Microtubule catastrophe under force: mathematical and computational results from a Brownian ratchet model.

In the intracellular environment, the intrinsic dynamics of microtubule filaments is often hindered by the presence of barriers of various kind, such as kinetochore complexes and cell cortex, which impact their polymerisation force and dynamical properties such as catastrophe frequency. We present a theoretical study of the effect of a forced barrier, also subjected to thermal noise, on the statistics of catastrophe events in a single microtubule as well as a 'bundle' of two parallel microtubules. For microtubule dynamics, which includes growth, detachment, hydrolysis and the consequent dynamic instability, we employ a one-dimensional discrete stochastic model. The dynamics of the barrier is captured by over-damped Langevin equation, while its interaction with a growing filament is assumed to be hard-core repulsion. A unified treatment of the continuum dynamics of the barrier and the discrete dynamics of the filament is realized using a hybrid Fokker-Planck equation. An explicit mathematical formula for the force-dependent catastrophe frequency of a single microtubule is obtained by solving the above equation, under some assumptions. The prediction agrees well with results of numerical simulations in the appropriate parameter regime. More general situations are studied via numerical simulations. To investigate the extent of 'load-sharing' in a microtubule bundle, and its impact on the frequency of catastrophes, the dynamics of a two-filament bundle is also studied. Here, two parallel, non-interacting microtubules interact with a common, forced barrier. The equations for the two-filament model, when solved using a mean-field assumption, predicts equal sharing of load between the filaments. However, numerical results indicate the existence of a wide spectrum of load-sharing behaviour, which is characterized using a dimensionless parameter.

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

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 [Formula: see text] kinesins under stall conditions is a slowly decreasing function of [Formula: see text], while for dyneins, the time increases almost linearly with [Formula: see text]. 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 [Formula: see text] 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 [Formula: see text], which suggests non-cooperative behaviour. In the large [Formula: see text] limit, the emergence of cooperative behaviour is shown to be similar to a continuous phase transition.