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Deterministic and stochastic modeling of the Min system for cell division Jamieson-Lane, Alastair

Abstract

The Min system acts as a key regulator for cell division in E. coli, repressing cell division at either end of the cell via pole to pole oscillation. Recent in vitro experiments have demonstrated the Min system's tendency to create ``burst'' patterning under suitable concentration conditions, whereby high concentration `bursts' of Min proteins nucleate from an approximately homogeneous background, before ``freezing'' and fading away. I start this thesis by giving a quick review of some of the complexities involved in modeling chemical reactions via Partial Differential Equations - particularly in 2D surfaces such as the cell membrane. I consider a number of toy models, demonstrating discrepancies between classical Reaction-Diffusion representations of chemical systems, and the more foundational particle system. These discrepancies are in most cases minor, in some cases extreme. A simplified Min model is developed, demonstrating how particle models of Min dynamics can lead to burst formation, even in cases where differential equations predict a uniform solution. Next, I take a recently developed and parameterized ODE model of the Min system based on experimental data from Ivanov et al, and extend the model to consider finite space, both on the membrane and in the volume of the cell. This extended model allows me to map out a bifurcation diagram of the system's behavior for concentrations both higher and lower than those used in the original data fitting, and explore the conditions under which burst nucleation is predicted. Finally, I show that white noise can allow a spatially distributed reaction diffusion system to escape from a neutrally stable steady state at zero, passing to some fixed value u(0,T)>0 in finite time. The most probable path to such a state leads to a narrow sharp spike reminiscent of experimental observations. Dynamics of this kind are typical whenever a system loses stability by passing slowly through a saddle node bifurcation. Supplementary materials available at: http://hdl.handle.net/2429/68997

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