Near-Perfect Adaptation in E. coli Chemotaxis Signal Transduction Network
Yang Yang & Sima Setayeshgar
Jan, 2007
E. coli
•1-3 microns long
•1 micron in diameter
•4-6 flagella
• Small genome (4288 genes)
static.howstuffworks.com/gif/cell-ecoli.gif
www.hatetank.dk
Bacterial Chemotaxis
http://www.rowland.harvard.edu/labs/bacteria/index_movies.html
Increasing attractants or Decreasing repellents
Run Tumble
Chemotaxis signal transduction network in E. coli
well-characterized model system
CheW: Coupling CheA to MCPs CheB: CheBp demethylate MCPs CheR: Methylate MCPs CheY: flagella motor regulator protein CheZ: Dephosphoryte CheYp; CheA: Histidine kinase
ligand binding
Methylation
Phosphorylation
)()( )(7/7~5/5
)( CheRLTCheRTL pnkmkkmk
pn
ppnmkmk
ppn
pnckck
pn
CheBTLCheBTL
CheRTLCheRTL
)(14~1
)(
)(14~1
)(
)()(
)()(
PCheBCheB
PCheYCheZCheZCheY
CheBCheRTCheBCheRTL
CheYCheRTCheYCheRTL
ADPCheRTLATPCheRTL
kmbp
kmyp
pnkb
np
nky
np
npkk
n
)()()(
)()()(
)()()()( 9~7
T3 T4T2
T4pT2p T3p
LT3 LT4
LT4p
LT2
LT3pLT2p
phosphorylation
methylation
Lig
and
bind
ing
Full realistic model
Chemical reactions
Perfect adaptation
Steven M., et al. Journal of bacteriology. 1983
This property allows the system to compensate for the presence of continued stimulation and to be ready to respond to further stimuli
Robustness
U. Alon et al. Nature,1999
inputReaction rates
proteins outpu
t
Tau-Mu Yi* et al. Biophysics,2000
Can be achieved
by
Motivation
shedding light on biochemical steps and feedback mechanisms underlying robustness
shed light on values of unknown or partially known paramters
SOLVE: we develop a novel method for elucidating regions in parameter space allowing perfect adaptation.
QUESITON: basis of robustness of perfect adaptation?
START with a fine-tuned model of chemotaxis network that:
reproduces key features of experiments
is NOT robust
AUGMENT the model explicitly with the requirements that:
steady state value of CheYp
values of reaction rate constants, are independent of the external stimulus, s, thereby
achieving robustness of perfect adaptation.
s
k
F
u
skuFdt
ud
0);;(
: state variables
: reaction kinetics
: reaction constants
: external stimulus
Algorithm
Decretizing s
into H points
0
||
0);;(
ds
kdds
du
skuFdt
ud
N
02
|2
|
0);;(
)1(
11
11
s
kks
uu
skuFdt
ud
sjss
jm
jm
j
jN
jN
jjj
jlowj
Augmented system
The steady state concentration of proteins in the network must satisfy:
The steady state concentration of CheYp must satisfy:
At the same time, the reaction rate constants must be independent of stimulus:
: allows for near-perfect adaptation
= CheYp
0ds
kd
0);;( skuFdt
ud
N
N
u
ds
du
||
Implementation
Newton-Raphson, to solve for the steady state of augmented system:
multidimensional root finding method Efficient way of converging to a root with a sufficiently good initial guess.
x
f(x) 1
2 f(x)1
x
2
Works well unfortunate case fortunate case
x
f(x) 1
32
0
||
0);;(
ds
kdds
du
skuFdt
ud
N
Dsode (stiff ODE solver), to verify Time dependent behavior of proteins for different ranges of external stimulus by solving: 0);;( skuF
dt
ud
Implementation
Working progress
Exploring the parameter spaces of E. coli chemotaxis signaling transduction network
Exploring the unknown parameter ranges of chemotaxis signaling transduction network of species with multiple CheYs
E .coli
Relative change of CheYp:• less than 5%• less than 3%• less than 1%• pairwise trajectory
Pairwise result: 3D surface result:
parameter spaces of E. coli
E. coli
Relative change of CheYp:• less than 5%• less than 3%• less than 1%• pairwise trajectory
Pairwise result: 3D surface result:
parameter spaces of E. coli
Pairwise result: 3D surface result:
Relative change of CheYp:• less than 5%• less than 3%• less than 1%• pairwise
trajectory
E .coli
parameter spaces of E. coli
1%
k1c : 0.17 s-1 1 s-1
k8 : 15 s-1 12.7 s-1
Violating and restoring perfect adaptation 9%
k1c : 0.17 s-1 1 s-1(1,15)
(1,12.7)
At 250s, giving step stimulus from 0 to 1e-6M
Consistency with recent work done by Bernardo A. mello and Yuhai Tu
They list a series of conditions which allow near-perfect adaptations
They are a active-ependent model which the receptors are either in active or inactive state
Our parameter space remarkable shows the same consistency with their predictions about the relationships of the parameter values although we are using a active-independent model.
