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.