A first-passage-time theory for search and capture of chromosomes by microtubules in mitosis

The mitotic spindle is an important intermediate structure in eukaryotic cell division, in which each of a pair of duplicated chromosomes is attached through microtubules to centrosomal bodies located close to the two poles of the dividing cell. Several mechanisms are at work towards the formation of the spindle, one of which is the `capture’ of chromosome pairs, held together by kinetochores, by randomly searching microtubules. Although the entire cell cycle can be up to 24 hours long, the mitotic phase typically takes only less than an hour. How does the cell keep the duration of mitosis within this limit? Previous theoretical studies have suggested that the chromosome search and capture is optimized by tuning the microtubule dynamic parameters to minimize the search time. In this paper, we examine this conjecture. We compute the mean search time for a single target by microtubules from a single nucleating site, using a systematic and rigorous theoretical approach, for arbitrary kinetic parameters. The result is extended to multiple targets and nucleating sites by physical arguments. Estimates of mitotic time scales are then obtained for different cells using experimental data. In yeast and mammalian cells, the observed changes in microtubule kinetics between interphase and mitosis are beneficial in reducing the search time. In {\it Xenopus} extracts, by contrast, the opposite effect is observed, in agreement with the current understanding that large cells use additional mechanisms to regulate the duration of the mitotic phase.

💡 Research Summary

The mitotic spindle assembles by capturing duplicated chromosomes through dynamic microtubules (MTs) that grow from centrosomal nucleation sites. Although a full cell cycle may last up to 24 h, mitosis is completed in less than an hour, prompting the question of how the cell keeps the search‑and‑capture process fast enough. This paper addresses that question by formulating the capture of a kinetochore‑bound chromosome as a first‑passage‑time (FPT) problem for a stochastic MT.

Model formulation
A single MT emanates from a nucleation point (MTOC) and explores space in a random direction within a fixed solid angle. Its dynamics are fully described by four kinetic parameters: growth speed (v_g), shrinkage speed (v_s), catastrophe frequency (f_{cat}) (growth→shrinkage), and rescue frequency (f_{res}) (shrinkage→growth). The MT tip performs a drift‑diffusion process with drift velocities (+v_g) and (-v_s) in the two states, and stochastic switches governed by the two rates. The distance to a target kinetochore is denoted (L). The probability density (p(L,t)) that the tip first reaches (L) at time (t) satisfies a piecewise linear Fokker‑Planck equation with an absorbing boundary at (L) and a reflecting boundary at the origin.

Using Laplace transforms the authors obtain an exact expression for the mean first‑passage time (MFPT) for arbitrary kinetic parameters:

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