First passage times on a single molecule level and
leapovers of Lévy motions
Tal Koren (TAU) Iddo Eliazar (TAU)
Aleksei Chechkin (Kharkov)Ophir Flomenbom (MIT)Michael Urbakh (TAU)
Olga Dudko (NIH)
• Single molecule techniques offer a possibility to follow real-time dynamics of individual molecules.
• For some biological systems it is possible to probe the dynamics of conformational changes and follow reactivities.
• Distributions rather than ensemble averages (adhesion forces, translocation times, reactivities)
Processes on the level of a single molecule
• Dynamic Force Spectroscopy (DFS) of Adhesion Bonds
• Enzymatic activity(in collaboration with the groups of de Schryver and Nolte)
• Translocation of a ssDNA through a nanopore
• Search of a circular DNA
• Protein vibrations
2250
2350
2450
2550
2650
27500 20 40 60 80 100
Fo
rce
(pN
)
Distance (nm)
Distance (nm)
For
ce(p
N)
3200
3300
3400
3500
3600
0 20 40 60 80 100
Dynamic Force Spectroscopy:
Single Stranded DNA translocation through a Nanopore: One polymer at a time
A. Meller, L. Nivon, and D. Branton. Phys. Rev. Lett. 86 (2001)
J. J. Kasianowicz, E. Brandin, D. Branton and D. W. DeamerProc. Natl. Acad. Sci. USA 93 (1996)
Individual membrane channels: : ion flux & & biopolymers translocation
Relevant systems
O. Flomenbom and J. Klafter Biophys. J. 86 (2004).
Translocation and conformational fluctuation J. Li and H. A. Lester. Mol. Pharmacol. 55 (1999).
3 2 1
Lipase B From Candida Antarctica (CALB) Activity(The groups of de Schryver and Nolte)
• The enzyme (CALB) is immobilized.
• The substrate diffuses in the solution
• During the experiment, a laser beamis focused on the enzyme, and the fluorescent state of a single enzyme is monitored.
• The Michaelis-Menten reaction
Chemical activity
K. Velonia, et al., Angew. Chem. (2005)
O. Flomenbom, et al., PNAS (2005)
L. Edman, & R. Rigler, Proc. Natl. Acad. Sci. U.S.A., 97 (2000) H. Lu, L. Xun, X. S. Xie, Science, 282 (1998)
Relevant systems
1 2 N
1 2 N
rNr2r1k1 k2
kN
Single molecule experiments in proteins:Fractons in proteins
• Fluorescence resonant energy transfer (tens of angstroms).• Photo-induced electron transfer (a few angstroms).
eqXtXtx )()(
S. C. Kou and X. S. Xie, PRL (2004)W. Min et al., PRL (2005)R. Granek and J. Klafter, PRL (2005)
Autocorrelation function )0()()( xtxtCx
stt
stttCx
1
11~)(
2/1
2/1 const.
Small scale motion – VIBRATIONS?
2250
2350
2450
2550
2650
27500 20 40 60 80 100
Fo
rce
(pN
)
Distance (nm)
Distance (nm)
For
ce(p
N)
3200
3300
3400
3500
3600
0 20 40 60 80 100
Dynamic Force Spectroscopy:
Mechanical response:
Processes:
The main observable in DFS experiments is the velocity-dependent rupture force F(V)
An adhesion bond is driven away from its equilibrium by a spring pulled at
a given velocity. The barrier diminishes as the applied force increases.
Rupture occurs via thermally assisted escape from the bound state across
an activation barrier.
The rupture force is determined by interplay between the rate of escape in the absence of the force and the pulling velocity. The measured forces are not the intrinsic properties of molecules, but depend on the loading rate and mechanical setup.
To explore results of unbinding measurements we have to establish
relationships between equilibrium properties of the system and the
forces measured under nonequilibrium conditions.
