Stochastic modeling of cell growth with symmetric or asymmetric division

TL;DR

This paper develops a stochastic model for cell resource distribution during division, analyzing conditions for stable distributions, their properties, and the effects of asymmetry on stability and distribution width.

Contribution

It introduces a master equation for resource dynamics with symmetric/asymmetric division, deriving stability conditions and analyzing distribution tails and phase diagrams.

Findings

Stable resource distributions exist under certain conditions.

Asymmetry increases system stability but broadens the distribution.

Power-law tails can emerge in the resource distribution.

Abstract

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.