The Empirical Fluctuation Pattern of E. coli Division Control

Abstract

In physics, it is customary to represent the fluctuations of a stochastic system at steady state in terms of linear response to small random perturbations. Previous work has shown that the same framework describes effectively the trade-off between cell-to-cell variability and correction in the control of cell division of single E. coli cells. However, previous analyses were motivated by specific models and limited to a subset of the measured variables. For example, most analyses neglected the role of growth rate variability. Here, we take a comprehensive approach and consider several sets of available data from both microcolonies and microfluidic devices in different growth conditions. We evaluate all the coupling coefficients between the three main measured variables (interdivision times, cell sizes and individual-cell growth rates). The linear-response framework correctly predicts consistency relations between a priori independent experimental measurements, which confirms its validity. Additionally, the couplings between the cell-specific growth rate and the other variables are typically non zero. Finally, we use the framework to detect signatures of mechanisms in experimental data involving growth rate fluctuations, finding that (1) noise-generating coupling between size and growth rate is a consequence of inter-generation growth rate correlations and (2) the correlation patterns agree with a near-adder model where the added size has a dependence on the single-cell growth rate. Our findings define relevant constraints that any theoretical description should reproduce, and will help future studies aiming to falsify some of the competing models of the cell cycle existing today in the literature.

Keywords: control of cell division; data interpretation; fluctuation patterns; linear response theory; models; single-cell growth and division; statistical; theoretical.

Figure 1

Scaling of size fluctuations, quantified by log-initial size logV₀, and single-cell growth rate α for different datasets (symbols and colors) and conditions. Each symbol represents an average over all single cells available in the dataset (defined by shape and color) for a specific strain in a specific condition. (A) Shows that the standard deviation of the rescaled log-initial size log(V₀/〈V₀〉), is constant across conditions and experiments, and does not depend on the average log-initial size 〈logV₀〉. This fact is a consequence of the collapse of initial size distribution (Taheri-Araghi et al., ; Kennard et al., ; Grilli et al., 2017), since the standard deviation log(V₀/〈V₀〉) is a function of the coefficient of variation of the initial size only (Grilli et al., 2017). (B) Shows the standard deviation of the rescaled growth rate α/〈α〉 (which corresponds to the coefficient of variation of α). This quantity is approximately constant for some datasets (e.g., Taheri-Araghi et al., , blue squares), suggesting a collapse of the growth rate distribution (Taheri-Araghi et al., 2015), while it shows a decreasing trend in others (e.g., Kennard et al., , green circles).

Figure 2

Illustration of the linear-response framework of size control, including growth rate fluctuations. (A,B) Show the dependency of the interdivision time fluctuations from fluctuations of logarithmic initial size and growth rate for one dataset (Wallden et al., intermediate growth rate). Since growth rate and size fluctuations are not independent (C), the slopes observed in (A,B) are determined by a combination of the coupling between size fluctuation and interdivision time (with strength λ_τq, see Equation 2) and the effect of growth rate fluctuation on interdivision time (with strength λ_τα, see Equation 2). The method described in the text disentangles direct and indirect effect on the slopes of (A–C) to obtain the direct couplings between τ, α, and q, which are shown in (D–F), for different datasets (symbols and colors) and conditions. Interestingly, all the possible couplings are non-zero. Current models do not describe these nontrivial correlations, which poses a challenge to future models.

Figure 3

Alternative parameterizations of cell-size control linear response are consistent across different experimental datasets and conditions (symbols and colors are the same as in Figure 1). One can parametrize the effect of size and growth rate fluctuations on interdivision time, or, alternatively, their effect on the final size (or the elongation G). (A,C) Show the effect of log-size fluctuation on elongation (with strength λ_Gq, see Equation 3) and the effect of growth rate fluctuation on interdivision time (with strength λ_Gα). The linear framework predicts consistency relations between the alternative parameterizations (B,D), which are verified across datasets and conditions in the plots in the right panel.

Figure 4

A grower is not a timer. The absence of a correlation between elongation and initial size (A) does not necessarily imply a timer (B), where the interdivision time is independent of the initial size. (C) Shows the alternative scenario (the grower), where the growth rate depends on the initial size. This correlation, together with the independence of the elongation from the growth rate, implies a dependence of the interdivision time on both initial size and growth rate. Both panels are obtained from numerical simulations of the corresponding linearized model with σ_α/〈α〉 = 0.2 and ${\sigma }_q^2=0.2$.

Figure 5

Mother-daughter correlations in growth rate induce negative values of λ_αq. (A) Shows the correlation between mother and daughter growth rate across datasets (symbols and colors, see Figure 1) and conditions. All the datasets display a significant positive correlation. Panel shows the value of λ_αq predicted from our theory assuming the empirical mother-daughter growth rate correlations (see Equation 14), compared to the empirical ones for negative values of λ_αq. The assumed mechanism correctly predicts negative values of λ_αq, but the prediction cannot capture the datasets with positive values of λ_αq, which are shown in (C) (not visible in B).

Figure 6

Dependency of added size on single-cell growth rates and effect on the values of λ_Gα. (A) Shows that the average logarithmic added size is linear when plotted against the average growth rate (for all the conditions from Wallden et al., 2016). We consider two extreme scenarios for the logaritmic added size of individual cells: it could depend only on the average growth rate (gold line) or it could linearly grow with the single-cell growth rate (purple line), with the same slope observed in (A). (B) Shows the results of the two scenarios for one condition (intermediate growth rate data set from Wallden et al., 2016). These two scenarios translate into two different predictions for the value of λ_Gα, which are reported in panel (C) and tested in panel (D). The non-zero values of λ_Gα are therefore a consequences of the dependence of the added size on the the single-cell growth rate.