For unbinding along a single reaction coordinate, it is usually
assumed that Fmax(V) has the form:
where k0 is the spontaneous rate of bond dissociation, and x is the distance from the minimum to the activation barrier of the reaction potential U(x).
max0
( ) lnB
B
k T VK xF V const
x k k T
The logarithmic law has been derived assuming that the pulling force
produces a small constant bias which reduces the height of a potential barrier.
Unbinding is described as an activated crossing of a barrier, which is reduced
due to the pulling force Fpul
Then the average enforced rate of crossing the barrier is
These equations give logarithmic dependence of the force on the pulling velocity
0v pulG G xF
0 exp vv
B
Gk
k T
The Model
The dynamic response of the bound complex is governed by the Langevin equation
.. ( )( ) ( ) ( ) ( )x x
U xM x t x t K x Vt t
x
The molecule of mass M is pulled by a linker of a spring constant K moving at a
velocity V.
U(x) is the adhesion potential,
x is a dissipation constant and the effect of thermal fluctuations is given
by a random force x(t), which is -correlated
( ) (0) 2 ( )x x B xt k T t
PNAS, 100, 11378 (2003)
Analytical model
Total potential:
2( , ) ( ) ( )2
KX t U x x Vt
2
2 0cc
d d
dxdx
Φ(x)
In the absence of thermal fluctuations unbinding occurs only when the potential barrier vanishes:
At this point the measured force, reaches
its maximum value F=Fc.
1 2( ) ( ) ( )exp[ ( ) / ] ( )
2 Bx
dW t t tE t k T W t
dt
In the presence of fluctuations the escape from the potential well occurs earlier, and the probability W(t) that a molecule persists in its bound state is defined by the Kramer’s transition rate:
3/ 2
( ) ,cc
c
F FE t U
F
1/ 4
1,2 ( ) cc
c
F Ft
F
Close to the critical force Fc at which the barrier disappears completely the instantaneous barrier height and the oscillation frequencies can be written as
max maxmax
( ) ( )d
P F W FdF
' ' 'max max max max'
max0
( )d
F F W F dFdF
The experimentally measured mean maximal force,<Fmax>, and distribution of unbinding forces, P(Fmax), can be expressed in terms of W as
Maximal spring force:
2 32 3
max 2
31 ln c xB
cc B c c
U Kk TF F V
U k T MF
=> F(V) ~ (lnV)2/3
as compared with
( ) ln( )F V const V
3/ 2
max /cF F T with ln /V T
Theory predicts a universal scaling, independent of temperature, of
Results of numerical calculations supporting the scaling behavior of the ensemble averaged rupture force.
The inset shows a significantly worse scaling for the description <Fmax> ~ const – ln(V/T). (The units of velocity V are nm/sec, temperature is in degrees Kelvin.)
REBINDING
Time series of the spring force showing the rebinding events for T=293K, V=5.9 nm/sec.
Total potential,
2( ) ( )2
KU x x Vt
Distribution of unbinding forces
Normalized distribution of the unbinding force at temperature T=293K for two values of the velocity.
The result from the numerical simulation (points) is in a good agreementwith the analytical equation, for velocity V=117 nm/c. For velocity V=5.9 nm/sec, where the rebinding plays an essential role, the distribution function deviates from the one given above.
Lipase B From Candida Antarctica (CALB) Activity(The groups of de Schryver and Nolte)
• The enzyme (CALB) is immobilized.
• The substrate diffuses in the solution
• During the experiment, a laser beamis focused on the enzyme, and the fluorescent state of a single enzyme is monitored.
• The Michaelis-Menten reaction
E (off) [E+S) ]off( [E+P) ]on( E’ (off)S (off)
• ~30 minutes trajectories with detector resolution of 50 ns
• The signal is the photon count value per bin size, a(t).
