Microtubules Interacting with a Boundary: Mean Length and Mean First-Passage Times

TL;DR

This paper develops a mathematical model describing microtubule dynamics interacting with a boundary, calculating steady-state distributions and mean first-passage times, and applies it to chromosome search-and-capture mechanisms.

Contribution

It introduces a differential equation approach to explicitly solve for microtubule boundary interaction times and revisits a chromosome search model with a more direct solution.

Findings

Derived steady-state length distribution of microtubules.

Explicit solutions for mean first-passage times to boundary.

Applied the model to chromosome search-and-capture, simplifying previous approaches.

Abstract

We formulate a dynamical model for microtubules interacting with a catastrophe-inducing boundary. In this model microtubules are either waiting to be nucleated, actively growing or shrinking, or stalled at the boundary. We first determine the steady-state occupation of these various states and the resultant length distribution. Next, we formulate the problem of the Mean First-Passage Time to reach the boundary in terms of an appropriate set of splitting probabilities and conditional Mean First-Passage Times, and solve explicitly for these quantities using a differential equation approach. As an application, we revisit a recently proposed search-and-capture model for the interaction between microtubules and target chromosomes [Gopalakrishnan & Govindan, Bull. Math. Biol. 73:2483--506 (2011)]. We show how our approach leads to a direct and compact solution of this problem.