Bridging the Timescales of Single-Cell and Population Dynamics

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 replication-competent (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.

Popular Summary

Exponential growth processes are ubiquitous in both natural and everyday phenomena. Familiar examples include population growth of microorganisms, compounding of interest, nuclear fission, the inflation of the Universe, Moore’s law for the increase of computer processor power, and viral posts on social media. Despite the prevalence of exponential growth, there are still fundamental open questions concerning these phenomena. In particular, the earliest microbiology experiments revealed that population sizes of microorganisms increase exponentially. Yet this smooth growth masks the noisiness and randomness of the underlying growth and division of individual cells. How are the two perspectives to be reconciled? What signatures of the underlying probabilistic events remain in the coarse-grained exponential growth of populations?

Here, we take advantage of high-quality single-cell and bulk-population growth data to build a fundamental theory that is applicable at both scales and yields testable predictions. By combining data with our theoretical framework, we show that an emergent cellular unit of time governs growth phenomena at both scales. Owing to unexpected universality in growth dynamics under different growth conditions, a simple measurement of this timescale reveals a wealth of information about underlying probabilistic phenomena.

Our results serve as a starting point for future lines of inquiry involving time-dependent growth phenomena, such as the aging of cells and organisms.

Figure 1

Schematic representation of model and experimental system. In (a), we summarize the general model considered. Upon division, a normal (yellow) cell divides into $\nu$ normal cells and ${\nu }_q$ reproductively quiescent (blue) cells. Division occurs after a stochastic waiting time $\tau$, drawn from a distribution $P(\tau )$. The quiescent cells transition into normal cells with a waiting-time distribution $P_q(T_q)$. Model specification requires knowledge of $\nu$, ${\nu }_q$, $P(\tau )$, and $P_q(T_q)$. By setting ${\nu }_q=0$, one obtains a model of symmetric division (dotted cyan box). In (b), we show a schematic of the relevant timescales in the C. crescentus life cycle: the stalked (normal) cell division time $\tau$ and the swarmer (quiescent) cell to stalked cell transition time $T_q$. Upon division, a stalked cell gives birth to a stalked and a swarmer cell. Thus, the C. crescentus cell cycle reduces to the model in (a) with $\nu ={\nu }_q=1$. The single-cell technology introduced in Ref. [2] corresponds to a model with $\nu =1$ and ${\nu }_q=0$ (dotted cyan box).

Figure 2

Cell-age distributions, measured and predicted, from different temperatures. In (a), we show the age distributions from different temperatures (light green, $34°\mathrm{C}$; dark green, $31°\mathrm{C}$; cyan, $24°\mathrm{C}$; and blue, $17°\mathrm{C}$). The data from single-cell experiments are shown with circular symbols, and the predictions from the theory (with no adjustable parameters) are shown with corresponding dashed lines. In (b), we show the same age distributions as in (a), rescaled by their respective temperature-dependent mean ages. ${\tilde{G}}^{*}(\tau /{\mu }_a)\equiv {\mu }_a{G}^{*}(\tau )$ is the mean-rescaled probability density of cell ages. Evidently, once mean rescaled, the probability densities undergo a scaling collapse, consistent with a single condition-specific timescale dominating stochastic growth and division statistics (see text). For the measured cell division time, distributions corresponding to these age distributions; see Fig. 3B of [2].

Figure 3

Scaling of timescales in the C. crescentus life cycle. (a) The population growth rate $k$ (orange circles) and the single-cell mean division rate from [2] (cyan squares) are shown as functions of temperature on an Arrhenius plot. The measurement precision is better than the sizes of the plot markers. The dotted (dashed) line is the best fit Ratkowsky curve [2] for the single-cell (population) data. Both fits are found to be proportional to each other, with the Ratkowsky temperature 269 K; by finding the ratio of the two curves, we find that ${\mu }_{\tau }k=0.6$. (b) Combining the results from (a) with Eq. (6) and results from [2], we are able to characterize all timescales in the C. crescentus life cycle as fractions of mean single-cell division time or, equivalently, the inverse of the population growth rate (see the accompanying text). At different temperatures, all timescales change proportionally; the proportions are shown in (b). Single-cell measurements sample the dynamics of stalked cell division (cyan rectangle), whereas population growth measurements sample the dynamics of stalked cell division and swarmer cell differentiation (orange rectangle).

