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Anatoly B. Kolomeisky

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Anatoly B. Kolomeisky

UNDERSTANDING MECHANOCHEMICAL COUPLING IN KINESINS USING FIRST-PASSAGE PROCESSES

Collaboration:

Alex Popov, Evgeny Stukalin – Rice University

Prof. Michael E. Fisher -University of Maryland

Prof. Ben Widom – Cornell University

Financial Support:

National Science Foundation

Dreyfus Foundation

Welch Foundation

Rice University

- J. Stat. Phys., 93, 633 (1998).
- PNAS USA, 96, 6597 (1999).
- Physica A, 274, 241 (1999).
- Physica A, 279, 1 (2000).
- J. Chem. Phys., 113, 10867 (2000).
- PNAS USA, 98, 7748 (2001).
- J. Chem. Phys., 115, 7253 (2001).
- PNAS USA, 98, 7748 (2001).
- Biophys. J., 84, 1642 (2003).

Enzymes that convert the chemical energy into mechanical work

Functions: cell motility, cellular transport, cell division and growth, muscles, …

Courtesy of Marie Curie Research Institute, Molecular Motor Group

myosin-II

kinesin

RNA-polymeraze

F0F1-ATPase

There are many types:linear, rotational, processive, non-processive

Properties:

Non-equilibrium systems

Velocities: 0.01-100 mm/s

Step Sizes: 0.3-40 nm

Forces: 1-60 pN

Fuel: hydrolysis of ATP, or related compound,polymerization

Efficiency: 50-100% (!!!)

Main Problems:

What mechanism of motility? How many mechanisms?

- Thermal Ratchet Models

periodic spatially asymmetric potentials

2)Multi-State Chemical Kinetic (Stochastic) Models

sequence of discrete biochemical states

Idea: motor proteins are particles that move in periodic but asymmetric potentials, stochastically switching between them

Advantages:

1) continuum description, well developed formalism;

2) convenient for numerical calculations and simulations;

3) small number of parameters;

- Disadvantages:
- mainly numerical or simulations results;
- results depend on potentials used in calculations;
- hard to make quantitative comparisons with experiments;
- not flexible in description of complex biochemical networks;

Assumption: themotor proteinmoleculesteps through a sequence ofdiscrete biochemical states

- Advantages:
- Exact results
- Agreement with biochemical observations
- Flexibility in description of complex biochemical systems
- Agreement with experiments

- Disadvantages:
- Discreteness
- Mathematical complexity
- Large number of parameters

Optical Trap Experiment:

laser

bead

kinesin

microtubule

Optical trap works like an electronic spring

optical force clamp with a feedback-driven optical trap

Visscher,Schnitzer,Block (1999) Nature 400, 184-189

step-size d=8.2 nm

precise observations:

mean velocityV(F,[ATP])

stall force

dispersion D(F,[ATP])

mean run lengthL(F,[ATP])

- Description of biophysical properties of motor proteins (velocities, dispersions, stall forces, …) as the functions of concentrations and external loads
- Detailed mechanism of motor proteins motility
- coupling between ATP hydrolysis and the protein motion
- stepping mechanism – hand-over-hand versus inchworm
- conformational changes during the motion
- …

N=4model

j=0,1,2,…,N-1 – intermediate biochemical states

kinesin/

microtubule/ADP

kinesin/

microtubule/ATP

kinesin/

microtubule/ADP/Pi

kinesin/

microtubule

our model

periodic hopping model on 1D lattice

exact and explicit expressions for asymptotic (long-time) for any N!

Derrida, J. Stat. Phys. 31 (1983) 433-450

drift velocity

dispersion

x(t) – spatial displacement along the motor track

randomness

bound!r >1/N

stall force

Effect of an external loadF:

load distribution factors

activation barrier

F >0

F=0

j

j+1

j

j+1

stall forcedepends on [ATP]

Michaelis-Mentenplots

N=2model

F=3.59 pN

F=1.05 pN

force-velocitycurves

randomness

- How many molecules of ATP are consumed per kinesin step?
- Is ATP hydrolysis coupled to forward and/or backward steps?

