  The spatiotemporal oscillations of the Min proteins in the bacterium
Escherichia coli play an important role in cell division. A number of different
models have been proposed to explain the dynamics from the underlying
biochemistry. Here, we extend a previously described discrete polymer model
from a deterministic to a stochastic formulation. We express the stochastic
evolution of the oscillatory system as a map from the probability distribution
of maximum polymer length in one period of the oscillation to the probability
distribution of maximum polymer length half a period later and solve for the
fixed point of the map with a combined analytical and numerical technique. This
solution gives a theoretical prediction of the distributions of both lengths of
the polar MinD zones and periods of oscillations -- both of which are
experimentally measurable. The model provides an interesting example of a
stochastic hybrid system that is, in some limits, analytically tractable.
