We study the steady state of an assembly of microtubules in a confined volume, analogous to the situation inside a cell where the cell boundary forms a natural barrier to growth. We show that the dynamical equations for growing and shrinking microtubules predict the existence of two steady states, with either exponentially decaying or exponentially increasing distribution of microtubule lengths. We identify the regimes in parameter space corresponding to these steady states. In the latter case,

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... the apparent catastrophe frequency near the boundary was found to be significantly larger than that in the interior. Both the exponential distribution of lengths and the increase in the catastrophe frequency near the cell margin is in excellent agreement with recent experimental observations.