Molecular motors: thermodynamics and the random walk

Abstract

The biochemical cycle of a molecular motor provides the essential link between its thermodynamics and kinetics. The thermodynamics of the cycle determine the motor's ability to perform mechanical work, whilst the kinetics of the cycle govern its stochastic behaviour. We concentrate here on tightly coupled, processive molecular motors, such as kinesin and myosin V, which hydrolyse one molecule of ATP per forward step. Thermodynamics require that, when such a motor pulls against a constant load f, the ratio of the forward and backward products of the rate constants for its cycle is exp [-(DeltaG + u(0)f)/kT], where -DeltaG is the free energy available from ATP hydrolysis and u(0) is the motor's step size. A hypothetical one-state motor can therefore act as a chemically driven ratchet executing a biased random walk. Treating this random walk as a diffusion problem, we calculate the forward velocity v and the diffusion coefficient D and we find that its randomness parameter r is determined solely by thermodynamics. However, real molecular motors pass through several states at each attachment site. They satisfy a modified diffusion equation that follows directly from the rate equations for the biochemical cycle and their effective diffusion coefficient is reduced to D-v(2)tau, where tau is the time-constant for the motor to reach the steady state. Hence, the randomness of multistate motors is reduced compared with the one-state case and can be used for determining tau. Our analysis therefore demonstrates the intimate relationship between the biochemical cycle, the force-velocity relation and the random motion of molecular motors.

Kinesin: a molecular motor with a spring in its step

“…We may also simplify matters by neglecting dissociation of the kinesin-tubulin complex, since it has been shown experimentally that kinesin dimers can perform a hundred or more steps before they dissociate from a microtubule (Howard et al 1989;Block et al 1990;Svoboda et al 1993;Vale et al 1996). In that case, given that the ATP hydrolysis in figure 1c is tightly coupled to the kinesin stepping (Visscher et al 1999;Thomas et al 2001), the average velocity v = u 0 R. Since the Law of Mass Action requires that k 12 = K 12 [ATP], where K 12 is the molar rate constant for ATP-binding, we then find that the kinesin stepping velocity v obeys the Michaelis-Menten equation…”

“…This implies that the kinesin motor's internal time-constant must be somewhat longer than that predicted by the three-state model. Adding extra states to the model of a molecular motor tends to increase (Visscher et al 1999) and the randomness calculated for the three-state (dashed line) and four-state (solid line) kinesin models according to the theory of Thomas et al (2001). its internal time-constant; for instance, a hypothetical onestate motor has = 0, whilst a two-state motor has nonzero , which leads to a lower randomness (Thomas et al 2001). Hence, although the three-state model can account very well for both the force-velocity relation and the Michaelis-Menten behaviour of kinesin, four or more states may be required in order to account for the low randomness observed by Visscher et al (1999).…”

“…The forward motion of a tightly coupled processive molecular motor such as kinesin is inherently stochastic and constitutes a biased random walk (Berg 1993;Astumian 1997;Thomas et al 2001). Visscher et al (1999) determined the experimental randomness parameter r for kinesin, which may be defined as (Svoboda et al 1994) …”

“…We present an alternative physical analysis in which neck-linker docking is regarded as the thermally activated shortening of a spring. We show that the integration of this step into the kinesin cycle (Hackney 1994;Duke & Leibler 1996;Hancock & Howard 1999;Astumian & Derenyi 1999;Schnitzer et al 2000) produces a scheme that can account quantitatively for the kinesin force-velocity relation, the unusual tension-dependence of its Michaelis constant and the observed randomness of the kinesin motor (Visscher et al 1999;Thomas et al 2001). Furthermore, we discuss how neck-linker docking may lead to the occurrence of sub-8 nm steps (Coppin et al 1996;Nishiyama et al 2001).…”

“…Hence, we expect the rate constant for this reverse transition to be of the form factor k 43 in this case is proportional to exp[Ϫ (u 0 Ϫ d ) 2 /2kT] because elastic energy d 2 / 2 is required to compress the neck-linker spring of head A by d before this head can attach to site n in state 3.) For thermodynamic consistency, the rate constants for a three-state model of a tightly coupled processive molecular motor such as kinesin must satisfy the relation (Thomas et al 2001) …”