1. IPAM workshop IV: Molecular Machines. May 23-28, 2004
Chemical-Mechano Free-Energy Transduction: From Muscles to Kinesin Motors and Brownian Ratchets
Yi-der Chen, Laboratory of Biological Modeling, NIDDK, NIH
IPAM workshop IV: Molecular Machines. May 23-28, 2004

2. Processive motor: portor
Non-processive motor: rower

3. Chemical-Mechano Cycle (CMC): It describes how the biochemical and mechanical cycles of the motor are coupled at the molecular level
X-ray structure
Biochemistry
Cryo-electron microscopy
Molecular dynamics simulation
CMC of single motors
System mechanical properties
Formalism

4. M + T MT MT M + D + P AM AMT T DP M MT T DPk1 k2 k3 k4
Biochemical Cycle and Chemical-Mechano Cycle AM
AMT T DP M MT T DP

6. A. F. Huxley 1957 f g x F U A f g
Basic elements of a Chemical-Mechano Cycle:
Contains both attached and detached states;
Contains at least one asymmetric rate constants;
Contains at least one force-generating step.

7. Thermodynamic Consistency: Each chemical-mechano state is an equilibrium state and therefore has a unique Gibbs free energy. Thus the following two equations must be obeyed when assigning the rate constants to the cycle:

8. AM MT
AMDP
AMD AM

9. Structure and function of kinesin motor
Kinesin has been found:
Cell dividing
Heavy chain, 124KD Light chain, KD
Structure and function of kinesin motor
Microtubule
gliding
Kinesin pulls a cargo
MT binding site
In all types of cells At all stages of cell development
3kin

10. Vale and Milligan, Sci. 288, 88 (2000)

11. Linker-zipping Model
Lever-swinging Model F D T
D+Pi a
1-a F D T + Pi a
1-a
*Nucleotide-free: Head attached in 90o
*D state: Detached
*T and D+P states: Head attached in 45o
*Power stroke: 2 to 3 and 4 to 1’
*Nucleotide-free: Head attached; linker un-zipped
*D state: Head detached; Linker un-zipped.
*T and D+P states: Head attached; Linker zipped
*Power stroke: 2 to 3 and 4 to 1’. 1 2 3
Tight coupled model: One step forward for each ATP hydrolyzed 4 1’

12. Visscher et al., Nature 400:184-9 (1999)
Single Kinesin Motility Assay
Visscher et al., Nature 400:184-9 (1999)

13. 1 2 3
m = -1
Microtubule
lattice
FOT

14. Free kinesin in the absence of the bead:
The velocity of the motor is directly proportional to the flux of the cycle.
Kolomeisky, Fisher

15. Kinesin moving against a constant force: when the feedback of the optical trap in the motility assay of Visscher et al is infinitely fast.
Equations derived for the free-motor case are still applicable to this case except that the rate constants have to include the effect of force. F F
r12
r21 a
Kolomeisky, Fisher, Qian r = r e ( 1 - d ) Fa , 21 21
Chemical Kinetic Formalism

16. Kinesin moving against a randomly fluctuating force: Assay of Visscher et al
Question:
How does the Brownian motion of the bead affect the velocity of the motor? Slow it down or speed it up?

17. (A). Focus on the movement of the bead:
A Brownian motion problem with stochastic pulling
of the motor.
Fokker-Planck equation
(B). Monte Carlo simulation on the movement of the
motor and the Brownian motion of the bead
simultaneously

18. state 1 state 2 (A) (B) x coordinate m m-1 a ( b step ) ( a step ) -2+ Pi
(A)
(B) -2 -3 -1 1 2 3
m = -1
Microtubule
lattice -a
1-a
x2(1)
x1(0)
x2(0)
x1(-1)
x coordinate

19. Strain energy and rate constants
The strain energy obeys Hooke’s law:
Where x is the position of bead, and xi(m) is the position of the bead when the motor is attached to the lattice site m in state i and the spring is relaxed. Note: x1(m) = m and x2(m) = m - a.
x-dependent rate constants:

20. Diffusion-reaction equations of the bead movement
(3)
(4)
Sum Eqs. (3) and (4), we have
(6)
Diffusion-reaction equations of the bead movement
(5)
At steady-state, Thus, u = constant = mean velocity

21. into Eqs. (3)-(5), we get the final ODE at m=0: Substitute
etc.
into Eqs. (3)-(5), we get the final ODE at m=0:
Substitute

22. Chemical Kinetic formalism (CK):
L=8 nm
da=0.5
db =0.5
a=0.5
K=16 (1.03 pN/nm)
Our formalism CK
Chemical Kinetic formalism (CK):

23. Km Ke Kt
(Kt 0.037 pN/nm)

24. Chemical-Kinetic formalism (in the absence of the bead Brownian motion) always gives larger velocities: bead Brownian motion always reduces the velocity of the motor.
However, this effect becomes smaller as the elastic coefficient is reduced.

25. Model and Monte Carlo strategy
state 1
state 2 a
( b step )
( a step )
(A)
(B) MT x K
Strain energy of spring at z:
z-dependent rate constants:
(1) To evaluate the dwell time T:
(2) To determine the move direction:
Forward if
Backward otherwise.
Model and Monte Carlo strategy

29. A single one-headed kinesin can also execute directed movements on a microtubule, if it carries a cargo 1 2 3

30. Flashing Ratchets
Astumian and Bier. PRL 72, 1766 (1994) +

31. Flashing Brownian Particle+
It will not work, because it is against Thermodynamics.

32. A+B-
A+ + B-
A+B- A+ A+
A+B- B-

34. Zhou and Chen Phys. Rev. Lett. 77, 194 (1996).

35. SUMMARY
Chemical-Mechano Cycle is important in studying molecular motors. Future biochemical and structural experiments should focus on elucidating the “strain-dependent” rate constants of the cycle (by applying some sorts of force to the motor).
When studying the processive movement of a motor carrying a cargo (in vivo or in vitro), it is important to consider the effect of Brownian motion of the cargo.
An enzyme catalyzing a non-equilibrium chemical reaction can execute directed movement on an asymmetric static potential. The direction of enzyme movement depends not only on the asymmetry of the potential, but also on the direction of the catalytic cycle. The catalytic cycle of the enzyme can be identified as a working Chemical-Mechano Cycle.

36. Bo Yan, University of Georgia
Hwan-Xiang Zhou, University Florida
Robert Rubin, LBM, NIH
Terrell L. Hill