First passage times on a single molecule level and leapovers of Lvy motions Tal Koren TAU Iddo Eliaz

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First passage times on a single molecule level
andleapovers of Lévy motions Tal Koren (TAU)
Iddo Eliazar (TAU)Aleksei Chechkin
(Kharkov)Ophir Flomenbom (MIT)Michael Urbakh
(TAU)Olga Dudko (NIH)
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• 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)

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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

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Dynamic Force Spectroscopy
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Single Stranded DNA translocation through a
Nanopore One polymer at a time
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Relevant systems
Individual membrane channels ion flux
biopolymers translocation
A. Meller, L. Nivon, and D. Branton. Phys. Rev.
Lett. 86 (2001)
J. J. Kasianowicz, E. Brandin, D. Branton and D.
W. Deamer Proc. Natl. Acad. Sci. USA 93 (1996)
Translocation and conformational fluctuation
J. Li and H. A. Lester. Mol. Pharmacol. 55
(1999).
O. Flomenbom and J. Klafter Biophys. J. 86 (2004).
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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

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Relevant systems
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)
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Single molecule experiments in proteinsFractons
in proteins

• Fluorescence resonant energy transfer (tens of
  angstroms).
• Photo-induced electron transfer (a few
  angstroms).

S. C. Kou and X. S. Xie, PRL (2004) W. Min et
al., PRL (2005) R. Granek and J. Klafter, PRL
(2005)
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Autocorrelation function
Small scale motion VIBRATIONS?
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Dynamic Force Spectroscopy
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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.
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• To explore results of unbinding measurements we
  have to establish
• relationships between equilibrium properties of
  the system and the
• forces measured under nonequilibrium conditions.

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• 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).

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• 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

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The Model
The dynamic response of the bound complex is
governed by the Langevin equation
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
.
PNAS, 100, 11378 (2003)
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Analytical model
Total potential
In the absence of thermal fluctuations unbinding
occurs only when the potential barrier vanishes
F(x)
At this point the measured force, reaches its
maximum value FFc.
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• 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 Kramers transition
  rate

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• Close to the critical force Fc at which the
  barrier disappears completely the instantaneous
  barrier height and the oscillation frequencies
  can be written as

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• The experimentally measured mean maximal
  force,ltFmaxgt, and distribution of unbinding
  forces, P(Fmax), can be expressed in terms of W
  as

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Maximal spring force
gt
F(V) (lnV)2/3
as compared with
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• 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 ltFmaxgt const ln(V/T). (The
  units of velocity V are nm/sec, temperature is in
  degrees Kelvin.)

.
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REBINDING
Time series of the spring force showing the
rebinding events for T293K, V5.9 nm/sec.
Total potential,
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Distribution of unbinding forces
Normalized distribution of the unbinding force at
temperature T293K for two values of the
velocity. The result from the numerical
simulation (points) is in a good agreement with
the analytical equation, for velocity V117 nm/c.
For velocity V5.9 nm/sec, where the rebinding
plays an essential role, the distribution
function deviates from the one given above.
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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

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Experimental Output

• 30 minutes trajectories with detector
  resolution of 50 ns

• The signal is the photon count value per bin
  size, a(t).

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Analysis of The Digital Trajectory
Time series of on-off events (blinking)
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Analysis of the trajectory
Constructing the waiting time distributions by
building histograms from the off and the on time
durations
off
on
These functions can not be obtained from bulk
measurements
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Relationship between the scheme and the two-state
trajectory
An internal property of the system
Each substate belongs either to the on or the off
states. The number of substates in each of the
states can be different and the connectivity may
be complex
Random walk in a kinetic scheme (more generally
CTRW)
We wish to learn as much as possible about the
underlying kinetic scheme
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Analysis
Stretched exponential decay for the off waiting
time PDF for three different S
An independent check for the S effect regime.
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Modeling
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
For this model the off state waiting time
PDF is a peaked function with an exponential
tail. This does not describe the experimental
results.
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Correlation Function
A direct calculation of the correlation function
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Theoretical calculation of the correlation
function
Calculating the correlation function for a
two-state hopping process with arbitrary waiting
time PDFs (a CTRW model)

• C(t) for a stationary process

• The superscript 0 indicates the first event
  PDF (renewal theory).
• The propagator is calculated using exponential
  on time PDF

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Theoretical calculation of the correlation
function
This calculation supplies another check for the
validity of the experimental results.
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What can one learn from two-state single molecule
trajectories?
Reducible schemes (lack of correlations)

• The functional form of the waiting time PDFS

• Several possibilities for the on-off scheme
  connectivity

Irreducible schemes (existence of correlations)

• Calculating other functions from the trajectory,
  such as

• looking on the ordered waiting times trajectory

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Irreducible schemes
A full analysis should be preformed as more
information about the underlying kinetic scheme
can be obtained
Additional method
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Multi off sub-states exponential on-state
waiting times
Models
The Independent Channel Model
The Fluctuating Enzyme Model
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The ordered waiting time trajectory
Extracting the scheme parameters
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Discriminating between the models

• 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.

