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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 nature of the mechanism limiting the velocity of ATP-induced unidirectional movements of actin-myosin filaments in vitro is considered. In the sliding process two types of "cyclic" interactions between myosin heads and actin are involved, i.e., productive and nonproductive. In the productive interaction, myosin heads split ATP and generate a force which produces sliding between actin and myosin. In the nonproductive interaction "cycle," on the other hand, myosin heads rapidly attach to and detach from actin "reversibly," i.e., without splitting ATP or generating an active force. Such a nonproductive interaction "cycle" causes irreversible dissipation of sliding energy into heat, because the myosin cross-bridges during this interaction are passive elastic structures. This consideration has led us to postulate that such cross-bridges, in effect, exert viscous-like frictional drag on moving elements. Energetic considerations suggest that this frictional drag is much greater than the hydrodynamic viscous drag. We present a model in which the sliding velocity is limited by the balance between the force generated by myosin cross-bridges in the productive interaction and the frictional drag exerted by other myosin cross-bridges in the nonproductive interaction. The model is consistent with experimental findings of in vitro sliding, including the dependence of velocity on ATP concentration, as well as the sliding velocity of co-polymers of skeletal muscle myosin and phosphorylated and unphosphorylated smooth muscle myosins.

Spontaneous rotation and flapping oscillations of an actin filament driven by myosins have been observed in the in vitro setup of a quasi-two-dimensional motility experiment, where the head of the filament is spatially fixed. We present a simple phenomenological dynamical model that exhibits both the rotation and the oscillation of the filament as symmetry breaking instabilities of the filament conformation under pertinent boundary conditions. PACS numbers: 87.45.k, 02.30.Jr, 87.22.JbFrom the viewpoint of nonequilibrium dynamics, there are many biological systems of interest which fall into the category of systems of coupled active elements For. systems of this type, the origin of nonequilibrium motion is built into each of these elements, but the coupled nature of the system often gives rise to new dynamical behavior which is not inherent in the individual elements.In this Letter we consider as a representative of such systems a system consisting of a filament (either actin or microtubule) and many motor proteins (myosin, kinesin, or dynein) in the presence of an ATP solution. Many experiments have been done on this system using an in vitro setup [1], where the protein motors are firmly attached on the surface of a glass plate. These motors cause a directed sliding motion of the filaments with which they come into contact. Although the actual active elements are the protein motors, it suffices for our purpose that we regard a filament as a continuous train of (effective) active elements, each of which tends to move along the tangential direction of the filament.We shall focus on one phenomenon which is observed in the in vitro setup, but does not usually occur in actual in vivo biological systems.Here, the head of the filament is somehow pinned at a point on the glass plate, and the filament exhibits either rotation or Aapping oscillation [ Figs. 1(a) and 1(b)]. Our qualitative picture of this phenomenon is the following. (i) The driving force exerted by the active elements is accumulated along the filament, inducing a buckling instability of the otherwise straight filament [2]. (ii) The filament is thus bent, and a part of the motile force acts to displace the filament around the head. (iii) Different types of motion of the filament stem from different types of boundary conditions. Experimentally [3], a rotational motion is observed when the head of the filament is pinned down on the substrate and is free to rotate. We shall call the head in this case a torque fvee head-. On the other hand, a Ilapping motion is seen when the filament is pinned a short distance from the head, at the "neck. " The portion ahead of the pinned site is stretched, maintaining its orientation [4]. Hereafter, for simplicity we shall call this pinned neck the oriented head and ignore the portion ahead of the pinned point. FIG. 1. Consecutive video images of the in vitro motility experiment [3]: The pinned actin filaments (thick dark curve) undergo either rotation [series (a)] or Ilapping oscillation [series (b)]. The time intervals betwe...

The stiffness of glycerinated rabbit psoas fibers in the rigor state was measured at various sarcomere lengths in order to determine the distribution of the sarcomere compliance between the cross-bridge and other structures. The stiffness was determined by measuring the tension increment at one end of a fiber segment while stretching the other end of the fiber. The contribution of the end compliance to the rigor segments was checked both by laser diffractometry of the sarcomere length change and by measuring the length dependence of the Young's modulus; the contribution was found to be small. The stiffness in the rigor state was constant at sarcomere lengths of 2.4 microns or less; at greater sarcomere lengths the stiffness, when corrected for the contribution of resting stiffness, scaled with the amount of overlap between the thick and thin filaments. These results suggest that the source of the sarcomere compliance of the rigor fiber at the full overlapping of filaments is mostly the cross-bridge compliance.