Temporal cooperativity of motor proteins under constant force: insights from Kramers’ escape problem

Abstract

The microtubule-bound motors kinesin and dynein differ in many respects, a striking difference being that while kinesin is known to function mostly alone, dynein operates in large groups, much like myosinV in actin. Optical tweezer experiments in vitro have shown that the mean detachment time of a bead attached to kinesins under stall conditions is a slowly decreasing function of , while for dyneins, the time increases almost linearly with . This makes dynein a team worker, capable of producing and sustaining a large collective force without detaching. We characterize this phenomenon as ‘temporal cooperativity’ under load. In general, it is unclear which biophysical properties of a single motor determine whether it behaves cooperatively or not in a group. Our theoretical analysis shows that this is determined by two dimensionless parameters: (i) the ratio of single molecule, load-independent detachment and attachment rates and (ii) the ratio of the applied force per motor to the detachment force of a single motor. We show that the attachment-detachment dynamics of a motor assembly may be mapped to the motion of a hypothetical, overdamped Brownian particle in an effective potential, the form of which depends on the load-dependence of binding and unbinding rate of the motor. In this picture, the total number of motors is proportional to the inverse temperature and cooperative behaviour arises from the trapping of the particle in the minima of the potential, when present. In the latter case, application of results from Kramers’ theory predicts that the mean time of escape of the particle, equivalent to the mean detachment time of the bead under stall, increases exponentially with the number of motors, indicating cooperative behaviour. If the potential does not have minima, the detachment time depends only weakly on , which suggests non-cooperative behaviour. In the large limit, the emergence of cooperative behaviour is shown to be similar to a continuous phase transition.