Cell Size Regulation Induces Sustained Oscillations in the Population Growth Rate

We study the effect of correlations in generation times on the dynamics of population growth of microorganisms. We show that any nonzero correlation that is due to cell-size regulation, no matter how small, induces long-term oscillations in the population growth rate. The population only reaches its steady state when we include the often-neglected variability in the growth rates of individual cells. We discover that the relaxation timescale of the population to its steady state is determined by the distribution of single-cell growth rates and is surprisingly independent of details of the division process such as the noise in the timing of division and the mechanism of cell-size regulation. We validate the predictions of our model using existing experimental data and propose an experimental method to measure single-cell growth variability by observing how long it takes for the population to reach its steady state or balanced growth.

Figure 1

Lineage tree of populations starting from a single cell. (Left) In the absence of cell size control, the division times (circular markers) become less synchronized over time due to the accumulation of noise in their generation times. (Right) The division times of cells with cell size control stay synchronized due to correlations in the generation times of mother and daughter cells.

Figure 2

Log-scale plot of the expected value of the rate of change of the number of cells in a population starting with a single cell, calculated analytically (red solid curve) and compared with simulation (blue circles). The rate of change of the number of cells can be written as the sum of the division rates (parabolic dashed lines) of all generations [see Eq. (3)]. (Top) In the absence of cell size control, $\alpha =0$, the distribution of division times of higher generations get wider and start to overlap, damping out the oscillations in the growth rate. (Bottom) In the presence of even a small cell size control, $\alpha =0.1$, the distribution of successive division times quickly approach a steady state distribution with a finite variance [see Eq. (5)] leading to the persistence of oscillations in the growth of the population. The distribution of timing of the 7th and 18th generations are highlighted in both cases for comparison.

Figure 3

Oscillations in the population growth rate decay exponentially due to the stochasticity in growth rates of individual cells. The solid red line is the exponential fit used in Fig. 4, the horizontal dashed red line is the steady-state value of the population growth rate, and vertical dashed lines are the expected values of successive division times where the population growth rate peaks. Simulation parameters: $\alpha =0.5$, ${\sigma }_{\xi }=0.1$, $\overline{\kappa }=\ln (2)$, and ${\sigma }_{\kappa }=0.07\overline{\kappa }$.

Figure 4

Simulation results for the decay rate of oscillations in the population growth rate shown as functions of ${\sigma }_{\kappa }$, ${\sigma }_{\xi }$, and $\alpha$: (Top) decay rate increases linearly with the variance of single-cell growth rate distribution, ${\sigma }_{\kappa }^2$ (the red solid line is a parabolic fit; inset is the log-log plot with linear fit); (Middle) the noise in the division process has no effect on the damping of the oscillations in the population growth rate; (Bottom) the mechanism for cell-size control does not affect the decay rate either as long as there is a nonzero cell-size control, $\alpha \ne 0$. Simulation parameters: (top) $\alpha =0.5$ and ${\sigma }_{\xi }=0.1\overline{\tau }$, (middle) $\alpha =0.5$ and ${\sigma }_{\kappa }=0.07\overline{\kappa }$, and (bottom) ${\sigma }_{\xi }=0.1\overline{\tau }$ and ${\sigma }_{\kappa }=0.07\overline{\kappa }$.

Figure 5

Histogram of division times in a synchronized ensemble of single cell lineages from Ref. [2] (see Supplemental Material [25] for details). The decay rate of these oscillations is measured to be $\lambda \approx (0.27\pm 0.03){\mathrm{h}}^{-1}$ which predicts the variability in single-cell growth rates of ${\sigma }_{\kappa }=(0.130\pm 0.007){\mathrm{h}}^{-1}$ compared to direct measurement of the sample standard deviation ${\sigma }_{\kappa }=0.135{\mathrm{h}}^{-1}$.