Figure 4

The physics of transient oscillations. In (a), we sketch a simple example that illustrates how transient oscillations in population numbers arise. See the accompanying text for notation and definition of variables. Initially, $N_0$ normal cells are present, and population numbers increase as these cells divide and differentiate. The possible lineages of cells present in each demarcated interval are shown as the appropriate sequence of normal ($N$) and quiescent ($Q$) stages in their ancestry, starting at $t=0$, and including their current state. At $t=\tau$, all cells of lineage N divide, causing an increase by $N_0$ in $N_q(t)$, while $N(t)$ remains equal to $N_0$. At $t=\tau +T_q$, the quiescent cells differentiate synchronously, creating the lineage NQN. This results in a simultaneous decrease in $N_q(t)$ and increase in $N(t)$, each by $N_0$. The next change in numbers occurs with the division of all cells of NN lineage at $t=2\tau$. In this manner, the persistence of distinct lineage identities causes distinct fractions of the population to undergo division or differentiation synchronously, resulting in the observed population oscillations. In (b), we show the corresponding transient oscillations in population numbers for a realistic model where division and differentiation occur probabilistically. See the Fig. 6 legend for the parameter values used; for these parameter values, we find that ${\mu }_{\tau }=1$ and ${\mu }_{T_q}=0.4$. From the times at which the first oscillation in quiescent cell populations begins and ends, we can deduce the mean cell division (${\mu }_{\tau }$) and differentiation (${\mu }_{T_q}$) times [compare with panel (a)]. The age distributions at the transient times (i)–(iii), corresponding to stages 2–4 in (a), are shown in Fig. 5. All curves were calculated from our exact analytical time-dependent solutions (see the Appendix for details).

Figure 5

Time evolution of age distributions of normal and quiescent cells. The solid curves in (a) and (b) show transient normal and quiescent cell-age distributions, respectively. We have plotted these distributions at $t=1.2$, 1.6, and 2.05, corresponding to times labeled (i), (ii), and (iii), respectively, in Fig. 4. See the Fig. 6 legend for the parameter values used; for these parameter values, we find that ${\mu }_{\tau }=1$ and ${\mu }_{T_q}=0.4$. The initial population consists of newborn normal cells. Each mode of a time-dependent age distribution corresponds to a specific lineage, as labeled above. The separations between different modes of multimodal time-dependent age distributions contain information about underlying timescales of division and differentiation. For instance, in (a), curve (ii) shows a bimodal normal cell-age distribution observed at $t=1.6$. The separation between the two modes (corresponding to the lineages NN and NQN) is approximately equal to the differentiation timescale, ${\mu }_{T_q}=0.4$. The broadening of age distributions of each new lineage, because of fluctuations in division and differentiation times, results in steady-state age distributions (dashed curves) in which contributing lineages can no longer be distinguished. All curves were calculated from the exact analytical time-dependent solutions, which are presented in the Appendix. The $\delta$-function corresponding to the initial age distribution has not been shown in (a). See Supplementary Videos 1 and 2 in [24] for the full time evolution of the age distributions corresponding to panels (a) and (b), respectively.

Figure 6

Scaling of number fluctuations during stochastic population growth. In (a), we show analytical results for the time evolution of population sizes of normal (yellow) and quiescent (blue) cells. In the steady state, they have the same exponential growth rate. Oscillations in the numbers of normal and quiescent cells are observed during the transient phase. The bold curves show the mean behavior of population numbers, calculated using our exact analytical solution. Also shown are representative stochastic trajectories that account for fluctuations in population numbers. The stochastic trajectories were computed using exact numerical simulations (see the accompanying text). In (b), we show that the mean-rescaled population number distributions of normal and quiescent cells [from $t=9$ in panel (a)] undergo a scaling collapse. Here, $\tilde{N}(t)=N(t)/⟨N(t)⟩$ and $\widetilde{N_q}(t)=N_q(t)/⟨N_q(t)⟩$. The parameter values used for these simulations are $\nu ={\nu }_q=1$; $N(0)=10$; and $N_q(0)=0$. $P(\tau )$ is a gamma distribution with mean $=1$ and $\mathrm{COV}=0.13$, and $P_q(T_q)$ is a gamma distribution with mean $=0.4$ and $\mathrm{COV}=0.1$. All timescales are measured in units of ${\mu }_{\tau }$.

Figure 7

Region of integration for Eq. (a16). The shaded area corresponds to the region of integration in the $\tau -{\tau }^{\prime }$ plane, for the expression on the right-hand side of Eq. (a16).