A Moments-Based Analytical Approach for Cell Size Homeostasis

This contribution explores mechanisms that regulate the dynamics of single-cell size, maintaining equilibrium around a target set point. Using the formalism of Stochastic Hybrid Systems (SHS), we consider continuous exponential growth in cell size (as determined by volume/mass/surface area). This continuous-time evolution is interspersed by cell division events that occur randomly as per a given size-dependent rate, and upon division, only one of the two daughter cells is tracked. We show that a size-independent division rate does not provide cell size homeostasis, in the sense that the variance in cell size increases unboundedly over time. Next, we consider a division rate proportional to cell size that yields the adder size control observed in several bacteria – a constant size is added on average between birth and division regardless of the newborn size. For this scenario, we obtain exact formulas for the steady-state moments (mean, variance, and skewness) of cell size. Expanding the SHS model, we explore a biologically relevant scenario where the time between successive division events is further divided into multiple discrete stages with size-dependent stage transitions. Exact moment computations demonstrate that increasing the number of stages reduces cell size variability (noise). We also find formulas considering uneven size partitioning between daughters during division, and where the division rate follows a power law of the cell size leading to deviations from adder size control. This work provides a method for estimating model parameters from observed cell size distributions and enhances our understanding of mechanisms underlying cell size regulation.