A PDE-DDE model for cell polarization in fission yeast

Abstract

We consider a one-dimensional model of cell polarization in fission yeast consisting of a hybrid partial differential equation–delay differential equation system. The model describes bulk diffusion of the signaling molecule Rho GTPase Cdc42 in the cytoplasm, which is coupled to a pair of delay differential equations at the ends of the cell via boundary conditions. The latter represent the binding of Cdc42 to the cell membrane and rerelease into the cytoplasm via unbinding. The nontrivial nature of the dynamics arises from the fact that both the binding and unbinding rates at each end are taken to depend on the local membrane concentration of Cdc42. In particular, the association rate is regulated by positive feedback and the dissociation rate is regulated by delayed negative feedback. We use linear stability analysis and numerical simulations to investigate the onset of limit cycle oscillations at the end compartments for a cell of fixed length, distinguishing between symmetric solutions in which the mean concentration is identical at both ends and asymmetric solutions where the mean concentration at one end dominates. We find that the critical time delay for the onset of oscillations via a Hopf bifurcation increases as the diffusion coefficient D decreases. We then solve the diffusion equation on a growing domain under the additional assumption that the total amount Ctot of the signaling molecule increases as the cell length increases. We show that the system undergoes a transition from asymmetric to symmetric oscillations as the cell grows, consistent with experimental findings of “new-end-take-off” in fission yeast. (The latter refers to the switch from monopolar to bipolar growth as the cell grows.) The critical length where the switch occurs depends on D and the growth rate.