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

Abstract

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 ${v}_{\ensuremath{-}}$ and ${v}_{+},$ 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 $\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, where ${\ensuremath{\omega}}^{\ensuremath{-}1}$ 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-state 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 $F(t|{x}_{0})\ensuremath{\sim}{t}^{\ensuremath{-}3/2}{e}^{\ensuremath{-}t/{\ensuremath{\tau}}_{c}}$, where ${\ensuremath{\tau}}_{c}$ is the time scale at which the drift term and the diffusive term are comparable.