Uncovering the quantitative laws that govern the growth and division of single cells remains a major challenge. Using a unique combination of technologies that yields unprecedented statistical precision, we find that the sizes of individual Caulobacter crescentus cells increase exponentially in time. We also establish that they divide upon reaching a critical multiple (≈1.8) of their initial sizes, rather than an absolute size. We show that when the temperature is varied, the growth and division timescales scale proportionally with each other over the physiological temperature range. Strikingly, the cell-size and division-time distributions can both be rescaled by their mean values such that the conditionspecific distributions collapse to universal curves. We account for these observations with a minimal stochastic model that is based on an autocatalytic cycle. It predicts the scalings, as well as specific functional forms for the universal curves. Our experimental and theoretical analysis reveals a simple physical principle governing these complex biological processes: a single temperature-dependent scale of cellular time governs the stochastic dynamics of growth and division in balanced growth conditions. single-cell dynamics | cell-to-cell variability | exponential growth | Hinshelwood cycle | Arrhenius law

Cell size is specific to each species and impacts cell function. Various phenomenological models for cell size regulation have been proposed, but recent work in bacteria has suggested an 'adder' model, in which a cell increments its size by a constant amount between each division. However, the coupling between cell size, shape and constriction remains poorly understood. Here, we investigate size control and the cell cycle dependence of bacterial growth using multigenerational cell growth and shape data for single Caulobacter crescentus cells. Our analysis reveals a biphasic mode of growth: a relative timer phase before constriction where cell growth is correlated to its initial size, followed by a pure adder phase during constriction. Cell wall labelling measurements reinforce this biphasic model, in which a crossover from uniform lateral growth to localized septal growth is observed. We present a mathematical model that quantitatively explains this biphasic 'mixer' model for cell size control.

How are granular details of stochastic growth and division of individual cells reflected in smooth deterministic growth of population numbers? We provide an integrated, multiscale perspective of microbial growth dynamics by formulating a data-validated theoretical framework that accounts for observables at both single-cell and population scales. We derive exact analytical complete time-dependent solutions to cell-age distributions and population growth rates as functionals of the underlying interdivision time distributions, for symmetric and asymmetric cell division. These results provide insights into the surprising implications of stochastic single-cell dynamics for population growth. Using our results for asymmetric division, we deduce the time to transition from the reproductively quiescent (swarmer) to the replicationcompetent (stalked) stage of the Caulobacter crescentus life cycle. Remarkably, population numbers can spontaneously oscillate with time. We elucidate the physics leading to these population oscillations. For C. crescentus cells, we show that a simple measurement of the population growth rate, for a given growth condition, is sufficient to characterize the condition-specific cellular unit of time and, thus, yields the mean (single-cell) growth and division timescales, fluctuations in cell division times, the cell-age distribution, and the quiescence timescale.

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1 Cell size is specific to each species and impacts their ability to function. While 1 various phenomenological models for cell size regulation have been proposed, recent 2 work in bacteria have demonstrated an adder mechanism, in which a cell increments 3 its size by a constant amount between each division. However, the coupling between 4 cell size, shape and constriction, remain poorly understood. Here, we investigate size 5 control and the cell cycle dependence of bacterial growth, using multigenerational cell 6 growth and shape data for single Caulobacter crescentus cells. Our analysis reveals 7 a biphasic growth mechanism: a relative timer phase before constriction where cell 8 growth is correlated to its initial size, followed by a pure adder phase during constric-9 tion. Cell wall labeling measurements reinforce this biphasic behavior: a crossover 10 from uniform lateral growth to localized septal growth is observed. We develop a 11 mathematical model that quantitatively explains this mixer mechanism for size con-12 trol. 13 14 We recently introduced a technology that enables obtaining unprecedented amounts of precise 15 quantitative information about the shapes of single bacteria as they grow and divide under non-16 crowding and controllable environmental conditions [1, 2]. Others have developed complementary 17 methods [3-6]. These single-cell studies are generating great interest because they reveal unantic-18 ipated relationships between cell size and division control [5]. Recent work in bacteria that utilize 19 these technologies revealed that a constant size increment between successive generations [7] quan-20 titatively describes the strategy for bacterial size maintenance in E. coli [3-5], B. subtilis [5], C. 21 crescentus [4], P. aeruginosa [8] and even in the yeast S. cerevisiae [9]. This phenomenological ob-22 servation has been termed an adder model for cell size control. Competing models for size control 23 include cell division at a critical size (sizer model) [10] or at a constant interdivision time (timer 24 model) [1]. Analysis of single-cell data show that cell size at division is positively correlated with 25the cell size at birth [1, 4, 5, 11, 12], thus precluding a sizer model. In addition, a negative corre-26 lation between initial cell size and interdivision times, as reported here and in refs [1, 4, 5, 12, 13], 27 is inconsistent with the timer model. However, other studies have suggested mixed models of 28 size control, with diverse combinations of sizer, timer and adder mechanisms [14][15][16][17]. The spatial 29 resolution and statistically large size of our data now allow us to revisit these issues with greater 30 precision. 31While cell size serves as an important determinant of growth, the bacterial cell cycle is composed 32 of various coupled processes including DNA replication and cell wall constriction that have to be 33 2 faithfully coordinated for cells to successfully divide [18]. This raises the question of what other 34 cell cycle variables regulate growth and how the ...