  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.
