  The disappearance of the guanosine triphosphate (GTP)-tubulin cap is widely
believed to be the forerunner event for the growth-shrinkage transition
(`catastrophe') in microtubule filaments in eukaryotic cells. We study a
discrete version of a stochastic model of the GTP cap dynamics, originally
proposed by Flyvbjerg, Holy and Leibler (Flyvbjerg, Holy and Leibler, Phys.
Rev. Lett. 73, 2372, 1994). Our model includes both spontaneous and vectorial
hydrolysis, as well as dissociation of a non-hydrolyzed dimer from the filament
after incorporation. In the first part of the paper, we apply this model to a
single protofilament of a microtubule. A catastrophe transition is defined for
each protofilament, similar to the earlier one-dimensional models, the
frequency of occurrence of which is then calculated under various conditions,
but without explicit assumption of steady state conditions. Using a
perturbative approach, we show that the leading asymptotic behavior of the
protofilament catastrophe in the limit of large growth velocities is remarkably
similar across different models. In the second part of the paper, we extend our
analysis to the entire filament by making a conjecture that a minimum number of
such transitions are required to occur for the onset of microtubule
catastrophe. The frequency of microtubule catastrophe is then determined using
numerical simulations, and compared with analytical/semi-analytical estimates
made under steady state/quasi-steady state assumptions respectively for the
protofilament dynamics. A few relevant experimental results are analyzed in
detail, and compared with predictions from the model. Our results indicate that
loss of GTP cap in 2-3 protofilaments is necessary to trigger catastrophe in a
microtubule.
