  We investigate a simple model of microtubule dynamics in which a microtubule
evolves by: (i) attachment of guanosine triphosphate (GTP) to its end at rate
lambda, (ii) GTP converting irreversibly to guanosine diphosphate (GDP) at rate
1, and (iii) detachment of GDP from the end of a microtubule at rate mu. As a
function of these elemental rates, the microtubule can grow steadily or its
length can fluctuate wildly. A master equation approach is developed to
characterize these intriguing features. For mu=0, we find exact expressions for
tubule and GTP cap length distributions, as well as a power-law length
distributions of GTP and GDP islands. For mu=oo, we find the average time
between catastrophes, where the microtubule shrinks to zero length, and extend
this approach to also determine the size distribution of avalanches (sequence
of consecutive GDP detachment events). We obtain the phase diagram for general
rates and verify our predictions by numerical simulations.
