We studied the fluctuation in the translational sliding movement of microtubules driven by kinesin in a motility assay in vitro. By calculating the mean-square displacement deviation from the average as a function of time, we obtained motional diffusion coefficients for microtubules and analyzed the dependence of the coefficients on microtubule length. Our analyses suggest that 1) the motional diffusion coefficient consists of the sum of two terms, one that is proportional to the inverse of the microtubule length (as the longitudinal diffusion coefficient of a filament in Brownian movement is) and another that is independent of the length, and 2) the length-dependent term decreases with increasing kinesin concentration. This latter term almost vanishes within the length range we studied at high kinesin concentrations. From the length-dependence relationship, we evaluated the friction coefficient for sliding microtubules. This value is much larger than the solvent friction and thus consistent with protein friction. The length independence of the motional diffusion coefficient observed at sufficiently high kinesin concentrations indicates the presence of correlation in the sliding movement fluctuation. This places significant constraint on the possible mechanisms of the sliding movement generation by kinesin motors in vitro.

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.

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A key step in the processive motion of two-headed kinesin along a microtubule is the 'docking' of the neck linker that joins each kinesin head to the motor's dimerized coiled-coil neck. This process is similar to the folding of a protein β-hairpin, which starts in a highly mobile unfolded state that has significant entropic elasticity and finishes in a more rigid folded state. We therefore suggest that neck-linker docking is mechanically equivalent to the thermally activated shortening of a spring that has been stretched by an applied load. This critical tension-dependent step utilizes Brownian motion and it immediately follows the binding of ATP, the hydrolysis of which provides the free energy that drives the kinesin cycle. A simple three-state model incorporating neck-linker docking can account quantitatively for both the kinesin force-velocity relation and the unusual tension-dependence of its Michaelis constant. However, we find that the observed randomness of the kinesin motor requires a more detailed four-state model. Monte Carlo simulations of single-molecule stepping with this model illustrate the possibility of sub-8 nm steps, the size of which is predicted to vary linearly with the applied load.

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