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

PUBLICATIONS:1) J. Stat. Phys., 93, 633 (1998).

2) PNAS USA, 96, 6597 (1999).

3) Physica A, 274, 241 (1999).

3) Physica A, 279, 1 (2000).

4) J. Chem. Phys., 113, 10867 (2000).

5) PNAS USA, 98, 7748 (2001).

6) J. Chem. Phys., 115, 7253 (2001).

7) PNAS USA, 98, 7748 (2001).

8) Biophys. J., 84, 1642 (2003).

Motor ProteinsEnzymes 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

Motor Proteins:

kinesin myosin-II

RNA-polymerazeF0F1-ATPase

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

Motor Proteins

Properties:

Non-equilibrium systems

Velocities: 0.01-100 m/s

Step Sizes: 0.3-40 nm

Forces: 1-60 pN

Fuel: hydrolysis of ATP, or related compound, polymerization

Efficiency: 50-100% (!!!)

Motor ProteinsMain Problems: What mechanism of motility? How many mechanisms?

THEORETICAL MODELING1) Thermal Ratchet Models

periodic spatially asymmetric potentials

2) Multi-State Chemical Kinetic (Stochastic) Models

sequence of discrete biochemical states

Ratchet ModelsIdea: 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:

1) mainly numerical or simulations results;

2) results depend on potentials used in calculations;

3) hard to make quantitative comparisons with experiments;

4) not flexible in description of complex biochemical networks;

Multi-State Chemical Kinetic (Stochastic) Models

Assumption: the motor protein molecule steps through a sequence of discrete biochemical states

Multi-State Chemical Kinetic (Stochastic) Models

Advantages:

1) Exact results

2) Agreement with biochemical observations

3) Flexibility in description of complex biochemical systems

4) Agreement with experiments

Disadvantages:

1) Discreteness

2) Mathematical complexity

3) Large number of parameters

Single-Molecules Experiments

microtubule

bead

kinesin

Optical Trap Experiment:

laser

Optical trap works like an electronic spring

EXPERIMENTS ON KINESINoptical force clamp with a feedback-driven optical trap

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

precise observations:

mean velocity V(F,[ATP])

SFstall force

dispersion D(F,[ATP])

mean run length L(F,[ATP])

step-size d=8.2 nm

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

a) coupling between ATP hydrolysis and the protein motion

b) stepping mechanism – hand-over-hand versus inchworm

c) conformational changes during the motion

d) …

OUR THEORETICAL APPROACH

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

kinesin/

microtubule

kinesin/

microtubule/ATP

kinesin/

microtubule/ADP/Pi

kinesin/

microtubule/ADP

N=4 model

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) 433-450

22 )()(2

1}),({

,)(}),({

lim

lim

txtxdt

dwuDD

txdt

dwuVV

tjj

tjj

dispersion

x(t) – spatial displacement along the motor track

dV

Dr

2

drift velocity

randomness bound! r >1/N

stall force

1

0

B

)0(

)0(ln

N

j j

jS w

u

d

TkF 0)( SFFV

OUR THEORETICAL APPROACHEffect of an external load F:

TkFdjjj

TkFdjjj

jj ewFwweuFuu BB // )0()(,)0()(

jj and load distribution factors 1)(

1

0

N

jjj

activation barrier aE

Fdj

Fdj1

F=0F >0

j

j+1j

j+1

TkEj

aeu B/

RESULTS FOR KINESINSstall force depends on [ATP]

Michaelis-Menten plots

F=1.05 pNF=3.59 pN

N=2 model

)(

)(

1010

1010

wwuu

wwuudV

1

0

B

)0(

)0(ln

N

j j

jS w

u

d

TkF

RESULTS FOR KINESINS

force-velocity curves randomness

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, 790-797 (2002)

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, 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

Mechanochemical Coupling

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

Investigation of kinesin motor proteins motion using optical trapping nanometry system

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, 790-797 (2002)

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, 790-797 (2002)

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

)()(

111

3321 FkFkkk fb

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

Conclusion: kinesin hydrolyses ATP at any forward or backward step

Mechanochemical Coupling PROBLEMS:

1) Backward biochemical reactions are not taken into account

2) Asymmetric potential violates the periodic symmetry of the system and the principle of microscopic reversibility

3) Detachments are not explained

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

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

Our Approach

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

1,1,,

jN

jj

jjN

jj

jjN wu

w

wu

u

0 ,1 ,, NNNN Boundary conditions:

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

Our Approach

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

fraction of forward steps

0,0, 1 NN fraction of backward steps

1

0

0,

1

1N

j j

jN

u

w

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

eff

NN

eff

NN w

Tu

T 0,0,

0,0, ,

)1(

1 ,

1 1

1 11

0

N

k

kj

ji i

i

jjN

jj

eff u

w

ur

ru

with

1

00,

0,N

j j

j

N

N

eff

eff

w

u

w

u

Our Approach

0,0,0,0, but , NNNN TT

eff

NN

eff

NN w

Tu

T 0,0,

0,0, ,

Important observation:

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

)( effeff wudV Drift velocity

Our Approach

jWith irreversible detachments

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

1,,, jjNjN - fractions of steps forward, backward and detachments

1,1,,

jN

jjj

jjN

jjj

jjN wu

w

wu

u

Our ApproachjWith irreversible detachments

1*

1*

,*,

,*,

,

, ,

jjjjjj

j

jNjN

j

jNjN

uwuu

TT

Define new parameters:

j – the solution of matrix equation 0M* 1,,...,,...,,1 11 NjN -vector

;1for ,

;1for ,

;for ),(

1

1

iju

ijw

jiwu

M

i

i

jjj

ij

matrix elements

Our ApproachjWith irreversible detachments

Model with detachments jNjNjj Twu ,, ,,,

Model without detachments *

,*

,** ,,, jNjNjj Twu

N=1 case:

wuTTT

wuwu

w

wu

u

1

, , ,

0,0,10,1

0,0,10,1

Our ApproachjWith irreversible detachments

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

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

)exp()0()(

)exp()0()(

Tk

FduFw

Tk

FduFu

B

jjj

B

jjj

Load dependence of rates

Comparison with Experiments

Fractions of forward and backward steps, and detachments

[ATP]=10M [ATP]=1mM

Comparison with Experiments

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

APPLICATION FOR MYOSIN-V

N=2 model

mean forward-step first-passage time )(

)(

1010

1010

wwuu

wwuu

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

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