Cell morphology is an important adaptive trait that is crucial for bacterial growth, motility, nutrient uptake, and proliferation (Young, 2006). When rod-shaped bacteria grow in media with different nutrient availability, both cell length and width increase with growth rate (Schaechter et al., 1958; Si et al., 2017). At the single-cell level, control of cell volume in many rod-shaped cells is achieved via an adder mechanism, whereby cells elongate by a fixed length per division cycle (Amir, 2014; Campos et al., 2014; Taheri-Araghi et al., 2015; Deforet et al., 2015; Wallden et al., 2016; Banerjee et al., 2017). A recent study has linked the determination of cell size to a condition-dependent regulation of cell surface-to-volume ratio (Harris and Theriot, 2016). However, it remains largely unknown how cell length and width are coupled to regulate rod-like bacterial shapes in diverse growth conditions (Volkmer and Heinemann, 2011; Belgrave and Wolgemuth, 2013; Colavin et al., 2018; Shi et al., 2018).

Here we investigated the relation between cell surface area ($S$) and cell volume ($V$) for E. coli cells grown under different nutrient conditions, challenged with antibiotics, protein overexpression or depletion, and single gene deletions (Nonejuie et al., 2013; Harris and Theriot, 2016; Si et al., 2017; Vadia et al., 2017; Campos et al., 2018; Gray et al., 2019). Collected surface and volume data span two orders of magnitude and exhibit a single power law in this regime: $S=\gamma V^{2/3}$ (Figure 1A). Specifically, during steady-state growth (Si et al., 2017), γ = 6.24 ± 0.04, suggesting an elegant geometric relation: $S\approx 2\pi V^{2/3}$. This surface-to-volume scaling with a constant prefactor, $\gamma$, is a consequence of tight control of cell aspect ratio $\eta$ (length/width) (Figure 1D), whose mechanistic origin has been puzzling for almost half a century (Zaritsky, 1975; Zaritsky, 2015). Specifically, for a sphero-cylindrical bacterium, $S=\gamma V^{2/3}$ implies $\gamma =\eta \pi {\left(\frac{\eta \pi }{4}-\frac{\pi }{12}\right)}^{-2/3}$. A constant $\gamma$ thus defines a constant aspect ratio η = 4.14 ± 0.17 (Figure 1B-inset), with a coefficient of variation ∼14% (Figure 1B).

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The surface-to-volume relation for steady-state growth, $S\approx 2\pi V^{2/3}$, results in a simple expression for cell surface-to-volume ratio: $S/V\approx 2\pi V^{-1/3}$. Using the phenomenological nutrient growth law $V=V_0{e}^{\alpha \kappa }$ (Schaechter et al., 1958), where $\kappa$ is the population growth rate, a negative correlation emerges between $S/V$ and $\kappa$:

with $V_0$ the cell volume at $\kappa =0$, and $\alpha$ is the relative rate of increase in $V$ with $\kappa$ (Figure 1C). In Equation (1) underlies an adaptive feedback response of the cell — at low nutrient concentrations, cells increase their surface-to-volume ratio to promote nutrient influx (Si et al., 2017; Harris and Theriot, 2018). Prediction from Equation (1) is in excellent agreement with the best fit to the experimental data. Furthermore, a constant aspect ratio of ≈4 implies $V\approx \sqrt{8}{w}^3$ and $S\approx 4\pi w^2$, where $w$ is the cell width, suggesting stronger geometric constraints than recently proposed (Harris and Theriot, 2018; Shi et al., 2018). Thus, knowing cell volume as a function of cell cycle parameters (Si et al., 2017) we can directly predict cell width and length under changes in growth media, in agreement with experimental data (Figure 1—figure supplement 1A–B). We further analysed cell shape data for 48 rod-shaped bacteria, one rod-shaped Archaea (H. vulcanii), two long spiral Spirochete, and one coccoid bacteria (Figure 1E). Collected data for all rod-shaped cells follow closely the relationship $S\approx 2\pi V^{2/3}$, while the long Spirochetes deviate from this curve (Figure 1D–E). Coccoid S. aureus also follows the universal scaling relation $S=\gamma V^{2/3}$ (with γ = 4.92), but maintains a much lower aspect ratio η = 1.38 ± 0.18 (Quach et al., 2016) when exposed to different antibiotics (Figure 1D–E). Therefore, aspect-ratio preservation likely emerges from a mechanism that is common to diverse rod-shaped and coccoid bacterial species.