Nature Cell Biology, 4, 790-797 (2002)

- Kinesin molecules hydrolyze a single ATP molecule per 8-nm advance
- The hydrolysis of ATP molecule is coupled to either the forward or the backward movement (!!!!!!!!!!)

Schnitzer and Block, Nature, 388, 386-390 (1997)

Hua et al., Nature, 388, 390-394 (1997)

Coy et al., J. Biol. Chem., 274, 3667-3671 (1999)

Problem: back steps ignored in the analysis

Nishiyama et al., Nature Cell Biology, 4, 790-797 (2002)

Backward steps are taken into account

Investigation of kinesin motor proteins motion using optical trapping nanometry system

Nishiyama et al., Nature Cell Biology, 4, 790-797 (2002)

Fraction of 8-nmforward and backward steps, and detachments as a function of the force at different ATP concentrations

circles - forward steps;

triangles - backward steps;

squares – detachments

Stall force – when the ratio of forward to backward steps =1

Nishiyama et al., Nature Cell Biology, 4, 790-797 (2002)

Dwell times between the adjacent stepwise movements

Dwell times of the backward steps+detachments are the same as for the forward8-nm steps

Both forward and backward movements of kinesin molecules are coupled to ATP hydrolysis

Nishiyama et al., Nature Cell Biology, 4, 790-797 (2002)

Branched kinetic pathway modelwith asymmetricpotential of the activation energy

Idea: barrier to the forward motion is lower than for the backward motion

Conclusion:kinesin hydrolyses ATP at any forward or backward step

Nishiyama et al., Nature Cell Biology, 4, 790-797 (2002)

- PROBLEMS:
- Backward biochemical reactions are not taken into account
- Asymmetric potential violates the periodic symmetry of the system and the principle of microscopic reversibility
- Detachments are not explained

Nishiyama et al., Nature Cell Biology, 4, 790-797 (2002)

The protein molecule moves from one binding site to another one through the sequence of discrete biochemical states, i.e., only forward motions are coupled with ATP hydrolysis

Random walker hopping on a periodic random infinite 1D lattice

Dwell times – mean first-passage times;Fractions – splitting probabilities

N,j– the probability that Nis reached before –N, starting from the site j

Boundary conditions:

N.G. van Kampen, Stochastic Processes in Physics and Chemistry, Elseiver, 1992

-splitting probability to go to site N, starting from the site 0,

fraction of forward steps

fraction of backward steps

TN,j – mean first-passage time to reach N, starting from j

TN,0 – dwell time for the forward motion;

T-N,0 – dwell time for the backward motion

with

Important observation:

Dwell times for the forward and backward steps are the same, probabilities are different

Drift velocity

With irreversible detachments

j

-probability to dissociate before reaching N or -N, starting from j

- fractions of steps forward, backward and detachments

With irreversible detachments

j

Define new parameters:

j – the solution of matrix equation

-vector

matrix elements

With irreversible detachments

j

Model with detachments

Model without detachments

N=1case:

With irreversible detachments

j

Description of experimental data using N=2 model; reasonable for kinesins

Fisher and Kolomeisky, PNAS USA, 98, 7748 (2001).

Load dependence of rates

Fractions of forward and backward steps, and detachments

[ATP]=1mM

[ATP]=10M

Dwell times before forward and backward steps, and before the detachments at different ATP concentrations

N=2model

mean forward-step first-passage time

Kolomeisky and Fisher, Biophys. J., 84, 1642 (2003)

- Analysis of motor protein motility using first-passage processes is presented
- Effect of irreversible detachments is taken into account
- Our analysis of experimental data suggests that 1 ATP molecule is hydrolyzed when the kinesin moves forward 1 step