pore formation
Experiments which can determine the number of pores or the
steady state lsquomerrsquo distributions on the membrane surface will
shed light on the pathways for oligomerization and enable a
more denite conclusion of the underlying kinetics Furtherthe rupture kinetics model contains only one adjustable
parameter Lysis experiments conducted with Vibrio cholerae El
Tor cytolysin33 and Monalysin34 show similar lysis times (in the
order of 10s of minutes) as observed in our study suggesting
similarities in the underlying kinetic pathways that lead to pore
formation and rupture The model developed in this manu-
script is generic and could be recast with some variation to
study the dynamics of other PFTs
We brie y discuss some of the limitations of the model in its
present form The model is based on the mean eld approxi-
mation where all cells are assumed to be identical and for the
purpose of binding and oligomerization this is an adequatestarting point The more complicated process is the mechanics
of rupture with the correct functional dependence on the pore
density In general there could exist a distribution of cells with
diff erent densities of pores Preliminary experiments by varying
the number of RBCs at a xed toxin concentration led to an
increase in lysis suggesting that cell heterogeneity could be
playing a role Although a population balance model35 could
include these variations this is at an added cost of complexity A
second aspect inherent to the model is the presence of lysis
beyond the time at which steady state is observed in the
experiments (30 minutes) Once the number of pores has
Fig 11 Oligomerconcentrationsas a function of time areshownfor (a) 544 nM and(b) 147 nMSteady state oligomer concentrations areshown
for (c) 544 nM and (d) 147 nM At steady state 1ndash4 mer concentrations are zero for both 544 and 147 nM The 12-mer concentration at steady
state is quite large compared to the other oligomer concentrations present in the system
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reached a steady state which occurs within 30 s (Fig 7a) cell
lysis continues to occur at a xed number of pores proportional
to np npc Running the simulation to steady state would
eventually lead to lysis of all the cells in the system albeit at an
exceedingly slow rate at the lower and intermediate toxin
concentrations We have also assumed complete and irrevers-
ible binding of the monomer to the membrane and hence the
number of pores predicted represent an upper limit within the
proposed kinetic framework Despite the mean eld approxi-mation the model proposed in this manuscript is able to
capture the inherent time scales in the process of pore forma-
tion and rupture as well as predict the variation of hemoglobin
release as a function of the initial toxin concentration observed
in the experiments We nally point out that the eff ect of
temperature on the kinetics and lysis has not been investigated
in this work Preliminary experiments at 147 nM toxin
concentration carried out to steady state indicated a marked
drop in lytic activity for temperatures below 15 C and lysis was
not observed at 10 C Data on binding isotherms would be
required to understand temperature eff ects in these systems
Our work has implications in optimizing dosage and devel-oping novel drug delivery strategies based on PFTs The obser-
vation that lysis occurs in a given window of toxin concentration
has implications while developing PFT-based drug and gene
therapy protocols In recent studies E coli used in conjunction
with radiation therapy was found to retard the growth of cancer
tumors when compared with only radiation therapy protocols36
Quantifying the dynamics and extent of lysis is an important
aspect of developing appropriate treatment protocols in these
combination therapies In other applications where pores are
used for gene delivery and in the development of articial cells
lysis must be prevented and pore formation restricted to
concentrations below the lysis threshold
Acknowledgements
We acknowledge funding from Department of Science and
Technology (DST) India under the IRHPA grant The authors
thank Sanjeev Kumar Gupta and Jaydeep Basu for several useful
discussions on the development of the model as well as the
reviewers for their critical comments
References
1 S Bhakdi and J Tranum-Jensen Microbiol Rev 1991 55
733ndash
7512 J Thiery D Keefe S Boulant E Boucrot M Walch
D Martinvalet I S Goping R C Bleackley
T Kirchhausen and J Lieberman Nat Immunol 2011 12
770ndash777
3 A Ludwig C von Rhein S Bauer C Huttinger and
W Goebel J Bacteriol 2004 186 5311ndash5320
4 A W Bernheimer and L S Avigad Proc Natl Acad Sci U S
A 1976 73 467ndash471
5 M V Sousa M Richardson W Fontes and L Morhy
J Protein Chem 1994 13 659ndash667
6 G Anderluh and P Macek Toxicon 2002 40 111ndash124
7 I Iacovache F G van der Goot and L Pernot Biochim
Biophys Acta Biomembr 2008 1778 1611ndash1623
8 N R Wyborn A Clark R E Roberts S J Jamieson
S Tzokov P A Bullough T J Stillman P J Artymiuk
J E Galen L Zhao M M Levine and J Green
Microbiology 2004 150 1495ndash1505
9 A J Wallace T J Stillman A Atkins S J Jamieson
P A Bullough J Green and P J Artymiuk Cell 2000 100
265ndash27610 M Mueller U Grauschopf T Maier R Glockshuber and
N Ban Nature 2009 459 726ndash730
11 J Oscarsson Y Mizunoe L Li X-H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
12 A Atkins N R Wyborn A J Wallace T J Stillman
L K Black A B Fielding M Hisakado P J Artymiuk and
J Green J Biol Chem 2000 275 41150ndash41155
13 A Ludwig G Volkerink C von Rhein S Bauer E Maier
B Bergmann W Goebel and R Benz J Bacteriol 2010
192 4001ndash4011
14 N Eier M Vetsch M Gregorini P Ringler M Chami
A Philippsen A Fritz S A Muller R Glockshuber A Engel and U Grauschopf EMBO J 2006 25 2652ndash2661
15 L A Shepard O Shatursky A E Johnson and R K Tweten
Biochemistry 2000 39 10284ndash10293
16 A J Ratner K R Hippe J L Aguilar M H Bender
A L Nelson and J N Weiser J Biol Chem 2006 281
12994ndash12998
17 I Iacovache M Bischoerger and F G van der Goot Curr
Opin Struct Biol 2010 20 241ndash246
18 G Schwarz and C H Robert Biophys Chem 1992 42 291ndash296
19 G Schwarz R-t Zong and T Popescu Biochim Biophys
Acta Biomembr 1992 1110 97ndash104
20 D Bolintineanu E Hazrati H T Davis R I Lehrer and
Y N Kaznessis Peptides 2010 31 1ndash821 J M Alam T Kobayashi and M Yamazaki Biochemistry
2012 51 5160ndash5172
22 J Lemiere K Guevorkian C Campillo C Sykes and T Betz
So Matter 2013 9 3181ndash3187
23 V Noireaux and A Libchaber Proc Natl Acad Sci U S A
2004 101 17669ndash17674
24 M M Bradford Anal Biochem 1976 72 248ndash254
25 H Xu E K Bjerneld M Kall and L Borjesson Phys Rev
Lett 1999 83 4357ndash4360
26 J Oscarsson Y Mizunoe L Li X H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
27 T M Raschke J Tsai and M Levitt Proc Natl Acad SciU S A 2001 98 5965ndash5969
28 L A Rodrıguez-Guadarrama S K Talsania K K Mohanty
and R Rajagopalan Langmuir 1999 15 437ndash446
29 C E McLaren G M Brittenham and V Hasselblad Am J
Physiol 1987 252 3181ndash3187
30 J G Kim M Xia and H Liu IEEE Eng Med Biol Mag 2005
24 118ndash121
31 M Nagahama H Hara M Fernandez-Miyakawa
Y Itohayashi and J Sakurai Biochemistry 2006 45 296ndash302
32 J R Thompson B Cronin H Bayley and M I Wallace
Biophys J 2011 101 2679ndash2683
This journal is copy The Royal Society of Chemistry 2014 RSC Adv 2014 4 4930ndash4942 | 4941
Paper RSC Advances
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7252019 c3ra45159c
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33 A Zitzer I Walev M Palmer and S Bhakdi Med Microbiol
Immunol 1995 184 37ndash44
34 O Opota I Vallet-Gely R Vincentelli C Kellenberger
I Iacovache M R Gonzalez A Roussel F G van der Goot
and B Lemaitre PLoS Pathog 2011 7 e1002259
35 S Kumar and D Ramkrishna Chem Eng Sci 1996 51
1311ndash1332
36 S N Jiang T X Phan T K Nam V H Nguyen H S Kim
H S Bom H E Choy Y Hong and J J Min Mol Ther
2010 18 635ndash642
RSC Advances Paper
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the small eff ective pore diameter available for transport The
inner pore diameter exposed to the cytosol is 4 nm in the crystal
structure and in a fully solvated environment the eff ective
diameter is expected to decrease further Further osmoticprotection assays of ClyA conclude that the eff ective pore sizes
range from 20ndash35 nm26 The solution is assumed to remain
isotonic as lysis proceeds since lysis did not occur in the
absence of toxin when RBCs were incubated in buff er solution
made up of fully lysed (sonicating 1 RBC (vv)) RBCs We have
assumed that the conformational change follows rst order
irreversible kinetics since the conformational step involves a
transition from a water-soluble monomer to a membrane-
inserted protomer via a series of conformational changes in the
regions around the b-tongue region of the monomer and the N-
terminus10 This is succeeded by a fast oligomerization step to
form the pore complex
The rate equation for the membrane-bound monomer
whose surface molar concentration is denoted as m is
dm
dt frac14 k aC m
ms m
Xnl frac141
pl
k dm k cm (1)
where k a is the adsorption rate constant k d is the desorption
rate constant ms is the saturated surface molar concentration
and the last term represents the rate at which the membrane-
bound monomer (m) undergoes a conformational change
to the membrane-bound protomer ( p1) with a rate constant
k c and pl is the surface molar concentration of the oligomer
containing l -mers If the bulk concentration of the toxin
monomer is constant eqn (1) is similar in form to the
LangmuirndashHinshelwood equation traditionally used to
describe the concentration of surface species undergoing
both adsorption and reaction
Since the initial toxin concentration (C in) in the aliquots is in
the range of 294ndash147 nM an additional balance is used to
describe the concentration change of toxins in solution This
yields
V sol
dC m
dt frac14
k dm k aC m
ms m
Xnl frac141
pl
ARBCN RBC (2)
where V sol denotes the volume of solution in the aliquot ARBC is
the area of a single RBC and N RBC is the number of erythrocytes
present in V sol at any instant
32 Oligomerization kinetics
Oligomerization involves the formation of dimers trimers and
higher mers from the protomer until an n-mer complex (pore)
is formed Data obtained from scanning transmission electronmicroscopy (STEM) and single-wavelength anomalous diff rac-
tion (SAD) indicate that the Cytolysin A (ClyA) pore complex
consists of n frac14 12 and 13 mers respectively1014 Oligomerization
can occur in a number of distinct kinetic pathways In Fig 2
the two main mechanisms are illustrated In the sequential
mechanism the nth mer is formed by the addition of a 1 mer to
a (n 1) mer complex In the non-sequential mechanism the
nth mer can be formed by allowed integer combinations of the
smaller mers As an example a 4 mer can be formed by a
combination of 2 + 2 mers as well as a 3 + 1 mers as illustrated
in Fig 2
If oligomerization occurs sequentially and irreversibly thereaction mechanism is
p1 thorn pl k l pl thorn1 l frac14 1 n 1 (3)
where k l is the reaction rate constant for the l th oligomerization
step If the l th oligomer is formed in an irreversible non-
sequential process the reaction mechanism is
pr thorn pl rk l pl l frac14 2 n
r frac14 1 to l =2 l even
1 to ethl 1THORN=2 l odd
for the formation of the l th oligomer In the above non-
sequential mechanism for oligomerization the number of
distinct reaction rate constants for the formation of a 12-mer
pore complex is 66 In what follows we develop the model for
the irreversible sequential mechanism We are unaware of any
Fig 1 Schematic indicating the various steps leading to pore forma-
tion The water-soluble monomer adsorbs onto the cell membrane
and undergoes a conformational change to form the membrane-
bound protomer This is followed by an oligomerization step to form
the dodecameric pore complex
Fig 2 Two possible modes of oligomerization (top) sequential olig-
omerization and (bottom) non-sequential oligomerization are shown
In sequential oligomerization a protomer is necessary for the forma-
tion of a higher oligomer whereas in the non-sequential mechanism a
higher oligomer can be formed from an allowed combination of lower
oligomers
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experiments which shed light on either of these mechanisms
Results for the reversible sequential mechanism and non-
sequential irreversible kinetics are presented later in the text
In the sequential mechanism there are 11 rate constants
Molecular simulation of hydrophobic association of small
solutes in water27 reveal that a sequential aggregation procedure
is favored during cluster formation The sequential aggregation
mechanism is also used for modelling micellar aggregation28
We assume that all the rate constants for the sequentialmechanism are identical This assumption is widely used in
sequential polymerization reactions With the assumption that
all rate constants for oligomerization (k l ) are identical a balance
on the protomer yields
d p1
dt frac14 k cm k l
Xn1
l frac141
p1 pl (4)
where the rst term on the right hand side represents the
formation of the protomer from the monomer and the other
terms represent sequential oligomerization steps wherein the
protomer binds with the other lsquomersrsquo to form the higher lsquomersrsquo
with a rate constant k l From rate considerations for dimerformation (l frac14 2) a prefactor of 12 appears in the term which
corresponds to the formation of the dimer This is a necessary
condition for satisfying the species mass balance The govern-
ing equation for the dimer (l frac14 2) is given by
d p2
dt frac14
1
2k 2 p1 p1 k 2 p1 p2 (5)
where p2 represents the concentration of dimer and k 2 repre-
sents the rate constant for the reaction The equation for the
formation of the l th oligomer (l gt 2) is
d pl
dt frac14 k l p1 pethl 1THORN eth1 dl 12THORNk l p1 pl l frac14 3 12 (6)
where d l 12 represents the Kronecker delta function The corre-
sponding number of pores per RBC is obtained using
np frac14 p12N avARBC (7)
where N av is the Avogadro number
33 Cell lysis
Every dodecamer corresponds to a stable pore in the
membrane If the rate at which lysis occurs is directly propor-
tional to the number of cells that are present at any instant of
time then cell lysis follows a rst order process On physical
grounds we further assume that cell lysis occurs only when the
number of pores exceeds a critical number of pores in each cell
Since our experiments are carried out under isotonic condi-
tions lysis is associated with rupture Lysis can be described
using the following rst order process
dx
dt frac14 klxR
np npc
(8)
where
R frac14
0 npnpc
np npc np $npc
x represents the fraction of unlysed cells at any instant of time
np is number of pores per RBC at any instant and npc is the
critical number of pores per RBC above which cell lysis occurs
The constant kl represents the decay rate constant for cell lysis
In eqn (8) the ramp function R(np npc) incorporates the
increased lysis as a function of the excess pores np npc We
also investigate other functional forms for R such as a unit step
function and a higher power dependence on np npc The
inuence of these on the model predictions are discussed laterin the text As cells lyse the number of RBCs N RBC at any
instant is
N RBC frac14 N inRBCx (9)
N inRBC is the initial number of RBCs
The rate at which hemoglobin is released from the RBCs into
solution is
dH out
dt frac14 V hrhN
inRBCkl xR
np npc
thornDhAp
ms
h H out
N RBCnp
V soll p
(10)
where the rst term represents the contribution due to lysis
(rupture) and the second term is the diff usive ux contribution
from the pores of unlysed cells In the above equation V h is the
volume of hemoglobin present in a single RBC rh is the density
of hemoglobin Dh is the diff usivity of hemoglobin l p is the
diff usion length along the pore Ap is the average area of a pore
msh H out and rh represents the saturated hemoglobin mass in
one RBC amount of hemoglobin present in the solution at any
instant of time and the density of hemoglobin respectively
4 Solution procedure While analyzing the problem it is useful to recast the equations in
suitable dimensionless forms If t frac14 t s C m frac14 C mC in m frac14 mms
and pl frac14 pl ms then eqn (1) (2) and (4)ndash(6) in dimensionless forms
are
dm
dt frac14
s
sa
C m
1 m
Xnl frac141
pl
s
sd
m s
sc
m (11a)
dC m
dt frac14
ARBCN RBCm
s
V solC inb
s
sd
m s
sa
C m
1 m
Xnl frac141
pl
(11b)
d p1
dt frac14
s
sc
m s
sl
Xn1
l frac141
p1 pl (11c)
d p2
dt frac14
s
2s2
p1 p1 s
s2
p1 p2 (11d)
d pl dt frac14
s
sl
p1 pethl 1THORN eth1 dl 12THORNs
sl
p1 pl l frac14 3 12 (11e)
From eqn (11a)ndash(11e) we can extract the following set of time
constants
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sa frac14 1
k aC inis the adsorption time constant
sd frac14 1
k dis the desorption time constant
sc frac14 1
k cis the conformational time constant
sl frac14 1
k l ms is the reaction time constant
s frac14 1
k aC in thorn k c thorn k d
(12)
41 Model parameters
Since we do not have experimental data to independently
determine the various time constants it is more convenient to
dene a ratio between time constants We dene the ratio
between conformational and adsorption times as
l frac14sc
sa
(13)
It has been observed that the conformational times sc arelarger than the time for membrane binding and oligomeriza-
tion1014 suggesting that l gt 1 We can also rewrite the adsorption
time constant and conformational time constant in terms of l
provided we have an estimate of the time required for initiation
of pore formation Since the processes leading to pore forma-
tion occur in series the total time constant snet for pore
formation is the sum of the time constants for the individual
steps
snet frac14 sc + sa + stl + sd (14)
Using eqn (12) and (13) and with the added assumption that
desorption rate is negligible and the adsorption and reaction
time constants are similar (sa st
l frac14 (n 1)sl ) the constants k aand k c can be expressed in terms of l Hence
k a frac14 2 thorn l
snetC in(15a)
and
k c frac142 thorn l
lsnet
(15b)
Under these assumptions for a xed initial concentration of
toxins and cell mass specifying l snet and C in is sufficient to
make predictions for the rate at which pores are formed in thesublytic regime With these assumptions eqn (11) can be
expressed solely in terms of the constant l The values of various
system properties used in the simulation are given in Table 1
and the values of diff erent parameters are given in Table 2 For
snet the model predictions were tested for a range of values as
indicated The parameters related to the pore geometry radius
of the pore r p and length of the pore l p are obtained from the
crystal structure of the ClyA pore10 The initial number of RBCs
are counted using the hemocytometric technique Typical
liquid diff usivities are used for hemoglobin Since the diff u-
sivity only inuences the hemoglobin release in the sublytic
regime obtaining a precise value of the diff usivity is not of
special consequence
42 Simulation details
We used an explicit Euler scheme for discretizing the governing
ordinary diff erential equations and the equations were solved
with a reduced time step of 0005 (0016 s l frac14 4) Calculations
performed with a reduced time step of 0001 did not alter the
reported results We developed an in-house program using
Matlab 70 to solve the discretized equations Calculations werechecked with a mass balance on the monomers
5 Results and discussions51 Lysis experiments
The OD data from lysis experiments at 60 minutes are illus-
trated in Fig 3a as a function of the bulk monomer toxin
concentration C in The data represents an average over 5
independent experiments Based on the time evolution (Fig 3b)
data no further lysis was observed above 30 minutes for all the
toxin concentrations investigated in the study Hence data at 30
minutes is expected to represent the steady state in the systemFrom the cell counts in the hemocytometer we nd 98 lysis at
147 nM and about 10ndash15 lysis at 588 nM The data clearly
reveals that the RBC lysis occurs only above the critical toxin
concentration (npc) which we estimate at 544 nM Below this
critical concentration lies the regime of low hemolytic activity
where the absolute OD values are an order of magnitude below
values obtained with lysis indicating that leakage from pores is
not signicant Although pore formation occurs in this regime
the concentration of pores is not sufficient to initiate lysis To
further support this hypothesis we carried out turbidity assay
experiments (Fig 4) for initial toxin concentrations ranging
Table 1 Various system properties and parameters used in the
simulation In some cases only the range of parameters that were
tested are given
Area of RBC29 ( ARBC) 136 mm2
Volume of RBC29 (V h) 90 fLRadius of pore10 (r p) 35 nmLength of pore10 (l p) 13 nmDiff usivity of hemoglobin ( Dh) 109 m2 s1
Initial number of RBC ( N inRBC) 32 107 cells per ml Volume of lysis assay (V sol) 1 mlSaturated surface concentration (ms) 109 mol m2
Net reaction time constant (snet ) 1ndash25 s
Table 2 Values of constants obtained from sequential and non-
sequential oligomerization
ParameterSequentialoligomerization
Non-sequentialoligomerization
l 2ndash4 2ndash4npc 392ndash768 pores 5300ndash6300 poresDecay rate
constant (kl )
15ndash18 107 s1 125ndash135 107 s1
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from 294ndash294 nM The decrease in turbidity is seen to occur
simultaneously with an increase in the OD supporting the view that the increase in OD is due to cell lysis The turbidity
decrease is also mirrored with the corresponding OD data Since
the cell mass used in the turbidity experiments is lower than
that used in the lysis experiments sublytic toxin concentrations
lie below 294 nM
52 Model predictions
In this section model predictions for the sequential irreversible
kinetics are compared with the lysis data
521 Sublytic regime
In the sublytic regime (np lt npc) the evolution of np with time is
obtained by solving eqn (11a)ndash(e) The number of pores np is
obtained from eqn (7) From the simulations we observe that the number of pores per RBC for a given initial toxin concen-
tration saturates within 10 s (Fig 5a) This saturation in np is
due to the limiting amount of toxin present in solution In
Fig 5b we plot the variation in np at saturation (20 s) with
l frac14sc
sa for diff erent initial toxin concentration assuming that
np lt npc The curve corresponding to 544 nM is tted to the
form y frac14 axb and the relation npc frac14 20033l097 is obtained
which can be used to x the value of npc for a given value of l In
order to simulate the lysis data a value of snet (eqn (14)) which
is the time constant associated with the time required for
formation of the rst pore has to be specied Initial estimates
are in the range of 1ndash25 s and we use a value of 1 s in all our
simulations unless specied We show later that our results are
Fig 3 (a) The normalized optical density values as a function of the
initial toxin concentration observed after 60 minutes during the lysis
experiments A distinct jump is observed above a toxin concentration
of 544 nM (b) Time evolution data of optical density values during
RBC lysis At 147 nM 98 lysis is observed
Fig 4 Turbidity (left axis) and lysis data (right axis) show that the
decrease in turbidity occurs simultaneously with an increase in the
OD The sublytic initial toxin concentration is less than 294 nM
(100 ng ml1)
Fig 5 Relation between l and npc in the sublytic regime (a) Number
of pores per RBC as a function of time The data plotted are for l frac14 2
and snet frac14 1 s (b) A regression analysis of the curve at 544 nM (dash-
dotted line) yields the relation npc frac14 20033l097
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relatively insensitive to the value of snet in this range The
parameters l npc and snet are obtained in the sublytic regime as
discussed above
522 Lysis regime
In the lysis regime in addition to the kinetic eqn (11andashe) we also
solve the lysis and hemoglobin release equations eqn (8) and
eqn (10) Cell lysis occurs once np gt npc and the OD increases with increasing toxin concentration as shown in Fig 3a Once
values of l npc and snet are xed in the sub-lytic regime the only
unknown parameter in the model is the value of kl (eqn (8)) In
all cases kl is xed by matching the maximum extent of lysis of
98 obtained at 147 nM Subsequent simulations are run with
diff erent values of C in to compare with the experimental data
Fig 6 illustrates the comparison between the model prediction
and the experimental OD data Since the OD varies linearly with
the amount of hemoglobin released during lysis30 we scale both
the experimental and predicted data by their respective
maximum OD to facilitate a meaningful comparison We also
carried out independent lysis experiments to verify the linear
relationship between the OD versus hemoglobin data Hence we
normalize the long time data and dene H max as the ratio of the
mass of the steady state hemoglobin released at a given C in to
the corresponding value at C in frac14 147 nM which is the highest
C in considered in the study This facilitates a comparison of
H max predicted from the model directly with the normalized OD
values For l frac14 2 the corresponding npc frac14 392 and the
comparison of H max at a value of kl frac14 18 107 s1 is
illustrated in Fig 6a The comparison for lfrac14 4 (npcfrac14 768) and kl
frac14 15 107 s1 is illustrated in Fig 6c Comparison of the
hemoglobin release dynamics ( H out vs time) for the corre-
sponding set of parameters are illustrated in Fig 6b and d
respectively
We observe that in this range of l (2 l 4) values and
kl 15ndash18 107 s1 the OD vs C in data is captured quite
accurately Since l is the ratio of the ClyA monomer confor-
mation time to the monomer adsorption time this range of l value is consistent with experiments10 which indicates that
conformation is preceded by fast adsorption followed by rapid
oligomerization The H out dynamics predicted by the model is
seen to capture the experimental data quite well (Fig 6b and d)
Upon increasing l we nd that a lower value of kl is required to
match the OD vs C in data (Fig 6d) A value of snet frac14 1 s captures
the early time release in the H out data at 147 nM quite accu-
rately and increasing snet to 24 s results in a short delay at early
times Since snet represents the time taken to form the rst pore
in situ monitoring of the hemoglobin release dynamics would be
required to determine snet more precisely In our experiments the
time evolution of the lysis data is carried out by intermittently arresting lysis at diff erent times and quenching the aliquots in
ice for a period of 3ndash5 minutes while the OD is determined
Experiments carried out continuously for the diff erent time
points shown in Fig 6b and d did not alter the data obtained
from the intermittent experiments Fluorescence permeation
experiments by Yamazaki and co-workers21 by lysenin (334 kDa)
induced pore formation on single giant unilamellar vesicles show
that pore formation is complete within about 10 s for toxin
concentrations of 200 ng ml1 and 50 s at 40 ng ml1 These are
similar to the time scales deduced in our model
At a toxin concentration of 147 nM a steady distribution of
mers is achieved on the time scale of 10ndash20 s (for a snet frac14 1 s)
Within this time interval the pore density goes through a phaseof rapid increase to exceed the critical pore density (Fig 7)
during which very little lysis is observed (Fig 6a and d) Lysis is
predominantly observed a er a steady number of pores have
formed on the RBC surface This steady number of pores ranges
Fig 6 Comparison of model predictions (open circles) with experi-
mental results (open squares) The amount of hemoglobin released
(Hmax) atthe end of30min for(a) npcfrac14 392 (c) npcfrac14 768 Dynamics of
hemoglobin released (Hout) for (b) npcfrac14 392 (d) npcfrac14 768 A value of l
between 2 and 4 is seen to accurately capture the Hmax versus C in data
[(a) and (c)] The amount of hemoglobin released is scaled with the
maximum amount to facilitate a comparison Simulations corresponding
to snet frac14 24 s are shown in bold lines and snet frac14 1 s in dashed lines
Fig 7 The pore density is plotted as a function of time in thepost-lysis
regime The number of pores (np) per RBC ranges from 515 at 588 nM
to 11 657 at 147 nM l frac14 2 and kl frac14 18 107 s1 npc frac14 392
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from 515 at 588 nM to 11 657 at 147 nM Given this situation it
is instructive to dene an eff ective lysis time constant kleff frac14
kl (np npc) where np is the steady state value of the number of
pores at a given value of C in (Fig 7) The value of kleff at 147 nM
is 2027 103 s1 which results in an eff ective lysis time
constant of 493 s
523 Oligomer and pore concentration
In both the low (lt544 nM) and high toxin ($544 nM) regimes
the amount of toxin is found to be limiting Even at the highest
toxin concentration C in frac14 147 nM the toxin in bulk solution is
depleted within 10 s Selected oligomer concentrations as a
function of time are plotted in Fig 8a and b for both high and
low toxin concentrations as predicted by the kinetic model (eqn
(4) and (6)) The pore density is illustrated in Fig 7 The gov-
erning equations for the formation of an l -mer are given in eqn
(6) Since pore formation occurs via a sequential oligomeriza-
tion mechanism a protomer (1-mer) is necessary for the
formation of all other l mers Hence a steady monomer
concentration on the RBC is achieved once the 1-mer concen-
tration reduces to zero and lag times are observed for theformation of the l + 1-mer From the distribution of mers on the
membrane we observe that a large fraction of protomers remain
trapped as intermediate mers on the membrane At 147 nM the
number of monomers per ml is 8854 1012 The initial
number N RBC frac14 32 107 the monomers per RBC is 276
105 If all the monomers were converted to pores each RBC
would have 23 105 pores However the number of pores
formed per RBC at 147 nM is 11657 105 pores (Fig 7) indi-
cating that about 50 of the mers remain on the membrane
surface as intermediate n-mers (n frac14 1ndash11) At a sublytic
concentration of 544 nM only about 45 of the mers are
converted to pores resulting in 375 pores per RBC (Fig 5a)
The eff ect of l is more prominent at the higher toxin
concentration where both the life time and the maximum
concentration for 1-mers is found to decrease as l is increased
from 2 to 4 The steady state concentrations of various n-mers aspredicted by the model are illustrated in Fig 8c and d A change
in the value of l results in a shi in the distribution for a
particular initial toxin concentration An increase in l implies an
increase in the conformational time relative to the adsorption
and reaction times Hence as l is increased occurrence of the
lower mers on the surface decreases due to the faster reaction
time scales relative to conformation At low bulk toxin concen-
