1. Anatoly B. Kolomeisky
UNDERSTANDING MECHANOCHEMICAL COUPLING IN KINESINS USING FIRST-PASSAGE PROCESSES

2. 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

3. PUBLICATIONS: 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, (2000).
PNAS USA, 98, 7748 (2001).
J. Chem. Phys., 115, 7253 (2001).
Biophys. J., 84, 1642 (2003).

4. Motor Proteins
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

5. Motor Proteins: myosin-II kinesin RNA-polymeraze F0F1-ATPase
There are many types: linear, rotational, processive, non-processive

6. Motor Proteins Properties: Non-equilibrium systems
Velocities: mm/s
Step Sizes: nm
Forces: 1-60 pN
Fuel: hydrolysis of ATP, or related compound, polymerization
Efficiency: % (!!!)

7. Motor Proteins Main Problems:
What mechanism of motility? How many mechanisms?

8. THEORETICAL MODELING Thermal Ratchet Models
periodic spatially asymmetric potentials
2) Multi-State Chemical Kinetic (Stochastic) Models
sequence of discrete biochemical states

9. Ratchet Models
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;

10. Multi-State Chemical Kinetic (Stochastic) Models
Assumption: the motor protein molecule steps through a sequence of discrete biochemical states

11. Multi-State Chemical Kinetic (Stochastic) Models
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

12. Single-Molecules Experiments
Optical Trap Experiment:
laser
bead
kinesin
microtubule
Optical trap works like an electronic spring

13. EXPERIMENTS ON KINESIN
optical force clamp with a feedback-driven optical trap
Visscher,Schnitzer,Block (1999) Nature 400,
step-size d=8.2 nm
precise observations:
mean velocity V(F,[ATP])
stall force
dispersion D(F,[ATP])
mean run length L(F,[ATP])

14. Theoretical Problems:
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 …

15. OUR THEORETICAL APPROACH
N=4 model
j=0,1,2,…,N-1 – intermediate biochemical states
kinesin/
microtubule/ADP
kinesin/
microtubule/ATP
kinesin/
microtubule/ADP/Pi
kinesin/
microtubule

16. OUR THEORETICAL APPROACH
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)
drift velocity
dispersion
x(t) – spatial displacement along the motor track
randomness
bound! r >1/N
stall force

17. OUR THEORETICAL APPROACH
Effect of an external load F:
load distribution factors
activation barrier
F >0
F=0 j
j+1 j
j+1

18. RESULTS FOR KINESINS stall force depends on [ATP]
Michaelis-Menten plots
N=2 model
F=3.59 pN
F=1.05 pN

19. RESULTS FOR KINESINS
force-velocity curves
randomness

20. Mechanochemical Coupling in Kinesins
How many molecules of ATP are consumed per kinesin step?
Is ATP hydrolysis coupled to forward and/or backward steps?
Nature Cell Biology, 4, (2002)

21. Mechanochemical Coupling
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, (1997)
Hua et al., Nature, 388, (1997)
Coy et al., J. Biol. Chem., 274, (1999)
Problem: back steps ignored in the analysis
Nishiyama et al., Nature Cell Biology, 4, (2002)
Backward steps are taken into account

22. Mechanochemical Coupling
Investigation of kinesin motor proteins motion using optical trapping nanometry system
Nishiyama et al., Nature Cell Biology, 4, (2002)

23. Mechanochemical Coupling
Fraction of 8-nm forward 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, (2002)

24. Mechanochemical Coupling
Dwell times between the adjacent stepwise movements
Dwell times of the backward steps+detachments are the same as for the forward 8-nm steps
Both forward and backward movements of kinesin molecules are coupled to ATP hydrolysis
Nishiyama et al., Nature Cell Biology, 4, (2002)

25. Mechanochemical Coupling
Branched kinetic pathway model with asymmetric potential 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, (2002)

26. Mechanochemical Coupling
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, (2002)

27. Our Approach
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

28. Our Approach
N,j – the probability that N is reached before –N, starting from the site j
Boundary conditions:
N.G. van Kampen, Stochastic Processes in Physics and Chemistry, Elseiver, 1992

29. Our Approach
-splitting probability to go to site N, starting from the site 0,
fraction of forward steps
fraction of backward steps

30. Our Approach
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

31. Our Approach Important observation:
Dwell times for the forward and backward steps are the same, probabilities are different
Drift velocity

32. Our Approach With irreversible detachments j
-probability to dissociate before reaching N or -N, starting from j
- fractions of steps forward, backward and detachments

33. Our Approach With irreversible detachments j Define new parameters:
j – the solution of matrix equation
-vector
matrix elements

34. Our Approach With irreversible detachments j Model with detachments
Model without detachments
N=1 case:

35. Our Approach 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

36. Comparison with Experiments
Fractions of forward and backward steps, and detachments
[ATP]=1mM
[ATP]=10M

37. Comparison with Experiments
Dwell times before forward and backward steps, and before the detachments at different ATP concentrations

38. APPLICATION FOR MYOSIN-V
N=2 model
mean forward-step first-passage time
Kolomeisky and Fisher, Biophys. J., 84, 1642 (2003)

39. CONCLUSIONS
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