Super-exponential growth and stochastic size dynamics in rod-like bacteria

Abstract

Proliferating bacterial cells exhibit stochastic growth and size dynamics, but the regulation of noise in bacterial growth and morphogenesis remains poorly understood. A quantitative understanding of morphogenetic noise control, and how it changes under different growth conditions, would provide better insights into cell-to-cell variability and intergenerational fluctuations in cell physiology. Using multigenerational growth and width data of single Escherichia coli and Caulobacter crescentus cells, we deduce the equations governing growth and size dynamics of rod-like bacterial cells. Interestingly, we find that both E. coli and C. crescentus cells deviate from exponential growth within the cell cycle. In particular, the exponential growth rate increases during the cell cycle irrespective of nutrient or temperature conditions. We propose a mechanistic model that explains the emergence of super-exponential growth from autocatalytic production of ribosomes coupled to the rate of cell elongation and surface area synthesis. Using this new model and statistical inference on large datasets, we construct the Langevin equations governing cell growth and size dynamics of E. coli cells in different nutrient conditions. The single-cell level model predicts how noise in intragenerational and intergenerational processes regulate variability in cell morphology and generation times, revealing quantitative strategies for cellular resource allocation and morphogenetic noise control in different growth conditions.

Figure 1

Super-exponential growth in E. coli and C. crescentus cells in different conditions. (A) Ensemble-averaged instantaneous growth rate of E. coli cells grown in tryptic soy broth (TSB) media at $37°\mathrm{C}$ versus normalized time $t/\tau$, where τ is cell-cycle duration. Data are taken from (4). Error bars in all parts show $\pm$ 1 standard error of the mean. The solid green line shows a fit of exponential length growth Eq. 1, dot-dashed red line represents a prediction from exponential volume growth Eq. 2, dashed blue curve shows a fit to the super-exponential growth model Eq. 3, and the dotted orange curve shows fit to exponential volume growth model with constriction dynamics. Fitting parameters: model 1: $L_0=3.87\mu$m, $\kappa =0.041$ min${}^{-1}$; model 2: $L=3.89\mu$m, $k_V=0.044$ min${}^{-1}$; model 2 with constriction: $L_0=3.87\mu$m, $k_V=0.04$ min${}^{-1}$; model 3: $L_0=3.99\mu$m, $k=0.057$ min${}^{-1}$, $\lambda =1.59\mu$m. Cell width ($w=0.98\mu$m) value is taken directly from experimental data. Inset: a simplified cell shape schematic for E. coli, defining the size parameters. (B) Fits of super-exponential growth model Eq. 3 to average growth rate data for seven different growth conditions grown at $37°\mathrm{C}$, taken from (4). Error bars are negligible on the plotted scale. The values of k and λ for each condition are provided in Fig. 2B. (C) Fits of the super-exponential growth model Eq. 3 to average instantaneous growth rate versus $t/\tau$ of C. crescentus cells grown in PYE at three different temperatures. Data are taken from (11). (D) Time-dependent growth rate in filamentous E. coli cells grown in LB at different temperatures, presented in absolute time. Data are taken from (6). Dashed lines in (A)–(D) represent fits of the model in Eq. 3. To see this figure in color, go online.

Figure 2

Intergenerational fluctuations and correlations in E. coli cell growth parameters. (A) Scatterplot showing the correlation between k and λ obtained by fitting super-exponential growth model to E. coli cell length versus time data in TSB. The solid black curve represents a fit of Eq. 8 to the data. We remove outliers and the small number of generations in fast growth conditions with $\lambda <0$ from further analysis (see materials and methods). (B) Ensemble-averaged $⟨\lambda ⟩$ versus $⟨k⟩$ across different growth conditions. The black curve is a model prediction for $⟨\lambda ⟩$ as a function of $⟨k⟩$ according to Eq. 9, with $⟨L_0⟩=\left(1.50\mu \mathrm{m}\right)\exp \left(\left(16.19\min \right)⟨k⟩\right)$, $⟨\alpha R_0⟩=\left(0.84\mu m\right)⟨k⟩\exp \left(\left(16.19\min \right)⟨k⟩\right)$, and ${\sigma }_k^2=\left(11.14{\min }^2\right){⟨k⟩}^4+\left(2.67·{10}^{-5}\right){\min }^{-2}$. (C) Marginal probability distributions of $\delta \lambda =\lambda -⟨\lambda ⟩$ across different growth conditions. (D) Marginal probability distributions of $\delta k=k-⟨k⟩$ for each growth condition shown as solid color curves. Dashed curves of the same color depict fits of log-normal distributions. Experimental data presented in (A)–(D) are taken from (4). (E) Representative contour plots of the model predictions for the joint distribution $P\left(k,\lambda \right)$, corresponding to mean growth rates in glucose, MOPS, and TSB media. Darker blue indicates higher probability. To see this figure in color, go online.