Experimental Output
Time series of on-off events (blinking)
Analysis of The Digital Trajectory
Analysis of the trajectory
These functions can not be obtained from bulk measurements
Constructing the waiting time distributions by building histograms fromConstructing the waiting time distributions by building histograms fromthe off and the on time durationsthe off and the on time durations
)(ton
)(toff
off
ton et )(
toff et )(
on
Each substate belongs either to the on or the off states .The number of substates in each of the states can bedifferent and the connectivity may be complex
Relationship between the schemeand the two-state trajectory
An internal property of the ‘system’
We wish to learn as much as possible about the underlying kinetic scheme
1
5
2
3
4
1
2
3
4
Random walk in a kinetic scheme)more generally CTRW(
Stretched exponential decay for the off waiting time PDF for three different [S]
15.0;)/1(
/)( )/(
toff et
An independent check for the [S] effect regime.
Analysis
Modeling
)()( _ tataoff eeAt
taon eat 1,3
1,3)(
2,11,22
3,21,22,13,21,22,1 4)(2
1aaaaaaaaa
Single-Molecule Michaelis-Menten
[S] independent regime
The enzyme can not catalyze the backward reaction: P S.
The product molecules diffusion back to theconfocal focus can beneglected (as shown by control experiments).
The waiting time PDFs for the scheme
)/(A aaaa
For this model the off state waiting time PDF is a peaked function with an exponential tail. This does not describe the experimental results.
E (off); state 1 [E+S) ]off ;(state 2 [E+P) ]on ;(state 3S (off)
3,2a2,1a
1,3a
1,2a
trajectory
trajectoryttC
2)]0([
)0()()(
)()( tt
085.0~;)(~
)~/( tetC
A direct calculation of the correlation function
Correlation Function
)(ton)(toff Calculating the correlation function for a two-statehopping process with arbitrary waiting time PDFs(a CTRW model)
oneq
oneq
P
PonontPtC
,
,
1
),0|,()(
• C(t) for a stationary process:
t x
ononon dxdyyxyxttonontP0 0
00 )()()()(),0|,(
)()(1
)()()()()(
0 ss
sssss
onoff
off
N
N
onoffoff
• The superscript ‘0’ indicates the first event PDF (renewal theory).
• The propagator is calculated using exponential on time PDF
t
onon dxxt )()(
1
1;)(
mset ton
Theoretical calculation of the correlation function
087.0~;)(~
)~/( tetC
)(ton)(toff
)]()(1[
)](1)][(1[11
1)(
ss
ss
tt
tt
sssC
offon
offon
offon
offon
This calculation supplies another check for the validity of the experimental results.
Theoretical calculation of the correlation function
Reducible schemes (lack of correlations):Reducible schemes (lack of correlations):
What can one learn from two-state single moleculeWhat can one learn from two-state single molecule trajectories?trajectories?
• The functional form of the waiting time PDFS The functional form of the waiting time PDFS
• Several possibilities for the on-off scheme connectivitySeveral possibilities for the on-off scheme connectivity
Irreducible schemes (existence of correlations):Irreducible schemes (existence of correlations):
• Calculating other functions from the trajectory, such as Calculating other functions from the trajectory, such as
• looking on the ordered waiting times trajectorylooking on the ordered waiting times trajectory
)(tyx
Irreducible schemes
)y,0 |,() y,|,( xPttxP
),( 21, ttyx
0 2121,0 21 ),()()( dtdttttttt yxyx
),,( 321,, tttzyx
)z,0 |,,,( tytxP
32 41
iii ttt ,2,1
A full analysis should be preformed as more information about the underlyingA full analysis should be preformed as more information about the underlying kinetic scheme can be obtainedkinetic scheme can be obtained
Additional method
Multi off sub-states & exponential on-state waiting times
The Fluctuating Enzyme Model The Independent Channel Model
Models
The ordered waiting time trajectory
32 41A
1B 432
)/(1 4323, kkt fastoff
)/(1 3212, kkt slowoff 2343, / kkn fastoff
3212, /2 kkn slowoff
)()( tt Bon
Aon )()( tt B
offA
off
Bunched fast eventsBunched fast events
Extracting the scheme parametersExtracting the scheme parameters
• The trajectory of off duration times contains 104 events.
• Local trends of grouped fast events (faster than 30 ms), with an average of8 ms per event. Each group contains onaverage 3-4 events with an overallduration time of 28 ms per group.