To investigate how aspect ratio is regulated at the single cell level we analysed the morphologies of E. coli cells grown in the mother machine (Taheri-Araghi et al., 2015) (Figure 2A,B). For five different growth media, mean volume and surface area of newborn cells also follow the relationship $S=2\pi V^{2/3}$, suggesting that a fixed aspect ratio is maintained on average. In the single-cell data, slight deviation from the 2/3 scaling is a consequence of large fluctuations in newborn cell lengths for a given cell width (Figure 2—figure supplement 1A–B). Importantly, the probability distribution of aspect ratio is independent of the growth media (Figure 2B), implying that cellular aspect ratio is independent of cell size as well as growth rate.

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To explain the origin of aspect ratio homeostasis we developed a quantitative model for cell shape dynamics that accounts for the coupling between cell elongation and the accumulation of cell division proteins FtsZ (Figure 2C). Our model is thus only applicable to bacteria that divide using the FtsZ machinery. E. coli and other rod-like bacteria maintain a constant width during their cell cycle while elongating exponentially in length $L$ (Donachie et al., 1976; Taheri-Araghi et al., 2015): $\mathrm{d}L/\mathrm{d}t=kL$, with $k$ the elongation rate. Cell division is triggered when a constant length is added per division cycle — a mechanism that is captured by a model for threshold accumulation of division initiator proteins, produced at a rate proportional to cell size (Basan et al., 2015; Deforet et al., 2015; Ghusinga et al., 2016). While many molecular candidates have been suggested as initiators of division (Adams and Errington, 2009), a recent study (Si et al., 2019) has identified FtsZ as the key initiator protein that assembles a ring-like structure in the mid-cell region to trigger septation.

Dynamics of division protein accumulation can be described using a two-component model. First, a cytoplasmic component with abundance $P_{\mathrm{c}}$ grows in proportion to cell size ($\propto L$), as ribosome content increases with cell size (Marguerat and Bähler, 2012). Second, a ring-bound component, $P_{\mathrm{r}}$, is assembled from the cytoplasmic pool at a constant rate. Dynamics of the cytoplasmic and ring-bound FtsZ are given by:

where $k_P$ is the constant production rate of cytoplasmic FtsZ, $k_{\mathrm{b}}$ is the rate of binding of cytoplasmic FtsZ to the Z-ring, and $k_{\mathrm{d}}$ is the rate of disassembly of Z-ring bound FtsZ. At the start of the cell cycle, we have $P_{\mathrm{c}}=P^{*}$ (a constant) and $P_r=0$. Cell divides when $P_{\mathrm{r}}$ reaches a threshold amount, $P_0$, required for the completion of ring assembly. A key ingredient of our model is that $P_0$ scales linearly with the cell circumference, $P_0=\rho \pi w$, preserving the density $\rho$ of FtsZ in the ring. This is consistent with experimental findings that the total FtsZ scales with the cell width (Shi et al., 2017). Accumulation of FtsZ proteins, $P=P_{\mathrm{c}}+P_{\mathrm{r}}-P^{*}$, follows the equation: $\mathrm{d}P/\mathrm{d}t=k_{P}L$, where $k_P$ is the production rate of division proteins, with $P=0$ at the start of the division cycle. We assume that $k_{\mathrm{b}}\gg k_{\mathrm{d}}$, such that all the newly synthesized cytoplasmic proteins are recruited to the Z-ring at a rate much faster than growth rate (Söderström et al., 2018). As a result, cell division occurs when $P=P_0$ (Figure 2C). Upon division $P$ is reset to 0 for the two daughter cells. It is reasonable to assume that all the FtsZ proteins are in filamentous form at cell division, as the concentration of FtsZ in an average E. coli cell is in the range 4–10 μM, much higher than the critical concentration 1 μM (Erickson et al., 2010).

