  We consider a class of biologically-motivated stochastic processes in which a
unicellular organism divides its resources (volume or damaged proteins, in
particular) symmetrically or asymmetrically between its progeny. Assuming the
final amount of the resource is controlled by a growth policy and subject to
additive and multiplicative noise, we derive the "master equation" describing
how the resource distribution evolves over subsequent generations and use it to
study the properties of stable resource distributions. We find conditions under
which a unique stable resource distribution exists and calculate its moments
for the class of affine linear growth policies. Moreover, we apply an
asymptotic analysis to elucidate the conditions under which the stable
distribution (when it exists) has a power-law tail. Finally, we use the results
of this asymptotic analysis along with the moment equations to draw a stability
phase diagram for the system that reveals the counterintuitive result that
asymmetry serves to increase stability while at the same time widening the
stable distribution. We also briefly discuss how cells can divide damaged
proteins asymmetrically between their progeny as a form of damage control. In
the appendix, motivated by the asymmetric division of cell volume in
Saccharomyces cerevisiae, we extend our results to the case wherein mother and
daughter cells follow different growth policies.