trations (Fig 8c) the distribution of higher mers and conse-
quently the number of pores (12 mers) is very low due to the
limited supply of monomers in the system However at higher
concentration (Fig 8d) the number of monomers is no longer
the limiting factor and the distribution shi
s towards the highermers thereby increasing the number of pores on the surface
524 Parameter sensitivity
We brie y summarize the results of simulations carried out to
test the inuence of the estimated parameters on the model
predictions In the absence of monomer membrane binding
equilibria the value of saturated surface concentration ( ms) is
Fig 8 Model predictions for oligomer concentration pro1047297les ( pl) for l frac14 2 and kl frac14 18 107 s1 (a) Initial bulk concentration frac14 544 nM (b)
Initial bulk concentration frac14 147 nM Steady state distribution of oligomers for (c) C in frac14 544 nM (d) C in frac14 147 nM
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unknown In order to test the inuence of ms on the model
prediction we carried out a few simulations for ms frac14 1 108
mol m2 and msfrac14 1 1010 mol m2 for various C in values For
ms frac14 1 108 mol m2 a negligible number of pores were
formed and the H max ndashC in data (Fig 3a) is underpredicted At
ms frac14 1 1010 mol m2 pore formation was extremely rapid
and little variation in pore density between C in frac14 882 nM and
147 nM was observed As a consequence H max ndashC in data (Fig 3a)
is grossly overpredicted Hence a value of ms frac14 1 10
9 molm2 was used in the simulations (Fig 6) We further note
that the amount of saturated surface concentration ms
implicitly changes the reaction rate constant k l (eqn (12))
Increasing ms eff ectively decreases the reaction rate constant
(eqn (12))
Once snet is xed npc is related to l through the relation npcfrac14
alb with the constants a and b being xed for a given initial toxin
concentration C in (Fig 5) We have found that 2 l 4 ts the
hemoglobin release data very closely (Fig 6) and although the
hemoglobin released as a function of time is slightly under-
estimated by the model the agreement is reasonable Upon
increasing l
to 7 and keeping m
s
frac14 10
9
mol m
s1
the corre-sponding value of npc is 1349 With a value of kl frac14 13 107 s1
although H max versus C in data is accurately predicted the
hemoglobin versus time data is grossly underpredicted Varying
ms between 108 and 1010 mol m2 further deteriorated the
prediction Finally we point out that other functional forms of the
dependence on np npc in the cell lysis equation eqn (8) such as
the unit step function or a quadratic dependence (np npc)2 only
overestimated the H max versus C in data
53 Sequential oligomerization with reversible kinetics
In the previous discussion we present the results for the oligo-
merization kinetics which was assumed to be irreversible Thesequential oligomerization model with reversible kinetics is
p1 thorn pl ) k f
k b pl thorn1 l frac14 1 n 1 (16)
where k f and k b represent the forward and backward reaction
rate constants The kinetic equations are
d p1
dt frac14 k cm k f
Xn1
l frac141
p1 pl thorn k bXn1
l frac142
pl (17a)
d p2
dt frac14
1
2k f p1 p1 k f p1 p2 thorn k b p3
1
2 p2 (17b)
d pl
dt frac14 k f p1 pethl 1THORN eth1 dl 12THORNk f p1 pl
thorn k beth1 dl 12THORNetheth1 dl 11THORN pl thorn1 pl THORN l frac14 3 12 (17c)
A reversible time constant can be dened from eqn (17) as
sb frac14 1
k b The ratio R is dened as the ratio of forward to back-
ward time constants ( R frac14 sf sb) to study the eff ect of revers-
ibility Upon examining the number of pores as a function of
time we observe that the time taken to reach a steady number of
pores is signicantly larger than the time taken to reach steady
state in the lysis experiments In order to make comparisons
with the irreversible mechanism we evaluated the number of
pores at the threshold concentration of 544 nM It is observed
that the number of pores required for lysis initially increases
and then decreases for increments in R values The distribution
of oligomers at steady state are shown for diff erent R values in
Fig 9a and b for C in frac14 147 nM The steady state concentrations
of lsquomersrsquo change from a predominantly 11-mer concentration to
a predominant 1-mer concentration as R is varied between0 and 1 For R lt 1 we nd that the concentration of pores (12
mers) are signicantly higher than the intermediate lsquomerrsquo
concentrations shown in Fig 9b and range from 0142 nmol
Fig 9 Oligomer distribution as a function of the ratio of forward and
backward time constants R frac14 sfsb for C in frac14 147 nM (a) R varies
between 0 and 1 (b) R varies between 0 and 01 (c) Hemoglobin
release data as a function of toxin concentration is shown forvarious R
R frac14 0 has the closest agreement with the experimental data (open
squares)
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m2 ( R frac14 0) to 028 nmol m2 ( R frac14 001) For R frac14 1 the
concentration of 12 mers is 024 nmol m2 and decreases with a
further increase in R The predictions using reversible sequen-
tial kinetics for the H max vs C in data (Fig 9c) indicate
greater deviation from the experimental data when compared
with R frac14 0 These results indicate that the irreversible mecha-
nism provides the best agreement with the experimental data
We point out that the critical number of pores npc in
the reversible framework is non-monotonic ranging from 392( R frac14 0) to 5128 ( R frac14 1) for l frac14 2
54 Non-sequential oligomerization
In contrast to the 392 pores obtained for the critical number of
pores via the sequential mechanism a substantially larger
critical number of pores are observed via the non-sequential
oligomerization (6000 pores per cell) The mass balance for
the protomer ( p1) concentration remains identical to that of the
sequential oligomerization mechanism (eqn (4)) The governing
equations for pl (l lt l n) oligomer undergoing non-sequential
irreversible oligomerization are
d pl
dt frac14
1
2k l Xl 1
ufrac141
pu pl u eth1 dl 12THORNXn1
ufrac141
k l pl pu for 1l n (18)
where n represents the number of monomers in a pore The
relation between the critical number of pores and npc for non-
sequential oligomerization is npc frac14 5439l015 The critical
number of pores for l frac14 2 is 6035 pores which is about 20 times
greater than that obtained from a sequential oligomerization
mechanism In this scheme the hemoglobin release (Fig 10a)
data is overpredicted at intermediate toxin concentrations when
compared with the sequential oligomerization The predictions
of the H out vs time data at 147 nM (Fig 10b) are similar whencompared with the sequential oligomerization (Fig 6b and d)
A comparison of the tted parameters between the sequential
and non-sequential oligomerization mechanisms are given in
Table 2
Oligomer distributions obtained from the non-sequential
mechanism (Fig 11b) show an entirely diff erent trend when
compared to that obtained from the sequential mechanism
(Fig 8) In the sequential mechanism the higher lsquomersrsquo
attained a steady state once the protomer was depleted In the
non-sequential mechanism the contribution to the dodeca-
mers (12 mer) can be obtained from a large number of combi-
nations of the lower oligomers leading to a larger dodecamer orpore concentration The time scale required to attain the olig-
omer steady state concentration (Fig 11a) is about 20 s for an
initial concentration of 147 nM Similar time scales are
observed in the sequential mechanism as well
Further experiments are required to distinguish between the
various mechanisms Western Blot experiments conducted on
Hemolysin E12 and Clostridium perfringens 3 toxins31 showed the
presence of intermediate oligomers On the other hand single-
molecule uorescence imaging of a-hemolysin on a droplet
interface bilayer showed the presence of only monomers and
heptamers (pores)32
6 Discussion and conclusions
Lysis experiments on RBCs with the ClyA pore-forming toxin
show that a threshold initial toxin concentration is required to
initiate lysis From this observation we analyzed the problem in
two regimes a low toxin concentration regime where rupture of
cells is absent and a high toxin concentration regime where
lysis occurs and hemoglobin is released Kinetic models which
accounts for monomer binding conformation (membrane-
bound monomer to protomer) and oligomerization to form the
dodecameric pore complex are developed Models which
account for sequential and non-sequential oligomerization are
tested Cell rupture is assumed to be
rst order in the number of live cells and directly proportional to the pores in excess of the
critical number of pores npc In the sublytic regime the number
of pores is found to have a power law dependence on l which is
the ratio of conformational time to the reaction time This leads
to the construction of a ldquophase diagramrdquo between the number
of pores np and l for diff erent values of the initial toxin
concentration Comparing simulations with experimental data
the range of npc was 392ndash768 for the sequential mechanism and
5300ndash6300 pores for the non-sequential mechanism for 2 l 4
The range of l values is consistent with available experimental
data on ClyA which indicates that the membrane-bound
Fig 10 (a) Model predictions from the non-sequential mechanism
Simulated Hmax C in (open circles) curves grossly overpredicts the
experimental data (open squares) (b) Simulated hemoglobin release
compares well with the experimental data Dashed line snetfrac14 1 s solid
line snet frac14 24 s
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conformational step is slower than the preceding adsorption
and subsequent oligomerization steps10
From the model we are also able to comment on the time
constants for the various processes The time constant for poreformation is about 1 s indicating that pore formation itself is a
fast process relative to the time taken for the pore population on
a single RBC to reach steady state which is about 20ndash30 s Since
rupture kinetics is dynamic and depends on the fraction of live
cells as well as the number of pores on the cell lysis occurs in
the time scale of 10s of minutes Due to this separation of time
scales lysis is seen to occur once the number of pores has
reached a steady state This steady number of pores ranges from
515 to 11 657 as the toxin concentration ranges from 588ndash
147 nM For the non-sequential mechanism the critical
number of pores required to initiate lysis is about 20 times
higher when compared to that of sequential oligomerizationComparison of three diff erent kinetic models reveals that the
irreversible sequential kinetics provides the closest match with
the hemoglobin released as a function of the initial toxin
concentration Although we observe an overprediction of the
hemoglobin release data with the non-sequential mechanism
the hemoglobin release kinetics are similar to that of the
sequential mechanism The distribution of lower oligomers is
distinctly diff erent in both cases with a negligible numbers of
lower mers observed in the non-sequential oligomerization
These diff erences in the distribution of mers off er a ngerprint
to identify the underlying mechanism for pore formation
Experiments which can determine the number of pores or the
steady state lsquomerrsquo distributions on the membrane surface will
shed light on the pathways for oligomerization and enable a
more denite conclusion of the underlying kinetics Furtherthe rupture kinetics model contains only one adjustable
parameter Lysis experiments conducted with Vibrio cholerae El
Tor cytolysin33 and Monalysin34 show similar lysis times (in the
order of 10s of minutes) as observed in our study suggesting
similarities in the underlying kinetic pathways that lead to pore
formation and rupture The model developed in this manu-
script is generic and could be recast with some variation to
study the dynamics of other PFTs
We brie y discuss some of the limitations of the model in its
present form The model is based on the mean eld approxi-
mation where all cells are assumed to be identical and for the
purpose of binding and oligomerization this is an adequatestarting point The more complicated process is the mechanics
of rupture with the correct functional dependence on the pore
density In general there could exist a distribution of cells with
diff erent densities of pores Preliminary experiments by varying
the number of RBCs at a xed toxin concentration led to an
increase in lysis suggesting that cell heterogeneity could be
playing a role Although a population balance model35 could
include these variations this is at an added cost of complexity A
second aspect inherent to the model is the presence of lysis
beyond the time at which steady state is observed in the
experiments (30 minutes) Once the number of pores has
Fig 11 Oligomerconcentrationsas a function of time areshownfor (a) 544 nM and(b) 147 nMSteady state oligomer concentrations areshown
for (c) 544 nM and (d) 147 nM At steady state 1ndash4 mer concentrations are zero for both 544 and 147 nM The 12-mer concentration at steady
state is quite large compared to the other oligomer concentrations present in the system
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reached a steady state which occurs within 30 s (Fig 7a) cell
lysis continues to occur at a xed number of pores proportional
to np npc Running the simulation to steady state would
eventually lead to lysis of all the cells in the system albeit at an
exceedingly slow rate at the lower and intermediate toxin
concentrations We have also assumed complete and irrevers-
ible binding of the monomer to the membrane and hence the
number of pores predicted represent an upper limit within the
proposed kinetic framework Despite the mean eld approxi-mation the model proposed in this manuscript is able to
capture the inherent time scales in the process of pore forma-
tion and rupture as well as predict the variation of hemoglobin
release as a function of the initial toxin concentration observed
in the experiments We nally point out that the eff ect of
temperature on the kinetics and lysis has not been investigated
in this work Preliminary experiments at 147 nM toxin
concentration carried out to steady state indicated a marked
drop in lytic activity for temperatures below 15 C and lysis was
not observed at 10 C Data on binding isotherms would be
required to understand temperature eff ects in these systems
Our work has implications in optimizing dosage and devel-oping novel drug delivery strategies based on PFTs The obser-
vation that lysis occurs in a given window of toxin concentration
has implications while developing PFT-based drug and gene
therapy protocols In recent studies E coli used in conjunction
with radiation therapy was found to retard the growth of cancer
tumors when compared with only radiation therapy protocols36
Quantifying the dynamics and extent of lysis is an important
aspect of developing appropriate treatment protocols in these
combination therapies In other applications where pores are
used for gene delivery and in the development of articial cells
lysis must be prevented and pore formation restricted to
concentrations below the lysis threshold
Acknowledgements
We acknowledge funding from Department of Science and
Technology (DST) India under the IRHPA grant The authors
thank Sanjeev Kumar Gupta and Jaydeep Basu for several useful
discussions on the development of the model as well as the
reviewers for their critical comments
References
1 S Bhakdi and J Tranum-Jensen Microbiol Rev 1991 55
733ndash
7512 J Thiery D Keefe S Boulant E Boucrot M Walch
D Martinvalet I S Goping R C Bleackley
T Kirchhausen and J Lieberman Nat Immunol 2011 12
770ndash777
3 A Ludwig C von Rhein S Bauer C Huttinger and
W Goebel J Bacteriol 2004 186 5311ndash5320
4 A W Bernheimer and L S Avigad Proc Natl Acad Sci U S
A 1976 73 467ndash471
5 M V Sousa M Richardson W Fontes and L Morhy
J Protein Chem 1994 13 659ndash667
6 G Anderluh and P Macek Toxicon 2002 40 111ndash124
7 I Iacovache F G van der Goot and L Pernot Biochim
Biophys Acta Biomembr 2008 1778 1611ndash1623
8 N R Wyborn A Clark R E Roberts S J Jamieson
S Tzokov P A Bullough T J Stillman P J Artymiuk
J E Galen L Zhao M M Levine and J Green
Microbiology 2004 150 1495ndash1505
9 A J Wallace T J Stillman A Atkins S J Jamieson
P A Bullough J Green and P J Artymiuk Cell 2000 100
265ndash27610 M Mueller U Grauschopf T Maier R Glockshuber and
N Ban Nature 2009 459 726ndash730
11 J Oscarsson Y Mizunoe L Li X-H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
12 A Atkins N R Wyborn A J Wallace T J Stillman
L K Black A B Fielding M Hisakado P J Artymiuk and
J Green J Biol Chem 2000 275 41150ndash41155
13 A Ludwig G Volkerink C von Rhein S Bauer E Maier
B Bergmann W Goebel and R Benz J Bacteriol 2010
192 4001ndash4011
14 N Eier M Vetsch M Gregorini P Ringler M Chami
A Philippsen A Fritz S A Muller R Glockshuber A Engel and U Grauschopf EMBO J 2006 25 2652ndash2661
15 L A Shepard O Shatursky A E Johnson and R K Tweten
Biochemistry 2000 39 10284ndash10293
16 A J Ratner K R Hippe J L Aguilar M H Bender
A L Nelson and J N Weiser J Biol Chem 2006 281
12994ndash12998
17 I Iacovache M Bischoerger and F G van der Goot Curr
Opin Struct Biol 2010 20 241ndash246
18 G Schwarz and C H Robert Biophys Chem 1992 42 291ndash296
19 G Schwarz R-t Zong and T Popescu Biochim Biophys
Acta Biomembr 1992 1110 97ndash104
20 D Bolintineanu E Hazrati H T Davis R I Lehrer and
Y N Kaznessis Peptides 2010 31 1ndash821 J M Alam T Kobayashi and M Yamazaki Biochemistry
2012 51 5160ndash5172
22 J Lemiere K Guevorkian C Campillo C Sykes and T Betz
So Matter 2013 9 3181ndash3187
23 V Noireaux and A Libchaber Proc Natl Acad Sci U S A
2004 101 17669ndash17674
24 M M Bradford Anal Biochem 1976 72 248ndash254
25 H Xu E K Bjerneld M Kall and L Borjesson Phys Rev
Lett 1999 83 4357ndash4360
26 J Oscarsson Y Mizunoe L Li X H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
27 T M Raschke J Tsai and M Levitt Proc Natl Acad SciU S A 2001 98 5965ndash5969
28 L A Rodrıguez-Guadarrama S K Talsania K K Mohanty
and R Rajagopalan Langmuir 1999 15 437ndash446
29 C E McLaren G M Brittenham and V Hasselblad Am J
Physiol 1987 252 3181ndash3187
30 J G Kim M Xia and H Liu IEEE Eng Med Biol Mag 2005
24 118ndash121
31 M Nagahama H Hara M Fernandez-Miyakawa
Y Itohayashi and J Sakurai Biochemistry 2006 45 296ndash302
32 J R Thompson B Cronin H Bayley and M I Wallace
Biophys J 2011 101 2679ndash2683
This journal is copy The Royal Society of Chemistry 2014 RSC Adv 2014 4 4930ndash4942 | 4941
Paper RSC Advances
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7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1313
33 A Zitzer I Walev M Palmer and S Bhakdi Med Microbiol
Immunol 1995 184 37ndash44
34 O Opota I Vallet-Gely R Vincentelli C Kellenberger
I Iacovache M R Gonzalez A Roussel F G van der Goot
and B Lemaitre PLoS Pathog 2011 7 e1002259
35 S Kumar and D Ramkrishna Chem Eng Sci 1996 51
1311ndash1332
36 S N Jiang T X Phan T K Nam V H Nguyen H S Kim
H S Bom H E Choy Y Hong and J J Min Mol Ther
2010 18 635ndash642
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experiments which shed light on either of these mechanisms
Results for the reversible sequential mechanism and non-
sequential irreversible kinetics are presented later in the text
In the sequential mechanism there are 11 rate constants
Molecular simulation of hydrophobic association of small
solutes in water27 reveal that a sequential aggregation procedure
is favored during cluster formation The sequential aggregation
mechanism is also used for modelling micellar aggregation28
We assume that all the rate constants for the sequentialmechanism are identical This assumption is widely used in
sequential polymerization reactions With the assumption that
all rate constants for oligomerization (k l ) are identical a balance
on the protomer yields
d p1
dt frac14 k cm k l
Xn1
l frac141
p1 pl (4)
where the rst term on the right hand side represents the
formation of the protomer from the monomer and the other
terms represent sequential oligomerization steps wherein the
protomer binds with the other lsquomersrsquo to form the higher lsquomersrsquo
with a rate constant k l From rate considerations for dimerformation (l frac14 2) a prefactor of 12 appears in the term which
corresponds to the formation of the dimer This is a necessary
condition for satisfying the species mass balance The govern-
ing equation for the dimer (l frac14 2) is given by
d p2
dt frac14
1
2k 2 p1 p1 k 2 p1 p2 (5)
where p2 represents the concentration of dimer and k 2 repre-
sents the rate constant for the reaction The equation for the
formation of the l th oligomer (l gt 2) is
d pl
dt frac14 k l p1 pethl 1THORN eth1 dl 12THORNk l p1 pl l frac14 3 12 (6)
where d l 12 represents the Kronecker delta function The corre-
sponding number of pores per RBC is obtained using
np frac14 p12N avARBC (7)
where N av is the Avogadro number
33 Cell lysis
Every dodecamer corresponds to a stable pore in the
membrane If the rate at which lysis occurs is directly propor-
tional to the number of cells that are present at any instant of
time then cell lysis follows a rst order process On physical
grounds we further assume that cell lysis occurs only when the
number of pores exceeds a critical number of pores in each cell
Since our experiments are carried out under isotonic condi-
tions lysis is associated with rupture Lysis can be described
using the following rst order process
dx
dt frac14 klxR
np npc
(8)
where
R frac14
0 npnpc
np npc np $npc
x represents the fraction of unlysed cells at any instant of time
np is number of pores per RBC at any instant and npc is the
critical number of pores per RBC above which cell lysis occurs
The constant kl represents the decay rate constant for cell lysis
In eqn (8) the ramp function R(np npc) incorporates the
increased lysis as a function of the excess pores np npc We
also investigate other functional forms for R such as a unit step
function and a higher power dependence on np npc The
inuence of these on the model predictions are discussed laterin the text As cells lyse the number of RBCs N RBC at any
instant is
N RBC frac14 N inRBCx (9)
N inRBC is the initial number of RBCs
The rate at which hemoglobin is released from the RBCs into
solution is
dH out
dt frac14 V hrhN
inRBCkl xR
np npc
thornDhAp
ms
h H out
N RBCnp
V soll p
(10)
where the rst term represents the contribution due to lysis
(rupture) and the second term is the diff usive ux contribution
from the pores of unlysed cells In the above equation V h is the
volume of hemoglobin present in a single RBC rh is the density
of hemoglobin Dh is the diff usivity of hemoglobin l p is the
diff usion length along the pore Ap is the average area of a pore
msh H out and rh represents the saturated hemoglobin mass in
one RBC amount of hemoglobin present in the solution at any
instant of time and the density of hemoglobin respectively
4 Solution procedure While analyzing the problem it is useful to recast the equations in
suitable dimensionless forms If t frac14 t s C m frac14 C mC in m frac14 mms
and pl frac14 pl ms then eqn (1) (2) and (4)ndash(6) in dimensionless forms
are
dm
dt frac14
s
sa
C m
1 m
Xnl frac141
pl
s
sd
m s
sc
m (11a)
dC m
dt frac14
ARBCN RBCm
s
V solC inb
s
sd
m s
sa
C m
1 m
Xnl frac141
pl
(11b)
d p1
dt frac14
s
sc
m s
sl
Xn1
l frac141
p1 pl (11c)
d p2
dt frac14
s
2s2
p1 p1 s
s2
p1 p2 (11d)
d pl dt frac14
s
sl
p1 pethl 1THORN eth1 dl 12THORNs
sl
p1 pl l frac14 3 12 (11e)
From eqn (11a)ndash(11e) we can extract the following set of time
constants
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sa frac14 1
k aC inis the adsorption time constant
sd frac14 1
k dis the desorption time constant
sc frac14 1
k cis the conformational time constant
sl frac14 1
k l ms is the reaction time constant
s frac14 1
k aC in thorn k c thorn k d
(12)
41 Model parameters
Since we do not have experimental data to independently
determine the various time constants it is more convenient to
dene a ratio between time constants We dene the ratio
between conformational and adsorption times as
l frac14sc
sa
(13)
It has been observed that the conformational times sc arelarger than the time for membrane binding and oligomeriza-
tion1014 suggesting that l gt 1 We can also rewrite the adsorption
time constant and conformational time constant in terms of l
provided we have an estimate of the time required for initiation
of pore formation Since the processes leading to pore forma-
tion occur in series the total time constant snet for pore
formation is the sum of the time constants for the individual
steps
snet frac14 sc + sa + stl + sd (14)
Using eqn (12) and (13) and with the added assumption that
desorption rate is negligible and the adsorption and reaction
time constants are similar (sa st
l frac14 (n 1)sl ) the constants k aand k c can be expressed in terms of l Hence
k a frac14 2 thorn l
snetC in(15a)
and
k c frac142 thorn l
lsnet
(15b)
Under these assumptions for a xed initial concentration of
toxins and cell mass specifying l snet and C in is sufficient to
make predictions for the rate at which pores are formed in thesublytic regime With these assumptions eqn (11) can be
expressed solely in terms of the constant l The values of various
system properties used in the simulation are given in Table 1
and the values of diff erent parameters are given in Table 2 For
snet the model predictions were tested for a range of values as
indicated The parameters related to the pore geometry radius
of the pore r p and length of the pore l p are obtained from the
crystal structure of the ClyA pore10 The initial number of RBCs
are counted using the hemocytometric technique Typical
liquid diff usivities are used for hemoglobin Since the diff u-
sivity only inuences the hemoglobin release in the sublytic
regime obtaining a precise value of the diff usivity is not of
special consequence
42 Simulation details
We used an explicit Euler scheme for discretizing the governing
ordinary diff erential equations and the equations were solved
with a reduced time step of 0005 (0016 s l frac14 4) Calculations
performed with a reduced time step of 0001 did not alter the
reported results We developed an in-house program using
Matlab 70 to solve the discretized equations Calculations werechecked with a mass balance on the monomers
5 Results and discussions51 Lysis experiments
The OD data from lysis experiments at 60 minutes are illus-
trated in Fig 3a as a function of the bulk monomer toxin
concentration C in The data represents an average over 5
independent experiments Based on the time evolution (Fig 3b)
data no further lysis was observed above 30 minutes for all the
toxin concentrations investigated in the study Hence data at 30
minutes is expected to represent the steady state in the systemFrom the cell counts in the hemocytometer we nd 98 lysis at
147 nM and about 10ndash15 lysis at 588 nM The data clearly
reveals that the RBC lysis occurs only above the critical toxin
concentration (npc) which we estimate at 544 nM Below this
critical concentration lies the regime of low hemolytic activity
where the absolute OD values are an order of magnitude below
values obtained with lysis indicating that leakage from pores is
not signicant Although pore formation occurs in this regime
the concentration of pores is not sufficient to initiate lysis To
further support this hypothesis we carried out turbidity assay
experiments (Fig 4) for initial toxin concentrations ranging
Table 1 Various system properties and parameters used in the
simulation In some cases only the range of parameters that were
tested are given
Area of RBC29 ( ARBC) 136 mm2
Volume of RBC29 (V h) 90 fLRadius of pore10 (r p) 35 nmLength of pore10 (l p) 13 nmDiff usivity of hemoglobin ( Dh) 109 m2 s1
Initial number of RBC ( N inRBC) 32 107 cells per ml Volume of lysis assay (V sol) 1 mlSaturated surface concentration (ms) 109 mol m2
Net reaction time constant (snet ) 1ndash25 s
Table 2 Values of constants obtained from sequential and non-
sequential oligomerization
ParameterSequentialoligomerization
Non-sequentialoligomerization
l 2ndash4 2ndash4npc 392ndash768 pores 5300ndash6300 poresDecay rate
constant (kl )
15ndash18 107 s1 125ndash135 107 s1
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from 294ndash294 nM The decrease in turbidity is seen to occur
simultaneously with an increase in the OD supporting the view that the increase in OD is due to cell lysis The turbidity
decrease is also mirrored with the corresponding OD data Since
the cell mass used in the turbidity experiments is lower than
that used in the lysis experiments sublytic toxin concentrations
lie below 294 nM
52 Model predictions
In this section model predictions for the sequential irreversible
kinetics are compared with the lysis data
521 Sublytic regime
In the sublytic regime (np lt npc) the evolution of np with time is
obtained by solving eqn (11a)ndash(e) The number of pores np is
obtained from eqn (7) From the simulations we observe that the number of pores per RBC for a given initial toxin concen-
tration saturates within 10 s (Fig 5a) This saturation in np is
due to the limiting amount of toxin present in solution In
Fig 5b we plot the variation in np at saturation (20 s) with
l frac14sc
sa for diff erent initial toxin concentration assuming that
np lt npc The curve corresponding to 544 nM is tted to the
form y frac14 axb and the relation npc frac14 20033l097 is obtained
which can be used to x the value of npc for a given value of l In
order to simulate the lysis data a value of snet (eqn (14)) which
is the time constant associated with the time required for
formation of the rst pore has to be specied Initial estimates
are in the range of 1ndash25 s and we use a value of 1 s in all our
simulations unless specied We show later that our results are
Fig 3 (a) The normalized optical density values as a function of the
initial toxin concentration observed after 60 minutes during the lysis
experiments A distinct jump is observed above a toxin concentration
of 544 nM (b) Time evolution data of optical density values during
RBC lysis At 147 nM 98 lysis is observed
Fig 4 Turbidity (left axis) and lysis data (right axis) show that the
decrease in turbidity occurs simultaneously with an increase in the
OD The sublytic initial toxin concentration is less than 294 nM
(100 ng ml1)
Fig 5 Relation between l and npc in the sublytic regime (a) Number
of pores per RBC as a function of time The data plotted are for l frac14 2
and snet frac14 1 s (b) A regression analysis of the curve at 544 nM (dash-
dotted line) yields the relation npc frac14 20033l097
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relatively insensitive to the value of snet in this range The
parameters l npc and snet are obtained in the sublytic regime as
discussed above
522 Lysis regime
In the lysis regime in addition to the kinetic eqn (11andashe) we also
solve the lysis and hemoglobin release equations eqn (8) and
