A model of oscillatory protein dynamics in bacteria

Modeling the Dynamics of Cdc42 Oscillation in Fission Yeast

Regulation of polarized cell growth is essential for many cellular processes, including spatial coordination of cell morphology changes during growth and division. We present a mathematical model of the core mechanism responsible for the regulation of polarized growth dynamics by the small GTPase Cdc42. The model is based on the competition of growth zones of Cdc42 localized at the cell tips for a common substrate (inactive Cdc42) that diffuses in the cytosol. We consider several potential ways of implementing negative feedback between Cd42 and its GEF in this model that would be consistent with the observed oscillations of Cdc42 in fission yeast. We analyze the bifurcations in this model as the cell length increases, and total amount of Cdc42 and GEF increase. Symmetric antiphase oscillations at two tips emerge via saddle-homoclinic bifurcations or Hopf bifurcations. We find that a stable oscillation and a stable steady state can coexist, which is consistent with the experimental finding that only 50% of bipolar cells oscillate. The mean amplitude and period can be tuned by parameters involved in the negative feedback. We link modifications in the parameters of the model to observed mutant phenotypes. Our model suggests that negative feedback is more likely to be acting through inhibition of GEF association rather than upregulation of GEF dissociation.

A model for spatio-temporal dynamics in a regulatory network for cell polarity

Cell polarity in Myxococcus xanthus is crucial for the directed motility of individual cells. The polarity system is characterised by a dynamic spatio-temporal localisation of the regulatory proteins MglA and MglB at opposite cell poles. In response to signalling by the Frz chemosensory system, MglA and MglB are released from the poles and then rebind at the opposite poles. Thus, over time MglA and MglB oscillate irregularly between the poles in synchrony but out of phase. A minimal macroscopic model of the Mgl/Frz regulatory system based on a reaction-diffusion PDE system is presented. Mathematical analysis of the steady states derives conditions on the reaction terms for formation of dynamic localisation patterns of the regulatory proteins under different biologically-relevant regimes, i.e. with and without Frz signalling. Numerical simulations of the model system produce either a stationary pattern in time (fixed polarity), periodic solutions in time (oscillating polarity), or excitable behaviour (irregular switching of polarity).