  During cell division, the mitotic spindle moves dynamically through the cell
to position the chromosomes and determine the ultimate spatial position of the
two daughter cells. These movements have been attributed to the action of
cortical force generators which pull on the astral microtubules to position the
spindle, as well as pushing events by these same microtubules against the cell
cortex and plasma membrane. Attachment and detachment of cortical force
generators working antagonistically against centring forces of microtubules
have been modelled previously (Grill et al. 2005, Phys. Rev. Lett. 94:108104)
via stochastic simulations and mean-field Fokker-Planck equations (describing
random motion of force generators) to predict oscillations of a spindle pole in
one spatial dimension. Using systematic asymptotic methods, we reduce the
Fokker-Planck system to a set of ordinary differential equations (ODEs),
consistent with a set proposed by Grill et al., which can provide accurate
predictions of the conditions for the Fokker-Planck system to exhibit
oscillations. In the limit of small restoring forces, we derive an algebraic
prediction of the amplitude of spindle-pole oscillations and demonstrate the
relaxation structure of nonlinear oscillations. We also show how noise-induced
oscillations can arise in stochastic simulations for conditions in which the
mean-field Fokker-Planck system predicts stability, but for which the period
can be estimated directly by the ODE model and the amplitude by a related
stochastic differential equation that incorporates random binding kinetics.