eqn (10) Cell lysis occurs once np gt npc and the OD increases with increasing toxin concentration as shown in Fig 3a Once
values of l npc and snet are xed in the sub-lytic regime the only
unknown parameter in the model is the value of kl (eqn (8)) In
all cases kl is xed by matching the maximum extent of lysis of
98 obtained at 147 nM Subsequent simulations are run with
diff erent values of C in to compare with the experimental data
Fig 6 illustrates the comparison between the model prediction
and the experimental OD data Since the OD varies linearly with
the amount of hemoglobin released during lysis30 we scale both
the experimental and predicted data by their respective
maximum OD to facilitate a meaningful comparison We also
carried out independent lysis experiments to verify the linear
relationship between the OD versus hemoglobin data Hence we
normalize the long time data and dene H max as the ratio of the
mass of the steady state hemoglobin released at a given C in to
the corresponding value at C in frac14 147 nM which is the highest
C in considered in the study This facilitates a comparison of
H max predicted from the model directly with the normalized OD
values For l frac14 2 the corresponding npc frac14 392 and the
comparison of H max at a value of kl frac14 18 107 s1 is
illustrated in Fig 6a The comparison for lfrac14 4 (npcfrac14 768) and kl
frac14 15 107 s1 is illustrated in Fig 6c Comparison of the
hemoglobin release dynamics ( H out vs time) for the corre-
sponding set of parameters are illustrated in Fig 6b and d
respectively
We observe that in this range of l (2 l 4) values and
kl 15ndash18 107 s1 the OD vs C in data is captured quite
accurately Since l is the ratio of the ClyA monomer confor-
mation time to the monomer adsorption time this range of l value is consistent with experiments10 which indicates that
conformation is preceded by fast adsorption followed by rapid
oligomerization The H out dynamics predicted by the model is
seen to capture the experimental data quite well (Fig 6b and d)
Upon increasing l we nd that a lower value of kl is required to
match the OD vs C in data (Fig 6d) A value of snet frac14 1 s captures
the early time release in the H out data at 147 nM quite accu-
rately and increasing snet to 24 s results in a short delay at early
times Since snet represents the time taken to form the rst pore
in situ monitoring of the hemoglobin release dynamics would be
required to determine snet more precisely In our experiments the
time evolution of the lysis data is carried out by intermittently arresting lysis at diff erent times and quenching the aliquots in
ice for a period of 3ndash5 minutes while the OD is determined
Experiments carried out continuously for the diff erent time
points shown in Fig 6b and d did not alter the data obtained
from the intermittent experiments Fluorescence permeation
experiments by Yamazaki and co-workers21 by lysenin (334 kDa)
induced pore formation on single giant unilamellar vesicles show
that pore formation is complete within about 10 s for toxin
concentrations of 200 ng ml1 and 50 s at 40 ng ml1 These are
similar to the time scales deduced in our model
At a toxin concentration of 147 nM a steady distribution of
mers is achieved on the time scale of 10ndash20 s (for a snet frac14 1 s)
Within this time interval the pore density goes through a phaseof rapid increase to exceed the critical pore density (Fig 7)
during which very little lysis is observed (Fig 6a and d) Lysis is
predominantly observed a er a steady number of pores have
formed on the RBC surface This steady number of pores ranges
Fig 6 Comparison of model predictions (open circles) with experi-
mental results (open squares) The amount of hemoglobin released
(Hmax) atthe end of30min for(a) npcfrac14 392 (c) npcfrac14 768 Dynamics of
hemoglobin released (Hout) for (b) npcfrac14 392 (d) npcfrac14 768 A value of l
between 2 and 4 is seen to accurately capture the Hmax versus C in data
[(a) and (c)] The amount of hemoglobin released is scaled with the
maximum amount to facilitate a comparison Simulations corresponding
to snet frac14 24 s are shown in bold lines and snet frac14 1 s in dashed lines
Fig 7 The pore density is plotted as a function of time in thepost-lysis
regime The number of pores (np) per RBC ranges from 515 at 588 nM
to 11 657 at 147 nM l frac14 2 and kl frac14 18 107 s1 npc frac14 392
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from 515 at 588 nM to 11 657 at 147 nM Given this situation it
is instructive to dene an eff ective lysis time constant kleff frac14
kl (np npc) where np is the steady state value of the number of
pores at a given value of C in (Fig 7) The value of kleff at 147 nM
is 2027 103 s1 which results in an eff ective lysis time
constant of 493 s
523 Oligomer and pore concentration
In both the low (lt544 nM) and high toxin ($544 nM) regimes
the amount of toxin is found to be limiting Even at the highest
toxin concentration C in frac14 147 nM the toxin in bulk solution is
depleted within 10 s Selected oligomer concentrations as a
function of time are plotted in Fig 8a and b for both high and
low toxin concentrations as predicted by the kinetic model (eqn
(4) and (6)) The pore density is illustrated in Fig 7 The gov-
erning equations for the formation of an l -mer are given in eqn
(6) Since pore formation occurs via a sequential oligomeriza-
tion mechanism a protomer (1-mer) is necessary for the
formation of all other l mers Hence a steady monomer
concentration on the RBC is achieved once the 1-mer concen-
tration reduces to zero and lag times are observed for theformation of the l + 1-mer From the distribution of mers on the
membrane we observe that a large fraction of protomers remain
trapped as intermediate mers on the membrane At 147 nM the
number of monomers per ml is 8854 1012 The initial
number N RBC frac14 32 107 the monomers per RBC is 276
105 If all the monomers were converted to pores each RBC
would have 23 105 pores However the number of pores
formed per RBC at 147 nM is 11657 105 pores (Fig 7) indi-
cating that about 50 of the mers remain on the membrane
surface as intermediate n-mers (n frac14 1ndash11) At a sublytic
concentration of 544 nM only about 45 of the mers are
converted to pores resulting in 375 pores per RBC (Fig 5a)
The eff ect of l is more prominent at the higher toxin
concentration where both the life time and the maximum
concentration for 1-mers is found to decrease as l is increased
from 2 to 4 The steady state concentrations of various n-mers aspredicted by the model are illustrated in Fig 8c and d A change
in the value of l results in a shi in the distribution for a
particular initial toxin concentration An increase in l implies an
increase in the conformational time relative to the adsorption
and reaction times Hence as l is increased occurrence of the
lower mers on the surface decreases due to the faster reaction
time scales relative to conformation At low bulk toxin concen-
trations (Fig 8c) the distribution of higher mers and conse-
quently the number of pores (12 mers) is very low due to the
limited supply of monomers in the system However at higher
concentration (Fig 8d) the number of monomers is no longer
the limiting factor and the distribution shi
s towards the highermers thereby increasing the number of pores on the surface
524 Parameter sensitivity
We brie y summarize the results of simulations carried out to
test the inuence of the estimated parameters on the model
predictions In the absence of monomer membrane binding
equilibria the value of saturated surface concentration ( ms) is
Fig 8 Model predictions for oligomer concentration pro1047297les ( pl) for l frac14 2 and kl frac14 18 107 s1 (a) Initial bulk concentration frac14 544 nM (b)
Initial bulk concentration frac14 147 nM Steady state distribution of oligomers for (c) C in frac14 544 nM (d) C in frac14 147 nM
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unknown In order to test the inuence of ms on the model
prediction we carried out a few simulations for ms frac14 1 108
mol m2 and msfrac14 1 1010 mol m2 for various C in values For
ms frac14 1 108 mol m2 a negligible number of pores were
formed and the H max ndashC in data (Fig 3a) is underpredicted At
ms frac14 1 1010 mol m2 pore formation was extremely rapid
and little variation in pore density between C in frac14 882 nM and
147 nM was observed As a consequence H max ndashC in data (Fig 3a)
is grossly overpredicted Hence a value of ms frac14 1 10
9 molm2 was used in the simulations (Fig 6) We further note
that the amount of saturated surface concentration ms
implicitly changes the reaction rate constant k l (eqn (12))
Increasing ms eff ectively decreases the reaction rate constant
(eqn (12))
Once snet is xed npc is related to l through the relation npcfrac14
alb with the constants a and b being xed for a given initial toxin
concentration C in (Fig 5) We have found that 2 l 4 ts the
hemoglobin release data very closely (Fig 6) and although the
hemoglobin released as a function of time is slightly under-
estimated by the model the agreement is reasonable Upon
increasing l
to 7 and keeping m
s
frac14 10
9
mol m
s1
the corre-sponding value of npc is 1349 With a value of kl frac14 13 107 s1
although H max versus C in data is accurately predicted the
hemoglobin versus time data is grossly underpredicted Varying
ms between 108 and 1010 mol m2 further deteriorated the
prediction Finally we point out that other functional forms of the
dependence on np npc in the cell lysis equation eqn (8) such as
the unit step function or a quadratic dependence (np npc)2 only
overestimated the H max versus C in data
53 Sequential oligomerization with reversible kinetics
In the previous discussion we present the results for the oligo-
merization kinetics which was assumed to be irreversible Thesequential oligomerization model with reversible kinetics is
p1 thorn pl ) k f
k b pl thorn1 l frac14 1 n 1 (16)
where k f and k b represent the forward and backward reaction
rate constants The kinetic equations are
d p1
dt frac14 k cm k f
Xn1
l frac141
p1 pl thorn k bXn1
l frac142
pl (17a)
d p2
dt frac14
1
2k f p1 p1 k f p1 p2 thorn k b p3
1
2 p2 (17b)
d pl
dt frac14 k f p1 pethl 1THORN eth1 dl 12THORNk f p1 pl
thorn k beth1 dl 12THORNetheth1 dl 11THORN pl thorn1 pl THORN l frac14 3 12 (17c)
A reversible time constant can be dened from eqn (17) as
sb frac14 1
k b The ratio R is dened as the ratio of forward to back-
ward time constants ( R frac14 sf sb) to study the eff ect of revers-
ibility Upon examining the number of pores as a function of
time we observe that the time taken to reach a steady number of
pores is signicantly larger than the time taken to reach steady
state in the lysis experiments In order to make comparisons
with the irreversible mechanism we evaluated the number of
pores at the threshold concentration of 544 nM It is observed
that the number of pores required for lysis initially increases
and then decreases for increments in R values The distribution
of oligomers at steady state are shown for diff erent R values in
Fig 9a and b for C in frac14 147 nM The steady state concentrations
of lsquomersrsquo change from a predominantly 11-mer concentration to
a predominant 1-mer concentration as R is varied between0 and 1 For R lt 1 we nd that the concentration of pores (12
mers) are signicantly higher than the intermediate lsquomerrsquo
concentrations shown in Fig 9b and range from 0142 nmol
Fig 9 Oligomer distribution as a function of the ratio of forward and
backward time constants R frac14 sfsb for C in frac14 147 nM (a) R varies
between 0 and 1 (b) R varies between 0 and 01 (c) Hemoglobin
release data as a function of toxin concentration is shown forvarious R
R frac14 0 has the closest agreement with the experimental data (open
squares)
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m2 ( R frac14 0) to 028 nmol m2 ( R frac14 001) For R frac14 1 the
concentration of 12 mers is 024 nmol m2 and decreases with a
further increase in R The predictions using reversible sequen-
tial kinetics for the H max vs C in data (Fig 9c) indicate
greater deviation from the experimental data when compared
with R frac14 0 These results indicate that the irreversible mecha-
nism provides the best agreement with the experimental data
We point out that the critical number of pores npc in
the reversible framework is non-monotonic ranging from 392( R frac14 0) to 5128 ( R frac14 1) for l frac14 2
54 Non-sequential oligomerization
In contrast to the 392 pores obtained for the critical number of
pores via the sequential mechanism a substantially larger
critical number of pores are observed via the non-sequential
oligomerization (6000 pores per cell) The mass balance for
the protomer ( p1) concentration remains identical to that of the
sequential oligomerization mechanism (eqn (4)) The governing
equations for pl (l lt l n) oligomer undergoing non-sequential
irreversible oligomerization are
d pl
dt frac14
1
2k l Xl 1
ufrac141
pu pl u eth1 dl 12THORNXn1
ufrac141
k l pl pu for 1l n (18)
where n represents the number of monomers in a pore The
relation between the critical number of pores and npc for non-
sequential oligomerization is npc frac14 5439l015 The critical
number of pores for l frac14 2 is 6035 pores which is about 20 times
greater than that obtained from a sequential oligomerization
mechanism In this scheme the hemoglobin release (Fig 10a)
data is overpredicted at intermediate toxin concentrations when
compared with the sequential oligomerization The predictions
of the H out vs time data at 147 nM (Fig 10b) are similar whencompared with the sequential oligomerization (Fig 6b and d)
A comparison of the tted parameters between the sequential
and non-sequential oligomerization mechanisms are given in
Table 2
Oligomer distributions obtained from the non-sequential
mechanism (Fig 11b) show an entirely diff erent trend when
compared to that obtained from the sequential mechanism
(Fig 8) In the sequential mechanism the higher lsquomersrsquo
attained a steady state once the protomer was depleted In the
non-sequential mechanism the contribution to the dodeca-
mers (12 mer) can be obtained from a large number of combi-
nations of the lower oligomers leading to a larger dodecamer orpore concentration The time scale required to attain the olig-
omer steady state concentration (Fig 11a) is about 20 s for an
initial concentration of 147 nM Similar time scales are
observed in the sequential mechanism as well
Further experiments are required to distinguish between the
various mechanisms Western Blot experiments conducted on
Hemolysin E12 and Clostridium perfringens 3 toxins31 showed the
presence of intermediate oligomers On the other hand single-
molecule uorescence imaging of a-hemolysin on a droplet
interface bilayer showed the presence of only monomers and
heptamers (pores)32
6 Discussion and conclusions
Lysis experiments on RBCs with the ClyA pore-forming toxin
show that a threshold initial toxin concentration is required to
initiate lysis From this observation we analyzed the problem in
two regimes a low toxin concentration regime where rupture of
cells is absent and a high toxin concentration regime where
lysis occurs and hemoglobin is released Kinetic models which
accounts for monomer binding conformation (membrane-
bound monomer to protomer) and oligomerization to form the
dodecameric pore complex are developed Models which
account for sequential and non-sequential oligomerization are
tested Cell rupture is assumed to be
rst order in the number of live cells and directly proportional to the pores in excess of the
critical number of pores npc In the sublytic regime the number
of pores is found to have a power law dependence on l which is
the ratio of conformational time to the reaction time This leads
to the construction of a ldquophase diagramrdquo between the number
of pores np and l for diff erent values of the initial toxin
concentration Comparing simulations with experimental data
the range of npc was 392ndash768 for the sequential mechanism and
5300ndash6300 pores for the non-sequential mechanism for 2 l 4
The range of l values is consistent with available experimental
data on ClyA which indicates that the membrane-bound
Fig 10 (a) Model predictions from the non-sequential mechanism
Simulated Hmax C in (open circles) curves grossly overpredicts the
experimental data (open squares) (b) Simulated hemoglobin release
compares well with the experimental data Dashed line snetfrac14 1 s solid
line snet frac14 24 s
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conformational step is slower than the preceding adsorption
and subsequent oligomerization steps10
From the model we are also able to comment on the time
constants for the various processes The time constant for poreformation is about 1 s indicating that pore formation itself is a
fast process relative to the time taken for the pore population on
a single RBC to reach steady state which is about 20ndash30 s Since
rupture kinetics is dynamic and depends on the fraction of live
cells as well as the number of pores on the cell lysis occurs in
the time scale of 10s of minutes Due to this separation of time
scales lysis is seen to occur once the number of pores has
reached a steady state This steady number of pores ranges from
515 to 11 657 as the toxin concentration ranges from 588ndash
147 nM For the non-sequential mechanism the critical
number of pores required to initiate lysis is about 20 times
higher when compared to that of sequential oligomerizationComparison of three diff erent kinetic models reveals that the
irreversible sequential kinetics provides the closest match with
the hemoglobin released as a function of the initial toxin
concentration Although we observe an overprediction of the
hemoglobin release data with the non-sequential mechanism
the hemoglobin release kinetics are similar to that of the
sequential mechanism The distribution of lower oligomers is
distinctly diff erent in both cases with a negligible numbers of
lower mers observed in the non-sequential oligomerization
These diff erences in the distribution of mers off er a ngerprint
to identify the underlying mechanism for pore formation
Experiments which can determine the number of pores or the
steady state lsquomerrsquo distributions on the membrane surface will
shed light on the pathways for oligomerization and enable a
more denite conclusion of the underlying kinetics Furtherthe rupture kinetics model contains only one adjustable
parameter Lysis experiments conducted with Vibrio cholerae El
Tor cytolysin33 and Monalysin34 show similar lysis times (in the
order of 10s of minutes) as observed in our study suggesting
similarities in the underlying kinetic pathways that lead to pore
formation and rupture The model developed in this manu-
script is generic and could be recast with some variation to
study the dynamics of other PFTs
We brie y discuss some of the limitations of the model in its
present form The model is based on the mean eld approxi-
mation where all cells are assumed to be identical and for the
purpose of binding and oligomerization this is an adequatestarting point The more complicated process is the mechanics
of rupture with the correct functional dependence on the pore
density In general there could exist a distribution of cells with
diff erent densities of pores Preliminary experiments by varying
the number of RBCs at a xed toxin concentration led to an
increase in lysis suggesting that cell heterogeneity could be
playing a role Although a population balance model35 could
include these variations this is at an added cost of complexity A
second aspect inherent to the model is the presence of lysis
beyond the time at which steady state is observed in the
experiments (30 minutes) Once the number of pores has
Fig 11 Oligomerconcentrationsas a function of time areshownfor (a) 544 nM and(b) 147 nMSteady state oligomer concentrations areshown
for (c) 544 nM and (d) 147 nM At steady state 1ndash4 mer concentrations are zero for both 544 and 147 nM The 12-mer concentration at steady
state is quite large compared to the other oligomer concentrations present in the system
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reached a steady state which occurs within 30 s (Fig 7a) cell
lysis continues to occur at a xed number of pores proportional
to np npc Running the simulation to steady state would
eventually lead to lysis of all the cells in the system albeit at an
exceedingly slow rate at the lower and intermediate toxin
concentrations We have also assumed complete and irrevers-
ible binding of the monomer to the membrane and hence the
number of pores predicted represent an upper limit within the
proposed kinetic framework Despite the mean eld approxi-mation the model proposed in this manuscript is able to
capture the inherent time scales in the process of pore forma-
tion and rupture as well as predict the variation of hemoglobin
release as a function of the initial toxin concentration observed
in the experiments We nally point out that the eff ect of
temperature on the kinetics and lysis has not been investigated
in this work Preliminary experiments at 147 nM toxin
concentration carried out to steady state indicated a marked
drop in lytic activity for temperatures below 15 C and lysis was
not observed at 10 C Data on binding isotherms would be
required to understand temperature eff ects in these systems
Our work has implications in optimizing dosage and devel-oping novel drug delivery strategies based on PFTs The obser-
vation that lysis occurs in a given window of toxin concentration
has implications while developing PFT-based drug and gene
therapy protocols In recent studies E coli used in conjunction
with radiation therapy was found to retard the growth of cancer
tumors when compared with only radiation therapy protocols36
Quantifying the dynamics and extent of lysis is an important
aspect of developing appropriate treatment protocols in these
combination therapies In other applications where pores are
used for gene delivery and in the development of articial cells
lysis must be prevented and pore formation restricted to
concentrations below the lysis threshold
Acknowledgements
We acknowledge funding from Department of Science and
Technology (DST) India under the IRHPA grant The authors
thank Sanjeev Kumar Gupta and Jaydeep Basu for several useful
discussions on the development of the model as well as the
reviewers for their critical comments
References
1 S Bhakdi and J Tranum-Jensen Microbiol Rev 1991 55
733ndash
7512 J Thiery D Keefe S Boulant E Boucrot M Walch
D Martinvalet I S Goping R C Bleackley
T Kirchhausen and J Lieberman Nat Immunol 2011 12
770ndash777
3 A Ludwig C von Rhein S Bauer C Huttinger and
W Goebel J Bacteriol 2004 186 5311ndash5320
4 A W Bernheimer and L S Avigad Proc Natl Acad Sci U S
A 1976 73 467ndash471
5 M V Sousa M Richardson W Fontes and L Morhy
J Protein Chem 1994 13 659ndash667
6 G Anderluh and P Macek Toxicon 2002 40 111ndash124
7 I Iacovache F G van der Goot and L Pernot Biochim
Biophys Acta Biomembr 2008 1778 1611ndash1623
8 N R Wyborn A Clark R E Roberts S J Jamieson
S Tzokov P A Bullough T J Stillman P J Artymiuk
J E Galen L Zhao M M Levine and J Green
Microbiology 2004 150 1495ndash1505
9 A J Wallace T J Stillman A Atkins S J Jamieson
P A Bullough J Green and P J Artymiuk Cell 2000 100
265ndash27610 M Mueller U Grauschopf T Maier R Glockshuber and
N Ban Nature 2009 459 726ndash730
11 J Oscarsson Y Mizunoe L Li X-H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
12 A Atkins N R Wyborn A J Wallace T J Stillman
L K Black A B Fielding M Hisakado P J Artymiuk and
J Green J Biol Chem 2000 275 41150ndash41155
13 A Ludwig G Volkerink C von Rhein S Bauer E Maier
B Bergmann W Goebel and R Benz J Bacteriol 2010
192 4001ndash4011
14 N Eier M Vetsch M Gregorini P Ringler M Chami
A Philippsen A Fritz S A Muller R Glockshuber A Engel and U Grauschopf EMBO J 2006 25 2652ndash2661
15 L A Shepard O Shatursky A E Johnson and R K Tweten
Biochemistry 2000 39 10284ndash10293
16 A J Ratner K R Hippe J L Aguilar M H Bender
A L Nelson and J N Weiser J Biol Chem 2006 281
12994ndash12998
17 I Iacovache M Bischoerger and F G van der Goot Curr
Opin Struct Biol 2010 20 241ndash246
18 G Schwarz and C H Robert Biophys Chem 1992 42 291ndash296
19 G Schwarz R-t Zong and T Popescu Biochim Biophys
Acta Biomembr 1992 1110 97ndash104
20 D Bolintineanu E Hazrati H T Davis R I Lehrer and
Y N Kaznessis Peptides 2010 31 1ndash821 J M Alam T Kobayashi and M Yamazaki Biochemistry
2012 51 5160ndash5172
22 J Lemiere K Guevorkian C Campillo C Sykes and T Betz
So Matter 2013 9 3181ndash3187
23 V Noireaux and A Libchaber Proc Natl Acad Sci U S A
2004 101 17669ndash17674
24 M M Bradford Anal Biochem 1976 72 248ndash254
25 H Xu E K Bjerneld M Kall and L Borjesson Phys Rev
Lett 1999 83 4357ndash4360
26 J Oscarsson Y Mizunoe L Li X H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
27 T M Raschke J Tsai and M Levitt Proc Natl Acad SciU S A 2001 98 5965ndash5969
28 L A Rodrıguez-Guadarrama S K Talsania K K Mohanty
and R Rajagopalan Langmuir 1999 15 437ndash446
29 C E McLaren G M Brittenham and V Hasselblad Am J
Physiol 1987 252 3181ndash3187
30 J G Kim M Xia and H Liu IEEE Eng Med Biol Mag 2005
24 118ndash121
31 M Nagahama H Hara M Fernandez-Miyakawa
Y Itohayashi and J Sakurai Biochemistry 2006 45 296ndash302
32 J R Thompson B Cronin H Bayley and M I Wallace
Biophys J 2011 101 2679ndash2683
This journal is copy The Royal Society of Chemistry 2014 RSC Adv 2014 4 4930ndash4942 | 4941
Paper RSC Advances
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33 A Zitzer I Walev M Palmer and S Bhakdi Med Microbiol
Immunol 1995 184 37ndash44
34 O Opota I Vallet-Gely R Vincentelli C Kellenberger
I Iacovache M R Gonzalez A Roussel F G van der Goot
and B Lemaitre PLoS Pathog 2011 7 e1002259
35 S Kumar and D Ramkrishna Chem Eng Sci 1996 51
1311ndash1332
36 S N Jiang T X Phan T K Nam V H Nguyen H S Kim
H S Bom H E Choy Y Hong and J J Min Mol Ther
2010 18 635ndash642
RSC Advances Paper
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sa frac14 1
k aC inis the adsorption time constant
sd frac14 1
k dis the desorption time constant
sc frac14 1
k cis the conformational time constant
sl frac14 1
k l ms is the reaction time constant
s frac14 1
k aC in thorn k c thorn k d
(12)
41 Model parameters
Since we do not have experimental data to independently
determine the various time constants it is more convenient to
dene a ratio between time constants We dene the ratio
between conformational and adsorption times as
l frac14sc
sa
(13)
It has been observed that the conformational times sc arelarger than the time for membrane binding and oligomeriza-
tion1014 suggesting that l gt 1 We can also rewrite the adsorption
time constant and conformational time constant in terms of l
provided we have an estimate of the time required for initiation
of pore formation Since the processes leading to pore forma-
tion occur in series the total time constant snet for pore
formation is the sum of the time constants for the individual
steps
snet frac14 sc + sa + stl + sd (14)
Using eqn (12) and (13) and with the added assumption that
desorption rate is negligible and the adsorption and reaction
time constants are similar (sa st
l frac14 (n 1)sl ) the constants k aand k c can be expressed in terms of l Hence
k a frac14 2 thorn l
snetC in(15a)
and
k c frac142 thorn l
lsnet
(15b)
Under these assumptions for a xed initial concentration of
toxins and cell mass specifying l snet and C in is sufficient to
make predictions for the rate at which pores are formed in thesublytic regime With these assumptions eqn (11) can be
expressed solely in terms of the constant l The values of various
system properties used in the simulation are given in Table 1
and the values of diff erent parameters are given in Table 2 For
snet the model predictions were tested for a range of values as
indicated The parameters related to the pore geometry radius
of the pore r p and length of the pore l p are obtained from the
crystal structure of the ClyA pore10 The initial number of RBCs
are counted using the hemocytometric technique Typical
liquid diff usivities are used for hemoglobin Since the diff u-
sivity only inuences the hemoglobin release in the sublytic
regime obtaining a precise value of the diff usivity is not of
special consequence
42 Simulation details
We used an explicit Euler scheme for discretizing the governing
ordinary diff erential equations and the equations were solved
with a reduced time step of 0005 (0016 s l frac14 4) Calculations
performed with a reduced time step of 0001 did not alter the
reported results We developed an in-house program using
Matlab 70 to solve the discretized equations Calculations werechecked with a mass balance on the monomers
5 Results and discussions51 Lysis experiments
The OD data from lysis experiments at 60 minutes are illus-
trated in Fig 3a as a function of the bulk monomer toxin
concentration C in The data represents an average over 5
independent experiments Based on the time evolution (Fig 3b)
data no further lysis was observed above 30 minutes for all the
toxin concentrations investigated in the study Hence data at 30
minutes is expected to represent the steady state in the systemFrom the cell counts in the hemocytometer we nd 98 lysis at
147 nM and about 10ndash15 lysis at 588 nM The data clearly
reveals that the RBC lysis occurs only above the critical toxin
concentration (npc) which we estimate at 544 nM Below this
critical concentration lies the regime of low hemolytic activity
where the absolute OD values are an order of magnitude below
values obtained with lysis indicating that leakage from pores is
not signicant Although pore formation occurs in this regime
the concentration of pores is not sufficient to initiate lysis To
further support this hypothesis we carried out turbidity assay
experiments (Fig 4) for initial toxin concentrations ranging
Table 1 Various system properties and parameters used in the
simulation In some cases only the range of parameters that were
tested are given
Area of RBC29 ( ARBC) 136 mm2
Volume of RBC29 (V h) 90 fLRadius of pore10 (r p) 35 nmLength of pore10 (l p) 13 nmDiff usivity of hemoglobin ( Dh) 109 m2 s1
Initial number of RBC ( N inRBC) 32 107 cells per ml Volume of lysis assay (V sol) 1 mlSaturated surface concentration (ms) 109 mol m2
Net reaction time constant (snet ) 1ndash25 s
Table 2 Values of constants obtained from sequential and non-
sequential oligomerization
ParameterSequentialoligomerization
Non-sequentialoligomerization
l 2ndash4 2ndash4npc 392ndash768 pores 5300ndash6300 poresDecay rate
constant (kl )
15ndash18 107 s1 125ndash135 107 s1
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from 294ndash294 nM The decrease in turbidity is seen to occur
simultaneously with an increase in the OD supporting the view that the increase in OD is due to cell lysis The turbidity
decrease is also mirrored with the corresponding OD data Since
the cell mass used in the turbidity experiments is lower than
