GREENS-FUNCTIONS AND FIRST PASSAGE TIME DISTRIBUTIONS FOR DYNAMIC INSTABILITY OF MICROTUBULES

It is shown that the dynamic instability process describing the self-a ssembly and/or disassembly of microtubules is a continuous version of a variant of persistent random walks described by the generalized tele grapher's equation. That is to say, a microtubule is likely to undergo stochastic traveling waves in which catastrophe and rescue events can not propagate faster than upsilon(-) and upsilon(+), respectively. For this stochastic process, analytic expressions for Green's functions o f position and velocity of a microtubule and exact solutions for the f irst passage time distributions of a microtubule to the nucleating sit e are obtained. It is shown that, in the omega-->infinity limit, where omega(-1) is the persistence time, the dynamic instability process ca n 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 r esult, the catastrophe time distribution (i.e., the distribution of mi crotubule 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 te rm are comparable.