Transcript protws4 4739

Anatoly B. Kolomeisky
UNDERSTANDING MECHANOCHEMICAL
COUPLING IN KINESINS USING FIRSTPASSAGE 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:
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2)
3)
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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).
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
Motor Proteins:
kinesin
F0F1-ATPase
myosin-II
RNA-polymeraze
There are many types: linear, rotational, processive, non-processive
Motor Proteins
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% (!!!)
Motor Proteins
Main Problems:
What mechanism of motility? How many mechanisms?
THEORETICAL MODELING
1) Thermal Ratchet Models
periodic spatially
asymmetric potentials
2) Multi-State Chemical Kinetic (Stochastic) Models
sequence of discrete
biochemical states
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:
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:
Disadvantages:
1) Exact results
1) Discreteness
2) Agreement with biochemical
observations
2) Mathematical complexity
3) Flexibility in description of
complex biochemical systems
4) Agreement with experiments
3) Large number of parameters
Single-Molecules Experiments
Optical Trap Experiment:
laser
bead
microtubule
kinesin
Optical trap works like an electronic spring
EXPERIMENTS ON KINESIN
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 velocity V(F,[ATP])
stall force FS
dispersion D(F,[ATP])
mean run length L(F,[ATP])
Theoretical Problems:
•
•
a)
b)
c)
d)
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
…
OUR THEORETICAL APPROACH
j=0,1,2,…,N-1 – intermediate biochemical states
kinesin/
microtubule
N=4 model
kinesin/
kinesin/
kinesin/
microtubule/
ATP
microtubule/
ADP/Pi
microtubule/
ADP
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
d
x(t ) ,
drift velocity V  V ({u j , w j })  lim
t  dt
1
d
dispersion
D  D({u j , w j })  lim
x 2 (t )  x(t )
2 t  dt
x(t) – spatial displacement along the motor track

randomness
stall force
r 
2D
dV
N 1 u (0)
k BT
j
FS 
ln 
d
j  0 w j (0)
bound!
2

r >1/N
V ( F  FS )  0
OUR THEORETICAL APPROACH
Effect of an external load F:
u j  u j ( F )  u j (0)e
 and 

j

j
 j Fd / kBT
, w j  w j ( F )  w j (0)e
N 1

(

load distribution factors 
j 
j 0
activation
barrier Ea
F=0
j
uj  e

j
) 1
F >0
 j Fd
j+1
 j Fd / kBT
j
 Ea / k BT
j+1
 j1Fd
RESULTS FOR KINESINS
stall force depends on [ATP]
k BT N 1 u j (0)
FS 
ln 
d
j  0 w j (0)
Michaelis-Menten plots
N=2 model
F=3.59 pN
F=1.05 pN
(u0u1  w0 w1 )
V d
(u0  u1  w0  w1 )
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
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
• The hydrolysis of ATP molecule is coupled to either
the forward or the backward movement (!!!!!!!!!!)
Nishiyama et al., Nature Cell Biology, 4, 790-797 (2002)
Backward steps are taken into account
Mechanochemical Coupling
Investigation of kinesin motor
proteins motion using optical
trapping nanometry system
Nishiyama et al., Nature Cell Biology, 4, 790-797 (2002)
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
1
1
1
 


k1 k2 k3b ( F )  k3 f ( F )
Conclusion: kinesin hydrolyses ATP at any
forward or backward step
Nishiyama et al., Nature Cell Biology, 4,
790-797 (2002)
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
 N,j
uj
wj

 N , j 1 
 N , j 1
uj  wj
uj  wj
Boundary conditions:  N , N  1,  N ,  N  0
N.G. van Kampen, Stochastic Processes in Physics and Chemistry,
Elseiver, 1992
Our Approach
 N ,0 -splitting probability to go to site N, starting from the site 0,
  N ,0
fraction of forward steps
 1   N ,0 fraction of backward steps
 N ,0 
1
N 1
1 
j 0
wj
uj
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
 N ,0
  N ,0
TN ,0 
, T N ,0 
with
ueff
weff
ueff
 N ,0


weff
  N ,0
N 1
uj

j 0 w j
ueff 
1
N 1
r
j 0
j
N 1 j  k
1
w
, rj 
(1    i )
uj
k 1 i  j 1 ui
Our Approach
TN ,0 
 N ,0
ueff
, T N , 0 
  N ,0
Important observation:
weff
Dwell times for the
forward and backward
steps are the same,
probabilities are different
TN ,0  T N ,0 , but  N ,0    N ,0
Drift velocity
V  d (ueff  weff )
Our Approach
With irreversible detachments
j
  , j -probability to dissociate before reaching N or -N, starting from j
 N , j   N , j    , j  1
- fractions of steps forward, backward and
detachments
 N,j
uj
wj

 N , j 1 
 N , j 1
uj  wj  j
uj  wj  j
Our Approach
j
With irreversible detachments

Define new parameters:
*
N,j
 N,j

,
j
u*j  u j j 1 ,
*
N,j
T

TN , j
j
,
w *j  u j j 1
 *M  0
  1,  N 1 ,..., j ,..., N 1 ,1 -vector
j – the solution of matrix equation
matrix elements
 (u j  w j   j ), for i  j;

M ij  
wi 1 , for j  i  1;

ui 1 , for j  i  1;

Our Approach
With irreversible detachments
Model with detachments
u j , w j ,  N , j , TN , j 
N=1 case:
j
Model without detachments
u*j , w*j , N* , j , TN* , j 
u
w

 1,0 
,  1,0 
,   ,0 
,
u  w 
u  w 
u  w 
1
T1,0  T1,0  T ,0 
u  w 
Our Approach
With irreversible detachments
Description of experimental data using
N=2 model; reasonable for kinesins
Fisher and Kolomeisky, PNAS USA,
98, 7748 (2001).
j
u j ( F )  u j (0) exp( 
w j ( F )  u j (0) exp(
 j Fd
k BT
 j Fd
k BT
)
Load dependence of rates
)
Comparison with Experiments
Fractions of forward and backward steps, and detachments
[ATP]=10mM
[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
(u0  u1  w0  w1 )
 
(u0u1  w0 w1 )
Kolomeisky and Fisher, Biophys. J., 84, 1642 (2003)
CONCLUSIONS
• Analysis of motor protein motility using firstpassage 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