1. The timescale for ligand binding is much shorter than the methylation and phosphorylation timescale. This condition allows us to neglect ligand-binding/unbinding kinetics.
2. The association rates between the receptor and the methylation/demethylation enzymes, CheR and CheB-P, are linearly related to the activity of the receptor and are zero for n = 4 and n = 0, respectively: and : The dissociation rates of the enzyme receptor bound states are independent of λ.
3. The receptor activities of the nonmethylated and the maximally methylated receptors are independent of l: P0v = P0o, P4v = P4o.
4. The ratios between the CheR catalytic rate kR n and the CheB-P catalytic rate of the next methylation level kB n+1 are the same for all methylation states n: kB n+1 / kR n = const:
5. The phosphate transfer rates from CheA to CheB or CheY are proportional to CheA autophosphorylation rate:
6. The explicit dependence on [TFn] distribution can be
removed from the expression
this condition can only be strictly satisfied when
nRn PPK 4 0PPK n
Bn
nPYnn
PBn PkPk ;
4
0
2 ][)][][
(n
FnnB
F
R
F
TPK
B
K
R
B
F
R
F
KB
KR ][][
List of conditions:
Condition 2
nancnRn kkPPK 4
ckbkakPPK nanamnnBn 2
0 )(
Condition 3 P0v = P0o, P4v =
P4o
Condition 4 kB n+1 / kR n =
constan
nc
nm kkk
)1()1(
Condition 5 n
PYnn
PBn PkPk ;
kb /k8-13 and ky/k8-13 are linearly related
*The parameter value are normalized to the literature value( Peter A. S., John S.P. and Hans G.O. , A model of excitation and adaptation in bacterial chemotaxis, biochemistry 1997) while the inset is not since the literature value is zero for k11.
Condition 6 B
F
R
F
KB
KR ][][
kb /k8-13 and ky/k8-13 are linearly related
*The parameter value are normalized to the literature value( Peter A. S., John S.P. and Hans G.O. , A model of excitation and adaptation in bacterial chemotaxis, biochemistry 1997) while the inset is not since the literature value is zero for k11.
Two CheY system
•Rhodobacter sphaeroides, Caulobacter crescentus have multiple CheYs while lack of CheZ protein.•Similar chemotaxis behaviors.
Two CheY system
Our work:Reproduce the key feature of chemotaxis behavior in two CheY system by replacing CheZ with CheY2.
Parameter spaces of two CheY system
Introducing [CheY2] and CheY2 (de-)phosphorylation rates.Exploring the parameter values which can give perfect adaptation.
Two CheY
Relative change of CheYp:• less than 5%• less than 3%• less than 1%
Other parameter value were set as the literature value except Kb= 1e+6 M-1s-1 instead of 8e+5 M-1s-1.
parameter spaces comparison of two and single CheY case
The parameter space for single CheY case seems more restrict than the two CheY case
Conclusions
Successful implementation of the augmented model of the chemotaxis signal transduction network in E. coli that explicitly takes into account robust perfect adaption.
Preliminary results on projections of robustness manifolds in parameter space of E. coli and two CheY system
Complete construction of manifolds in parameter space, allowing insight into parameter dependence giving rise to robustness
Work in progress
Future workApplying the method to other cellular signal
transduction networks exhibiting robust homeostasis, such as phototransduction
Signal flow in visual transduction, Leon Lagnado and Denis Baylor,Neuron,1992
Thanks and comments!
Physics limitation in signal sensing
25 years ago Berg and Purcell had showed that the physics limitation of the single celled organism.The derivation is mainly assumed a perfect measurement device and they determined the relative measurement accuracy is :
But for multiple and noninterating receptors shaped as a ring, the formula is derived by Willam and Sima recently as:
With know parameter value, we can get the actual physics limit to measurements of CheYp concentration corresponds to
cDac
C 1 : diffusion constant
: device size: average concentration: sampling time
210 )2
1(
1
a
g
mbcDc
C
C
a
D
0g
b
m : receptor numbers: single receptor size: geometric factor of order unity
c
M
c
C 035.3
*Computational models of chemotaxis signal transduction network
Activity dependent model
Barkai & Leibler (1997)
The concept of robustness in biochemical networks introduced, showing how it may arise in bacterial chemotaxis through activity-dependent kinetics. The chemoreceptor is either in active state or inactive state.Simulations show that precision of adaptation is a robust property, while adaptation time is not, and that adaptation time is inversely proportional to receptor-complex activity.It did not show how the parameter space will change which is very important for understanding the robustness mechanism.
Activity independent model
Spiro et al. (1997)
Simplified three-methylation-state model, fine-tuned by trial and error, simulates ramp, step and saturation responses to aspartate.Although it can not achieve the robust perfect adaptation, but it is a more realistic model without assuming any of the activity dependent parameter values.And our work is start from implementing this fine-tuned model.
k1c/km1, k2c/km2, k3c/km3, k4c/km4 are linearly related:
*The parameter value are normalized to the literature value( Peter A. S., John S.P. and Hans G.O. , A model of excitation and adaptation in bacterial chemotaxis, biochemistry 1997).