that used in the lysis experiments sublytic toxin concentrations
lie below 294 nM
52 Model predictions
In this section model predictions for the sequential irreversible
kinetics are compared with the lysis data
521 Sublytic regime
In the sublytic regime (np lt npc) the evolution of np with time is
obtained by solving eqn (11a)ndash(e) The number of pores np is
obtained from eqn (7) From the simulations we observe that the number of pores per RBC for a given initial toxin concen-
tration saturates within 10 s (Fig 5a) This saturation in np is
due to the limiting amount of toxin present in solution In
Fig 5b we plot the variation in np at saturation (20 s) with
l frac14sc
sa for diff erent initial toxin concentration assuming that
np lt npc The curve corresponding to 544 nM is tted to the
form y frac14 axb and the relation npc frac14 20033l097 is obtained
which can be used to x the value of npc for a given value of l In
order to simulate the lysis data a value of snet (eqn (14)) which
is the time constant associated with the time required for
formation of the rst pore has to be specied Initial estimates
are in the range of 1ndash25 s and we use a value of 1 s in all our
simulations unless specied We show later that our results are
Fig 3 (a) The normalized optical density values as a function of the
initial toxin concentration observed after 60 minutes during the lysis
experiments A distinct jump is observed above a toxin concentration
of 544 nM (b) Time evolution data of optical density values during
RBC lysis At 147 nM 98 lysis is observed
Fig 4 Turbidity (left axis) and lysis data (right axis) show that the
decrease in turbidity occurs simultaneously with an increase in the
OD The sublytic initial toxin concentration is less than 294 nM
(100 ng ml1)
Fig 5 Relation between l and npc in the sublytic regime (a) Number
of pores per RBC as a function of time The data plotted are for l frac14 2
and snet frac14 1 s (b) A regression analysis of the curve at 544 nM (dash-
dotted line) yields the relation npc frac14 20033l097
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relatively insensitive to the value of snet in this range The
parameters l npc and snet are obtained in the sublytic regime as
discussed above
522 Lysis regime
In the lysis regime in addition to the kinetic eqn (11andashe) we also
solve the lysis and hemoglobin release equations eqn (8) and
eqn (10) Cell lysis occurs once np gt npc and the OD increases with increasing toxin concentration as shown in Fig 3a Once
values of l npc and snet are xed in the sub-lytic regime the only
unknown parameter in the model is the value of kl (eqn (8)) In
all cases kl is xed by matching the maximum extent of lysis of
98 obtained at 147 nM Subsequent simulations are run with
diff erent values of C in to compare with the experimental data
Fig 6 illustrates the comparison between the model prediction
and the experimental OD data Since the OD varies linearly with
the amount of hemoglobin released during lysis30 we scale both
the experimental and predicted data by their respective
maximum OD to facilitate a meaningful comparison We also
carried out independent lysis experiments to verify the linear
relationship between the OD versus hemoglobin data Hence we
normalize the long time data and dene H max as the ratio of the
mass of the steady state hemoglobin released at a given C in to
the corresponding value at C in frac14 147 nM which is the highest
C in considered in the study This facilitates a comparison of
H max predicted from the model directly with the normalized OD
values For l frac14 2 the corresponding npc frac14 392 and the
comparison of H max at a value of kl frac14 18 107 s1 is
illustrated in Fig 6a The comparison for lfrac14 4 (npcfrac14 768) and kl
frac14 15 107 s1 is illustrated in Fig 6c Comparison of the
hemoglobin release dynamics ( H out vs time) for the corre-
sponding set of parameters are illustrated in Fig 6b and d
respectively
We observe that in this range of l (2 l 4) values and
kl 15ndash18 107 s1 the OD vs C in data is captured quite
accurately Since l is the ratio of the ClyA monomer confor-
mation time to the monomer adsorption time this range of l value is consistent with experiments10 which indicates that
conformation is preceded by fast adsorption followed by rapid
oligomerization The H out dynamics predicted by the model is
seen to capture the experimental data quite well (Fig 6b and d)
Upon increasing l we nd that a lower value of kl is required to
match the OD vs C in data (Fig 6d) A value of snet frac14 1 s captures
the early time release in the H out data at 147 nM quite accu-
rately and increasing snet to 24 s results in a short delay at early
times Since snet represents the time taken to form the rst pore
in situ monitoring of the hemoglobin release dynamics would be
required to determine snet more precisely In our experiments the
time evolution of the lysis data is carried out by intermittently arresting lysis at diff erent times and quenching the aliquots in
ice for a period of 3ndash5 minutes while the OD is determined
Experiments carried out continuously for the diff erent time
points shown in Fig 6b and d did not alter the data obtained
from the intermittent experiments Fluorescence permeation
experiments by Yamazaki and co-workers21 by lysenin (334 kDa)
induced pore formation on single giant unilamellar vesicles show
that pore formation is complete within about 10 s for toxin
concentrations of 200 ng ml1 and 50 s at 40 ng ml1 These are
similar to the time scales deduced in our model
At a toxin concentration of 147 nM a steady distribution of
mers is achieved on the time scale of 10ndash20 s (for a snet frac14 1 s)
Within this time interval the pore density goes through a phaseof rapid increase to exceed the critical pore density (Fig 7)
during which very little lysis is observed (Fig 6a and d) Lysis is
predominantly observed a er a steady number of pores have
formed on the RBC surface This steady number of pores ranges
Fig 6 Comparison of model predictions (open circles) with experi-
mental results (open squares) The amount of hemoglobin released
(Hmax) atthe end of30min for(a) npcfrac14 392 (c) npcfrac14 768 Dynamics of
hemoglobin released (Hout) for (b) npcfrac14 392 (d) npcfrac14 768 A value of l
between 2 and 4 is seen to accurately capture the Hmax versus C in data
[(a) and (c)] The amount of hemoglobin released is scaled with the
maximum amount to facilitate a comparison Simulations corresponding
to snet frac14 24 s are shown in bold lines and snet frac14 1 s in dashed lines
Fig 7 The pore density is plotted as a function of time in thepost-lysis
regime The number of pores (np) per RBC ranges from 515 at 588 nM
to 11 657 at 147 nM l frac14 2 and kl frac14 18 107 s1 npc frac14 392
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from 515 at 588 nM to 11 657 at 147 nM Given this situation it
is instructive to dene an eff ective lysis time constant kleff frac14
kl (np npc) where np is the steady state value of the number of
pores at a given value of C in (Fig 7) The value of kleff at 147 nM
is 2027 103 s1 which results in an eff ective lysis time
constant of 493 s
523 Oligomer and pore concentration
In both the low (lt544 nM) and high toxin ($544 nM) regimes
the amount of toxin is found to be limiting Even at the highest
toxin concentration C in frac14 147 nM the toxin in bulk solution is
depleted within 10 s Selected oligomer concentrations as a
function of time are plotted in Fig 8a and b for both high and
low toxin concentrations as predicted by the kinetic model (eqn
(4) and (6)) The pore density is illustrated in Fig 7 The gov-
erning equations for the formation of an l -mer are given in eqn
(6) Since pore formation occurs via a sequential oligomeriza-
tion mechanism a protomer (1-mer) is necessary for the
formation of all other l mers Hence a steady monomer
concentration on the RBC is achieved once the 1-mer concen-
tration reduces to zero and lag times are observed for theformation of the l + 1-mer From the distribution of mers on the
membrane we observe that a large fraction of protomers remain
trapped as intermediate mers on the membrane At 147 nM the
number of monomers per ml is 8854 1012 The initial
number N RBC frac14 32 107 the monomers per RBC is 276
105 If all the monomers were converted to pores each RBC
would have 23 105 pores However the number of pores
formed per RBC at 147 nM is 11657 105 pores (Fig 7) indi-
cating that about 50 of the mers remain on the membrane
surface as intermediate n-mers (n frac14 1ndash11) At a sublytic
concentration of 544 nM only about 45 of the mers are
converted to pores resulting in 375 pores per RBC (Fig 5a)
The eff ect of l is more prominent at the higher toxin
concentration where both the life time and the maximum
concentration for 1-mers is found to decrease as l is increased
from 2 to 4 The steady state concentrations of various n-mers aspredicted by the model are illustrated in Fig 8c and d A change
in the value of l results in a shi in the distribution for a
particular initial toxin concentration An increase in l implies an
increase in the conformational time relative to the adsorption
and reaction times Hence as l is increased occurrence of the
lower mers on the surface decreases due to the faster reaction
time scales relative to conformation At low bulk toxin concen-
trations (Fig 8c) the distribution of higher mers and conse-
quently the number of pores (12 mers) is very low due to the
limited supply of monomers in the system However at higher
concentration (Fig 8d) the number of monomers is no longer
the limiting factor and the distribution shi
s towards the highermers thereby increasing the number of pores on the surface
524 Parameter sensitivity
We brie y summarize the results of simulations carried out to
test the inuence of the estimated parameters on the model
predictions In the absence of monomer membrane binding
equilibria the value of saturated surface concentration ( ms) is
Fig 8 Model predictions for oligomer concentration pro1047297les ( pl) for l frac14 2 and kl frac14 18 107 s1 (a) Initial bulk concentration frac14 544 nM (b)
Initial bulk concentration frac14 147 nM Steady state distribution of oligomers for (c) C in frac14 544 nM (d) C in frac14 147 nM
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unknown In order to test the inuence of ms on the model
prediction we carried out a few simulations for ms frac14 1 108
mol m2 and msfrac14 1 1010 mol m2 for various C in values For
ms frac14 1 108 mol m2 a negligible number of pores were
formed and the H max ndashC in data (Fig 3a) is underpredicted At
ms frac14 1 1010 mol m2 pore formation was extremely rapid
and little variation in pore density between C in frac14 882 nM and
147 nM was observed As a consequence H max ndashC in data (Fig 3a)
is grossly overpredicted Hence a value of ms frac14 1 10
9 molm2 was used in the simulations (Fig 6) We further note
that the amount of saturated surface concentration ms
implicitly changes the reaction rate constant k l (eqn (12))
Increasing ms eff ectively decreases the reaction rate constant
(eqn (12))
Once snet is xed npc is related to l through the relation npcfrac14
alb with the constants a and b being xed for a given initial toxin
concentration C in (Fig 5) We have found that 2 l 4 ts the
hemoglobin release data very closely (Fig 6) and although the
hemoglobin released as a function of time is slightly under-
estimated by the model the agreement is reasonable Upon
increasing l
to 7 and keeping m
s
frac14 10
9
mol m
s1
the corre-sponding value of npc is 1349 With a value of kl frac14 13 107 s1
although H max versus C in data is accurately predicted the
hemoglobin versus time data is grossly underpredicted Varying
ms between 108 and 1010 mol m2 further deteriorated the
prediction Finally we point out that other functional forms of the
dependence on np npc in the cell lysis equation eqn (8) such as
the unit step function or a quadratic dependence (np npc)2 only
overestimated the H max versus C in data
53 Sequential oligomerization with reversible kinetics
In the previous discussion we present the results for the oligo-
merization kinetics which was assumed to be irreversible Thesequential oligomerization model with reversible kinetics is
p1 thorn pl ) k f
k b pl thorn1 l frac14 1 n 1 (16)
where k f and k b represent the forward and backward reaction
rate constants The kinetic equations are
d p1
dt frac14 k cm k f
Xn1
l frac141
p1 pl thorn k bXn1
l frac142
pl (17a)
d p2
dt frac14
1
2k f p1 p1 k f p1 p2 thorn k b p3
1
2 p2 (17b)
d pl
dt frac14 k f p1 pethl 1THORN eth1 dl 12THORNk f p1 pl
thorn k beth1 dl 12THORNetheth1 dl 11THORN pl thorn1 pl THORN l frac14 3 12 (17c)
A reversible time constant can be dened from eqn (17) as
sb frac14 1
k b The ratio R is dened as the ratio of forward to back-
ward time constants ( R frac14 sf sb) to study the eff ect of revers-
ibility Upon examining the number of pores as a function of
time we observe that the time taken to reach a steady number of
pores is signicantly larger than the time taken to reach steady
state in the lysis experiments In order to make comparisons
with the irreversible mechanism we evaluated the number of
pores at the threshold concentration of 544 nM It is observed
that the number of pores required for lysis initially increases
and then decreases for increments in R values The distribution
of oligomers at steady state are shown for diff erent R values in
Fig 9a and b for C in frac14 147 nM The steady state concentrations
of lsquomersrsquo change from a predominantly 11-mer concentration to
a predominant 1-mer concentration as R is varied between0 and 1 For R lt 1 we nd that the concentration of pores (12
mers) are signicantly higher than the intermediate lsquomerrsquo
concentrations shown in Fig 9b and range from 0142 nmol
Fig 9 Oligomer distribution as a function of the ratio of forward and
backward time constants R frac14 sfsb for C in frac14 147 nM (a) R varies
between 0 and 1 (b) R varies between 0 and 01 (c) Hemoglobin
release data as a function of toxin concentration is shown forvarious R
R frac14 0 has the closest agreement with the experimental data (open
squares)
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m2 ( R frac14 0) to 028 nmol m2 ( R frac14 001) For R frac14 1 the
concentration of 12 mers is 024 nmol m2 and decreases with a
further increase in R The predictions using reversible sequen-
tial kinetics for the H max vs C in data (Fig 9c) indicate
greater deviation from the experimental data when compared
with R frac14 0 These results indicate that the irreversible mecha-
nism provides the best agreement with the experimental data
We point out that the critical number of pores npc in
the reversible framework is non-monotonic ranging from 392( R frac14 0) to 5128 ( R frac14 1) for l frac14 2
54 Non-sequential oligomerization
In contrast to the 392 pores obtained for the critical number of
pores via the sequential mechanism a substantially larger
critical number of pores are observed via the non-sequential
oligomerization (6000 pores per cell) The mass balance for
the protomer ( p1) concentration remains identical to that of the
sequential oligomerization mechanism (eqn (4)) The governing
equations for pl (l lt l n) oligomer undergoing non-sequential
irreversible oligomerization are
d pl
dt frac14
1
2k l Xl 1
ufrac141
pu pl u eth1 dl 12THORNXn1
ufrac141
k l pl pu for 1l n (18)
where n represents the number of monomers in a pore The
relation between the critical number of pores and npc for non-
sequential oligomerization is npc frac14 5439l015 The critical
number of pores for l frac14 2 is 6035 pores which is about 20 times
greater than that obtained from a sequential oligomerization
mechanism In this scheme the hemoglobin release (Fig 10a)
data is overpredicted at intermediate toxin concentrations when
compared with the sequential oligomerization The predictions
of the H out vs time data at 147 nM (Fig 10b) are similar whencompared with the sequential oligomerization (Fig 6b and d)
A comparison of the tted parameters between the sequential
and non-sequential oligomerization mechanisms are given in
Table 2
Oligomer distributions obtained from the non-sequential
mechanism (Fig 11b) show an entirely diff erent trend when
compared to that obtained from the sequential mechanism
(Fig 8) In the sequential mechanism the higher lsquomersrsquo
attained a steady state once the protomer was depleted In the
non-sequential mechanism the contribution to the dodeca-
mers (12 mer) can be obtained from a large number of combi-
nations of the lower oligomers leading to a larger dodecamer orpore concentration The time scale required to attain the olig-
omer steady state concentration (Fig 11a) is about 20 s for an
initial concentration of 147 nM Similar time scales are
observed in the sequential mechanism as well
Further experiments are required to distinguish between the
various mechanisms Western Blot experiments conducted on
Hemolysin E12 and Clostridium perfringens 3 toxins31 showed the
presence of intermediate oligomers On the other hand single-
molecule uorescence imaging of a-hemolysin on a droplet
interface bilayer showed the presence of only monomers and
heptamers (pores)32
6 Discussion and conclusions
Lysis experiments on RBCs with the ClyA pore-forming toxin
show that a threshold initial toxin concentration is required to
initiate lysis From this observation we analyzed the problem in
two regimes a low toxin concentration regime where rupture of
cells is absent and a high toxin concentration regime where
lysis occurs and hemoglobin is released Kinetic models which
accounts for monomer binding conformation (membrane-
bound monomer to protomer) and oligomerization to form the
dodecameric pore complex are developed Models which
account for sequential and non-sequential oligomerization are
tested Cell rupture is assumed to be
rst order in the number of live cells and directly proportional to the pores in excess of the
critical number of pores npc In the sublytic regime the number
of pores is found to have a power law dependence on l which is
the ratio of conformational time to the reaction time This leads
to the construction of a ldquophase diagramrdquo between the number
of pores np and l for diff erent values of the initial toxin
concentration Comparing simulations with experimental data
the range of npc was 392ndash768 for the sequential mechanism and
5300ndash6300 pores for the non-sequential mechanism for 2 l 4
The range of l values is consistent with available experimental
data on ClyA which indicates that the membrane-bound
Fig 10 (a) Model predictions from the non-sequential mechanism
Simulated Hmax C in (open circles) curves grossly overpredicts the
experimental data (open squares) (b) Simulated hemoglobin release
compares well with the experimental data Dashed line snetfrac14 1 s solid
line snet frac14 24 s
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conformational step is slower than the preceding adsorption
and subsequent oligomerization steps10
From the model we are also able to comment on the time
constants for the various processes The time constant for poreformation is about 1 s indicating that pore formation itself is a
fast process relative to the time taken for the pore population on
a single RBC to reach steady state which is about 20ndash30 s Since
rupture kinetics is dynamic and depends on the fraction of live
cells as well as the number of pores on the cell lysis occurs in
the time scale of 10s of minutes Due to this separation of time
scales lysis is seen to occur once the number of pores has
reached a steady state This steady number of pores ranges from
515 to 11 657 as the toxin concentration ranges from 588ndash
147 nM For the non-sequential mechanism the critical
number of pores required to initiate lysis is about 20 times
higher when compared to that of sequential oligomerizationComparison of three diff erent kinetic models reveals that the
irreversible sequential kinetics provides the closest match with
the hemoglobin released as a function of the initial toxin
concentration Although we observe an overprediction of the
hemoglobin release data with the non-sequential mechanism
the hemoglobin release kinetics are similar to that of the
sequential mechanism The distribution of lower oligomers is
distinctly diff erent in both cases with a negligible numbers of
lower mers observed in the non-sequential oligomerization
These diff erences in the distribution of mers off er a ngerprint
to identify the underlying mechanism for pore formation
Experiments which can determine the number of pores or the
steady state lsquomerrsquo distributions on the membrane surface will
shed light on the pathways for oligomerization and enable a
more denite conclusion of the underlying kinetics Furtherthe rupture kinetics model contains only one adjustable
parameter Lysis experiments conducted with Vibrio cholerae El
Tor cytolysin33 and Monalysin34 show similar lysis times (in the
order of 10s of minutes) as observed in our study suggesting
similarities in the underlying kinetic pathways that lead to pore
formation and rupture The model developed in this manu-
script is generic and could be recast with some variation to
study the dynamics of other PFTs
We brie y discuss some of the limitations of the model in its
present form The model is based on the mean eld approxi-
mation where all cells are assumed to be identical and for the
purpose of binding and oligomerization this is an adequatestarting point The more complicated process is the mechanics
of rupture with the correct functional dependence on the pore
density In general there could exist a distribution of cells with
diff erent densities of pores Preliminary experiments by varying
the number of RBCs at a xed toxin concentration led to an
increase in lysis suggesting that cell heterogeneity could be
playing a role Although a population balance model35 could
include these variations this is at an added cost of complexity A
second aspect inherent to the model is the presence of lysis
beyond the time at which steady state is observed in the
experiments (30 minutes) Once the number of pores has
Fig 11 Oligomerconcentrationsas a function of time areshownfor (a) 544 nM and(b) 147 nMSteady state oligomer concentrations areshown
for (c) 544 nM and (d) 147 nM At steady state 1ndash4 mer concentrations are zero for both 544 and 147 nM The 12-mer concentration at steady
state is quite large compared to the other oligomer concentrations present in the system
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reached a steady state which occurs within 30 s (Fig 7a) cell
lysis continues to occur at a xed number of pores proportional
to np npc Running the simulation to steady state would
eventually lead to lysis of all the cells in the system albeit at an
exceedingly slow rate at the lower and intermediate toxin
concentrations We have also assumed complete and irrevers-
ible binding of the monomer to the membrane and hence the
number of pores predicted represent an upper limit within the
proposed kinetic framework Despite the mean eld approxi-mation the model proposed in this manuscript is able to
capture the inherent time scales in the process of pore forma-
tion and rupture as well as predict the variation of hemoglobin
release as a function of the initial toxin concentration observed
in the experiments We nally point out that the eff ect of
temperature on the kinetics and lysis has not been investigated
in this work Preliminary experiments at 147 nM toxin
concentration carried out to steady state indicated a marked
drop in lytic activity for temperatures below 15 C and lysis was
not observed at 10 C Data on binding isotherms would be
required to understand temperature eff ects in these systems
Our work has implications in optimizing dosage and devel-oping novel drug delivery strategies based on PFTs The obser-
vation that lysis occurs in a given window of toxin concentration
has implications while developing PFT-based drug and gene
therapy protocols In recent studies E coli used in conjunction
with radiation therapy was found to retard the growth of cancer
tumors when compared with only radiation therapy protocols36
Quantifying the dynamics and extent of lysis is an important
aspect of developing appropriate treatment protocols in these
combination therapies In other applications where pores are
used for gene delivery and in the development of articial cells
lysis must be prevented and pore formation restricted to
concentrations below the lysis threshold
Acknowledgements
We acknowledge funding from Department of Science and
Technology (DST) India under the IRHPA grant The authors
thank Sanjeev Kumar Gupta and Jaydeep Basu for several useful
discussions on the development of the model as well as the
reviewers for their critical comments
References
1 S Bhakdi and J Tranum-Jensen Microbiol Rev 1991 55
733ndash
7512 J Thiery D Keefe S Boulant E Boucrot M Walch
D Martinvalet I S Goping R C Bleackley
T Kirchhausen and J Lieberman Nat Immunol 2011 12
770ndash777
3 A Ludwig C von Rhein S Bauer C Huttinger and
W Goebel J Bacteriol 2004 186 5311ndash5320
4 A W Bernheimer and L S Avigad Proc Natl Acad Sci U S
A 1976 73 467ndash471
5 M V Sousa M Richardson W Fontes and L Morhy
J Protein Chem 1994 13 659ndash667
6 G Anderluh and P Macek Toxicon 2002 40 111ndash124
7 I Iacovache F G van der Goot and L Pernot Biochim
Biophys Acta Biomembr 2008 1778 1611ndash1623
8 N R Wyborn A Clark R E Roberts S J Jamieson
S Tzokov P A Bullough T J Stillman P J Artymiuk
J E Galen L Zhao M M Levine and J Green
Microbiology 2004 150 1495ndash1505
9 A J Wallace T J Stillman A Atkins S J Jamieson
P A Bullough J Green and P J Artymiuk Cell 2000 100
265ndash27610 M Mueller U Grauschopf T Maier R Glockshuber and
N Ban Nature 2009 459 726ndash730
11 J Oscarsson Y Mizunoe L Li X-H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
12 A Atkins N R Wyborn A J Wallace T J Stillman
L K Black A B Fielding M Hisakado P J Artymiuk and
J Green J Biol Chem 2000 275 41150ndash41155
13 A Ludwig G Volkerink C von Rhein S Bauer E Maier
B Bergmann W Goebel and R Benz J Bacteriol 2010
192 4001ndash4011
14 N Eier M Vetsch M Gregorini P Ringler M Chami
A Philippsen A Fritz S A Muller R Glockshuber A Engel and U Grauschopf EMBO J 2006 25 2652ndash2661
15 L A Shepard O Shatursky A E Johnson and R K Tweten
Biochemistry 2000 39 10284ndash10293
16 A J Ratner K R Hippe J L Aguilar M H Bender
A L Nelson and J N Weiser J Biol Chem 2006 281
12994ndash12998
17 I Iacovache M Bischoerger and F G van der Goot Curr
Opin Struct Biol 2010 20 241ndash246
18 G Schwarz and C H Robert Biophys Chem 1992 42 291ndash296
19 G Schwarz R-t Zong and T Popescu Biochim Biophys
Acta Biomembr 1992 1110 97ndash104
20 D Bolintineanu E Hazrati H T Davis R I Lehrer and
Y N Kaznessis Peptides 2010 31 1ndash821 J M Alam T Kobayashi and M Yamazaki Biochemistry
2012 51 5160ndash5172
22 J Lemiere K Guevorkian C Campillo C Sykes and T Betz
So Matter 2013 9 3181ndash3187
23 V Noireaux and A Libchaber Proc Natl Acad Sci U S A
2004 101 17669ndash17674
24 M M Bradford Anal Biochem 1976 72 248ndash254
25 H Xu E K Bjerneld M Kall and L Borjesson Phys Rev
Lett 1999 83 4357ndash4360
26 J Oscarsson Y Mizunoe L Li X H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
27 T M Raschke J Tsai and M Levitt Proc Natl Acad SciU S A 2001 98 5965ndash5969
28 L A Rodrıguez-Guadarrama S K Talsania K K Mohanty
and R Rajagopalan Langmuir 1999 15 437ndash446
29 C E McLaren G M Brittenham and V Hasselblad Am J
Physiol 1987 252 3181ndash3187
30 J G Kim M Xia and H Liu IEEE Eng Med Biol Mag 2005
24 118ndash121
31 M Nagahama H Hara M Fernandez-Miyakawa
Y Itohayashi and J Sakurai Biochemistry 2006 45 296ndash302
32 J R Thompson B Cronin H Bayley and M I Wallace
Biophys J 2011 101 2679ndash2683
This journal is copy The Royal Society of Chemistry 2014 RSC Adv 2014 4 4930ndash4942 | 4941
Paper RSC Advances
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7252019 c3ra45159c
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33 A Zitzer I Walev M Palmer and S Bhakdi Med Microbiol
Immunol 1995 184 37ndash44
34 O Opota I Vallet-Gely R Vincentelli C Kellenberger
I Iacovache M R Gonzalez A Roussel F G van der Goot
and B Lemaitre PLoS Pathog 2011 7 e1002259
35 S Kumar and D Ramkrishna Chem Eng Sci 1996 51
1311ndash1332
36 S N Jiang T X Phan T K Nam V H Nguyen H S Kim
H S Bom H E Choy Y Hong and J J Min Mol Ther
2010 18 635ndash642
RSC Advances Paper
View Article Online
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from 294ndash294 nM The decrease in turbidity is seen to occur
simultaneously with an increase in the OD supporting the view that the increase in OD is due to cell lysis The turbidity
decrease is also mirrored with the corresponding OD data Since
the cell mass used in the turbidity experiments is lower than
that used in the lysis experiments sublytic toxin concentrations
lie below 294 nM
52 Model predictions
In this section model predictions for the sequential irreversible
kinetics are compared with the lysis data
521 Sublytic regime
In the sublytic regime (np lt npc) the evolution of np with time is
obtained by solving eqn (11a)ndash(e) The number of pores np is
obtained from eqn (7) From the simulations we observe that the number of pores per RBC for a given initial toxin concen-
tration saturates within 10 s (Fig 5a) This saturation in np is
due to the limiting amount of toxin present in solution In
Fig 5b we plot the variation in np at saturation (20 s) with
l frac14sc
sa for diff erent initial toxin concentration assuming that
np lt npc The curve corresponding to 544 nM is tted to the
form y frac14 axb and the relation npc frac14 20033l097 is obtained
which can be used to x the value of npc for a given value of l In
order to simulate the lysis data a value of snet (eqn (14)) which
is the time constant associated with the time required for
formation of the rst pore has to be specied Initial estimates
are in the range of 1ndash25 s and we use a value of 1 s in all our
simulations unless specied We show later that our results are
Fig 3 (a) The normalized optical density values as a function of the
initial toxin concentration observed after 60 minutes during the lysis
experiments A distinct jump is observed above a toxin concentration
of 544 nM (b) Time evolution data of optical density values during
RBC lysis At 147 nM 98 lysis is observed
Fig 4 Turbidity (left axis) and lysis data (right axis) show that the
decrease in turbidity occurs simultaneously with an increase in the
OD The sublytic initial toxin concentration is less than 294 nM
(100 ng ml1)
Fig 5 Relation between l and npc in the sublytic regime (a) Number
of pores per RBC as a function of time The data plotted are for l frac14 2
and snet frac14 1 s (b) A regression analysis of the curve at 544 nM (dash-
dotted line) yields the relation npc frac14 20033l097
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relatively insensitive to the value of snet in this range The
parameters l npc and snet are obtained in the sublytic regime as
