Conditions for Cell Size Homeostasis: A Stochastic Hybrid System Approach

How isogenic cell populations maintain size homeostasis, i.e., a narrow distribution of cell size, is an intriguing fundamental problem. We model cell size using a stochastic hybrid system, where a cell grows exponentially in size (volume) over time and probabilistic division events are triggered at discrete-time intervals. Moreover, whenever division occurs, size is randomly partitioned among daughter cells. We first consider a scenario where a timer (cell-cycle clock) that measures the time elapsed since the last division event regulates both the cellular growth and division rates. The analysis reveals that such a timer-controlled system cannot achieve size homeostasis, in the sense that the cell-to-cell size variation grows unboundedly with time. To explore biologically meaningful mechanisms for controlling size, we consider two classes of regulation: a size-dependent growth rate and a size-dependent division rate. Our results show that these strategies can provide bounded intercellular variation in cell size and exact mathematical conditions on the form of regulation needed for size homeostasis are derived. Different known forms of size control strategies, such as the adder and the sizer, are shown to be consistent with these results. Finally, we discuss how organisms ranging from bacteria to mammalian cells have adopted different control approaches for maintaining size homeostasis.