Dionn Hargreaves1, Sarah Woolner1 and Oliver E. Jensen21 Faculty of Biology, Medicine and Health, University of Manchester, Oxford Road, Manchester, M13 9PL, UK.2 Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK.* Corresponding author(s). E-mail(s):dionn.hargreaves@manchester.ac.uk;Contributing authors: sarah.woolner@manchester.ac.uk;oliver.jensen@manchester.ac.uk;AbstractDuring 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 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 Fokker–Planck equations 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 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 Fokker–Planck system predicts stability, but for which the period can be estimated directly by the ODE model.
Thus, for large N , the upper branch of the stability boundary defined by (19b) in Figure 5(a) approaches on = on in (22), confirming that a necessary condition for oscillations is that the tension-sensitivity parameter satisfies > 1, i.e. that linkers exhibit slip-bond behaviour. Indeed, removal of the tension-sensitivity of the unbinding 19 (20a) (20b) (21) (22) (23) (24) (25) rate in the stochastic simulations leads to a reduction of the coherence of the oscillatory behaviour of the spindle pole (Figure 3d). The upper-branch asymptote on = on appears to be shared also by PDE solutions (which suggests an upper stability threshold between 0.006 < on < 0.007 for N 80, within 80% of on = 0.0074). Also in the large-N limit, the lower branch of (19b) is captured by (24), consistent with PDE solutions in this limit. This limit shows explicitly how increasing the restoring force K has a stabilising effect.
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We also recall that, in the FokkerPlanck model, decreasing the restoring force parameter K promotes oscillations at smaller N (Figure 6c, where N = 15). This behaviour is conserved in the ODE system, where the lowK approximation (21) shown in Figure 5(a), predicts oscillations in a greater region of the (N, on)-plane. Evaluating don/dN using (21) gives dN don = 0 on
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Thus for the neutral curve to lie in N > 0 requires N > 1 v0 f0( 1) , providing a lower bound on the number of linkers needed for oscillations in terms of the walking speed and stall force of a linker, and the drag on the spindle. The period of oscillations along the neutral stability curve predicted using (19a) increases as K decreases (Figure 5b); thus a reduction of restoring forces corresponds to longer periods of oscillation. The rapid increase of the period as on on coincides with N . (19a) is well matched with (20b) determined by Grill et al (2005), as well as with the periods along the approximate stability curve identified by numerical solutions of the PDEs.
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Finally, we can use the ODE solution to provide further evidence that the oscillations in Figure 3(a) are noise-induced. Despite lying outside the neutral curve (Figure 5a), the period of the stochastic oscillations is well approximated by (18b) (Figure 3a, red bar), indicating that noise due to the relatively small number of linkers is sufficient to overcome the damping evident in the FokkerPlanck description (Figure 5c) and in the ODE model. As explained in Appendix B, the Fokker Planck system (7, 10) proposed by Grill et al (2005) is a simplified form of the high-dimensional chemical FokkerPlanck equation associated with the full stochastic model; we attribute the failure of (7, 10) to predict the oscillations in Figure 3(a) to this simplification.
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