discussed above
522 Lysis regime
In the lysis regime in addition to the kinetic eqn (11andashe) we also
solve the lysis and hemoglobin release equations eqn (8) and
eqn (10) Cell lysis occurs once np gt npc and the OD increases with increasing toxin concentration as shown in Fig 3a Once
values of l npc and snet are xed in the sub-lytic regime the only
unknown parameter in the model is the value of kl (eqn (8)) In
all cases kl is xed by matching the maximum extent of lysis of
98 obtained at 147 nM Subsequent simulations are run with
diff erent values of C in to compare with the experimental data
Fig 6 illustrates the comparison between the model prediction
and the experimental OD data Since the OD varies linearly with
the amount of hemoglobin released during lysis30 we scale both
the experimental and predicted data by their respective
maximum OD to facilitate a meaningful comparison We also
carried out independent lysis experiments to verify the linear
relationship between the OD versus hemoglobin data Hence we
normalize the long time data and dene H max as the ratio of the
mass of the steady state hemoglobin released at a given C in to
the corresponding value at C in frac14 147 nM which is the highest
C in considered in the study This facilitates a comparison of
H max predicted from the model directly with the normalized OD
values For l frac14 2 the corresponding npc frac14 392 and the
comparison of H max at a value of kl frac14 18 107 s1 is
illustrated in Fig 6a The comparison for lfrac14 4 (npcfrac14 768) and kl
frac14 15 107 s1 is illustrated in Fig 6c Comparison of the
hemoglobin release dynamics ( H out vs time) for the corre-
sponding set of parameters are illustrated in Fig 6b and d
respectively
We observe that in this range of l (2 l 4) values and
kl 15ndash18 107 s1 the OD vs C in data is captured quite
accurately Since l is the ratio of the ClyA monomer confor-
mation time to the monomer adsorption time this range of l value is consistent with experiments10 which indicates that
conformation is preceded by fast adsorption followed by rapid
oligomerization The H out dynamics predicted by the model is
seen to capture the experimental data quite well (Fig 6b and d)
Upon increasing l we nd that a lower value of kl is required to
match the OD vs C in data (Fig 6d) A value of snet frac14 1 s captures
the early time release in the H out data at 147 nM quite accu-
rately and increasing snet to 24 s results in a short delay at early
times Since snet represents the time taken to form the rst pore
in situ monitoring of the hemoglobin release dynamics would be
required to determine snet more precisely In our experiments the
time evolution of the lysis data is carried out by intermittently arresting lysis at diff erent times and quenching the aliquots in
ice for a period of 3ndash5 minutes while the OD is determined
Experiments carried out continuously for the diff erent time
points shown in Fig 6b and d did not alter the data obtained
from the intermittent experiments Fluorescence permeation
experiments by Yamazaki and co-workers21 by lysenin (334 kDa)
induced pore formation on single giant unilamellar vesicles show
that pore formation is complete within about 10 s for toxin
concentrations of 200 ng ml1 and 50 s at 40 ng ml1 These are
similar to the time scales deduced in our model
At a toxin concentration of 147 nM a steady distribution of
mers is achieved on the time scale of 10ndash20 s (for a snet frac14 1 s)
Within this time interval the pore density goes through a phaseof rapid increase to exceed the critical pore density (Fig 7)
during which very little lysis is observed (Fig 6a and d) Lysis is
predominantly observed a er a steady number of pores have
formed on the RBC surface This steady number of pores ranges
Fig 6 Comparison of model predictions (open circles) with experi-
mental results (open squares) The amount of hemoglobin released
(Hmax) atthe end of30min for(a) npcfrac14 392 (c) npcfrac14 768 Dynamics of
hemoglobin released (Hout) for (b) npcfrac14 392 (d) npcfrac14 768 A value of l
between 2 and 4 is seen to accurately capture the Hmax versus C in data
[(a) and (c)] The amount of hemoglobin released is scaled with the
maximum amount to facilitate a comparison Simulations corresponding
to snet frac14 24 s are shown in bold lines and snet frac14 1 s in dashed lines
Fig 7 The pore density is plotted as a function of time in thepost-lysis
regime The number of pores (np) per RBC ranges from 515 at 588 nM
to 11 657 at 147 nM l frac14 2 and kl frac14 18 107 s1 npc frac14 392
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from 515 at 588 nM to 11 657 at 147 nM Given this situation it
is instructive to dene an eff ective lysis time constant kleff frac14
kl (np npc) where np is the steady state value of the number of
pores at a given value of C in (Fig 7) The value of kleff at 147 nM
is 2027 103 s1 which results in an eff ective lysis time
constant of 493 s
523 Oligomer and pore concentration
In both the low (lt544 nM) and high toxin ($544 nM) regimes
the amount of toxin is found to be limiting Even at the highest
toxin concentration C in frac14 147 nM the toxin in bulk solution is
depleted within 10 s Selected oligomer concentrations as a
function of time are plotted in Fig 8a and b for both high and
low toxin concentrations as predicted by the kinetic model (eqn
(4) and (6)) The pore density is illustrated in Fig 7 The gov-
erning equations for the formation of an l -mer are given in eqn
(6) Since pore formation occurs via a sequential oligomeriza-
tion mechanism a protomer (1-mer) is necessary for the
formation of all other l mers Hence a steady monomer
concentration on the RBC is achieved once the 1-mer concen-
tration reduces to zero and lag times are observed for theformation of the l + 1-mer From the distribution of mers on the
membrane we observe that a large fraction of protomers remain
trapped as intermediate mers on the membrane At 147 nM the
number of monomers per ml is 8854 1012 The initial
number N RBC frac14 32 107 the monomers per RBC is 276
105 If all the monomers were converted to pores each RBC
would have 23 105 pores However the number of pores
formed per RBC at 147 nM is 11657 105 pores (Fig 7) indi-
cating that about 50 of the mers remain on the membrane
surface as intermediate n-mers (n frac14 1ndash11) At a sublytic
concentration of 544 nM only about 45 of the mers are
converted to pores resulting in 375 pores per RBC (Fig 5a)
The eff ect of l is more prominent at the higher toxin
concentration where both the life time and the maximum
concentration for 1-mers is found to decrease as l is increased
from 2 to 4 The steady state concentrations of various n-mers aspredicted by the model are illustrated in Fig 8c and d A change
in the value of l results in a shi in the distribution for a
particular initial toxin concentration An increase in l implies an
increase in the conformational time relative to the adsorption
and reaction times Hence as l is increased occurrence of the
lower mers on the surface decreases due to the faster reaction
time scales relative to conformation At low bulk toxin concen-
trations (Fig 8c) the distribution of higher mers and conse-
quently the number of pores (12 mers) is very low due to the
limited supply of monomers in the system However at higher
concentration (Fig 8d) the number of monomers is no longer
the limiting factor and the distribution shi
s towards the highermers thereby increasing the number of pores on the surface
524 Parameter sensitivity
We brie y summarize the results of simulations carried out to
test the inuence of the estimated parameters on the model
predictions In the absence of monomer membrane binding
equilibria the value of saturated surface concentration ( ms) is
Fig 8 Model predictions for oligomer concentration pro1047297les ( pl) for l frac14 2 and kl frac14 18 107 s1 (a) Initial bulk concentration frac14 544 nM (b)
Initial bulk concentration frac14 147 nM Steady state distribution of oligomers for (c) C in frac14 544 nM (d) C in frac14 147 nM
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unknown In order to test the inuence of ms on the model
prediction we carried out a few simulations for ms frac14 1 108
mol m2 and msfrac14 1 1010 mol m2 for various C in values For
ms frac14 1 108 mol m2 a negligible number of pores were
formed and the H max ndashC in data (Fig 3a) is underpredicted At
ms frac14 1 1010 mol m2 pore formation was extremely rapid
and little variation in pore density between C in frac14 882 nM and
147 nM was observed As a consequence H max ndashC in data (Fig 3a)
is grossly overpredicted Hence a value of ms frac14 1 10
9 molm2 was used in the simulations (Fig 6) We further note
that the amount of saturated surface concentration ms
implicitly changes the reaction rate constant k l (eqn (12))
Increasing ms eff ectively decreases the reaction rate constant
(eqn (12))
Once snet is xed npc is related to l through the relation npcfrac14
alb with the constants a and b being xed for a given initial toxin
concentration C in (Fig 5) We have found that 2 l 4 ts the
hemoglobin release data very closely (Fig 6) and although the
hemoglobin released as a function of time is slightly under-
estimated by the model the agreement is reasonable Upon
increasing l
to 7 and keeping m
s
frac14 10
9
mol m
s1
the corre-sponding value of npc is 1349 With a value of kl frac14 13 107 s1
although H max versus C in data is accurately predicted the
hemoglobin versus time data is grossly underpredicted Varying
ms between 108 and 1010 mol m2 further deteriorated the
prediction Finally we point out that other functional forms of the
dependence on np npc in the cell lysis equation eqn (8) such as
the unit step function or a quadratic dependence (np npc)2 only
overestimated the H max versus C in data
53 Sequential oligomerization with reversible kinetics
In the previous discussion we present the results for the oligo-
merization kinetics which was assumed to be irreversible Thesequential oligomerization model with reversible kinetics is
p1 thorn pl ) k f
k b pl thorn1 l frac14 1 n 1 (16)
where k f and k b represent the forward and backward reaction
rate constants The kinetic equations are
d p1
dt frac14 k cm k f
Xn1
l frac141
p1 pl thorn k bXn1
l frac142
pl (17a)
d p2
dt frac14
1
2k f p1 p1 k f p1 p2 thorn k b p3
1
2 p2 (17b)
d pl
dt frac14 k f p1 pethl 1THORN eth1 dl 12THORNk f p1 pl
thorn k beth1 dl 12THORNetheth1 dl 11THORN pl thorn1 pl THORN l frac14 3 12 (17c)
A reversible time constant can be dened from eqn (17) as
sb frac14 1
k b The ratio R is dened as the ratio of forward to back-
ward time constants ( R frac14 sf sb) to study the eff ect of revers-
ibility Upon examining the number of pores as a function of
time we observe that the time taken to reach a steady number of
pores is signicantly larger than the time taken to reach steady
state in the lysis experiments In order to make comparisons
with the irreversible mechanism we evaluated the number of
pores at the threshold concentration of 544 nM It is observed
that the number of pores required for lysis initially increases
and then decreases for increments in R values The distribution
of oligomers at steady state are shown for diff erent R values in
Fig 9a and b for C in frac14 147 nM The steady state concentrations
of lsquomersrsquo change from a predominantly 11-mer concentration to
a predominant 1-mer concentration as R is varied between0 and 1 For R lt 1 we nd that the concentration of pores (12
mers) are signicantly higher than the intermediate lsquomerrsquo
concentrations shown in Fig 9b and range from 0142 nmol
Fig 9 Oligomer distribution as a function of the ratio of forward and
backward time constants R frac14 sfsb for C in frac14 147 nM (a) R varies
between 0 and 1 (b) R varies between 0 and 01 (c) Hemoglobin
release data as a function of toxin concentration is shown forvarious R
R frac14 0 has the closest agreement with the experimental data (open
squares)
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m2 ( R frac14 0) to 028 nmol m2 ( R frac14 001) For R frac14 1 the
concentration of 12 mers is 024 nmol m2 and decreases with a
further increase in R The predictions using reversible sequen-
tial kinetics for the H max vs C in data (Fig 9c) indicate
greater deviation from the experimental data when compared
with R frac14 0 These results indicate that the irreversible mecha-
nism provides the best agreement with the experimental data
We point out that the critical number of pores npc in
the reversible framework is non-monotonic ranging from 392( R frac14 0) to 5128 ( R frac14 1) for l frac14 2
54 Non-sequential oligomerization
In contrast to the 392 pores obtained for the critical number of
pores via the sequential mechanism a substantially larger
critical number of pores are observed via the non-sequential
oligomerization (6000 pores per cell) The mass balance for
the protomer ( p1) concentration remains identical to that of the
sequential oligomerization mechanism (eqn (4)) The governing
equations for pl (l lt l n) oligomer undergoing non-sequential
irreversible oligomerization are
d pl
dt frac14
1
2k l Xl 1
ufrac141
pu pl u eth1 dl 12THORNXn1
ufrac141
k l pl pu for 1l n (18)
where n represents the number of monomers in a pore The
relation between the critical number of pores and npc for non-
sequential oligomerization is npc frac14 5439l015 The critical
number of pores for l frac14 2 is 6035 pores which is about 20 times
greater than that obtained from a sequential oligomerization
mechanism In this scheme the hemoglobin release (Fig 10a)
data is overpredicted at intermediate toxin concentrations when
compared with the sequential oligomerization The predictions
of the H out vs time data at 147 nM (Fig 10b) are similar whencompared with the sequential oligomerization (Fig 6b and d)
A comparison of the tted parameters between the sequential
and non-sequential oligomerization mechanisms are given in
Table 2
Oligomer distributions obtained from the non-sequential
mechanism (Fig 11b) show an entirely diff erent trend when
compared to that obtained from the sequential mechanism
(Fig 8) In the sequential mechanism the higher lsquomersrsquo
attained a steady state once the protomer was depleted In the
non-sequential mechanism the contribution to the dodeca-
mers (12 mer) can be obtained from a large number of combi-
nations of the lower oligomers leading to a larger dodecamer orpore concentration The time scale required to attain the olig-
omer steady state concentration (Fig 11a) is about 20 s for an
initial concentration of 147 nM Similar time scales are
observed in the sequential mechanism as well
Further experiments are required to distinguish between the
various mechanisms Western Blot experiments conducted on
Hemolysin E12 and Clostridium perfringens 3 toxins31 showed the
presence of intermediate oligomers On the other hand single-
molecule uorescence imaging of a-hemolysin on a droplet
interface bilayer showed the presence of only monomers and
heptamers (pores)32
6 Discussion and conclusions
Lysis experiments on RBCs with the ClyA pore-forming toxin
show that a threshold initial toxin concentration is required to
initiate lysis From this observation we analyzed the problem in
two regimes a low toxin concentration regime where rupture of
cells is absent and a high toxin concentration regime where
lysis occurs and hemoglobin is released Kinetic models which
accounts for monomer binding conformation (membrane-
bound monomer to protomer) and oligomerization to form the
dodecameric pore complex are developed Models which
account for sequential and non-sequential oligomerization are
tested Cell rupture is assumed to be
rst order in the number of live cells and directly proportional to the pores in excess of the
critical number of pores npc In the sublytic regime the number
of pores is found to have a power law dependence on l which is
the ratio of conformational time to the reaction time This leads
to the construction of a ldquophase diagramrdquo between the number
of pores np and l for diff erent values of the initial toxin
concentration Comparing simulations with experimental data
the range of npc was 392ndash768 for the sequential mechanism and
5300ndash6300 pores for the non-sequential mechanism for 2 l 4
The range of l values is consistent with available experimental
data on ClyA which indicates that the membrane-bound
Fig 10 (a) Model predictions from the non-sequential mechanism
Simulated Hmax C in (open circles) curves grossly overpredicts the
experimental data (open squares) (b) Simulated hemoglobin release
compares well with the experimental data Dashed line snetfrac14 1 s solid
line snet frac14 24 s
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conformational step is slower than the preceding adsorption
and subsequent oligomerization steps10
From the model we are also able to comment on the time
constants for the various processes The time constant for poreformation is about 1 s indicating that pore formation itself is a
fast process relative to the time taken for the pore population on
a single RBC to reach steady state which is about 20ndash30 s Since
rupture kinetics is dynamic and depends on the fraction of live
cells as well as the number of pores on the cell lysis occurs in
the time scale of 10s of minutes Due to this separation of time
scales lysis is seen to occur once the number of pores has
reached a steady state This steady number of pores ranges from
515 to 11 657 as the toxin concentration ranges from 588ndash
147 nM For the non-sequential mechanism the critical
number of pores required to initiate lysis is about 20 times
higher when compared to that of sequential oligomerizationComparison of three diff erent kinetic models reveals that the
irreversible sequential kinetics provides the closest match with
the hemoglobin released as a function of the initial toxin
concentration Although we observe an overprediction of the
hemoglobin release data with the non-sequential mechanism
the hemoglobin release kinetics are similar to that of the
sequential mechanism The distribution of lower oligomers is
distinctly diff erent in both cases with a negligible numbers of
lower mers observed in the non-sequential oligomerization
These diff erences in the distribution of mers off er a ngerprint
to identify the underlying mechanism for pore formation
Experiments which can determine the number of pores or the
steady state lsquomerrsquo distributions on the membrane surface will
shed light on the pathways for oligomerization and enable a
more denite conclusion of the underlying kinetics Furtherthe rupture kinetics model contains only one adjustable
parameter Lysis experiments conducted with Vibrio cholerae El
Tor cytolysin33 and Monalysin34 show similar lysis times (in the
order of 10s of minutes) as observed in our study suggesting
similarities in the underlying kinetic pathways that lead to pore
formation and rupture The model developed in this manu-
script is generic and could be recast with some variation to
study the dynamics of other PFTs
We brie y discuss some of the limitations of the model in its
present form The model is based on the mean eld approxi-
mation where all cells are assumed to be identical and for the
purpose of binding and oligomerization this is an adequatestarting point The more complicated process is the mechanics
of rupture with the correct functional dependence on the pore
density In general there could exist a distribution of cells with
diff erent densities of pores Preliminary experiments by varying
the number of RBCs at a xed toxin concentration led to an
increase in lysis suggesting that cell heterogeneity could be
playing a role Although a population balance model35 could
include these variations this is at an added cost of complexity A
second aspect inherent to the model is the presence of lysis
beyond the time at which steady state is observed in the
experiments (30 minutes) Once the number of pores has
Fig 11 Oligomerconcentrationsas a function of time areshownfor (a) 544 nM and(b) 147 nMSteady state oligomer concentrations areshown
for (c) 544 nM and (d) 147 nM At steady state 1ndash4 mer concentrations are zero for both 544 and 147 nM The 12-mer concentration at steady
state is quite large compared to the other oligomer concentrations present in the system
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reached a steady state which occurs within 30 s (Fig 7a) cell
lysis continues to occur at a xed number of pores proportional
to np npc Running the simulation to steady state would
eventually lead to lysis of all the cells in the system albeit at an
exceedingly slow rate at the lower and intermediate toxin
concentrations We have also assumed complete and irrevers-
ible binding of the monomer to the membrane and hence the
number of pores predicted represent an upper limit within the
proposed kinetic framework Despite the mean eld approxi-mation the model proposed in this manuscript is able to
capture the inherent time scales in the process of pore forma-
tion and rupture as well as predict the variation of hemoglobin
release as a function of the initial toxin concentration observed
in the experiments We nally point out that the eff ect of
temperature on the kinetics and lysis has not been investigated
in this work Preliminary experiments at 147 nM toxin
concentration carried out to steady state indicated a marked
drop in lytic activity for temperatures below 15 C and lysis was
not observed at 10 C Data on binding isotherms would be
required to understand temperature eff ects in these systems
Our work has implications in optimizing dosage and devel-oping novel drug delivery strategies based on PFTs The obser-
vation that lysis occurs in a given window of toxin concentration
has implications while developing PFT-based drug and gene
therapy protocols In recent studies E coli used in conjunction
with radiation therapy was found to retard the growth of cancer
tumors when compared with only radiation therapy protocols36
Quantifying the dynamics and extent of lysis is an important
aspect of developing appropriate treatment protocols in these
combination therapies In other applications where pores are
used for gene delivery and in the development of articial cells
lysis must be prevented and pore formation restricted to
concentrations below the lysis threshold
Acknowledgements
We acknowledge funding from Department of Science and
Technology (DST) India under the IRHPA grant The authors
thank Sanjeev Kumar Gupta and Jaydeep Basu for several useful
discussions on the development of the model as well as the
reviewers for their critical comments
References
1 S Bhakdi and J Tranum-Jensen Microbiol Rev 1991 55
733ndash
7512 J Thiery D Keefe S Boulant E Boucrot M Walch
D Martinvalet I S Goping R C Bleackley
T Kirchhausen and J Lieberman Nat Immunol 2011 12
770ndash777
3 A Ludwig C von Rhein S Bauer C Huttinger and
W Goebel J Bacteriol 2004 186 5311ndash5320
4 A W Bernheimer and L S Avigad Proc Natl Acad Sci U S
A 1976 73 467ndash471
5 M V Sousa M Richardson W Fontes and L Morhy
J Protein Chem 1994 13 659ndash667
6 G Anderluh and P Macek Toxicon 2002 40 111ndash124
7 I Iacovache F G van der Goot and L Pernot Biochim
Biophys Acta Biomembr 2008 1778 1611ndash1623
8 N R Wyborn A Clark R E Roberts S J Jamieson
S Tzokov P A Bullough T J Stillman P J Artymiuk
J E Galen L Zhao M M Levine and J Green
Microbiology 2004 150 1495ndash1505
9 A J Wallace T J Stillman A Atkins S J Jamieson
P A Bullough J Green and P J Artymiuk Cell 2000 100
265ndash27610 M Mueller U Grauschopf T Maier R Glockshuber and
N Ban Nature 2009 459 726ndash730
11 J Oscarsson Y Mizunoe L Li X-H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
12 A Atkins N R Wyborn A J Wallace T J Stillman
L K Black A B Fielding M Hisakado P J Artymiuk and
J Green J Biol Chem 2000 275 41150ndash41155
13 A Ludwig G Volkerink C von Rhein S Bauer E Maier
B Bergmann W Goebel and R Benz J Bacteriol 2010
192 4001ndash4011
14 N Eier M Vetsch M Gregorini P Ringler M Chami
A Philippsen A Fritz S A Muller R Glockshuber A Engel and U Grauschopf EMBO J 2006 25 2652ndash2661
15 L A Shepard O Shatursky A E Johnson and R K Tweten
Biochemistry 2000 39 10284ndash10293
16 A J Ratner K R Hippe J L Aguilar M H Bender
A L Nelson and J N Weiser J Biol Chem 2006 281
12994ndash12998
17 I Iacovache M Bischoerger and F G van der Goot Curr
Opin Struct Biol 2010 20 241ndash246
18 G Schwarz and C H Robert Biophys Chem 1992 42 291ndash296
19 G Schwarz R-t Zong and T Popescu Biochim Biophys
Acta Biomembr 1992 1110 97ndash104
20 D Bolintineanu E Hazrati H T Davis R I Lehrer and
Y N Kaznessis Peptides 2010 31 1ndash821 J M Alam T Kobayashi and M Yamazaki Biochemistry
2012 51 5160ndash5172
22 J Lemiere K Guevorkian C Campillo C Sykes and T Betz
So Matter 2013 9 3181ndash3187
23 V Noireaux and A Libchaber Proc Natl Acad Sci U S A
2004 101 17669ndash17674
24 M M Bradford Anal Biochem 1976 72 248ndash254
25 H Xu E K Bjerneld M Kall and L Borjesson Phys Rev
Lett 1999 83 4357ndash4360
26 J Oscarsson Y Mizunoe L Li X H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
27 T M Raschke J Tsai and M Levitt Proc Natl Acad SciU S A 2001 98 5965ndash5969
28 L A Rodrıguez-Guadarrama S K Talsania K K Mohanty
and R Rajagopalan Langmuir 1999 15 437ndash446
29 C E McLaren G M Brittenham and V Hasselblad Am J
Physiol 1987 252 3181ndash3187
30 J G Kim M Xia and H Liu IEEE Eng Med Biol Mag 2005
24 118ndash121
31 M Nagahama H Hara M Fernandez-Miyakawa
Y Itohayashi and J Sakurai Biochemistry 2006 45 296ndash302
32 J R Thompson B Cronin H Bayley and M I Wallace
Biophys J 2011 101 2679ndash2683
This journal is copy The Royal Society of Chemistry 2014 RSC Adv 2014 4 4930ndash4942 | 4941
Paper RSC Advances
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7252019 c3ra45159c
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33 A Zitzer I Walev M Palmer and S Bhakdi Med Microbiol
Immunol 1995 184 37ndash44
34 O Opota I Vallet-Gely R Vincentelli C Kellenberger
I Iacovache M R Gonzalez A Roussel F G van der Goot
and B Lemaitre PLoS Pathog 2011 7 e1002259
35 S Kumar and D Ramkrishna Chem Eng Sci 1996 51
1311ndash1332
36 S N Jiang T X Phan T K Nam V H Nguyen H S Kim
H S Bom H E Choy Y Hong and J J Min Mol Ther
2010 18 635ndash642
RSC Advances Paper
View Article Online
7252019 c3ra45159c
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relatively insensitive to the value of snet in this range The
parameters l npc and snet are obtained in the sublytic regime as
discussed above
522 Lysis regime
In the lysis regime in addition to the kinetic eqn (11andashe) we also
solve the lysis and hemoglobin release equations eqn (8) and
eqn (10) Cell lysis occurs once np gt npc and the OD increases with increasing toxin concentration as shown in Fig 3a Once
values of l npc and snet are xed in the sub-lytic regime the only
unknown parameter in the model is the value of kl (eqn (8)) In
all cases kl is xed by matching the maximum extent of lysis of
98 obtained at 147 nM Subsequent simulations are run with
diff erent values of C in to compare with the experimental data
Fig 6 illustrates the comparison between the model prediction
and the experimental OD data Since the OD varies linearly with
the amount of hemoglobin released during lysis30 we scale both
the experimental and predicted data by their respective
maximum OD to facilitate a meaningful comparison We also
carried out independent lysis experiments to verify the linear
relationship between the OD versus hemoglobin data Hence we
normalize the long time data and dene H max as the ratio of the
mass of the steady state hemoglobin released at a given C in to
the corresponding value at C in frac14 147 nM which is the highest
C in considered in the study This facilitates a comparison of
H max predicted from the model directly with the normalized OD
values For l frac14 2 the corresponding npc frac14 392 and the
comparison of H max at a value of kl frac14 18 107 s1 is
illustrated in Fig 6a The comparison for lfrac14 4 (npcfrac14 768) and kl
frac14 15 107 s1 is illustrated in Fig 6c Comparison of the
hemoglobin release dynamics ( H out vs time) for the corre-
sponding set of parameters are illustrated in Fig 6b and d
respectively
We observe that in this range of l (2 l 4) values and
kl 15ndash18 107 s1 the OD vs C in data is captured quite
accurately Since l is the ratio of the ClyA monomer confor-
mation time to the monomer adsorption time this range of l value is consistent with experiments10 which indicates that
conformation is preceded by fast adsorption followed by rapid
oligomerization The H out dynamics predicted by the model is
seen to capture the experimental data quite well (Fig 6b and d)
Upon increasing l we nd that a lower value of kl is required to
match the OD vs C in data (Fig 6d) A value of snet frac14 1 s captures
the early time release in the H out data at 147 nM quite accu-
rately and increasing snet to 24 s results in a short delay at early
times Since snet represents the time taken to form the rst pore
in situ monitoring of the hemoglobin release dynamics would be
required to determine snet more precisely In our experiments the
time evolution of the lysis data is carried out by intermittently arresting lysis at diff erent times and quenching the aliquots in
ice for a period of 3ndash5 minutes while the OD is determined
Experiments carried out continuously for the diff erent time
points shown in Fig 6b and d did not alter the data obtained
from the intermittent experiments Fluorescence permeation
experiments by Yamazaki and co-workers21 by lysenin (334 kDa)
induced pore formation on single giant unilamellar vesicles show
that pore formation is complete within about 10 s for toxin
concentrations of 200 ng ml1 and 50 s at 40 ng ml1 These are
similar to the time scales deduced in our model
At a toxin concentration of 147 nM a steady distribution of
mers is achieved on the time scale of 10ndash20 s (for a snet frac14 1 s)
Within this time interval the pore density goes through a phaseof rapid increase to exceed the critical pore density (Fig 7)
during which very little lysis is observed (Fig 6a and d) Lysis is
predominantly observed a er a steady number of pores have
formed on the RBC surface This steady number of pores ranges
Fig 6 Comparison of model predictions (open circles) with experi-
mental results (open squares) The amount of hemoglobin released
(Hmax) atthe end of30min for(a) npcfrac14 392 (c) npcfrac14 768 Dynamics of
hemoglobin released (Hout) for (b) npcfrac14 392 (d) npcfrac14 768 A value of l
between 2 and 4 is seen to accurately capture the Hmax versus C in data
[(a) and (c)] The amount of hemoglobin released is scaled with the
maximum amount to facilitate a comparison Simulations corresponding
to snet frac14 24 s are shown in bold lines and snet frac14 1 s in dashed lines
Fig 7 The pore density is plotted as a function of time in thepost-lysis
regime The number of pores (np) per RBC ranges from 515 at 588 nM
to 11 657 at 147 nM l frac14 2 and kl frac14 18 107 s1 npc frac14 392
