Universal surface-to-volume scaling and aspect ratio homeostasis in rod-shaped bacteria

Rod-shaped bacterial cells can readily adapt their lengths and widths in response to environmental changes. While many recent studies have focused on the mechanisms underlying bacterial cell size control, it remains largely unknown how the coupling between cell length and width results in robust control of rod-like bacterial shapes. In this study we uncover a universal surface-to-volume scaling relation in Escherichia coli and other rod-shaped bacteria, resulting from the preservation of cell aspect ratio. To explain the mechanistic origin of aspect-ratio control, we propose a quantitative model for the coupling between bacterial cell elongation and the accumulation of an essential division protein, FtsZ. This model reveals a mechanism for why bacterial aspect ratio is independent of cell size and growth conditions, and predicts cell morphological changes in response to nutrient perturbations, antibiotics, MreB or FtsZ depletion, in quantitative agreement with experimental data.

, where κ is the population growth rate, we predict a negative correlation 35 between S/V and κ:

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with V 0 the cell volume at κ = 0, and α is the relative rate of increase in V with κ (Fig. 1C). Eq. supplement 1C-D). Importantly, the probability distribution of aspect ratio is independent of the 51 growth media (Fig. 1E), implying that cellular aspect ratio is independent of cell size as well as 52 growth rate (Fig. 1F). We further analysed cell shape data for seven additional rod-shaped and 53 one coccoid bacteria (Fig. 1G). Surprisingly, all rod-like cells follow the same universal surface-to-54 volume scaling, while the coccoid S. aureus maintains a much lower aspect ratio η = Aspect ratio homeostasis is thus achieved via a balance between the rates of cell elongation and Growth rate (h -1 )   ( Figure 2-figure supplement 1B). For a sphero-cylinder shaped bacterium, we have such that w = 4k/β at steady-state. When simulated cells are exposed to new nutrient conditions supplement 1F). In our model, cell shape changes are controlled by two parameters: the ratio k/k P 99 that determines cell aspect ratio, and k/β that controls cell width (Fig. 2D). Nutrient upshift or 100 downshift only changes the ratio k/β while keeping the steady-state aspect ratio (∝ k/k P ) constant. 101 We further used our model to predict drastic shape changes, leading to deviations from the home-102 ostatic aspect ratio, when cells are perturbed by FtsZ knockdown, MreB depletion, and antibiotic 103 treatments that induce non steady state filamentation (Fig. 2E). First, FtsZ depletion results in 104 long cells while the width stays approximately constant [29]. We modelled FtsZ knockdown by 105 decreasing k P and simulations quantitatively agree with experimental data. Second, MreB deple-106 tion increases the cell width and slightly decreases cell length while keeping growth rate constant 107 [29]. We modelled MreB knockdown by decreasing β as expected for disruption in cell wall synthe-108 sis machinery, while simultaneously increasing k P . This increase in k P is consistent with a prior where aspect-ratio control is the consequence of a constant ratio between the rate of cell elongation 121 (k) and division protein accumulation (k P ). Deviation from the homeostatic aspect ratio is a 122 consequence of altered k/k P , as observed in filamentous cells or MreB depleted cells. Aspect ratio 123 control may have several adaptive benefits. For instance, increasing cell surface-to-volume ratio 124 under low nutrient conditions can result in an increased nutrient influx to promote cell growth (Fig.   125   1C). Under translation inhibition by ribosome-targeting antibiotics, bacterial cells increase their 126 volume while preserving aspect ratio [3,9]. This leads to a reduction in surface-to-volume ratio   where P = 0 at the start of the cell cycle. Assuming k b k d all the newly synthesized division prevent, which can be realised in the limit that k b is much smaller than the cell elongation rate.
Cell growth simulations. We simulated the single-cell model using coupled equations for the 156 dynamics of cell length (L), division protein (P ) production, and cell width (w) ( Fig. 2A) [18, 26].

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In simulations, when P reaches the threshold, P 0 , the mother cell divides into two daughter cells 158 whose lengths are 0.5 ± δ fractions of the mother cell. Parameter δ is picked from Gaussian 159 distribution (µ = 0, σ = 0.05). For nutrient shift simulations we simulated 10 5 asynchronous cells 160 growing with k = 0.75 h −1 (Fig. 2C). In Equation 3, parameter β = 4k/w is obtained from the fit  [29] was simulated for w = 1 µm while k P was reduced to 40 % of its initial value.

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Best fit for MreB depletion experiment [29] was obtained for η ≈ 2.7, by simulating reduction in 169 division protein production rate, k P , and by varying β so that width spans range from 0.9 to 1.8 170 µm. The best fit for long filamentous cells (resulting from DNA or cell-wall targeting antibiotics) 171 was obtained for η ≈ 11.0. Filamentation was simulated by decreasing k P and β so that w spans 172 the range from 0.9 to 1.4 µm as experimentally observed [12]. 173 We thank Suckjoon Jun lab (UCSD) for providing single cell shape data for E. coli, and Javier For each pair of values (σ(w), σ(L)) we pick 10 4 random numbers from corresponding distributions and computed surface-to-volume scaling exponent. Total of 2500 pairs (σ(w), σ(L)) were used. We obtained σ(w) and σ(L) of newborn cells grown in mother machine by fitting experimental distributions. These values are shown by coloured points that correspond to different growth media (data from Taheri-Araghi et al. Cell-to-cell or intergenerational variability results in scaling exponents slightly above 2/3, as expected. Newborn cells with aspect ratio between 5-6 or 3-3.5 were tracked over generations. Population average for given generation number over 737-2843 cells for different growth condition is shown.