From the model it follows that during one division cycle cells grow by adding a length $\mathrm{\Delta }L=P_{0}k/k_P$, which equals the homeostatic length of newborn cells. Furthermore, recent experiments suggest that the amount of FtsZ synthesised per unit cell length, $\mathrm{d}P/\mathrm{d}L$, is constant (Si et al., 2019). This implies,

Aspect ratio homeostasis is thus achieved via a balance between the rates of cell elongation and division protein production, consistent with observations that FtsZ overexpression leads to minicells and FtsZ depletion induces elongated phenotypes (Potluri et al., 2012; Zheng et al., 2016). Indeed single cell E. coli data (Taheri-Araghi et al., 2015) show that $\mathrm{\Delta }L/w$ is constant on average and independent of growth conditions (Figure 2D). Furthermore, added length correlates with cell width during one cell cycle implying that the cell width is a good predictor for added cell length (Figure 2—figure supplement 1C–D).

To predict cell-shape dynamics under perturbations to growth conditions we simulated our single-cell model (Figure 3, Materials and methods) with an additional equation for cell width that we derived from a recent model proposed by Harris and Theriot (2016): $\mathrm{d}S/\mathrm{d}t=\beta V$, where $\beta$ is the rate of surface area synthesis relative to volume and is a linearly increasing function of $k$ (Figure 3—figure supplement 1A). This model leads to an equation for the control of cell width for a sphero-cylinder shaped bacterium,

such that $w=4k/\beta$ at steady-state. It then follows from Equation (4) that the added cell length $\mathrm{\Delta }L\propto k^2/\beta k_{\mathrm{P}}$. However, our model for division control is mechanistically different from Harris and Theriot (2016). In the latter, cells accumulate a threshold amount of excess surface area material to trigger septation, which does not lead to aspect ratio preservation. By contrast, we propose that cells divide when they accumulate a threshold amount of division proteins in the Z-ring, proportional to the cell diameter.

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We simulated nutrient shift experiments using the coupled equations for cell length, width and division protein production (Materials and methods). When simulated cells are exposed to new nutrient conditions (Figure 3—figure supplement 1B–E), changes in cell width result in a transient increase in aspect ratio ($\eta =L/w$) during nutrient downshift, or a transient decrease in $\eta$ during nutrient upshift (Figure 3C). After nutrient shift, aspect ratio reaches its pre-stimulus homeostatic value over multiple generations. Typical timescale for transition to the new steady-state is controlled by the growth rate of the new medium ($\propto k^{-1}$), such that the cell shape parameters reach a steady state faster in media with higher growth rate. This result is consistent with the experimental observation that newborn aspect ratio reaches equilibrium faster in fast growing media (Taheri-Araghi et al., 2015) (Figure 3—figure supplement 1F). In our model, cell shape changes are controlled by two parameters: the ratio $k/k_P$ that determines cell aspect ratio, and $k/\beta$ that controls cell width (Figure 4A). Nutrient upshift or downshift only changes the ratio $k/\beta$ while keeping the steady-state aspect ratio $(\propto k/k_P)$ constant.

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We further used our model to predict drastic shape changes leading to deviations from the homeostatic aspect ratio when cells are perturbed by FtsZ knockdown, MreB depletion, and antibiotic treatments that induce non steady state filamentation (Figure 4B). First, FtsZ depletion results in long cells while the width stays approximately constant, $S\propto V^{0.95}$ (Figure 4—figure supplement 1C), data from Zheng et al. (2016). We modelled FtsZ knockdown by decreasing $k_P$ and simulations quantitatively agree with experimental data. Second, MreB depletion increases the cell width and slightly decreases cell length while keeping growth rate constant (Zheng et al., 2016). We modelled MreB knockdown by decreasing $\beta$ as expected for disruption in cell wall synthesis machinery, while simultaneously increasing $k_P$ (Materials and methods). This increase in $k_P$ is consistent with a prior finding that in MreB mutant cells of various sizes, the total FtsZ scales with the cell width (Shi et al., 2017). Furthermore, cells treated with MreB inhibitor A22 induce envelope stress response system (Rcs) that in turn activates FtsZ overproduction (Carballès et al., 1999; Cho et al., 2014). Third, transient long filamentous cells result from exposure to high dosages of cell-wall targeting antibiotics that prevent cell division, or DNA-targeting antibiotics that induce filamentation via SOS response (Nonejuie et al., 2013). Cell-wall targeting antibiotics inhibit the activity of essential septum forming penicillin binding proteins, preventing cell septation. We modelled this response as an effective reduction in $k_P$, while slightly decreasing surface synthesis rate $\beta$ (Materials and methods). For DNA targeting antibiotics, FtsZ is directly sequestered during SOS response resulting in delayed ring formation and septation (Chen et al., 2012). Surprisingly all filamentous cells have a similar aspect ratio of 11.0 ± 1.4, represented by a single curve in the $S$-$V$ plane (Figure 4B).

