  Many unicellular organisms allocate their key proteins asymmetrically between
the mother and daughter cells, especially in a stressed environment. A recent
theoretical model is able to predict when the asymmetry in segregation of key
proteins enhances the population fitness, extrapolating the solution at two
limits where the segregation is perfectly asymmetric (asymmetry $a$ = 1) and
when the asymmetry is small ($0 \leq a \ll 1$). We generalize the model by
introducing stochasticity and use a transport equation to obtain a
self-consistent equation for the population growth rate and the distribution of
the amount of key proteins. We provide two ways of solving the self-consistent
equation: numerically by updating the solution for the self-consistent equation
iteratively and analytically by expanding moments of the distribution. With
these more powerful tools, we can extend the previous model by Lin et al. to
include stochasticity to the segregation asymmetry. We show the stochastic
model is equivalent to the deterministic one with a modified effective
asymmetry parameter ($a_{\rm eff}$). We discuss the biological implication of
our models and compare with other theoretical models.
