Random hydrolysis controls the dynamic instability of microtubules

Abstract

Uncovering mechanisms that control the dynamics of microtubules is fundamental for our understanding of multiple cellular processes such as chromosome separation and cell motility. Building on previous theoretical work on the dynamic instability of microtubules, we propose here a stochastic model that includes all relevant biochemical processes that affect the dynamics of microtubule plus-end, namely, the binding of GTP-bound monomers, unbinding of GTP- and GDP-bound monomers, and hydrolysis of GTP monomers. The inclusion of dissociation processes, present in our approach but absent from many previous studies, is essential to guarantee the thermodynamic consistency of the model. Our theoretical method allows us to compute all dynamic properties of microtubules explicitly. Using experimentally determined rates, it is found that the cap size is ∼3.6 layers, an estimate that is compatible with several experimental observations. In the end, our model provides a comprehensive description of the dynamic instability of microtubules that includes not only the statistics of catastrophes but also the statistics of rescues.

Figure 1

(a) Representation of the various elementary transitions considered in the model with their corresponding rates, U the on-rate of GTP-subunits, W_T the off-rate of GTP-subunits, W_D the off-rate of GDP-subunits, and r the hydrolysis rate for each unhydrolyzed unit within the filament. (b) Pattern for a catastrophe with N terminal units in the GDP state.

Figure 2

Average cap size in number of subunits as function of the free tubulin concentration c in μM. The line is the mean-field analytical solution and (solid squares) are simulation points.

Figure 3

(a) Catastrophe time T_c versus growth velocity for the N = 2 case, with the theoretical prediction (solid line) together with simulation points (solid squares) and experimental data points taken from Janson et al. (21) for constrained growth and free growth (solid circles). (Error bars) Simulation points have been obtained using the block-averaging method (41). (b) Catastrophe frequency (f_cat = 1/T_c) versus velocity: our mean-field theory result (continuous curve) is compared with experiments of Drechsel et al. (34) (solid squares with error bars) and theory of Flyvbjerg et al. (9) (dotted curve).

Figure 4

(Left) Distribution of catastrophe time (N = 1) for different concentration values. C = 9 μM (solid squares) and C = 12 μM (open circles). (Right) The distribution of catastrophe time (N = 2) for different concentration values C = 9 μM (solid squares) and C = 12 μM (open circles). The distributions are normalized such that area under the curve is 1.

Figure 5

Distributions of the first passage time of the cap for an initial cap of k = 2 units, F₂(t) as function of the time t, for various initial concentration of free monomers. (Solid lines) Theoretical predictions deduced from Eq. 16 of the Appendix after numerically inverting the Laplace transform. (Symbols) Simulations. (Circles) Dilution into a medium with no free monomers; (squares) dilution into a medium with a concentration of free monomers of 2 μM; (diamonds) 5 μM; and (triangles) 9 μM.

Figure 6

Mean first-passage time 〈T(k)〉 as function of k for three values of the monomer concentration from bottom to top 0, 2, and 4 μM. The presence of steps in these curves is due to the fact that 〈T(k)〉 is only defined on integer values of k. Note the existence of a plateau for all values of the monomer concentration.

Figure 7

Delay time before catastrophe (s) as function of free tubulin concentration (in μM) before dilution in the case that the postdilution tubulin concentration is zero. (Solid line) Mean-field prediction based on Eq. 9; (symbols) simulation points. As found experimentally, the delay time is essentially independent of the concentration of tubulin in the predilution state, and the time to observe the first catastrophe is of the order of seconds or less.

Figure 8

Rescue time (N = 1) as function of concentration. (Solid line) Obtained from the mean-field theory. Data points (solid squares) are obtained from the simulations.