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Stochastic multi-step cell size homeostasis model for cycling human cells

Measurements of cell size dynamics have established the adder principle as a robust mechanism of cell size homeostasis. In this framework, cells add a nearly constant amount of size during each cell cycle, independent of their size at birth. Theoretical studies have shown that the adder principle can be achieved when cell-cycle progression is coupled to cell size. Here, we extend this framework by considering a general growth law modeled as a Hill-type function of cell size. This assumption introduces growth saturation to the model, such that very large cells grow approximately linearly rather than exponentially. Additionally, to capture the sequential nature of division, we implement a stochastic multi-step adder model in which cells progress through internal regulatory stages before dividing. From this model, we derive exact analytical expressions for the moments of cell size distributions. Our results show that stronger growth saturation increases the mean cell size in steady state, while slightly reducing fluctuations compared to exponential growth. Importantly, despite these changes, the adder property is preserved. This emphasizes that the reduction in size variability is a consequence of~the growth law rather than simple scaling with mean size. Finally, we analyze stochastic clonal proliferation and find that growth saturation influences both single-cell size statistics and variability across populations. Our results provide a generalized framework for connecting multi-step adder mechanisms with proliferation dynamics, extending size control theory beyond exponential growth.