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from 515 at 588 nM to 11 657 at 147 nM Given this situation it
is instructive to dene an eff ective lysis time constant kleff frac14
kl (np npc) where np is the steady state value of the number of
pores at a given value of C in (Fig 7) The value of kleff at 147 nM
is 2027 103 s1 which results in an eff ective lysis time
constant of 493 s
523 Oligomer and pore concentration
In both the low (lt544 nM) and high toxin ($544 nM) regimes
the amount of toxin is found to be limiting Even at the highest
toxin concentration C in frac14 147 nM the toxin in bulk solution is
depleted within 10 s Selected oligomer concentrations as a
function of time are plotted in Fig 8a and b for both high and
low toxin concentrations as predicted by the kinetic model (eqn
(4) and (6)) The pore density is illustrated in Fig 7 The gov-
erning equations for the formation of an l -mer are given in eqn
(6) Since pore formation occurs via a sequential oligomeriza-
tion mechanism a protomer (1-mer) is necessary for the
formation of all other l mers Hence a steady monomer
concentration on the RBC is achieved once the 1-mer concen-
tration reduces to zero and lag times are observed for theformation of the l + 1-mer From the distribution of mers on the
membrane we observe that a large fraction of protomers remain
trapped as intermediate mers on the membrane At 147 nM the
number of monomers per ml is 8854 1012 The initial
number N RBC frac14 32 107 the monomers per RBC is 276
105 If all the monomers were converted to pores each RBC
would have 23 105 pores However the number of pores
formed per RBC at 147 nM is 11657 105 pores (Fig 7) indi-
cating that about 50 of the mers remain on the membrane
surface as intermediate n-mers (n frac14 1ndash11) At a sublytic
concentration of 544 nM only about 45 of the mers are
converted to pores resulting in 375 pores per RBC (Fig 5a)
The eff ect of l is more prominent at the higher toxin
concentration where both the life time and the maximum
concentration for 1-mers is found to decrease as l is increased
from 2 to 4 The steady state concentrations of various n-mers aspredicted by the model are illustrated in Fig 8c and d A change
in the value of l results in a shi in the distribution for a
particular initial toxin concentration An increase in l implies an
increase in the conformational time relative to the adsorption
and reaction times Hence as l is increased occurrence of the
lower mers on the surface decreases due to the faster reaction
time scales relative to conformation At low bulk toxin concen-
trations (Fig 8c) the distribution of higher mers and conse-
quently the number of pores (12 mers) is very low due to the
limited supply of monomers in the system However at higher
concentration (Fig 8d) the number of monomers is no longer
the limiting factor and the distribution shi
s towards the highermers thereby increasing the number of pores on the surface
524 Parameter sensitivity
We brie y summarize the results of simulations carried out to
test the inuence of the estimated parameters on the model
predictions In the absence of monomer membrane binding
equilibria the value of saturated surface concentration ( ms) is
Fig 8 Model predictions for oligomer concentration pro1047297les ( pl) for l frac14 2 and kl frac14 18 107 s1 (a) Initial bulk concentration frac14 544 nM (b)
Initial bulk concentration frac14 147 nM Steady state distribution of oligomers for (c) C in frac14 544 nM (d) C in frac14 147 nM
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unknown In order to test the inuence of ms on the model
prediction we carried out a few simulations for ms frac14 1 108
mol m2 and msfrac14 1 1010 mol m2 for various C in values For
ms frac14 1 108 mol m2 a negligible number of pores were
formed and the H max ndashC in data (Fig 3a) is underpredicted At
ms frac14 1 1010 mol m2 pore formation was extremely rapid
and little variation in pore density between C in frac14 882 nM and
147 nM was observed As a consequence H max ndashC in data (Fig 3a)
is grossly overpredicted Hence a value of ms frac14 1 10
9 molm2 was used in the simulations (Fig 6) We further note
that the amount of saturated surface concentration ms
implicitly changes the reaction rate constant k l (eqn (12))
Increasing ms eff ectively decreases the reaction rate constant
(eqn (12))
Once snet is xed npc is related to l through the relation npcfrac14
alb with the constants a and b being xed for a given initial toxin
concentration C in (Fig 5) We have found that 2 l 4 ts the
hemoglobin release data very closely (Fig 6) and although the
hemoglobin released as a function of time is slightly under-
estimated by the model the agreement is reasonable Upon
increasing l
to 7 and keeping m
s
frac14 10
9
mol m
s1
the corre-sponding value of npc is 1349 With a value of kl frac14 13 107 s1
although H max versus C in data is accurately predicted the
hemoglobin versus time data is grossly underpredicted Varying
ms between 108 and 1010 mol m2 further deteriorated the
prediction Finally we point out that other functional forms of the
dependence on np npc in the cell lysis equation eqn (8) such as
the unit step function or a quadratic dependence (np npc)2 only
overestimated the H max versus C in data
53 Sequential oligomerization with reversible kinetics
In the previous discussion we present the results for the oligo-
merization kinetics which was assumed to be irreversible Thesequential oligomerization model with reversible kinetics is
p1 thorn pl ) k f
k b pl thorn1 l frac14 1 n 1 (16)
where k f and k b represent the forward and backward reaction
rate constants The kinetic equations are
d p1
dt frac14 k cm k f
Xn1
l frac141
p1 pl thorn k bXn1
l frac142
pl (17a)
d p2
dt frac14
1
2k f p1 p1 k f p1 p2 thorn k b p3
1
2 p2 (17b)
d pl
dt frac14 k f p1 pethl 1THORN eth1 dl 12THORNk f p1 pl
thorn k beth1 dl 12THORNetheth1 dl 11THORN pl thorn1 pl THORN l frac14 3 12 (17c)
A reversible time constant can be dened from eqn (17) as
sb frac14 1
k b The ratio R is dened as the ratio of forward to back-
ward time constants ( R frac14 sf sb) to study the eff ect of revers-
ibility Upon examining the number of pores as a function of
time we observe that the time taken to reach a steady number of
pores is signicantly larger than the time taken to reach steady
state in the lysis experiments In order to make comparisons
with the irreversible mechanism we evaluated the number of
pores at the threshold concentration of 544 nM It is observed
that the number of pores required for lysis initially increases
and then decreases for increments in R values The distribution
of oligomers at steady state are shown for diff erent R values in
Fig 9a and b for C in frac14 147 nM The steady state concentrations
of lsquomersrsquo change from a predominantly 11-mer concentration to
a predominant 1-mer concentration as R is varied between0 and 1 For R lt 1 we nd that the concentration of pores (12
mers) are signicantly higher than the intermediate lsquomerrsquo
concentrations shown in Fig 9b and range from 0142 nmol
Fig 9 Oligomer distribution as a function of the ratio of forward and
backward time constants R frac14 sfsb for C in frac14 147 nM (a) R varies
between 0 and 1 (b) R varies between 0 and 01 (c) Hemoglobin
release data as a function of toxin concentration is shown forvarious R
R frac14 0 has the closest agreement with the experimental data (open
squares)
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m2 ( R frac14 0) to 028 nmol m2 ( R frac14 001) For R frac14 1 the
concentration of 12 mers is 024 nmol m2 and decreases with a
further increase in R The predictions using reversible sequen-
tial kinetics for the H max vs C in data (Fig 9c) indicate
greater deviation from the experimental data when compared
with R frac14 0 These results indicate that the irreversible mecha-
nism provides the best agreement with the experimental data
We point out that the critical number of pores npc in
the reversible framework is non-monotonic ranging from 392( R frac14 0) to 5128 ( R frac14 1) for l frac14 2
54 Non-sequential oligomerization
In contrast to the 392 pores obtained for the critical number of
pores via the sequential mechanism a substantially larger
critical number of pores are observed via the non-sequential
oligomerization (6000 pores per cell) The mass balance for
the protomer ( p1) concentration remains identical to that of the
sequential oligomerization mechanism (eqn (4)) The governing
equations for pl (l lt l n) oligomer undergoing non-sequential
irreversible oligomerization are
d pl
dt frac14
1
2k l Xl 1
ufrac141
pu pl u eth1 dl 12THORNXn1
ufrac141
k l pl pu for 1l n (18)
where n represents the number of monomers in a pore The
relation between the critical number of pores and npc for non-
sequential oligomerization is npc frac14 5439l015 The critical
number of pores for l frac14 2 is 6035 pores which is about 20 times
greater than that obtained from a sequential oligomerization
mechanism In this scheme the hemoglobin release (Fig 10a)
data is overpredicted at intermediate toxin concentrations when
compared with the sequential oligomerization The predictions
of the H out vs time data at 147 nM (Fig 10b) are similar whencompared with the sequential oligomerization (Fig 6b and d)
A comparison of the tted parameters between the sequential
and non-sequential oligomerization mechanisms are given in
Table 2
Oligomer distributions obtained from the non-sequential
mechanism (Fig 11b) show an entirely diff erent trend when
compared to that obtained from the sequential mechanism
(Fig 8) In the sequential mechanism the higher lsquomersrsquo
attained a steady state once the protomer was depleted In the
non-sequential mechanism the contribution to the dodeca-
mers (12 mer) can be obtained from a large number of combi-
nations of the lower oligomers leading to a larger dodecamer orpore concentration The time scale required to attain the olig-
omer steady state concentration (Fig 11a) is about 20 s for an
initial concentration of 147 nM Similar time scales are
observed in the sequential mechanism as well
Further experiments are required to distinguish between the
various mechanisms Western Blot experiments conducted on
Hemolysin E12 and Clostridium perfringens 3 toxins31 showed the
presence of intermediate oligomers On the other hand single-
molecule uorescence imaging of a-hemolysin on a droplet
interface bilayer showed the presence of only monomers and
heptamers (pores)32
6 Discussion and conclusions
Lysis experiments on RBCs with the ClyA pore-forming toxin
show that a threshold initial toxin concentration is required to
initiate lysis From this observation we analyzed the problem in
two regimes a low toxin concentration regime where rupture of
cells is absent and a high toxin concentration regime where
lysis occurs and hemoglobin is released Kinetic models which
accounts for monomer binding conformation (membrane-
bound monomer to protomer) and oligomerization to form the
dodecameric pore complex are developed Models which
account for sequential and non-sequential oligomerization are
tested Cell rupture is assumed to be
rst order in the number of live cells and directly proportional to the pores in excess of the
critical number of pores npc In the sublytic regime the number
of pores is found to have a power law dependence on l which is
the ratio of conformational time to the reaction time This leads
to the construction of a ldquophase diagramrdquo between the number
of pores np and l for diff erent values of the initial toxin
concentration Comparing simulations with experimental data
the range of npc was 392ndash768 for the sequential mechanism and
5300ndash6300 pores for the non-sequential mechanism for 2 l 4
The range of l values is consistent with available experimental
data on ClyA which indicates that the membrane-bound
Fig 10 (a) Model predictions from the non-sequential mechanism
Simulated Hmax C in (open circles) curves grossly overpredicts the
experimental data (open squares) (b) Simulated hemoglobin release
compares well with the experimental data Dashed line snetfrac14 1 s solid
line snet frac14 24 s
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Paper RSC Advances
View Article Online
7252019 c3ra45159c
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conformational step is slower than the preceding adsorption
and subsequent oligomerization steps10
From the model we are also able to comment on the time
constants for the various processes The time constant for poreformation is about 1 s indicating that pore formation itself is a
fast process relative to the time taken for the pore population on
a single RBC to reach steady state which is about 20ndash30 s Since
rupture kinetics is dynamic and depends on the fraction of live
cells as well as the number of pores on the cell lysis occurs in
the time scale of 10s of minutes Due to this separation of time
scales lysis is seen to occur once the number of pores has
reached a steady state This steady number of pores ranges from
515 to 11 657 as the toxin concentration ranges from 588ndash
147 nM For the non-sequential mechanism the critical
number of pores required to initiate lysis is about 20 times
higher when compared to that of sequential oligomerizationComparison of three diff erent kinetic models reveals that the
irreversible sequential kinetics provides the closest match with
the hemoglobin released as a function of the initial toxin
concentration Although we observe an overprediction of the
hemoglobin release data with the non-sequential mechanism
the hemoglobin release kinetics are similar to that of the
sequential mechanism The distribution of lower oligomers is
distinctly diff erent in both cases with a negligible numbers of
lower mers observed in the non-sequential oligomerization
These diff erences in the distribution of mers off er a ngerprint
to identify the underlying mechanism for pore formation
Experiments which can determine the number of pores or the
steady state lsquomerrsquo distributions on the membrane surface will
shed light on the pathways for oligomerization and enable a
more denite conclusion of the underlying kinetics Furtherthe rupture kinetics model contains only one adjustable
parameter Lysis experiments conducted with Vibrio cholerae El
Tor cytolysin33 and Monalysin34 show similar lysis times (in the
order of 10s of minutes) as observed in our study suggesting
similarities in the underlying kinetic pathways that lead to pore
formation and rupture The model developed in this manu-
script is generic and could be recast with some variation to
study the dynamics of other PFTs
We brie y discuss some of the limitations of the model in its
present form The model is based on the mean eld approxi-
mation where all cells are assumed to be identical and for the
purpose of binding and oligomerization this is an adequatestarting point The more complicated process is the mechanics
of rupture with the correct functional dependence on the pore
density In general there could exist a distribution of cells with
diff erent densities of pores Preliminary experiments by varying
the number of RBCs at a xed toxin concentration led to an
increase in lysis suggesting that cell heterogeneity could be
playing a role Although a population balance model35 could
include these variations this is at an added cost of complexity A
second aspect inherent to the model is the presence of lysis
beyond the time at which steady state is observed in the
experiments (30 minutes) Once the number of pores has
Fig 11 Oligomerconcentrationsas a function of time areshownfor (a) 544 nM and(b) 147 nMSteady state oligomer concentrations areshown
for (c) 544 nM and (d) 147 nM At steady state 1ndash4 mer concentrations are zero for both 544 and 147 nM The 12-mer concentration at steady
state is quite large compared to the other oligomer concentrations present in the system
4940 | RSC Adv 2014 4 4930ndash4942 This journal is copy The Royal Society of Chemistry 2014
RSC Advances Paper
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7252019 c3ra45159c
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reached a steady state which occurs within 30 s (Fig 7a) cell
lysis continues to occur at a xed number of pores proportional
to np npc Running the simulation to steady state would
eventually lead to lysis of all the cells in the system albeit at an
exceedingly slow rate at the lower and intermediate toxin
concentrations We have also assumed complete and irrevers-
ible binding of the monomer to the membrane and hence the
number of pores predicted represent an upper limit within the
proposed kinetic framework Despite the mean eld approxi-mation the model proposed in this manuscript is able to
capture the inherent time scales in the process of pore forma-
tion and rupture as well as predict the variation of hemoglobin
release as a function of the initial toxin concentration observed
in the experiments We nally point out that the eff ect of
temperature on the kinetics and lysis has not been investigated
in this work Preliminary experiments at 147 nM toxin
concentration carried out to steady state indicated a marked
drop in lytic activity for temperatures below 15 C and lysis was
not observed at 10 C Data on binding isotherms would be
required to understand temperature eff ects in these systems
Our work has implications in optimizing dosage and devel-oping novel drug delivery strategies based on PFTs The obser-
vation that lysis occurs in a given window of toxin concentration
has implications while developing PFT-based drug and gene
therapy protocols In recent studies E coli used in conjunction
with radiation therapy was found to retard the growth of cancer
tumors when compared with only radiation therapy protocols36
Quantifying the dynamics and extent of lysis is an important
aspect of developing appropriate treatment protocols in these
combination therapies In other applications where pores are
used for gene delivery and in the development of articial cells
lysis must be prevented and pore formation restricted to
concentrations below the lysis threshold
Acknowledgements
We acknowledge funding from Department of Science and
Technology (DST) India under the IRHPA grant The authors
thank Sanjeev Kumar Gupta and Jaydeep Basu for several useful
discussions on the development of the model as well as the
reviewers for their critical comments
References
1 S Bhakdi and J Tranum-Jensen Microbiol Rev 1991 55
733ndash
7512 J Thiery D Keefe S Boulant E Boucrot M Walch
D Martinvalet I S Goping R C Bleackley
T Kirchhausen and J Lieberman Nat Immunol 2011 12
770ndash777
3 A Ludwig C von Rhein S Bauer C Huttinger and
W Goebel J Bacteriol 2004 186 5311ndash5320
4 A W Bernheimer and L S Avigad Proc Natl Acad Sci U S
A 1976 73 467ndash471
5 M V Sousa M Richardson W Fontes and L Morhy
J Protein Chem 1994 13 659ndash667
6 G Anderluh and P Macek Toxicon 2002 40 111ndash124
7 I Iacovache F G van der Goot and L Pernot Biochim
Biophys Acta Biomembr 2008 1778 1611ndash1623
8 N R Wyborn A Clark R E Roberts S J Jamieson
S Tzokov P A Bullough T J Stillman P J Artymiuk
J E Galen L Zhao M M Levine and J Green
Microbiology 2004 150 1495ndash1505
9 A J Wallace T J Stillman A Atkins S J Jamieson
P A Bullough J Green and P J Artymiuk Cell 2000 100
265ndash27610 M Mueller U Grauschopf T Maier R Glockshuber and
N Ban Nature 2009 459 726ndash730
11 J Oscarsson Y Mizunoe L Li X-H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
12 A Atkins N R Wyborn A J Wallace T J Stillman
L K Black A B Fielding M Hisakado P J Artymiuk and
J Green J Biol Chem 2000 275 41150ndash41155
13 A Ludwig G Volkerink C von Rhein S Bauer E Maier
B Bergmann W Goebel and R Benz J Bacteriol 2010
192 4001ndash4011
14 N Eier M Vetsch M Gregorini P Ringler M Chami
A Philippsen A Fritz S A Muller R Glockshuber A Engel and U Grauschopf EMBO J 2006 25 2652ndash2661
15 L A Shepard O Shatursky A E Johnson and R K Tweten
Biochemistry 2000 39 10284ndash10293
16 A J Ratner K R Hippe J L Aguilar M H Bender
A L Nelson and J N Weiser J Biol Chem 2006 281
12994ndash12998
17 I Iacovache M Bischoerger and F G van der Goot Curr
Opin Struct Biol 2010 20 241ndash246
18 G Schwarz and C H Robert Biophys Chem 1992 42 291ndash296
19 G Schwarz R-t Zong and T Popescu Biochim Biophys
Acta Biomembr 1992 1110 97ndash104
20 D Bolintineanu E Hazrati H T Davis R I Lehrer and
Y N Kaznessis Peptides 2010 31 1ndash821 J M Alam T Kobayashi and M Yamazaki Biochemistry
2012 51 5160ndash5172
22 J Lemiere K Guevorkian C Campillo C Sykes and T Betz
So Matter 2013 9 3181ndash3187
23 V Noireaux and A Libchaber Proc Natl Acad Sci U S A
2004 101 17669ndash17674
24 M M Bradford Anal Biochem 1976 72 248ndash254
25 H Xu E K Bjerneld M Kall and L Borjesson Phys Rev
Lett 1999 83 4357ndash4360
26 J Oscarsson Y Mizunoe L Li X H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
27 T M Raschke J Tsai and M Levitt Proc Natl Acad SciU S A 2001 98 5965ndash5969
28 L A Rodrıguez-Guadarrama S K Talsania K K Mohanty
and R Rajagopalan Langmuir 1999 15 437ndash446
29 C E McLaren G M Brittenham and V Hasselblad Am J
Physiol 1987 252 3181ndash3187
30 J G Kim M Xia and H Liu IEEE Eng Med Biol Mag 2005
24 118ndash121
31 M Nagahama H Hara M Fernandez-Miyakawa
Y Itohayashi and J Sakurai Biochemistry 2006 45 296ndash302
32 J R Thompson B Cronin H Bayley and M I Wallace
Biophys J 2011 101 2679ndash2683
This journal is copy The Royal Society of Chemistry 2014 RSC Adv 2014 4 4930ndash4942 | 4941
Paper RSC Advances
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1313
33 A Zitzer I Walev M Palmer and S Bhakdi Med Microbiol
Immunol 1995 184 37ndash44
34 O Opota I Vallet-Gely R Vincentelli C Kellenberger
I Iacovache M R Gonzalez A Roussel F G van der Goot
and B Lemaitre PLoS Pathog 2011 7 e1002259
35 S Kumar and D Ramkrishna Chem Eng Sci 1996 51
1311ndash1332
36 S N Jiang T X Phan T K Nam V H Nguyen H S Kim
H S Bom H E Choy Y Hong and J J Min Mol Ther
2010 18 635ndash642
RSC Advances Paper
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7252019 c3ra45159c
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from 515 at 588 nM to 11 657 at 147 nM Given this situation it
is instructive to dene an eff ective lysis time constant kleff frac14
kl (np npc) where np is the steady state value of the number of
pores at a given value of C in (Fig 7) The value of kleff at 147 nM
is 2027 103 s1 which results in an eff ective lysis time
constant of 493 s
523 Oligomer and pore concentration
In both the low (lt544 nM) and high toxin ($544 nM) regimes
the amount of toxin is found to be limiting Even at the highest
toxin concentration C in frac14 147 nM the toxin in bulk solution is
depleted within 10 s Selected oligomer concentrations as a
function of time are plotted in Fig 8a and b for both high and
low toxin concentrations as predicted by the kinetic model (eqn
(4) and (6)) The pore density is illustrated in Fig 7 The gov-
erning equations for the formation of an l -mer are given in eqn
(6) Since pore formation occurs via a sequential oligomeriza-
tion mechanism a protomer (1-mer) is necessary for the
formation of all other l mers Hence a steady monomer
concentration on the RBC is achieved once the 1-mer concen-
tration reduces to zero and lag times are observed for theformation of the l + 1-mer From the distribution of mers on the
membrane we observe that a large fraction of protomers remain
trapped as intermediate mers on the membrane At 147 nM the
number of monomers per ml is 8854 1012 The initial
number N RBC frac14 32 107 the monomers per RBC is 276
105 If all the monomers were converted to pores each RBC
would have 23 105 pores However the number of pores
formed per RBC at 147 nM is 11657 105 pores (Fig 7) indi-
cating that about 50 of the mers remain on the membrane
surface as intermediate n-mers (n frac14 1ndash11) At a sublytic
concentration of 544 nM only about 45 of the mers are
converted to pores resulting in 375 pores per RBC (Fig 5a)
The eff ect of l is more prominent at the higher toxin
concentration where both the life time and the maximum
concentration for 1-mers is found to decrease as l is increased
from 2 to 4 The steady state concentrations of various n-mers aspredicted by the model are illustrated in Fig 8c and d A change
in the value of l results in a shi in the distribution for a
particular initial toxin concentration An increase in l implies an
increase in the conformational time relative to the adsorption
and reaction times Hence as l is increased occurrence of the
lower mers on the surface decreases due to the faster reaction
time scales relative to conformation At low bulk toxin concen-
trations (Fig 8c) the distribution of higher mers and conse-
quently the number of pores (12 mers) is very low due to the
limited supply of monomers in the system However at higher
concentration (Fig 8d) the number of monomers is no longer
the limiting factor and the distribution shi
s towards the highermers thereby increasing the number of pores on the surface
524 Parameter sensitivity
We brie y summarize the results of simulations carried out to
test the inuence of the estimated parameters on the model
predictions In the absence of monomer membrane binding
equilibria the value of saturated surface concentration ( ms) is
Fig 8 Model predictions for oligomer concentration pro1047297les ( pl) for l frac14 2 and kl frac14 18 107 s1 (a) Initial bulk concentration frac14 544 nM (b)
Initial bulk concentration frac14 147 nM Steady state distribution of oligomers for (c) C in frac14 544 nM (d) C in frac14 147 nM
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unknown In order to test the inuence of ms on the model
prediction we carried out a few simulations for ms frac14 1 108
mol m2 and msfrac14 1 1010 mol m2 for various C in values For
ms frac14 1 108 mol m2 a negligible number of pores were
formed and the H max ndashC in data (Fig 3a) is underpredicted At
ms frac14 1 1010 mol m2 pore formation was extremely rapid
and little variation in pore density between C in frac14 882 nM and
147 nM was observed As a consequence H max ndashC in data (Fig 3a)
is grossly overpredicted Hence a value of ms frac14 1 10
9 molm2 was used in the simulations (Fig 6) We further note
that the amount of saturated surface concentration ms
implicitly changes the reaction rate constant k l (eqn (12))
Increasing ms eff ectively decreases the reaction rate constant
(eqn (12))
Once snet is xed npc is related to l through the relation npcfrac14
alb with the constants a and b being xed for a given initial toxin
concentration C in (Fig 5) We have found that 2 l 4 ts the
hemoglobin release data very closely (Fig 6) and although the
hemoglobin released as a function of time is slightly under-
estimated by the model the agreement is reasonable Upon
increasing l
to 7 and keeping m
s
frac14 10
9
mol m
s1
the corre-sponding value of npc is 1349 With a value of kl frac14 13 107 s1
although H max versus C in data is accurately predicted the
hemoglobin versus time data is grossly underpredicted Varying
ms between 108 and 1010 mol m2 further deteriorated the
prediction Finally we point out that other functional forms of the
dependence on np npc in the cell lysis equation eqn (8) such as
the unit step function or a quadratic dependence (np npc)2 only
overestimated the H max versus C in data
53 Sequential oligomerization with reversible kinetics
In the previous discussion we present the results for the oligo-
merization kinetics which was assumed to be irreversible Thesequential oligomerization model with reversible kinetics is
p1 thorn pl ) k f
k b pl thorn1 l frac14 1 n 1 (16)
where k f and k b represent the forward and backward reaction
rate constants The kinetic equations are
d p1
dt frac14 k cm k f
Xn1
l frac141
p1 pl thorn k bXn1
l frac142
pl (17a)
d p2
dt frac14
1
2k f p1 p1 k f p1 p2 thorn k b p3
1
2 p2 (17b)
d pl
dt frac14 k f p1 pethl 1THORN eth1 dl 12THORNk f p1 pl
thorn k beth1 dl 12THORNetheth1 dl 11THORN pl thorn1 pl THORN l frac14 3 12 (17c)
A reversible time constant can be dened from eqn (17) as
sb frac14 1
k b The ratio R is dened as the ratio of forward to back-
ward time constants ( R frac14 sf sb) to study the eff ect of revers-
ibility Upon examining the number of pores as a function of
time we observe that the time taken to reach a steady number of
pores is signicantly larger than the time taken to reach steady
state in the lysis experiments In order to make comparisons
with the irreversible mechanism we evaluated the number of
pores at the threshold concentration of 544 nM It is observed
that the number of pores required for lysis initially increases
and then decreases for increments in R values The distribution
of oligomers at steady state are shown for diff erent R values in
Fig 9a and b for C in frac14 147 nM The steady state concentrations
of lsquomersrsquo change from a predominantly 11-mer concentration to
a predominant 1-mer concentration as R is varied between0 and 1 For R lt 1 we nd that the concentration of pores (12
mers) are signicantly higher than the intermediate lsquomerrsquo
concentrations shown in Fig 9b and range from 0142 nmol
Fig 9 Oligomer distribution as a function of the ratio of forward and
backward time constants R frac14 sfsb for C in frac14 147 nM (a) R varies
between 0 and 1 (b) R varies between 0 and 01 (c) Hemoglobin
release data as a function of toxin concentration is shown forvarious R
R frac14 0 has the closest agreement with the experimental data (open
squares)
4938 | RSC Adv 2014 4 4930ndash4942 This journal is copy The Royal Society of Chemistry 2014
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7252019 c3ra45159c
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m2 ( R frac14 0) to 028 nmol m2 ( R frac14 001) For R frac14 1 the
concentration of 12 mers is 024 nmol m2 and decreases with a
further increase in R The predictions using reversible sequen-
tial kinetics for the H max vs C in data (Fig 9c) indicate
greater deviation from the experimental data when compared
with R frac14 0 These results indicate that the irreversible mecha-
nism provides the best agreement with the experimental data
We point out that the critical number of pores npc in
the reversible framework is non-monotonic ranging from 392( R frac14 0) to 5128 ( R frac14 1) for l frac14 2
54 Non-sequential oligomerization
In contrast to the 392 pores obtained for the critical number of
pores via the sequential mechanism a substantially larger
critical number of pores are observed via the non-sequential
oligomerization (6000 pores per cell) The mass balance for
the protomer ( p1) concentration remains identical to that of the
sequential oligomerization mechanism (eqn (4)) The governing
equations for pl (l lt l n) oligomer undergoing non-sequential
irreversible oligomerization are
d pl
dt frac14
1
2k l Xl 1
ufrac141
pu pl u eth1 dl 12THORNXn1
ufrac141
k l pl pu for 1l n (18)
where n represents the number of monomers in a pore The
relation between the critical number of pores and npc for non-
sequential oligomerization is npc frac14 5439l015 The critical
number of pores for l frac14 2 is 6035 pores which is about 20 times