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The resulting off waiting time trajectories
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• The fluctuating enzyme model accounts for the
  stretched exponential form
• of the off-state waiting time PDF.

• 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.

• Setting rmr, the on-state waiting time PDF is
  a single exponential.
• No coupling between the on sub-states.

The fluctuating enzyme model can account for
local trends in the off duration time trajectory.
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On the first passage times and leapovers of Lévy
flights
0
a
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 target3. The question of First Passage
Leapover (FPL) - Leapover how far from the
target the particle lands.
The FPL problem has been hardly investigated.
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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

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Lévy stable probability laws
Examples
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I. General Lévy motions
Typical Trajectories
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Sparre Anderson FPT density decays as
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Leapover problem What are the distributions of
the FPL ?
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II. One-sided Lévy motions
Typical trajectories
I. Eliazar, J. Klafter, Physica A 336, 219-244
(2004)
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First passage of one-sided Lèvy flight
3 rangesa?0 FPT is an
exponential distr.a?0.5 a?1

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Mean first passage of one-sided Lèvy flight
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Leapover for one-sided Lévy flight
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III. Two-sided Lévy flight
Typical trajectories
Leapover is Zero!
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First passage of two sided Lèvy flight
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Confined Lévy flights in bistable potential
Kramers problem
Fig.1. Escape of the trajectory over the barrier,
schematic view.
Fig.2. Typical trajectories for different ?.
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Power-law dependence of the MFPT
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How do proteins find their specific target on a
DNA?
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Intersegmental transfer
Intradomain assoc/dissoc or hopping
Bustamante, C. et al. J. Biol. Chem.
199927416665-16668
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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 it to move between
two-sites via an intermediate loop.
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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 ngtgtl

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Model

• The enzyme starts scanning at a random position
  along the DNA
• Upon the timers 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

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directional sliding
Target
1D search
3D search
Brownian motion
3D search
Target
1D search
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The working equations

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Parallel search

• Search by m independent enzymes acting in
  parallel
• Total search time
• Massively parallel

• The meaning of the limiting ratio ? is
  enzyme-concentration per DNA base-pairs (1D
  concentration). The limiting distribution

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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

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• 
  
• 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 ?
• Case 3
• The asymptotically optimal search strategy is to
• continue the local scan--no relocation until
• finding the target

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Brownian search

• In this case, the relocation mechanism reduces
  the search time significantly

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Sub-diffusive motion

• 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 ? (0lt ?lt1).
• In this case
• (i) (ii)
• The motions 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

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Example of rate function based on heavy-tailed
halting durations
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Heavy-tailed relocation times

• assume that the relocation times admit the
  algebraic decay
• Infinite mean relocation time
• For parallel search

Stretched exponential
Here we need an ensemble of independent enzymes
acting in parallel
The parallel search is powerful enough to
overcome the infinite-mean relocation times
governed by heavy-tailed probability distributions
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Conclusions

• A Stochastic model of searching a circular DNA
  strand for a target-site, using general
  local-scan and relocation mechanisms.
• Closed form formulae for the mean of the overall
  search duration obtained.
• 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

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Lévy noises
Lévy motion Lévy index ? ? outliers ?
Lévy index ? ? flights become
longer
Lévy noises Lévy motion Lévy index ?
? outliers ? Lévy index ? ? flights
become longer
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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 noise
• 0 lt ? lt 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 lt ? lt 2 Fractional FPE (FFPE)

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Definition of the Riesz fractional derivativevia
its Fourier representation

• .
• Examples
• 1.
• 2.

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Lévy flights in a harmonic potential, 1? ? lt 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

(Remind pass to the Fourier space
)
symmetric stable law
Harmonic force is not strong enough to
confine Lévy flights
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Part I. Confined Lévy flights. Confinement by
non-dissipative non-linearity. U ? x4
? Fractional FPE for the stationary PDF
Langevin equation
(1)
Equation for the characteristic function
(2)
normalization symmetry boundary conditions
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Confined Lévy flights. Quartic potential, U ? x4.
Cauchy case, ? 1

• Equation for the characteristic function
• PDF

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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 derivative
• Proposition 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