Intergenerational scaling law determines the precision kinematics of stochastic individual-cell-size homeostasis

Individual bacterial cells grow and divide stochastically. Yet they maintain their characteristic sizes across generations within a tightly controlled range. What rules ensure intergenerational stochastic homeostasis of individual cell sizes? Valuable clues have emerged from high-precision longterm tracking of individual statistically-identical Caulobacter crescentus cells as reported in [1, 2]: Intergenerational cell size homeostasis is an inherently stochastic phenomenon, follows Markovian or memory-free dynamics, and cells obey an intergenerational scaling law, which governs the stochastic map characterizing generational sequences of cell sizes. These observed emergent simplicities serve as essential building blocks of the data-informed principled theoretical framework we develop here. Our exact analytic fitting-parameter-free results for the predicted intergenerational stochastic map governing the precision kinematics of cell size homeostasis are remarkably well borne out by experimental data, including extant published data on other microorganisms, Escherichia coli and Bacillus subtilis. Furthermore, our framework naturally yields the general exact and analytic condition that is necessary and sufficient to ensure that stochastic homeostasis can be achieved and maintained. Significantly, this condition is more stringent than the known heuristic result from quasi-deterministic frameworks. In turn the fully stochastic treatment we present here extends and updates extant frameworks, and highlights the inherently stochastic behaviors of individual cells in homeostasis.