greater than that obtained from a sequential oligomerization
mechanism In this scheme the hemoglobin release (Fig 10a)
data is overpredicted at intermediate toxin concentrations when
compared with the sequential oligomerization The predictions
of the H out vs time data at 147 nM (Fig 10b) are similar whencompared with the sequential oligomerization (Fig 6b and d)
A comparison of the tted parameters between the sequential
and non-sequential oligomerization mechanisms are given in
Table 2
Oligomer distributions obtained from the non-sequential
mechanism (Fig 11b) show an entirely diff erent trend when
compared to that obtained from the sequential mechanism
(Fig 8) In the sequential mechanism the higher lsquomersrsquo
attained a steady state once the protomer was depleted In the
non-sequential mechanism the contribution to the dodeca-
mers (12 mer) can be obtained from a large number of combi-
nations of the lower oligomers leading to a larger dodecamer orpore concentration The time scale required to attain the olig-
omer steady state concentration (Fig 11a) is about 20 s for an
initial concentration of 147 nM Similar time scales are
observed in the sequential mechanism as well
Further experiments are required to distinguish between the
various mechanisms Western Blot experiments conducted on
Hemolysin E12 and Clostridium perfringens 3 toxins31 showed the
presence of intermediate oligomers On the other hand single-
molecule uorescence imaging of a-hemolysin on a droplet
interface bilayer showed the presence of only monomers and
heptamers (pores)32
6 Discussion and conclusions
Lysis experiments on RBCs with the ClyA pore-forming toxin
show that a threshold initial toxin concentration is required to
initiate lysis From this observation we analyzed the problem in
two regimes a low toxin concentration regime where rupture of
cells is absent and a high toxin concentration regime where
lysis occurs and hemoglobin is released Kinetic models which
accounts for monomer binding conformation (membrane-
bound monomer to protomer) and oligomerization to form the
dodecameric pore complex are developed Models which
account for sequential and non-sequential oligomerization are
tested Cell rupture is assumed to be
rst order in the number of live cells and directly proportional to the pores in excess of the
critical number of pores npc In the sublytic regime the number
of pores is found to have a power law dependence on l which is
the ratio of conformational time to the reaction time This leads
to the construction of a ldquophase diagramrdquo between the number
of pores np and l for diff erent values of the initial toxin
concentration Comparing simulations with experimental data
the range of npc was 392ndash768 for the sequential mechanism and
5300ndash6300 pores for the non-sequential mechanism for 2 l 4
The range of l values is consistent with available experimental
data on ClyA which indicates that the membrane-bound
Fig 10 (a) Model predictions from the non-sequential mechanism
Simulated Hmax C in (open circles) curves grossly overpredicts the
experimental data (open squares) (b) Simulated hemoglobin release
compares well with the experimental data Dashed line snetfrac14 1 s solid
line snet frac14 24 s
This journal is copy The Royal Society of Chemistry 2014 RSC Adv 2014 4 4930ndash4942 | 4939
Paper RSC Advances
View Article Online
7252019 c3ra45159c
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conformational step is slower than the preceding adsorption
and subsequent oligomerization steps10
From the model we are also able to comment on the time
constants for the various processes The time constant for poreformation is about 1 s indicating that pore formation itself is a
fast process relative to the time taken for the pore population on
a single RBC to reach steady state which is about 20ndash30 s Since
rupture kinetics is dynamic and depends on the fraction of live
cells as well as the number of pores on the cell lysis occurs in
the time scale of 10s of minutes Due to this separation of time
scales lysis is seen to occur once the number of pores has
reached a steady state This steady number of pores ranges from
515 to 11 657 as the toxin concentration ranges from 588ndash
147 nM For the non-sequential mechanism the critical
number of pores required to initiate lysis is about 20 times
higher when compared to that of sequential oligomerizationComparison of three diff erent kinetic models reveals that the
irreversible sequential kinetics provides the closest match with
the hemoglobin released as a function of the initial toxin
concentration Although we observe an overprediction of the
hemoglobin release data with the non-sequential mechanism
the hemoglobin release kinetics are similar to that of the
sequential mechanism The distribution of lower oligomers is
distinctly diff erent in both cases with a negligible numbers of
lower mers observed in the non-sequential oligomerization
These diff erences in the distribution of mers off er a ngerprint
to identify the underlying mechanism for pore formation
Experiments which can determine the number of pores or the
steady state lsquomerrsquo distributions on the membrane surface will
shed light on the pathways for oligomerization and enable a
more denite conclusion of the underlying kinetics Furtherthe rupture kinetics model contains only one adjustable
parameter Lysis experiments conducted with Vibrio cholerae El
Tor cytolysin33 and Monalysin34 show similar lysis times (in the
order of 10s of minutes) as observed in our study suggesting
similarities in the underlying kinetic pathways that lead to pore
formation and rupture The model developed in this manu-
script is generic and could be recast with some variation to
study the dynamics of other PFTs
We brie y discuss some of the limitations of the model in its
present form The model is based on the mean eld approxi-
mation where all cells are assumed to be identical and for the
purpose of binding and oligomerization this is an adequatestarting point The more complicated process is the mechanics
of rupture with the correct functional dependence on the pore
density In general there could exist a distribution of cells with
diff erent densities of pores Preliminary experiments by varying
the number of RBCs at a xed toxin concentration led to an
increase in lysis suggesting that cell heterogeneity could be
playing a role Although a population balance model35 could
include these variations this is at an added cost of complexity A
second aspect inherent to the model is the presence of lysis
beyond the time at which steady state is observed in the
experiments (30 minutes) Once the number of pores has
Fig 11 Oligomerconcentrationsas a function of time areshownfor (a) 544 nM and(b) 147 nMSteady state oligomer concentrations areshown
for (c) 544 nM and (d) 147 nM At steady state 1ndash4 mer concentrations are zero for both 544 and 147 nM The 12-mer concentration at steady
state is quite large compared to the other oligomer concentrations present in the system
4940 | RSC Adv 2014 4 4930ndash4942 This journal is copy The Royal Society of Chemistry 2014
RSC Advances Paper
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1213
reached a steady state which occurs within 30 s (Fig 7a) cell
lysis continues to occur at a xed number of pores proportional
to np npc Running the simulation to steady state would
eventually lead to lysis of all the cells in the system albeit at an
exceedingly slow rate at the lower and intermediate toxin
concentrations We have also assumed complete and irrevers-
ible binding of the monomer to the membrane and hence the
number of pores predicted represent an upper limit within the
proposed kinetic framework Despite the mean eld approxi-mation the model proposed in this manuscript is able to
capture the inherent time scales in the process of pore forma-
tion and rupture as well as predict the variation of hemoglobin
release as a function of the initial toxin concentration observed
in the experiments We nally point out that the eff ect of
temperature on the kinetics and lysis has not been investigated
in this work Preliminary experiments at 147 nM toxin
concentration carried out to steady state indicated a marked
drop in lytic activity for temperatures below 15 C and lysis was
not observed at 10 C Data on binding isotherms would be
required to understand temperature eff ects in these systems
Our work has implications in optimizing dosage and devel-oping novel drug delivery strategies based on PFTs The obser-
vation that lysis occurs in a given window of toxin concentration
has implications while developing PFT-based drug and gene
therapy protocols In recent studies E coli used in conjunction
with radiation therapy was found to retard the growth of cancer
tumors when compared with only radiation therapy protocols36
Quantifying the dynamics and extent of lysis is an important
aspect of developing appropriate treatment protocols in these
combination therapies In other applications where pores are
used for gene delivery and in the development of articial cells
lysis must be prevented and pore formation restricted to
concentrations below the lysis threshold
Acknowledgements
We acknowledge funding from Department of Science and
Technology (DST) India under the IRHPA grant The authors
thank Sanjeev Kumar Gupta and Jaydeep Basu for several useful
discussions on the development of the model as well as the
reviewers for their critical comments
References
1 S Bhakdi and J Tranum-Jensen Microbiol Rev 1991 55
733ndash
7512 J Thiery D Keefe S Boulant E Boucrot M Walch
D Martinvalet I S Goping R C Bleackley
T Kirchhausen and J Lieberman Nat Immunol 2011 12
770ndash777
3 A Ludwig C von Rhein S Bauer C Huttinger and
W Goebel J Bacteriol 2004 186 5311ndash5320
4 A W Bernheimer and L S Avigad Proc Natl Acad Sci U S
A 1976 73 467ndash471
5 M V Sousa M Richardson W Fontes and L Morhy
J Protein Chem 1994 13 659ndash667
6 G Anderluh and P Macek Toxicon 2002 40 111ndash124
7 I Iacovache F G van der Goot and L Pernot Biochim
Biophys Acta Biomembr 2008 1778 1611ndash1623
8 N R Wyborn A Clark R E Roberts S J Jamieson
S Tzokov P A Bullough T J Stillman P J Artymiuk
J E Galen L Zhao M M Levine and J Green
Microbiology 2004 150 1495ndash1505
9 A J Wallace T J Stillman A Atkins S J Jamieson
P A Bullough J Green and P J Artymiuk Cell 2000 100
265ndash27610 M Mueller U Grauschopf T Maier R Glockshuber and
N Ban Nature 2009 459 726ndash730
11 J Oscarsson Y Mizunoe L Li X-H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
12 A Atkins N R Wyborn A J Wallace T J Stillman
L K Black A B Fielding M Hisakado P J Artymiuk and
J Green J Biol Chem 2000 275 41150ndash41155
13 A Ludwig G Volkerink C von Rhein S Bauer E Maier
B Bergmann W Goebel and R Benz J Bacteriol 2010
192 4001ndash4011
14 N Eier M Vetsch M Gregorini P Ringler M Chami
A Philippsen A Fritz S A Muller R Glockshuber A Engel and U Grauschopf EMBO J 2006 25 2652ndash2661
15 L A Shepard O Shatursky A E Johnson and R K Tweten
Biochemistry 2000 39 10284ndash10293
16 A J Ratner K R Hippe J L Aguilar M H Bender
A L Nelson and J N Weiser J Biol Chem 2006 281
12994ndash12998
17 I Iacovache M Bischoerger and F G van der Goot Curr
Opin Struct Biol 2010 20 241ndash246
18 G Schwarz and C H Robert Biophys Chem 1992 42 291ndash296
19 G Schwarz R-t Zong and T Popescu Biochim Biophys
Acta Biomembr 1992 1110 97ndash104
20 D Bolintineanu E Hazrati H T Davis R I Lehrer and
Y N Kaznessis Peptides 2010 31 1ndash821 J M Alam T Kobayashi and M Yamazaki Biochemistry
2012 51 5160ndash5172
22 J Lemiere K Guevorkian C Campillo C Sykes and T Betz
So Matter 2013 9 3181ndash3187
23 V Noireaux and A Libchaber Proc Natl Acad Sci U S A
2004 101 17669ndash17674
24 M M Bradford Anal Biochem 1976 72 248ndash254
25 H Xu E K Bjerneld M Kall and L Borjesson Phys Rev
Lett 1999 83 4357ndash4360
26 J Oscarsson Y Mizunoe L Li X H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
27 T M Raschke J Tsai and M Levitt Proc Natl Acad SciU S A 2001 98 5965ndash5969
28 L A Rodrıguez-Guadarrama S K Talsania K K Mohanty
and R Rajagopalan Langmuir 1999 15 437ndash446
29 C E McLaren G M Brittenham and V Hasselblad Am J
Physiol 1987 252 3181ndash3187
30 J G Kim M Xia and H Liu IEEE Eng Med Biol Mag 2005
24 118ndash121
31 M Nagahama H Hara M Fernandez-Miyakawa
Y Itohayashi and J Sakurai Biochemistry 2006 45 296ndash302
32 J R Thompson B Cronin H Bayley and M I Wallace
Biophys J 2011 101 2679ndash2683
This journal is copy The Royal Society of Chemistry 2014 RSC Adv 2014 4 4930ndash4942 | 4941
Paper RSC Advances
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1313
33 A Zitzer I Walev M Palmer and S Bhakdi Med Microbiol
Immunol 1995 184 37ndash44
34 O Opota I Vallet-Gely R Vincentelli C Kellenberger
I Iacovache M R Gonzalez A Roussel F G van der Goot
and B Lemaitre PLoS Pathog 2011 7 e1002259
35 S Kumar and D Ramkrishna Chem Eng Sci 1996 51
1311ndash1332
36 S N Jiang T X Phan T K Nam V H Nguyen H S Kim
H S Bom H E Choy Y Hong and J J Min Mol Ther
2010 18 635ndash642
RSC Advances Paper
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 913
unknown In order to test the inuence of ms on the model
prediction we carried out a few simulations for ms frac14 1 108
mol m2 and msfrac14 1 1010 mol m2 for various C in values For
ms frac14 1 108 mol m2 a negligible number of pores were
formed and the H max ndashC in data (Fig 3a) is underpredicted At
ms frac14 1 1010 mol m2 pore formation was extremely rapid
and little variation in pore density between C in frac14 882 nM and
147 nM was observed As a consequence H max ndashC in data (Fig 3a)
is grossly overpredicted Hence a value of ms frac14 1 10
9 molm2 was used in the simulations (Fig 6) We further note
that the amount of saturated surface concentration ms
implicitly changes the reaction rate constant k l (eqn (12))
Increasing ms eff ectively decreases the reaction rate constant
(eqn (12))
Once snet is xed npc is related to l through the relation npcfrac14
alb with the constants a and b being xed for a given initial toxin
concentration C in (Fig 5) We have found that 2 l 4 ts the
hemoglobin release data very closely (Fig 6) and although the
hemoglobin released as a function of time is slightly under-
estimated by the model the agreement is reasonable Upon
increasing l
to 7 and keeping m
s
frac14 10
9
mol m
s1
the corre-sponding value of npc is 1349 With a value of kl frac14 13 107 s1
although H max versus C in data is accurately predicted the
hemoglobin versus time data is grossly underpredicted Varying
ms between 108 and 1010 mol m2 further deteriorated the
prediction Finally we point out that other functional forms of the
dependence on np npc in the cell lysis equation eqn (8) such as
the unit step function or a quadratic dependence (np npc)2 only
overestimated the H max versus C in data
53 Sequential oligomerization with reversible kinetics
In the previous discussion we present the results for the oligo-
merization kinetics which was assumed to be irreversible Thesequential oligomerization model with reversible kinetics is
p1 thorn pl ) k f
k b pl thorn1 l frac14 1 n 1 (16)
where k f and k b represent the forward and backward reaction
rate constants The kinetic equations are
d p1
dt frac14 k cm k f
Xn1
l frac141
p1 pl thorn k bXn1
l frac142
pl (17a)
d p2
dt frac14
1
2k f p1 p1 k f p1 p2 thorn k b p3
1
2 p2 (17b)
d pl
dt frac14 k f p1 pethl 1THORN eth1 dl 12THORNk f p1 pl
thorn k beth1 dl 12THORNetheth1 dl 11THORN pl thorn1 pl THORN l frac14 3 12 (17c)
A reversible time constant can be dened from eqn (17) as
sb frac14 1
k b The ratio R is dened as the ratio of forward to back-
ward time constants ( R frac14 sf sb) to study the eff ect of revers-
ibility Upon examining the number of pores as a function of
time we observe that the time taken to reach a steady number of
pores is signicantly larger than the time taken to reach steady
state in the lysis experiments In order to make comparisons
with the irreversible mechanism we evaluated the number of
pores at the threshold concentration of 544 nM It is observed
that the number of pores required for lysis initially increases
and then decreases for increments in R values The distribution
of oligomers at steady state are shown for diff erent R values in
Fig 9a and b for C in frac14 147 nM The steady state concentrations
of lsquomersrsquo change from a predominantly 11-mer concentration to
a predominant 1-mer concentration as R is varied between0 and 1 For R lt 1 we nd that the concentration of pores (12
mers) are signicantly higher than the intermediate lsquomerrsquo
concentrations shown in Fig 9b and range from 0142 nmol
Fig 9 Oligomer distribution as a function of the ratio of forward and
backward time constants R frac14 sfsb for C in frac14 147 nM (a) R varies
between 0 and 1 (b) R varies between 0 and 01 (c) Hemoglobin
release data as a function of toxin concentration is shown forvarious R
R frac14 0 has the closest agreement with the experimental data (open
squares)
4938 | RSC Adv 2014 4 4930ndash4942 This journal is copy The Royal Society of Chemistry 2014
RSC Advances Paper
View Article Online
7252019 c3ra45159c
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m2 ( R frac14 0) to 028 nmol m2 ( R frac14 001) For R frac14 1 the
concentration of 12 mers is 024 nmol m2 and decreases with a
further increase in R The predictions using reversible sequen-
tial kinetics for the H max vs C in data (Fig 9c) indicate
greater deviation from the experimental data when compared
with R frac14 0 These results indicate that the irreversible mecha-
nism provides the best agreement with the experimental data
We point out that the critical number of pores npc in
the reversible framework is non-monotonic ranging from 392( R frac14 0) to 5128 ( R frac14 1) for l frac14 2
54 Non-sequential oligomerization
In contrast to the 392 pores obtained for the critical number of
pores via the sequential mechanism a substantially larger
critical number of pores are observed via the non-sequential
oligomerization (6000 pores per cell) The mass balance for
the protomer ( p1) concentration remains identical to that of the
sequential oligomerization mechanism (eqn (4)) The governing
equations for pl (l lt l n) oligomer undergoing non-sequential
irreversible oligomerization are
d pl
dt frac14
1
2k l Xl 1
ufrac141
pu pl u eth1 dl 12THORNXn1
ufrac141
k l pl pu for 1l n (18)
where n represents the number of monomers in a pore The
relation between the critical number of pores and npc for non-
sequential oligomerization is npc frac14 5439l015 The critical
number of pores for l frac14 2 is 6035 pores which is about 20 times
greater than that obtained from a sequential oligomerization
mechanism In this scheme the hemoglobin release (Fig 10a)
data is overpredicted at intermediate toxin concentrations when
compared with the sequential oligomerization The predictions
of the H out vs time data at 147 nM (Fig 10b) are similar whencompared with the sequential oligomerization (Fig 6b and d)
A comparison of the tted parameters between the sequential
and non-sequential oligomerization mechanisms are given in
Table 2
Oligomer distributions obtained from the non-sequential
mechanism (Fig 11b) show an entirely diff erent trend when
compared to that obtained from the sequential mechanism
(Fig 8) In the sequential mechanism the higher lsquomersrsquo
attained a steady state once the protomer was depleted In the
non-sequential mechanism the contribution to the dodeca-
mers (12 mer) can be obtained from a large number of combi-
nations of the lower oligomers leading to a larger dodecamer orpore concentration The time scale required to attain the olig-
omer steady state concentration (Fig 11a) is about 20 s for an
initial concentration of 147 nM Similar time scales are
observed in the sequential mechanism as well
Further experiments are required to distinguish between the
various mechanisms Western Blot experiments conducted on
Hemolysin E12 and Clostridium perfringens 3 toxins31 showed the
presence of intermediate oligomers On the other hand single-
molecule uorescence imaging of a-hemolysin on a droplet
interface bilayer showed the presence of only monomers and
heptamers (pores)32
6 Discussion and conclusions
Lysis experiments on RBCs with the ClyA pore-forming toxin
show that a threshold initial toxin concentration is required to
initiate lysis From this observation we analyzed the problem in
two regimes a low toxin concentration regime where rupture of
cells is absent and a high toxin concentration regime where
lysis occurs and hemoglobin is released Kinetic models which
accounts for monomer binding conformation (membrane-
bound monomer to protomer) and oligomerization to form the
dodecameric pore complex are developed Models which
account for sequential and non-sequential oligomerization are
tested Cell rupture is assumed to be
rst order in the number of live cells and directly proportional to the pores in excess of the
critical number of pores npc In the sublytic regime the number
of pores is found to have a power law dependence on l which is
the ratio of conformational time to the reaction time This leads
to the construction of a ldquophase diagramrdquo between the number
of pores np and l for diff erent values of the initial toxin
concentration Comparing simulations with experimental data
the range of npc was 392ndash768 for the sequential mechanism and
5300ndash6300 pores for the non-sequential mechanism for 2 l 4
The range of l values is consistent with available experimental
data on ClyA which indicates that the membrane-bound
Fig 10 (a) Model predictions from the non-sequential mechanism
Simulated Hmax C in (open circles) curves grossly overpredicts the
experimental data (open squares) (b) Simulated hemoglobin release
compares well with the experimental data Dashed line snetfrac14 1 s solid
line snet frac14 24 s
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Paper RSC Advances
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1113
conformational step is slower than the preceding adsorption
and subsequent oligomerization steps10
From the model we are also able to comment on the time
constants for the various processes The time constant for poreformation is about 1 s indicating that pore formation itself is a
fast process relative to the time taken for the pore population on
a single RBC to reach steady state which is about 20ndash30 s Since
rupture kinetics is dynamic and depends on the fraction of live
cells as well as the number of pores on the cell lysis occurs in
the time scale of 10s of minutes Due to this separation of time
scales lysis is seen to occur once the number of pores has
reached a steady state This steady number of pores ranges from
515 to 11 657 as the toxin concentration ranges from 588ndash
147 nM For the non-sequential mechanism the critical
number of pores required to initiate lysis is about 20 times
higher when compared to that of sequential oligomerizationComparison of three diff erent kinetic models reveals that the
irreversible sequential kinetics provides the closest match with
the hemoglobin released as a function of the initial toxin
concentration Although we observe an overprediction of the
hemoglobin release data with the non-sequential mechanism
the hemoglobin release kinetics are similar to that of the
sequential mechanism The distribution of lower oligomers is
distinctly diff erent in both cases with a negligible numbers of
lower mers observed in the non-sequential oligomerization
These diff erences in the distribution of mers off er a ngerprint
to identify the underlying mechanism for pore formation
Experiments which can determine the number of pores or the
steady state lsquomerrsquo distributions on the membrane surface will
shed light on the pathways for oligomerization and enable a
more denite conclusion of the underlying kinetics Furtherthe rupture kinetics model contains only one adjustable
parameter Lysis experiments conducted with Vibrio cholerae El
Tor cytolysin33 and Monalysin34 show similar lysis times (in the
order of 10s of minutes) as observed in our study suggesting
similarities in the underlying kinetic pathways that lead to pore
formation and rupture The model developed in this manu-
script is generic and could be recast with some variation to
study the dynamics of other PFTs
We brie y discuss some of the limitations of the model in its
present form The model is based on the mean eld approxi-
mation where all cells are assumed to be identical and for the
purpose of binding and oligomerization this is an adequatestarting point The more complicated process is the mechanics
of rupture with the correct functional dependence on the pore
density In general there could exist a distribution of cells with
diff erent densities of pores Preliminary experiments by varying
the number of RBCs at a xed toxin concentration led to an
increase in lysis suggesting that cell heterogeneity could be
playing a role Although a population balance model35 could
include these variations this is at an added cost of complexity A
second aspect inherent to the model is the presence of lysis
beyond the time at which steady state is observed in the
experiments (30 minutes) Once the number of pores has
Fig 11 Oligomerconcentrationsas a function of time areshownfor (a) 544 nM and(b) 147 nMSteady state oligomer concentrations areshown
for (c) 544 nM and (d) 147 nM At steady state 1ndash4 mer concentrations are zero for both 544 and 147 nM The 12-mer concentration at steady
state is quite large compared to the other oligomer concentrations present in the system
4940 | RSC Adv 2014 4 4930ndash4942 This journal is copy The Royal Society of Chemistry 2014
RSC Advances Paper
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1213
reached a steady state which occurs within 30 s (Fig 7a) cell
lysis continues to occur at a xed number of pores proportional
to np npc Running the simulation to steady state would
eventually lead to lysis of all the cells in the system albeit at an
exceedingly slow rate at the lower and intermediate toxin
concentrations We have also assumed complete and irrevers-
ible binding of the monomer to the membrane and hence the
number of pores predicted represent an upper limit within the
proposed kinetic framework Despite the mean eld approxi-mation the model proposed in this manuscript is able to
capture the inherent time scales in the process of pore forma-
tion and rupture as well as predict the variation of hemoglobin
release as a function of the initial toxin concentration observed
in the experiments We nally point out that the eff ect of
temperature on the kinetics and lysis has not been investigated
in this work Preliminary experiments at 147 nM toxin
concentration carried out to steady state indicated a marked
drop in lytic activity for temperatures below 15 C and lysis was
not observed at 10 C Data on binding isotherms would be
required to understand temperature eff ects in these systems
Our work has implications in optimizing dosage and devel-oping novel drug delivery strategies based on PFTs The obser-
vation that lysis occurs in a given window of toxin concentration
has implications while developing PFT-based drug and gene
therapy protocols In recent studies E coli used in conjunction
with radiation therapy was found to retard the growth of cancer
tumors when compared with only radiation therapy protocols36
Quantifying the dynamics and extent of lysis is an important
aspect of developing appropriate treatment protocols in these
combination therapies In other applications where pores are
used for gene delivery and in the development of articial cells
lysis must be prevented and pore formation restricted to
concentrations below the lysis threshold
Acknowledgements
We acknowledge funding from Department of Science and
Technology (DST) India under the IRHPA grant The authors
thank Sanjeev Kumar Gupta and Jaydeep Basu for several useful
discussions on the development of the model as well as the
reviewers for their critical comments
References
1 S Bhakdi and J Tranum-Jensen Microbiol Rev 1991 55
733ndash
7512 J Thiery D Keefe S Boulant E Boucrot M Walch
D Martinvalet I S Goping R C Bleackley
T Kirchhausen and J Lieberman Nat Immunol 2011 12
770ndash777
3 A Ludwig C von Rhein S Bauer C Huttinger and
W Goebel J Bacteriol 2004 186 5311ndash5320
4 A W Bernheimer and L S Avigad Proc Natl Acad Sci U S
A 1976 73 467ndash471
5 M V Sousa M Richardson W Fontes and L Morhy
J Protein Chem 1994 13 659ndash667
6 G Anderluh and P Macek Toxicon 2002 40 111ndash124
7 I Iacovache F G van der Goot and L Pernot Biochim
Biophys Acta Biomembr 2008 1778 1611ndash1623
8 N R Wyborn A Clark R E Roberts S J Jamieson
S Tzokov P A Bullough T J Stillman P J Artymiuk
J E Galen L Zhao M M Levine and J Green
Microbiology 2004 150 1495ndash1505
9 A J Wallace T J Stillman A Atkins S J Jamieson
P A Bullough J Green and P J Artymiuk Cell 2000 100
265ndash27610 M Mueller U Grauschopf T Maier R Glockshuber and
N Ban Nature 2009 459 726ndash730
11 J Oscarsson Y Mizunoe L Li X-H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
12 A Atkins N R Wyborn A J Wallace T J Stillman
L K Black A B Fielding M Hisakado P J Artymiuk and
J Green J Biol Chem 2000 275 41150ndash41155
13 A Ludwig G Volkerink C von Rhein S Bauer E Maier
B Bergmann W Goebel and R Benz J Bacteriol 2010
192 4001ndash4011
14 N Eier M Vetsch M Gregorini P Ringler M Chami
A Philippsen A Fritz S A Muller R Glockshuber A Engel and U Grauschopf EMBO J 2006 25 2652ndash2661
15 L A Shepard O Shatursky A E Johnson and R K Tweten
Biochemistry 2000 39 10284ndash10293
16 A J Ratner K R Hippe J L Aguilar M H Bender
A L Nelson and J N Weiser J Biol Chem 2006 281
12994ndash12998
17 I Iacovache M Bischoerger and F G van der Goot Curr
Opin Struct Biol 2010 20 241ndash246
18 G Schwarz and C H Robert Biophys Chem 1992 42 291ndash296
19 G Schwarz R-t Zong and T Popescu Biochim Biophys
Acta Biomembr 1992 1110 97ndash104
20 D Bolintineanu E Hazrati H T Davis R I Lehrer and
Y N Kaznessis Peptides 2010 31 1ndash821 J M Alam T Kobayashi and M Yamazaki Biochemistry
2012 51 5160ndash5172
22 J Lemiere K Guevorkian C Campillo C Sykes and T Betz
So Matter 2013 9 3181ndash3187
23 V Noireaux and A Libchaber Proc Natl Acad Sci U S A
2004 101 17669ndash17674
24 M M Bradford Anal Biochem 1976 72 248ndash254
25 H Xu E K Bjerneld M Kall and L Borjesson Phys Rev
Lett 1999 83 4357ndash4360
26 J Oscarsson Y Mizunoe L Li X H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
27 T M Raschke J Tsai and M Levitt Proc Natl Acad SciU S A 2001 98 5965ndash5969
28 L A Rodrıguez-Guadarrama S K Talsania K K Mohanty
and R Rajagopalan Langmuir 1999 15 437ndash446
29 C E McLaren G M Brittenham and V Hasselblad Am J
Physiol 1987 252 3181ndash3187
30 J G Kim M Xia and H Liu IEEE Eng Med Biol Mag 2005
24 118ndash121
31 M Nagahama H Hara M Fernandez-Miyakawa
Y Itohayashi and J Sakurai Biochemistry 2006 45 296ndash302
32 J R Thompson B Cronin H Bayley and M I Wallace
Biophys J 2011 101 2679ndash2683
This journal is copy The Royal Society of Chemistry 2014 RSC Adv 2014 4 4930ndash4942 | 4941
