Microtubule length distributions in the presence of protein-induced severing

Microtubules are highly regulated dynamic elements of the cytoskeleton of eukaryotic cells. One of the regulation mechanisms observed in living cells is the severing by the proteins katanin and spastin. We introduce a model for the dynamics of microtubules in the presence of randomly occurring severing events. Under the biologically motivated assumption that the newly created plus end undergoes a catastrophe, we investigate the steady state length distribution. We show that the presence of severing does not affect the number of microtubules, regardless of the distribution of severing events. In the special case in which the microtubules cannot recover from the depolymerizing state (no rescue events) we derive an analytical expression for the length distribution. In the general case we transform the problem into a single ODE that is solved numerically.

💡 Research Summary

Microtubules are dynamic polymers whose length is regulated by stochastic switching between a growing state and a shrinking state. Classical models describe this behavior using the growth speed (v_g), shrinkage speed (v_s), catastrophe rate (\omega_c) (transition from growth to shrinkage) and rescue rate (\omega_r) (the reverse transition). In living cells, however, severing proteins such as katanin and spastin cut microtubules at random positions, creating two new filament ends. The present paper introduces a mathematically tractable model that explicitly incorporates these severing events.

The authors assume that severing occurs with a constant rate per unit length, (k_{sev}), so that the probability of a cut on a filament of length (L) is proportional to (k_{sev}L). When a cut happens, the newly generated plus end is immediately placed into a catastrophe state, i.e., it begins to depolymerize right away. This biologically motivated assumption reflects experimental observations that severed plus ends are highly unstable.

Starting from a master equation for the length probability density (P(L,t)), the model adds a loss term (-k_{sev}L P) (representing the disappearance of a filament due to cutting) and a gain term (2k_{sev}\int_L^\infty P(L’),dL’) (representing the two daughter filaments produced from any longer filament). The resulting equation couples growth, shrinkage, state switching, and severing in a single framework.

A striking analytical result is that the total number of microtubules, (N=\int_0^\infty P(L),dL), is independent of the severing rate. By integrating the master equation over all lengths, the severing contributions cancel out, showing that severing merely redistributes filament length without changing filament count. Consequently, changes in (k_{sev}) affect the average length but not the population size.

In the special case where rescue events are absent ((\omega_r=0)), the master equation reduces to a linear first‑order differential equation. The authors solve it exactly, obtaining a closed‑form steady‑state distribution:
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