The conserved surface-to-volume scaling in diverse bacterial species, $S\sim V^{2/3}$, is a direct consequence of aspect-ratio homeostasis at the single-cell level. We present a regulatory model (Figure 2C) where aspect-ratio control is the consequence of a constant ratio between the rate of cell elongation ($k$) and division protein accumulation ($k_P$). Deviation from the homeostatic aspect ratio is a consequence of altered $k/k_P$, as observed in filamentous cells, FtsZ or MreB depleted cells (Figure 4B). By contrast, drugs that target cell wall biogenesis, for example Fosfomycin, do not alter $k/k_P$ and maintain cellular aspect ratio (Figure 4—figure supplement 1C).

Our study suggests that cell width is an essential shape parameter for determining cell length in E. coli (Figure 2—figure supplement 1C–D). This is to be contrasted with B. subtilis, where cell width stays approximately constant across different media, while elongating in length (Sharpe et al., 1998). However, FtsZ recruitment in B. subtilis is additionally controlled by effector UgtP, which localises to the division site in a nutrient-dependent manner and prevents Z-ring assembly (Weart et al., 2007). This can be interpreted as a reduction in $k_P$ with increasing $k$, within the framework of our model. As a result, B. subtilis aspect ratio ($\propto k/k_P$) is predicted to increase with increasing growth rate.

Aspect ratio control may have several adaptive benefits. For instance, increasing cell surface-to-volume ratio under low nutrient conditions can result in an increased nutrient influx to promote cell growth (Figure 1C). Under translation inhibition by ribosome-targeting antibiotics, bacterial cells increase their volume while preserving aspect ratio (Harris and Theriot, 2016; Si et al., 2017). This leads to a reduction in surface-to-volume ratio to counter further antibiotic influx. Furthermore, recent studies have shown that the efficiency of swarming bacteria strongly depends on their aspect ratio (Ilkanaiv et al., 2017; Jeckel et al., 2019). The highest foraging speed has been observed for aspect ratios in the range 4–6 (Ilkanaiv et al., 2017), suggesting that the maintenance of an optimal aspect ratio may have evolutionary benefits for cell swarmers.

All data generated or analysed during this study are referenced in the manuscript and supporting files.

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Author details

1. Nikola Ojkic

   Department of Physics and Astronomy, Institute for the Physics of Living Systems, University College London, London, United Kingdom

   Contribution

   Conceptualization, Data curation, Formal analysis, Validation, Investigation, Visualization, Methodology, Writing—original draft

   Competing interests

   No competing interests declared

2. Diana Serbanescu

   Department of Physics and Astronomy, Institute for the Physics of Living Systems, University College London, London, United Kingdom

   Contribution

   Data curation, Investigation, Methodology

   Competing interests

   No competing interests declared

3. Shiladitya Banerjee

   Department of Physics and Astronomy, Institute for the Physics of Living Systems, University College London, London, United Kingdom

   Present address

   Department of Physics, Carnegie Mellon University, Pittsburgh, United States

   Contribution

   Conceptualization, Resources, Supervision, Funding acquisition, Validation, Investigation, Visualization, Writing—original draft, Project administration

   For correspondence

   shiladitya.banerjee@ucl.ac.uk

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-8000-2556

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