Paper RSC Advances
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1313
33 A Zitzer I Walev M Palmer and S Bhakdi Med Microbiol
Immunol 1995 184 37ndash44
34 O Opota I Vallet-Gely R Vincentelli C Kellenberger
I Iacovache M R Gonzalez A Roussel F G van der Goot
and B Lemaitre PLoS Pathog 2011 7 e1002259
35 S Kumar and D Ramkrishna Chem Eng Sci 1996 51
1311ndash1332
36 S N Jiang T X Phan T K Nam V H Nguyen H S Kim
H S Bom H E Choy Y Hong and J J Min Mol Ther
2010 18 635ndash642
RSC Advances Paper
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1013
m2 ( R frac14 0) to 028 nmol m2 ( R frac14 001) For R frac14 1 the
concentration of 12 mers is 024 nmol m2 and decreases with a
further increase in R The predictions using reversible sequen-
tial kinetics for the H max vs C in data (Fig 9c) indicate
greater deviation from the experimental data when compared
with R frac14 0 These results indicate that the irreversible mecha-
nism provides the best agreement with the experimental data
We point out that the critical number of pores npc in
the reversible framework is non-monotonic ranging from 392( R frac14 0) to 5128 ( R frac14 1) for l frac14 2
54 Non-sequential oligomerization
In contrast to the 392 pores obtained for the critical number of
pores via the sequential mechanism a substantially larger
critical number of pores are observed via the non-sequential
oligomerization (6000 pores per cell) The mass balance for
the protomer ( p1) concentration remains identical to that of the
sequential oligomerization mechanism (eqn (4)) The governing
equations for pl (l lt l n) oligomer undergoing non-sequential
irreversible oligomerization are
d pl
dt frac14
1
2k l Xl 1
ufrac141
pu pl u eth1 dl 12THORNXn1
ufrac141
k l pl pu for 1l n (18)
where n represents the number of monomers in a pore The
relation between the critical number of pores and npc for non-
sequential oligomerization is npc frac14 5439l015 The critical
number of pores for l frac14 2 is 6035 pores which is about 20 times
greater than that obtained from a sequential oligomerization
mechanism In this scheme the hemoglobin release (Fig 10a)
data is overpredicted at intermediate toxin concentrations when
compared with the sequential oligomerization The predictions
of the H out vs time data at 147 nM (Fig 10b) are similar whencompared with the sequential oligomerization (Fig 6b and d)
A comparison of the tted parameters between the sequential
and non-sequential oligomerization mechanisms are given in
Table 2
Oligomer distributions obtained from the non-sequential
mechanism (Fig 11b) show an entirely diff erent trend when
compared to that obtained from the sequential mechanism
(Fig 8) In the sequential mechanism the higher lsquomersrsquo
attained a steady state once the protomer was depleted In the
non-sequential mechanism the contribution to the dodeca-
mers (12 mer) can be obtained from a large number of combi-
nations of the lower oligomers leading to a larger dodecamer orpore concentration The time scale required to attain the olig-
omer steady state concentration (Fig 11a) is about 20 s for an
initial concentration of 147 nM Similar time scales are
observed in the sequential mechanism as well
Further experiments are required to distinguish between the
various mechanisms Western Blot experiments conducted on
Hemolysin E12 and Clostridium perfringens 3 toxins31 showed the
presence of intermediate oligomers On the other hand single-
molecule uorescence imaging of a-hemolysin on a droplet
interface bilayer showed the presence of only monomers and
heptamers (pores)32
6 Discussion and conclusions
Lysis experiments on RBCs with the ClyA pore-forming toxin
show that a threshold initial toxin concentration is required to
initiate lysis From this observation we analyzed the problem in
two regimes a low toxin concentration regime where rupture of
cells is absent and a high toxin concentration regime where
lysis occurs and hemoglobin is released Kinetic models which
accounts for monomer binding conformation (membrane-
bound monomer to protomer) and oligomerization to form the
dodecameric pore complex are developed Models which
account for sequential and non-sequential oligomerization are
tested Cell rupture is assumed to be
rst order in the number of live cells and directly proportional to the pores in excess of the
critical number of pores npc In the sublytic regime the number
of pores is found to have a power law dependence on l which is
the ratio of conformational time to the reaction time This leads
to the construction of a ldquophase diagramrdquo between the number
of pores np and l for diff erent values of the initial toxin
concentration Comparing simulations with experimental data
the range of npc was 392ndash768 for the sequential mechanism and
5300ndash6300 pores for the non-sequential mechanism for 2 l 4
The range of l values is consistent with available experimental
data on ClyA which indicates that the membrane-bound
Fig 10 (a) Model predictions from the non-sequential mechanism
Simulated Hmax C in (open circles) curves grossly overpredicts the
experimental data (open squares) (b) Simulated hemoglobin release
compares well with the experimental data Dashed line snetfrac14 1 s solid
line snet frac14 24 s
This journal is copy The Royal Society of Chemistry 2014 RSC Adv 2014 4 4930ndash4942 | 4939
Paper RSC Advances
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1113
conformational step is slower than the preceding adsorption
and subsequent oligomerization steps10
From the model we are also able to comment on the time
constants for the various processes The time constant for poreformation is about 1 s indicating that pore formation itself is a
fast process relative to the time taken for the pore population on
a single RBC to reach steady state which is about 20ndash30 s Since
rupture kinetics is dynamic and depends on the fraction of live
cells as well as the number of pores on the cell lysis occurs in
the time scale of 10s of minutes Due to this separation of time
scales lysis is seen to occur once the number of pores has
reached a steady state This steady number of pores ranges from
515 to 11 657 as the toxin concentration ranges from 588ndash
147 nM For the non-sequential mechanism the critical
number of pores required to initiate lysis is about 20 times
higher when compared to that of sequential oligomerizationComparison of three diff erent kinetic models reveals that the
irreversible sequential kinetics provides the closest match with
the hemoglobin released as a function of the initial toxin
concentration Although we observe an overprediction of the
hemoglobin release data with the non-sequential mechanism
the hemoglobin release kinetics are similar to that of the
sequential mechanism The distribution of lower oligomers is
distinctly diff erent in both cases with a negligible numbers of
lower mers observed in the non-sequential oligomerization
These diff erences in the distribution of mers off er a ngerprint
to identify the underlying mechanism for pore formation
Experiments which can determine the number of pores or the
steady state lsquomerrsquo distributions on the membrane surface will
shed light on the pathways for oligomerization and enable a
more denite conclusion of the underlying kinetics Furtherthe rupture kinetics model contains only one adjustable
parameter Lysis experiments conducted with Vibrio cholerae El
Tor cytolysin33 and Monalysin34 show similar lysis times (in the
order of 10s of minutes) as observed in our study suggesting
similarities in the underlying kinetic pathways that lead to pore
formation and rupture The model developed in this manu-
script is generic and could be recast with some variation to
study the dynamics of other PFTs
We brie y discuss some of the limitations of the model in its
present form The model is based on the mean eld approxi-
mation where all cells are assumed to be identical and for the
purpose of binding and oligomerization this is an adequatestarting point The more complicated process is the mechanics
of rupture with the correct functional dependence on the pore
density In general there could exist a distribution of cells with
diff erent densities of pores Preliminary experiments by varying
the number of RBCs at a xed toxin concentration led to an
increase in lysis suggesting that cell heterogeneity could be
playing a role Although a population balance model35 could
include these variations this is at an added cost of complexity A
second aspect inherent to the model is the presence of lysis
beyond the time at which steady state is observed in the
experiments (30 minutes) Once the number of pores has
Fig 11 Oligomerconcentrationsas a function of time areshownfor (a) 544 nM and(b) 147 nMSteady state oligomer concentrations areshown
for (c) 544 nM and (d) 147 nM At steady state 1ndash4 mer concentrations are zero for both 544 and 147 nM The 12-mer concentration at steady
state is quite large compared to the other oligomer concentrations present in the system
4940 | RSC Adv 2014 4 4930ndash4942 This journal is copy The Royal Society of Chemistry 2014
RSC Advances Paper
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1213
reached a steady state which occurs within 30 s (Fig 7a) cell
lysis continues to occur at a xed number of pores proportional
to np npc Running the simulation to steady state would
eventually lead to lysis of all the cells in the system albeit at an
exceedingly slow rate at the lower and intermediate toxin
concentrations We have also assumed complete and irrevers-
ible binding of the monomer to the membrane and hence the
number of pores predicted represent an upper limit within the
proposed kinetic framework Despite the mean eld approxi-mation the model proposed in this manuscript is able to
capture the inherent time scales in the process of pore forma-
tion and rupture as well as predict the variation of hemoglobin
release as a function of the initial toxin concentration observed
in the experiments We nally point out that the eff ect of
temperature on the kinetics and lysis has not been investigated
in this work Preliminary experiments at 147 nM toxin
concentration carried out to steady state indicated a marked
drop in lytic activity for temperatures below 15 C and lysis was
not observed at 10 C Data on binding isotherms would be
required to understand temperature eff ects in these systems
Our work has implications in optimizing dosage and devel-oping novel drug delivery strategies based on PFTs The obser-
vation that lysis occurs in a given window of toxin concentration
has implications while developing PFT-based drug and gene
therapy protocols In recent studies E coli used in conjunction
with radiation therapy was found to retard the growth of cancer
tumors when compared with only radiation therapy protocols36
Quantifying the dynamics and extent of lysis is an important
aspect of developing appropriate treatment protocols in these
combination therapies In other applications where pores are
used for gene delivery and in the development of articial cells
lysis must be prevented and pore formation restricted to
concentrations below the lysis threshold
Acknowledgements
We acknowledge funding from Department of Science and
Technology (DST) India under the IRHPA grant The authors
thank Sanjeev Kumar Gupta and Jaydeep Basu for several useful
discussions on the development of the model as well as the
reviewers for their critical comments
References
1 S Bhakdi and J Tranum-Jensen Microbiol Rev 1991 55
733ndash
7512 J Thiery D Keefe S Boulant E Boucrot M Walch
D Martinvalet I S Goping R C Bleackley
T Kirchhausen and J Lieberman Nat Immunol 2011 12
770ndash777
3 A Ludwig C von Rhein S Bauer C Huttinger and
W Goebel J Bacteriol 2004 186 5311ndash5320
4 A W Bernheimer and L S Avigad Proc Natl Acad Sci U S
A 1976 73 467ndash471
5 M V Sousa M Richardson W Fontes and L Morhy
J Protein Chem 1994 13 659ndash667
6 G Anderluh and P Macek Toxicon 2002 40 111ndash124
7 I Iacovache F G van der Goot and L Pernot Biochim
Biophys Acta Biomembr 2008 1778 1611ndash1623
8 N R Wyborn A Clark R E Roberts S J Jamieson
S Tzokov P A Bullough T J Stillman P J Artymiuk
J E Galen L Zhao M M Levine and J Green
Microbiology 2004 150 1495ndash1505
9 A J Wallace T J Stillman A Atkins S J Jamieson
P A Bullough J Green and P J Artymiuk Cell 2000 100
265ndash27610 M Mueller U Grauschopf T Maier R Glockshuber and
N Ban Nature 2009 459 726ndash730
11 J Oscarsson Y Mizunoe L Li X-H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
12 A Atkins N R Wyborn A J Wallace T J Stillman
L K Black A B Fielding M Hisakado P J Artymiuk and
J Green J Biol Chem 2000 275 41150ndash41155
13 A Ludwig G Volkerink C von Rhein S Bauer E Maier
B Bergmann W Goebel and R Benz J Bacteriol 2010
192 4001ndash4011
14 N Eier M Vetsch M Gregorini P Ringler M Chami
A Philippsen A Fritz S A Muller R Glockshuber A Engel and U Grauschopf EMBO J 2006 25 2652ndash2661
15 L A Shepard O Shatursky A E Johnson and R K Tweten
Biochemistry 2000 39 10284ndash10293
16 A J Ratner K R Hippe J L Aguilar M H Bender
A L Nelson and J N Weiser J Biol Chem 2006 281
12994ndash12998
17 I Iacovache M Bischoerger and F G van der Goot Curr
Opin Struct Biol 2010 20 241ndash246
18 G Schwarz and C H Robert Biophys Chem 1992 42 291ndash296
19 G Schwarz R-t Zong and T Popescu Biochim Biophys
Acta Biomembr 1992 1110 97ndash104
20 D Bolintineanu E Hazrati H T Davis R I Lehrer and
Y N Kaznessis Peptides 2010 31 1ndash821 J M Alam T Kobayashi and M Yamazaki Biochemistry
2012 51 5160ndash5172
22 J Lemiere K Guevorkian C Campillo C Sykes and T Betz
So Matter 2013 9 3181ndash3187
23 V Noireaux and A Libchaber Proc Natl Acad Sci U S A
2004 101 17669ndash17674
24 M M Bradford Anal Biochem 1976 72 248ndash254
25 H Xu E K Bjerneld M Kall and L Borjesson Phys Rev
Lett 1999 83 4357ndash4360
26 J Oscarsson Y Mizunoe L Li X H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
27 T M Raschke J Tsai and M Levitt Proc Natl Acad SciU S A 2001 98 5965ndash5969
28 L A Rodrıguez-Guadarrama S K Talsania K K Mohanty
and R Rajagopalan Langmuir 1999 15 437ndash446
29 C E McLaren G M Brittenham and V Hasselblad Am J
Physiol 1987 252 3181ndash3187
30 J G Kim M Xia and H Liu IEEE Eng Med Biol Mag 2005
24 118ndash121
31 M Nagahama H Hara M Fernandez-Miyakawa
Y Itohayashi and J Sakurai Biochemistry 2006 45 296ndash302
32 J R Thompson B Cronin H Bayley and M I Wallace
Biophys J 2011 101 2679ndash2683
This journal is copy The Royal Society of Chemistry 2014 RSC Adv 2014 4 4930ndash4942 | 4941
Paper RSC Advances
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1313
33 A Zitzer I Walev M Palmer and S Bhakdi Med Microbiol
Immunol 1995 184 37ndash44
34 O Opota I Vallet-Gely R Vincentelli C Kellenberger
I Iacovache M R Gonzalez A Roussel F G van der Goot
and B Lemaitre PLoS Pathog 2011 7 e1002259
35 S Kumar and D Ramkrishna Chem Eng Sci 1996 51
1311ndash1332
36 S N Jiang T X Phan T K Nam V H Nguyen H S Kim
H S Bom H E Choy Y Hong and J J Min Mol Ther
2010 18 635ndash642
RSC Advances Paper
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1113
conformational step is slower than the preceding adsorption
and subsequent oligomerization steps10
From the model we are also able to comment on the time
constants for the various processes The time constant for poreformation is about 1 s indicating that pore formation itself is a
fast process relative to the time taken for the pore population on
a single RBC to reach steady state which is about 20ndash30 s Since
rupture kinetics is dynamic and depends on the fraction of live
cells as well as the number of pores on the cell lysis occurs in
the time scale of 10s of minutes Due to this separation of time
scales lysis is seen to occur once the number of pores has
reached a steady state This steady number of pores ranges from
515 to 11 657 as the toxin concentration ranges from 588ndash
147 nM For the non-sequential mechanism the critical
number of pores required to initiate lysis is about 20 times
higher when compared to that of sequential oligomerizationComparison of three diff erent kinetic models reveals that the
irreversible sequential kinetics provides the closest match with
the hemoglobin released as a function of the initial toxin
concentration Although we observe an overprediction of the
hemoglobin release data with the non-sequential mechanism
the hemoglobin release kinetics are similar to that of the
sequential mechanism The distribution of lower oligomers is
distinctly diff erent in both cases with a negligible numbers of
lower mers observed in the non-sequential oligomerization
These diff erences in the distribution of mers off er a ngerprint
to identify the underlying mechanism for pore formation
Experiments which can determine the number of pores or the
steady state lsquomerrsquo distributions on the membrane surface will
shed light on the pathways for oligomerization and enable a
more denite conclusion of the underlying kinetics Furtherthe rupture kinetics model contains only one adjustable
parameter Lysis experiments conducted with Vibrio cholerae El
Tor cytolysin33 and Monalysin34 show similar lysis times (in the
order of 10s of minutes) as observed in our study suggesting
similarities in the underlying kinetic pathways that lead to pore
formation and rupture The model developed in this manu-
script is generic and could be recast with some variation to
study the dynamics of other PFTs
We brie y discuss some of the limitations of the model in its
present form The model is based on the mean eld approxi-
mation where all cells are assumed to be identical and for the
purpose of binding and oligomerization this is an adequatestarting point The more complicated process is the mechanics
of rupture with the correct functional dependence on the pore
density In general there could exist a distribution of cells with
diff erent densities of pores Preliminary experiments by varying
the number of RBCs at a xed toxin concentration led to an
increase in lysis suggesting that cell heterogeneity could be
playing a role Although a population balance model35 could
include these variations this is at an added cost of complexity A
second aspect inherent to the model is the presence of lysis
beyond the time at which steady state is observed in the
experiments (30 minutes) Once the number of pores has
Fig 11 Oligomerconcentrationsas a function of time areshownfor (a) 544 nM and(b) 147 nMSteady state oligomer concentrations areshown
for (c) 544 nM and (d) 147 nM At steady state 1ndash4 mer concentrations are zero for both 544 and 147 nM The 12-mer concentration at steady
state is quite large compared to the other oligomer concentrations present in the system
4940 | RSC Adv 2014 4 4930ndash4942 This journal is copy The Royal Society of Chemistry 2014
RSC Advances Paper
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1213
reached a steady state which occurs within 30 s (Fig 7a) cell
lysis continues to occur at a xed number of pores proportional
to np npc Running the simulation to steady state would
eventually lead to lysis of all the cells in the system albeit at an
exceedingly slow rate at the lower and intermediate toxin
concentrations We have also assumed complete and irrevers-
ible binding of the monomer to the membrane and hence the
number of pores predicted represent an upper limit within the
proposed kinetic framework Despite the mean eld approxi-mation the model proposed in this manuscript is able to
capture the inherent time scales in the process of pore forma-
tion and rupture as well as predict the variation of hemoglobin
release as a function of the initial toxin concentration observed
in the experiments We nally point out that the eff ect of
temperature on the kinetics and lysis has not been investigated
in this work Preliminary experiments at 147 nM toxin
concentration carried out to steady state indicated a marked
drop in lytic activity for temperatures below 15 C and lysis was
not observed at 10 C Data on binding isotherms would be
required to understand temperature eff ects in these systems
Our work has implications in optimizing dosage and devel-oping novel drug delivery strategies based on PFTs The obser-
vation that lysis occurs in a given window of toxin concentration
has implications while developing PFT-based drug and gene
therapy protocols In recent studies E coli used in conjunction
with radiation therapy was found to retard the growth of cancer
tumors when compared with only radiation therapy protocols36
Quantifying the dynamics and extent of lysis is an important
aspect of developing appropriate treatment protocols in these
combination therapies In other applications where pores are
used for gene delivery and in the development of articial cells
lysis must be prevented and pore formation restricted to
concentrations below the lysis threshold
Acknowledgements
We acknowledge funding from Department of Science and
Technology (DST) India under the IRHPA grant The authors
thank Sanjeev Kumar Gupta and Jaydeep Basu for several useful
discussions on the development of the model as well as the
reviewers for their critical comments
References
1 S Bhakdi and J Tranum-Jensen Microbiol Rev 1991 55
733ndash
7512 J Thiery D Keefe S Boulant E Boucrot M Walch
D Martinvalet I S Goping R C Bleackley
T Kirchhausen and J Lieberman Nat Immunol 2011 12
770ndash777
3 A Ludwig C von Rhein S Bauer C Huttinger and
W Goebel J Bacteriol 2004 186 5311ndash5320
4 A W Bernheimer and L S Avigad Proc Natl Acad Sci U S
A 1976 73 467ndash471
5 M V Sousa M Richardson W Fontes and L Morhy
J Protein Chem 1994 13 659ndash667
6 G Anderluh and P Macek Toxicon 2002 40 111ndash124
7 I Iacovache F G van der Goot and L Pernot Biochim
Biophys Acta Biomembr 2008 1778 1611ndash1623
8 N R Wyborn A Clark R E Roberts S J Jamieson
S Tzokov P A Bullough T J Stillman P J Artymiuk
J E Galen L Zhao M M Levine and J Green
Microbiology 2004 150 1495ndash1505
9 A J Wallace T J Stillman A Atkins S J Jamieson
P A Bullough J Green and P J Artymiuk Cell 2000 100
265ndash27610 M Mueller U Grauschopf T Maier R Glockshuber and
N Ban Nature 2009 459 726ndash730
11 J Oscarsson Y Mizunoe L Li X-H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
12 A Atkins N R Wyborn A J Wallace T J Stillman
L K Black A B Fielding M Hisakado P J Artymiuk and
J Green J Biol Chem 2000 275 41150ndash41155
13 A Ludwig G Volkerink C von Rhein S Bauer E Maier
B Bergmann W Goebel and R Benz J Bacteriol 2010
192 4001ndash4011
14 N Eier M Vetsch M Gregorini P Ringler M Chami
A Philippsen A Fritz S A Muller R Glockshuber A Engel and U Grauschopf EMBO J 2006 25 2652ndash2661
15 L A Shepard O Shatursky A E Johnson and R K Tweten
Biochemistry 2000 39 10284ndash10293
16 A J Ratner K R Hippe J L Aguilar M H Bender
A L Nelson and J N Weiser J Biol Chem 2006 281
12994ndash12998
17 I Iacovache M Bischoerger and F G van der Goot Curr
Opin Struct Biol 2010 20 241ndash246
18 G Schwarz and C H Robert Biophys Chem 1992 42 291ndash296
19 G Schwarz R-t Zong and T Popescu Biochim Biophys
Acta Biomembr 1992 1110 97ndash104
20 D Bolintineanu E Hazrati H T Davis R I Lehrer and
Y N Kaznessis Peptides 2010 31 1ndash821 J M Alam T Kobayashi and M Yamazaki Biochemistry
2012 51 5160ndash5172
22 J Lemiere K Guevorkian C Campillo C Sykes and T Betz
So Matter 2013 9 3181ndash3187
23 V Noireaux and A Libchaber Proc Natl Acad Sci U S A
2004 101 17669ndash17674
24 M M Bradford Anal Biochem 1976 72 248ndash254
25 H Xu E K Bjerneld M Kall and L Borjesson Phys Rev
Lett 1999 83 4357ndash4360
26 J Oscarsson Y Mizunoe L Li X H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
27 T M Raschke J Tsai and M Levitt Proc Natl Acad SciU S A 2001 98 5965ndash5969
28 L A Rodrıguez-Guadarrama S K Talsania K K Mohanty
and R Rajagopalan Langmuir 1999 15 437ndash446
29 C E McLaren G M Brittenham and V Hasselblad Am J
Physiol 1987 252 3181ndash3187
30 J G Kim M Xia and H Liu IEEE Eng Med Biol Mag 2005
24 118ndash121
31 M Nagahama H Hara M Fernandez-Miyakawa
Y Itohayashi and J Sakurai Biochemistry 2006 45 296ndash302
32 J R Thompson B Cronin H Bayley and M I Wallace
Biophys J 2011 101 2679ndash2683
This journal is copy The Royal Society of Chemistry 2014 RSC Adv 2014 4 4930ndash4942 | 4941
Paper RSC Advances
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1313
33 A Zitzer I Walev M Palmer and S Bhakdi Med Microbiol
Immunol 1995 184 37ndash44
34 O Opota I Vallet-Gely R Vincentelli C Kellenberger
I Iacovache M R Gonzalez A Roussel F G van der Goot
and B Lemaitre PLoS Pathog 2011 7 e1002259
35 S Kumar and D Ramkrishna Chem Eng Sci 1996 51
1311ndash1332
36 S N Jiang T X Phan T K Nam V H Nguyen H S Kim
H S Bom H E Choy Y Hong and J J Min Mol Ther
2010 18 635ndash642
RSC Advances Paper
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1213
reached a steady state which occurs within 30 s (Fig 7a) cell
lysis continues to occur at a xed number of pores proportional
to np npc Running the simulation to steady state would
eventually lead to lysis of all the cells in the system albeit at an
exceedingly slow rate at the lower and intermediate toxin
concentrations We have also assumed complete and irrevers-
ible binding of the monomer to the membrane and hence the
number of pores predicted represent an upper limit within the
proposed kinetic framework Despite the mean eld approxi-mation the model proposed in this manuscript is able to
capture the inherent time scales in the process of pore forma-
tion and rupture as well as predict the variation of hemoglobin
release as a function of the initial toxin concentration observed
in the experiments We nally point out that the eff ect of
temperature on the kinetics and lysis has not been investigated
in this work Preliminary experiments at 147 nM toxin
concentration carried out to steady state indicated a marked
drop in lytic activity for temperatures below 15 C and lysis was
not observed at 10 C Data on binding isotherms would be
required to understand temperature eff ects in these systems
Our work has implications in optimizing dosage and devel-oping novel drug delivery strategies based on PFTs The obser-
vation that lysis occurs in a given window of toxin concentration
has implications while developing PFT-based drug and gene
therapy protocols In recent studies E coli used in conjunction
with radiation therapy was found to retard the growth of cancer
tumors when compared with only radiation therapy protocols36
Quantifying the dynamics and extent of lysis is an important
aspect of developing appropriate treatment protocols in these
combination therapies In other applications where pores are
used for gene delivery and in the development of articial cells
lysis must be prevented and pore formation restricted to
concentrations below the lysis threshold
Acknowledgements
We acknowledge funding from Department of Science and
Technology (DST) India under the IRHPA grant The authors
thank Sanjeev Kumar Gupta and Jaydeep Basu for several useful
discussions on the development of the model as well as the
reviewers for their critical comments
References
1 S Bhakdi and J Tranum-Jensen Microbiol Rev 1991 55
733ndash
7512 J Thiery D Keefe S Boulant E Boucrot M Walch
D Martinvalet I S Goping R C Bleackley
T Kirchhausen and J Lieberman Nat Immunol 2011 12
770ndash777
3 A Ludwig C von Rhein S Bauer C Huttinger and
W Goebel J Bacteriol 2004 186 5311ndash5320
4 A W Bernheimer and L S Avigad Proc Natl Acad Sci U S
A 1976 73 467ndash471
5 M V Sousa M Richardson W Fontes and L Morhy
J Protein Chem 1994 13 659ndash667
6 G Anderluh and P Macek Toxicon 2002 40 111ndash124
7 I Iacovache F G van der Goot and L Pernot Biochim
Biophys Acta Biomembr 2008 1778 1611ndash1623
8 N R Wyborn A Clark R E Roberts S J Jamieson
S Tzokov P A Bullough T J Stillman P J Artymiuk
J E Galen L Zhao M M Levine and J Green
Microbiology 2004 150 1495ndash1505
9 A J Wallace T J Stillman A Atkins S J Jamieson
P A Bullough J Green and P J Artymiuk Cell 2000 100
265ndash27610 M Mueller U Grauschopf T Maier R Glockshuber and
N Ban Nature 2009 459 726ndash730
11 J Oscarsson Y Mizunoe L Li X-H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
12 A Atkins N R Wyborn A J Wallace T J Stillman
L K Black A B Fielding M Hisakado P J Artymiuk and
J Green J Biol Chem 2000 275 41150ndash41155
13 A Ludwig G Volkerink C von Rhein S Bauer E Maier
B Bergmann W Goebel and R Benz J Bacteriol 2010
192 4001ndash4011
14 N Eier M Vetsch M Gregorini P Ringler M Chami
A Philippsen A Fritz S A Muller R Glockshuber A Engel and U Grauschopf EMBO J 2006 25 2652ndash2661
15 L A Shepard O Shatursky A E Johnson and R K Tweten
Biochemistry 2000 39 10284ndash10293
16 A J Ratner K R Hippe J L Aguilar M H Bender
A L Nelson and J N Weiser J Biol Chem 2006 281
12994ndash12998
17 I Iacovache M Bischoerger and F G van der Goot Curr
Opin Struct Biol 2010 20 241ndash246
18 G Schwarz and C H Robert Biophys Chem 1992 42 291ndash296
19 G Schwarz R-t Zong and T Popescu Biochim Biophys
Acta Biomembr 1992 1110 97ndash104
20 D Bolintineanu E Hazrati H T Davis R I Lehrer and
Y N Kaznessis Peptides 2010 31 1ndash821 J M Alam T Kobayashi and M Yamazaki Biochemistry
2012 51 5160ndash5172
22 J Lemiere K Guevorkian C Campillo C Sykes and T Betz
So Matter 2013 9 3181ndash3187
23 V Noireaux and A Libchaber Proc Natl Acad Sci U S A
2004 101 17669ndash17674
24 M M Bradford Anal Biochem 1976 72 248ndash254
25 H Xu E K Bjerneld M Kall and L Borjesson Phys Rev
Lett 1999 83 4357ndash4360
26 J Oscarsson Y Mizunoe L Li X H Lai A Wieslander and
B E Uhlin Mol Microbiol 1999 32 1226ndash1238
27 T M Raschke J Tsai and M Levitt Proc Natl Acad SciU S A 2001 98 5965ndash5969
28 L A Rodrıguez-Guadarrama S K Talsania K K Mohanty
and R Rajagopalan Langmuir 1999 15 437ndash446
29 C E McLaren G M Brittenham and V Hasselblad Am J
Physiol 1987 252 3181ndash3187
30 J G Kim M Xia and H Liu IEEE Eng Med Biol Mag 2005
24 118ndash121
31 M Nagahama H Hara M Fernandez-Miyakawa
Y Itohayashi and J Sakurai Biochemistry 2006 45 296ndash302
32 J R Thompson B Cronin H Bayley and M I Wallace
Biophys J 2011 101 2679ndash2683
This journal is copy The Royal Society of Chemistry 2014 RSC Adv 2014 4 4930ndash4942 | 4941
Paper RSC Advances
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1313
33 A Zitzer I Walev M Palmer and S Bhakdi Med Microbiol
Immunol 1995 184 37ndash44
34 O Opota I Vallet-Gely R Vincentelli C Kellenberger
I Iacovache M R Gonzalez A Roussel F G van der Goot
and B Lemaitre PLoS Pathog 2011 7 e1002259
35 S Kumar and D Ramkrishna Chem Eng Sci 1996 51
1311ndash1332
36 S N Jiang T X Phan T K Nam V H Nguyen H S Kim
H S Bom H E Choy Y Hong and J J Min Mol Ther
2010 18 635ndash642
RSC Advances Paper
View Article Online
7252019 c3ra45159c
httpslidepdfcomreaderfullc3ra45159c 1313
33 A Zitzer I Walev M Palmer and S Bhakdi Med Microbiol
Immunol 1995 184 37ndash44
34 O Opota I Vallet-Gely R Vincentelli C Kellenberger
I Iacovache M R Gonzalez A Roussel F G van der Goot
and B Lemaitre PLoS Pathog 2011 7 e1002259
35 S Kumar and D Ramkrishna Chem Eng Sci 1996 51
1311ndash1332
36 S N Jiang T X Phan T K Nam V H Nguyen H S Kim
H S Bom H E Choy Y Hong and J J Min Mol Ther
2010 18 635ndash642
RSC Advances Paper
View Article Online