From single-cell variability to population growth

Single-cell experiments have revealed cell-to-cell variability in generation times and growth rates for genetically identical cells. Theoretical models relating the fluctuating generation times of single cells to the population growth rate are usually based on the assumption that the generation times of mother and daughter cells are uncorrelated. This assumption, however, is inconsistent with the exponential growth of cell volume in time observed for many cell types. Here we develop a more general and biologically relevant model in which cells grow exponentially and generation times are correlated in a manner which controls cell size. In addition to the fluctuating generation times, we also allow the single-cell growth rates to fluctuate and account for their correlations across the lineage tree. Surprisingly, we find that the population growth rate only depends on the distribution of single-cell growth rates and their correlations.

Figure 1

(a) Cell length vs. time along a single lineage. The black lines are the exponential fits of cell length in time. The dashed lines mark the boundaries between consecutive generations. Data from Ref. [24]. (b) A lineage tree starting from a single cell. After a long enough time, the number of cells increases as $N\left(t\right)\sim e^{{\mathrm{\Lambda }}_{p}t}$ in time.

Figure 2

In this figure, the variability in generation times arises only from the variability in single-cell growth rates. The figure shows ${\mathrm{\Lambda }}_p$ versus ${\sigma }_{\lambda }$, compared with the theoretical prediction of Eq. (9) (black lines). The correlation coefficient $C_{\lambda }$ is indicated in the legend. The inset shows the lineage distribution of generation times with ${\sigma }_{\lambda }=0.25$, which is non-Gaussian.

Figure 3

The dependence of the population growth rate on the growth rate noise is well captured by Eq. (9) (black lines) even in the presence of other sources of noise. Colors represent different $C_{\lambda }$ as shown in the legend. Both simulations with time-additive $\mathrm{noise}[{\sigma }_{\xi }=0.1$ with $\alpha =0.5$ ($◯$) and $\alpha =1$ ($△$)] and size-additive $\mathrm{noise}[{\sigma }_{\delta }=0.1$ with $\alpha =0.5$ ($\times$) and $\alpha =1$ ($+$)] are shown.

Figure 4

(a) Numerical simulations of the population growth based on the random-generation-time model. $C_{\tau }$ is the correlation coefficient between the generation times of mother and daughter cells. ${\sigma }_{\tau }$ is the standard deviation of generation times. The solids lines are the theoretical predictions, Eq. (5) in the main text. Here, the mean generation time is fixed to be 1. (b) The amplitude of the exponential growth of the population vs. the generation time of the first cell. The error bar is the standard deviation of 30 samples.

Figure 5

(a) The normalized population growth rate ${\mathrm{\Lambda }}_p\langle \tau \rangle /ln\left(2\right)$ vs. ${\sigma }_{\tau }^2/{\langle \tau \rangle }^2\frac{1+C_{\tau }}{1-C_{\tau }}$ for the random-growth-rate model. Each symbol represents a different $C_{\lambda }$ with changing ${\sigma }_{\lambda }$. All the data collapse on the solid line, which is the theoretical prediction from the random-generation-time model, Eq. (5) in the main text. The inset shows the lineage distribution of generation times at ${\sigma }_{\lambda }=0.25$, which is non-Gaussian. (b) The same analysis for the size-controlled model where the prediction of the random-generation-time model breaks down. Here ${\sigma }_{\delta }=0$, ${\sigma }_{\xi }=0.1$.

Figure 6

(a) ${\mathrm{\Lambda }}_p$ vs. the variance of time additive noise, ${\sigma }_{\xi }^2$. The single-cell growth rate variabilities are indicated in the legend. The black lines are linear fittings based on Eq. (E1). Here $C_{\lambda }=-0.5$, $\alpha =1$. (b) The fitted $C_2$ coefficients as functions of $C_{\lambda }$ and $\alpha$ for time-additive noise. (c) The fitted $C_2$ coefficients as functions of $C_{\lambda }$ and $\alpha$ for size-additive noise.