• Binning the over all time of each of the grouped fast events, and building a histogram.
• From this histogram an estimation for an average fluctuation rate can be extracted.
Discriminating between the models
The resulting off waiting time trajectories
• The fluctuating enzyme model accounts for the stretched exponential form of the off-state waiting time PDF.
• Setting rm=r, the on-state waiting time PDF is a single exponential. No coupling between the on sub-states.
• The model naturally gives the off-state waiting time PDF as a sum of weighted exponentials. A hierarchy of the reaction rates is then implied.
The fluctuating enzyme model can account for local trends in the off duration time trajectory.
On the first passage times and leapovers of Lévy flights
1. The question of First Passage Time (FPT) - first crossing of the target.
2. The question of First Arrival Time (FAT) - first arrival time at the target
3. The question of First Passage Leapover (FPL) - Leapover: how far from the target the particle lands.
0 a
a
The FPL problem has been hardly investigated.
Here, we focus on the question of the first passage time, and first passage leapover, but..
Sparre Anderson theorem - FPT density decays as t -3/2
for any symmetric random walk, independent of the PDF of the step length
• It has been shown recently (A. V. Chechkin et. al. (2003)) that the method of images is inconsistent with the universality of t -3/2
• The probability density of FAT differs from the density of FPT, namely:
1
1 1/ (2 1/ )( ) ~fa a
a
ap
D
1/ 1 1/( ) ~im aa
ap
D
Lévy stable probability laws
3/ 2
0.5, 0.5 :
1 1 1( ;1/ 2, 1/ 2,0,1) exp .
2 4L x x
x
, ( ; , ) exp | | 1 ( , ) , | |
kk k i k i k
k
tan , if 12( , )
2ln | | , if 1
kk
Examples:
2, 0 :
1, 0 :
2 2
1( ;1,0, , ) ,L x
x
3/ 2( ;1/ 2,0, , ) exp .
2 2( )L x x
x
2
2
1( ;2,0, , ) exp ,
42
xL x
0.5, 0 :
I. General Lévy motions
Typical Trajectories
Sparre Anderson – FPT density decays as 3/ 2~ t
Leapover problem – What are the distributions of the FPL ?
a
{ }a
a
1/
3/ 2
3/ 2 1
3 / 2 1 ( / 2 1)
( / 2 1)
~ ; ~
: ~
~ ( ) ( )
~
a a a
a a
aa a a a a
a
a a
a a
x a
Sparre Anderson p
df p
d
f
II. One-sided Lévy motions
I. Eliazar, J. Klafter, Physica A 336, 219-244 (2004)
Typical trajectories
(0,1), 1
First passage of one-sided Lèvy flight
3 ranges:
α→0 FPT is an exponential distr.
α→0.5
α→1
`
1~
(1 )aE ac
/a a c
2 2( ) exp / 4 ( 0),a a a
cp c a
a
Mean first passage of one-sided Lèvy flight
2 2
0
2exp / 4a
a a a
c ad c a
a c
Leapover for one-sided Lévy flight:
1
( ) ~ ( 0)( )
1~
a aa a
a
af
a
III. Two-sided Lévy flight (1,2), 1
Typical trajectories:
Leapover is Zero!
First passage of two sided Lèvy flight
(1/ 1)( ) ~a ap
Confined Lévy flights in bistable potential: Kramers problem
( )dx dU
Y tdt dx 2 4/ 2 / 4U x x x
1exp , 1
4GaussT D DD
Fig.2. Typical trajectories for different .
Fig.1. Escape of the trajectory over
the barrier, schematic view.
Power-law dependence of the MFPT
1.0 1.5 2.0 2.5 3.0 3.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
lg T
lg 1/D
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0.98
1.00
1.02
1.04
1.06
1.08
1.10
1.12
, 1
CT D D
D
Target site
Searching enzyme
DNA
How do proteins find their specific target on a DNA?
