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

Processive motor: portor Non-processive motor: rower

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

M + T MT MT M + D + P AM AMT T DP M MT T DP k1 k2 k3 k4 Biochemical Cycle and Chemical-Mechano Cycle AM AMT T DP M MT T DP

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.

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:

http://www.mpimf-heidelberg.mpg.de/~holmes/muscle/muscle1.html AM MT AMDP AMD AM

Structure and function of kinesin motor Kinesin has been found: Cell dividing Heavy chain, 124KD Light chain, 64KD 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

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

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’

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

1 2 3 m = -1 Microtubule lattice FOT

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

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

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?

(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

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

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:

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

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

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):

Km Ke Kt (Kt 0.037 pN/nm)

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.

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

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

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

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

A+B- A+ + B- A+B- A+ A+ A+B- B-

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

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.

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