Statistical properties of dynamical models underlying cell size homeostasis

In this contribution, we use the framework of discrete-event dynamical systems to model a fundamental aspect of life - the proliferation of cells. This process starts with a newborn cell, whose size (as quantified by its volume or mass) increases exponentially over time. At a prescribed time during cellular growth, a cell-division event is triggered that results in the mother cell dividing symmetrically/asymmetrically into newborn cells, and the cycle repeats. Division events are assumed to occur probabilistically at a rate that is a monotonically increasing function of cell size. We derive closed-form expressions for the steady-state moments (mean, variance, skewness) of cell size, and these results are exact when the cell-division events occur at a rate proportional to size. This latter case leads to the recently observed adder-based size control in several types of bacteria, where a fixed size is added between birth and division irrespective of the newborn size. Analytical moment formulas are extended to the biologically relevant scenario where the time between two successive division events is further divided into multiple discrete stages with size-dependent stage transitions. Exact moment computations show a decrease in cell size noise with the increasing number of stages, with a lesser degree of noise attenuation when there are partitioning errors in cell size during division. Development of these results will facilitate the estimation of model parameters from measured cell size distributions and drive an understanding of control mechanisms regulating cell size homeostasis.