Intersegmental transfer
Bustamante, C. et al. J. Biol. Chem. 1999;274:16665-16668
Intradomain assoc/dissoc or “hopping”
Facilitated Diffusion (Berg, Winter & von Hippel 1981)The basic idea: Reduction of Dimensionality
1) Scanning -1D diffusion along the DNA either directional (“energy dependent”)or random (Sliding).
2) Relocation - Dissociation from one site on the DNA is followed by re-association to another by a 3D hopping/jumping process.
3) ‘Intersegmental Transfer’- Protein has two-binding sites which enable itto move between two-sites via an intermediate loop.
Model
Search for a target by one enzyme
• Consider a Circular DNA strand composed of bps
l - target length
n – DNA length without target
L- overall DNA length
• The enzyme initiates a local scan on the DNA and an exponential timer is set with a rate
• We assume n>>l
l n L
Target l
L
Model
• The enzyme starts scanning at a random position along the DNA
• Upon the timer’s expiration, if it does not locate the target, the enzyme relocates (3D) to a random position along the DNA, and the search begins anew.
• The relocation time is R.
• The total search time is:
• We assume that probability for relocation to a new position has a well-defined mean.
• The enzyme can move by:
→ directional sliding → Brownian motion→ sub-diffusion→ …
'
:
T R T
timer duration
Target3D search
1D search
3D search
Target
1D search
directional sliding
Brownian motion
The working equations
lim exp nn
TP t t
n
1
1~n
n
l
r
E T n
mean search time
r E R
Parallel search
• Search by m independent enzymes acting in parallel
• Total search time:
• Massively parallel:
,lim ( / ) ,( 0) n m m n
1
~m
n nE T n
m
• The meaning of the limiting ratio κ is “enzyme-concentration per DNA base-pairs” (1D concentration). The limiting distribution:
,lim exp ( ) mn m nP T t t
( )( ) ~
1
t
lt t
r
The mean search time depends explicitly on the target length - l, relocation rate - and mean relocation time - r
Examples of local-scanning mechanisms:
• Linear directional sliding
• Brownian motion
• Sub-diffusive motion
1
~nn
E T n
v
2D
1 / 2~ , - constant, (0 1), 0c c
Directional Sliding
Case 1
The asymptotically optimal search strategy is
to repeatedly relocate, spending no time on
local scanning until finding the target
Case 2
The asymptotic search performance is
independent of relocation rate λ
Case 3
The asymptotically optimal search strategy is to
continue the local scan--no relocation until
finding the target
1
v l
r
rv l
( ) /l r
rv l
( ) v
rv l 0
( ) v
2
min 2 2
-1
min 2
11 1 2
( ) 1 1 2
l Dr
r Dr l
l D Dr
r l l
2( ) ( )O n O n
Brownian search
• In this case, the relocation mechanism reduces the search time significantly:
2
1
D l
r
The enzymes follow a Brownian motion with diffusion parameter D, but their motion is
occasionally interrupted by random halts of random durations. The halting durations
are heavy-tailed, i.e., their probability tails decay algebraically with exponent ν (0< ν<1).
In this case:
• (i) (ii)
• The motion’s asymptotic mean square displacement after ‘running’ for t “units of time”
is
• The scan duration is heavy tailed – its probability tails admitting the asymptotic form as
• In the presence of relocation:
1 / 2( )=c ( )= 2D 0
Sub-diffusive motion
2( )
( 1)
subD
lr c
( ) t D t
~ t ( ) t
Example of rate function based on heavy-tailed halting durations:
Heavy-tailed relocation times
The parallel search is “powerful” enough to “overcome” the infinite-mean relocation times governed by heavy-tailed probability distributions
assume that the relocation times admit the algebraic decay:
•
• Infinite mean relocation time• For parallel search:
( ) ~ / ( 0,0 1)
sin( ) ( )( ) ~
t
P R t b t b
tt l
b Stretched exponential
Here we need an ensemble of independent enzymes acting in parallel
Conclusions
1) A Stochastic model of searching a circular DNA strand for a target-site, using general local-scan and relocation mechanisms.
