Green's functions and first passage time distributions for dynamic instability of microtubules

It is shown that the dynamic instability process describing the self-assembly and/or disassembly of microtubules is a continuous version of a variant of persistent random walks described by the generalized telegrapher's equation. That is to say, a microtubule is likely to undergo stochastic traveling waves in which catastrophe and rescue events cannot propagate faster than upsilon(-) and upsilon(+), respectively. For this stochastic process, analytic expressions for Green's functions of position and velocity of a microtubule and exact solutions for the first passage time distributions of a microtubule to the nucleating site are obtained. It is shown that, in the omega-->infinity limit, where omega(-1) is the persistence time, the dynamic instability process can be described by a diffusion process in the presence of a drift term that, in fact, is the steady-stare velocity of the microtubule. As a result, the catastrophe time distribution (i.e., the distribution of microtubule lifetimes to the nucleating site) exhibits a power law with an exponential cutoff as F(t\x(0))similar to t(-3/2)e(-t/tau c), where tau(c) is the time scale at which the drift term and the diffusive term are comparable.