Conditions for Cell Size Homeostasis: A Stochastic Hybrid System Approach

Abstract

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.

Mitochondrial heterogeneity, metabolic scaling and cell death

“…We used a stochastic hybrid systems approach [52] to simulate a population of exponentially growing cells undergoing 'adder' dynamics [50,51], whereby cells add a constant amount of volume (K v ) to their initial volume (v 0 ) each cell cycle, which has found recent support in bacteria [53]. Cells are modelled to then divide by stochastically partitioning their volume, with the first daughter inheriting volume and Bj€ orklund as a non-heritable parabolic relationship between mitochondrial functionality (which we interpret as rDC) and cell volume v…”

Analysis of Cell Size Homeostasis at the Single-Cell and Population Level

“…These initial studies formulated sizer models based on the population balance equation, an integro-differential equation that is notoriously difficult to solve in practice [21][22][23][24]. With the advent of single-cell traps such as the mother machine [2], the theoretical focus moved toward describing single cells over many divisions [13,[25][26][27]. This path is more amenable to analysis because it considers only a single individual described by a discrete-time stochastic process or stochastic map [28].…”

Cell volume distributions in exponentially growing populations

“…Substituting (19) into (18) The condition α > 1 guarantees the (post-threshold) dominance of division over growth, which is critical for the maintenance of cell volume homeostasis. The recursive relation (19) implies that c n = c 0 e −µn , which, if substituted into (17), yields π n (ϕ) = c 0 e −µ(n+ϕ) , n ≥ 1.…”

“…Growth in cell volume is understood to occur continuously in time and is often assumed to be exponential. Cell division is typically represented as a discrete event at which the volume of a mother cell abruptly changes into the volume of either daughter cell [19]. Specifically, symmetric division means that each daughter obtains exactly one half of their mother's volume.…”