Relevant parameters in models of cell division control

A recent burst of dynamic single-cell data makes it possible to characterize the stochastic dynamics of cell division control in bacteria. Different models were used to propose specific mechanisms, but the links between them are poorly explored. The lack of comparative studies makes it difficult to appreciate how well any particular mechanism is supported by the data. Here, we describe a simple and generic framework in which two common formalisms can be used interchangeably: (i) a continuous-time division process described by a hazard function and (ii) a discrete-time equation describing cell size across generations (where the unit of time is a cell cycle). In our framework, this second process is a discrete-time Langevin equation with simple physical analogues. By perturbative expansion around the mean initial size (or interdivision time), we show how this framework describes a wide range of division control mechanisms, including combinations of time and size control, as well as the constant added size mechanism recently found to capture several aspects of the cell division behavior of different bacteria. As we show by analytical estimates and numerical simulations, the available data are described precisely by the first-order approximation of this expansion, i.e., by a “linear response” regime for the correction of size fluctuations. Hence, a single dimensionless parameter defines the strength and action of the division control against cell-to-cell variability (quantified by a single “noise” parameter). However, the same strength of linear response may emerge from several mechanisms, which are distinguished only by higher-order terms in the perturbative expansion. Our analytical estimate of the sample size needed to distinguish between second-order effects shows that this value is close to but larger than the values of the current datasets. These results provide a unified framework for future studies and clarify the relevant parameters at play in the control of cell division.

Figure 1

The division-control function follows finite-size scaling. (A) Collapse of the probability distribution of rescaled logarithmic cell size $log(x_0/{\langle x_0\rangle }_{\alpha })$ across different conditions and strains (colors) [equivalent to the condition in Eq. (3)]. (B) Collapse of the functions $f(·)$, defining the mechanism of division control in the discrete-time Langevin framework [Eq. (7)], obtained from experimental data in different conditions. The collapse of $f(·)$ is obtained, as predicted by Eq. (9). The function is evaluated as the conditional average of a cell's birth size ($y$ axis) given the mother's birth size [$x$ axis, see Eq. (7)]. Data from Ref. [1] refer to two strains, P5-ori (a BW25113 derivative strain) and MRR, grown on agarose pads in four nutrient conditions (Glc, CAA, RDM, and LB). Data from Ref. [19] (orange triangle) refer to MG1655 strain in a microfludic device with LB as growth medium.

Figure 2

Unified framework of division control and comparison with data. (a) Division control function $g(·)$, for an adder model (purple solid line) and linearized models with different values of the control parameter $\lambda$ (cyan, gray, and magenta solid lines). This function defines the control mechanism in the model [Eq. (10)]. The adder mechanism is near linear and closest to the linearization with $\lambda \sim 0.5$ [6] (see Appendix pp6). (b) Comparison between data (symbols) and the linearized discrete-time Langevin framework, for different values of the single control parameter $\lambda$. The function $g(·)$ can be estimated as the conditional average of the log-size fluctuation of a cell, given the log-size fluctuation of the mother [see Eq. (10)]. The roughly linear scaling of the symbols suggests that the data are close to a simple “linear response” scenario, and the collapse across different strains and conditions confirms the results of Fig. 1. Values of $\lambda$ around $1/2$ well reproduce the data, but deviations are visible. Data (from Refs. [1] and [19]) refer to different strains of dividing E. coli cells grown in different conditions (see Fig. 1).

Figure 3

Conditions for stability of the deterministic part of cell-size control. (a) Under linear control with negligible noise, $\mathrm{\Delta }q(i+1)=(1-\lambda )\mathrm{\Delta }q$. If $(1-\lambda )>1$ (green line) the system is unstable, while if $(1-\lambda )<1$ (red line) the system is stable. The black line represents the marginally stable case $\mathrm{\Delta }q(i+1)=\mathrm{\Delta }q$. By using a similar argument (see Appendix pp4), it is possible to show that if $\left|g\right(\mathrm{\Delta }q\left)\right|<\left|\mathrm{\Delta }q\right|$ for any $\left|\mathrm{\Delta }q\right|$, then the system is globally stable. (b) In the more general case of a locally stable point the basin of attraction can be obtained as the set of values of $\mathrm{\Delta }q$ such that $\left|g\right(\mathrm{\Delta }q\left)\right|<\left|\mathrm{\Delta }q\right|$.

Figure 4

Estimated threshold for detection of non-linear contributions to cell-division control. (a) The tradeoff between two competitive terms determines an optimal cell-size fluctuation to identify cell-size control. On one hand, large deviations of size from the average correspond to stronger corrections and make differences between mechanisms more detectable. On the other hand, fewer cells have large fluctuations, reducing the statistical power and increasing the sampling noise. The optimal fluctuation value is the one that minimizes the error on the inferred cell-size control mechanism. Our calculations [Eq. (17)] show that when fluctuations are rescaled by the variance of the distribution, the optimal value is independent of the cell-size control mechanism, as shown in the plots of the contributions described by Eqs. (15) (discriminatory power, green line) and (17) (sampling level, blue line). (b) Number of cells measured vs. the inverse of ${\sigma }_q$ [which is related to the coefficient of variation via equation (E15)] for the available datasets (points). The two gray lines represent the threshold of detectability obtained using Eq. (21), with $\epsilon =0.1$ or $\epsilon =0.5$. All the available data sets lay below the threshold of detectability (with one exception in the case of $\epsilon =0.5$), suggesting that a linear model of division control is sufficient to describe these data.

Figure 5

Detectability of adder mechanism from simulated models and direct test. Plotted data refer to simulations of models with realistic parameters and sampling of cells. (a) Conditional distribution of added size given initial size for different initial sizes (different colors) obtained with the linearized model [Eq. (14)] with $\lambda =0.5$. The linearized model reproduces visually the collapse expected for the adder model. The simulations consider conservatively a coefficient of variation of the added size equal to 0.3 (which is larger than the observed values [5]) and a total number of cells $N=50000$. (b) Error test on the collapse of the distribution of added size (estimated as the average $L_2$ distance between all pairs of curves) for different values of $N$ and $\lambda$. The horizontal dashed lines represent the expected error in collapse in the adder model due to fluctuations. For these parameter values, the error in the collapse for the model with $\lambda =0.5$ starts to be relevant when $N\sim 10000$. This test may be applied to empirical data: in order for an adder to be detectable, the solid line should stay above the dashed line.