Continuous Approximation of Stochastic Maps for Modeling Asymmetric Cell Division

Cell size control and homeostasis is a major topic in cell biology yet to be fully understood. Several growth laws like the timer, adder, and sizer were proposed, and mathematical approaches that model cell growth and division were developed. This study focuses on utilizing stochastic map modeling for investigating asymmetric cell division. We establish a mapping between the description of cell growth and division and the Ornstein-Uhlenbeck process with dichotomous noise. We leverage this mapping to achieve analytical solutions and derive a closed-form expression for the stable cell size distribution under asymmetric division. To validate our findings, we conduct numerical simulations encompassing several cell growth scenarios. Our approach allows us to obtain a precise criterion for a bi-phasic behavior of the cell size. While for the case of the sizer scenario, a transition from the uni-modal phase to bi-modal is always possible, given sufficiently large asymmetry at the division, the affine-linear approximation of the adder scenario invariably yields uni-modal distribution.