2) Closed form formulae for the mean of the overall search duration obtained.
3) Limiting distributions of the overall search duration in the following scenarios were presented:
• (i) Directional sliding• (ii) Brownian motion• (iii) Sub-diffusion• (iv) Heavy tailed relocation
Lévy noises Lévy motion Lévy index outliers Lévy index “flights” become longer
Lévy noises Lévy motion Lévy index outliers Lévy index “flights” become longer
Kinetic equations for the stochastic systems driven by white Lévy noise
assumptions: overdamped case, or strong friction limit, 1-dim, D = intensity of the noise = const
Langevin description, x(t)
U(x) : potential energy, Y(t) : white noise, : the Lévy index = 2 : white Gaussian noise0 < < 2 : white Lévy noise (stationary sequence of independent stationary increments of the Lévy stable process)
Kinetic description, f(x,t)
: Riesz fractional derivative: integrodifferential operator = 2 : Fokker - Planck equation (FPE)0 < < 2 : Fractional FPE (FFPE)
)(tYdx
dU
dt
dx
f dU ff D
t x dx x
x /
Definition of the Riesz fractional derivativevia its Fourier representation
.
Examples
1.
2.
)()()(ˆ xxedxkFT
ikx
)(ˆ kkxd
d FT
2 2
22 2
ˆ( )2
ikxd dk dk k e
dxd x
ˆ ˆ( ) ( )FT FTd d
k k ik kd x dx
)exp()(ˆ,1
1)(
2kk
xx
)(ˆ)exp(
2kkikx
dk
xd
d
x
xarctan1cos
1
)1(2/)1(2
2exp)( xx
211 ;
2
1;
2
1
2
12xF
xd
d
Lévy flights in a harmonic potential, 1 < 2
FFPE for the stationary PDF:
Equation for the characteristic function:
Two properties of stationary PDF:
1. Unimodality (one hump at the origin).
2. Slowly decaying tails:
0
xd
fdDf
dx
dU
dx
d
,2
)(2ax
xU FTd
kd x
(Remind: pass to the Fourier space )
)(ˆ)sgn(ˆ 1 kfkk
dk
fd
kkf exp)(ˆ : symmetric stable law
)(2/sin,)(
1
Cx
Cxf
)(22 xfxdxx
Harmonic force is not “strong” enough to “confine” Lévy flights
)(ˆ)( kfxfFT
Part I. Confined Lévy flights. Confinement by non-dissipative non-linearity. U x4
3 ( )dx
x Y tdt
3 0d d f
x f Ddx d x
k
xd
d FT:reminder )(ˆ)( kfxf
FT
)(ˆ)sgn(ˆ 13
3kfkk
dk
fd
Langevin equation: Fractional FPE for the stationary PDF:
(1)
Equation for the characteristic function:
+ normalization + symmetry + boundary conditions …
(2)
Confined Lévy flights. Quartic potential, U x4. Cauchy case, = 1
Equation for the characteristic function:
PDF:
)(ˆ)sgn(ˆ
3
3kfk
dk
fd
62
3cos
2exp
3
2)(ˆ kk
kf
)1(
1)(
42 xxxf
min 0x max 1/ 2x
4( )f x x2 2 ( )x dx x f x
Two properties of stationary PDF.1. Bimodality: local minimum at , two maxima at
.
2. Steep power law asymptotics with finite variance:
Two propositions
Proposition 1: Stationary PDF for the Lévy flights in external field is not unimodal. Proved with the use of the “hypersingular” representation of the Riesz derivativeProposition 2: Stationary PDF for the Lévy flights in external fieldhas power-law asymptotics,
is a “universal constant”, i.e., it does not depend on c.
Critical exponent
Proved with the use of the representation of the Riesz derivative in terms of left- and right
Liouville-Weyl derivatives
2, cxU c
2, cxU c
xx
Cxf
c,)(
1
1sin 2C
4crc
2xcc cr
2xcc cr : “confined”
