  A ubiquitous feature of living cells is their growth over time followed by
division into daughter cells. 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 events occur, size is randomly partitioned among daughter cells. We
first consider a scenario, where a timer (i.e., cell-cycle clock) that measures
the time since the last division event regulates both the cellular growth and
division rates. 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.
Interestingly, for timer-based division mechanisms, the mean cell size depends
on the noise in the cell-cycle duration but independent of errors incurred in
partitioning of volume among daughter cells. In contrast, the mean cell size
decreases with increasing partitioning errors for size-based division
mechanisms. Finally, we discuss how organisms ranging from bacteria to
mammalian cells have adopted different control approaches for maintaining size
homeostasis.
