  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.