Figure 3

Intergenerational variations in E. coli cell width across growth conditions. (A) Average width $⟨w_0⟩$ versus $⟨k⟩$ across growth conditions. The solid black line shows that $⟨w_0⟩$ increases exponentially with $⟨k⟩$, keeping a fixed length-width aspect ratio ($⟨w_0⟩=0.25⟨L_0⟩=\left(0.38\mu m\right)\exp \left(\left(16.19\min \right)⟨k⟩\right)$). (B) Probability distributions of the intergenerational fluctuations $\delta w_0=w_0-⟨w_0⟩$ across growth conditions scaled by their respective standard deviations (${\sigma }_{w_0}=0.06⟨w_0⟩=\left(0.02\mu m\right)\exp \left(\left(16.19\min \right)⟨k⟩\right)$). The dashed curve shows a universal Gaussian fit to the scaled data. Data are taken from (4). To see this figure in color, go online.

Figure 4

Langevin simulations for stochastic cell size dynamics in fast and slow growth conditions. For (A)–(D), blue trajectories correspond to a relatively fast-growing condition with $⟨k⟩=0.06{\min }^{-1}$ and red to a relatively slow-growing condition with $⟨k⟩=0.03{\min }^{-1}$. (A) Length versus normalized time $t/\tau$ for a single generation, where τ is cell-cycle duration. The dashed black curve is a fit of deterministic super-exponential growth Eq. 3. The transparent bands surrounding each deterministic curve represent the standard deviation in length fluctuations (${\sigma }_{\delta L}=0.066\mu \mathrm{m}$). (B) Length versus absolute time for several generations. (C) Width versus $t/\tau$ for a single generation in normalized time. The solid black shows individual cell mean width. The transparent band around each cell mean width line represents the standard deviation in width fluctuations (${\sigma }_{\delta w}=0.017\mu \mathrm{m}$). (D) Width versus absolute time for several generations. Cell mean width is represented with solid black lines, while population mean width is represented by dashed black. The transparent band around each population mean width line represents the standard deviation in intergenerational fluctuations (${\sigma }_{w_0}=\left(0.02\mu m\right)\exp \left(\left(16.19\min \right)⟨k⟩\right)$). To see this figure in color, go online.

Figure 5

Noise in cell-cycle time and cell size propagates from noise in growth parameters. (A) A colormap showing standard deviation in cell-cycle time τ (${\sigma }_{\tau }$) predicted by our stochastic simulations, as a function of intergenerational noise (${\sigma }_K,{\sigma }_{\mathrm{\Lambda }}$) and intragenerational noise (${\sigma }_{\nu k}$). Each axis is normalized by dividing the varied parameter(s) by the standard value(s) fitted to the data, and the scale of ${\sigma }_{\tau }$ is likewise normalized by the unperturbed value. Each parameter is varied $\pm 50\%$ along each axis. Bins sample 5000 generations for a single cell. (B) A colormap showing normalized standard deviation in initial cell size (${\sigma }_{L_0}/{\sigma }_{L_0}^{\text{fitted}}$), predicted by our simulations, as a function of intergenerational and intragenerational noise. To see this figure in color, go online.

Figure 6

Cellular resource allocation and noise control strategies across nutrient conditions. (A) A network diagram for the underlying protein synthesis model. Ribosomes (R) responsible for synthesizing new proteins do so in an autocatalytic process (rate k) while also producing proteins necessary for growth and division (rates α, β, and γ). (B) Normalized synthesis rates (${\alpha }^{\prime }=\alpha R_0/L_0$, ${\gamma }^{\prime }=\gamma R_0/S_0$, ${\beta }^{\prime }=\beta /\left(R_0{X}_0\right)$) as a function of the mean elongation rate $⟨k⟩$. Solid lines depict the mean values of the normalized rates, determined from fitting our model to experimental data (solid circles, showing mean values), and the transparent bands show one standard deviation of the corresponding distributions (see materials and methods). (C) Mean rates (as depicted in B) normalized by the sum of all rates for a given $⟨k⟩$. (D) The noise (standard deviation) in each rate corresponding to the transparent bands in (B) normalized by the sum of all rates at a given $⟨k⟩$. To see this figure in color, go online.