Abstract
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.
Main Text
Cell morphology is an important adaptive trait that is crucial for bacterial growth, motility, nutrient uptake, and proliferation [1]. When rod-shaped bacteria grow in media with different nutrient availability, both cell length and width increase exponentially with growth rate [2, 3]. 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 [4–8]. A recent study has linked the determination of cell size to a condition-dependent regulation of cell surface-to-volume ratio [9]. However, it remains largely unknown how cell length and width are coupled to regulate rod-like bacterial shapes in diverse growth conditions [10, 11].
Here we investigated the control of cell surface area (S) relative to cell volume (V) for E. coli cells grown under different nutrient conditions, challenged with antibiotics, protein overexpression or depletion, and single gene deletions [3, 9, 12–14]. Collected surface and volume data span two orders of magnitude and exhibit a single power law in this regime: S = µV2/3 (Fig. 1A). Specifically, during steady-state growth [3] µ = 6.24±0.04, suggesting an elegant geometric relation: S ≈ 2πV2/3. This universal surface-to-volume scaling with a constant prefactor, µ, is a consequence of tight control of cell aspect ratio η (length/width), whose mechanistic origin has been puzzling for almost half a century [15, 16]. Specifically, for a sphero-cylindrical shape, S = µV2/3 implies . A constant µ thus defines a constant aspect ratio η = 4.14 ± 0.17 (Fig. 1B-inset), with a coefficient of variation ∼ 14% (Fig. 1B).
This universal surface-to-volume relationship for steady-state growth results in a simple expression for cell surface-to-volume ratio: S/V ≈ 2πV −1/3. Using the phenomenological nutrient growth law V = V0eακ[2], where κ is the population growth rate, we predict a negative correlation between S/V and κ: with V0 the cell volume at κ = 0, and α is the relative rate of increase in V with κ (Fig. 1C). Eq. (1) underlies an adaptive feedback response of the cell — at low nutrient conditions, cells increase their surface-to-volume ratio to promote nutrient influx. Prediction from Eq. (1) is in excellent agreement with the best fit to the experimental data. Furthermore, a constant aspect ratio of ≈ 4 implies and S ≈ 4πw2, where w is the cell width, suggesting stronger geometric constraints than recently proposed [11, 24]. Thus, knowing cell volume as a function of cell cycle parameters [3] we can directly predict cell width and length in agreement with experimental data (Figure 1—figure supplement 1A-B).
To investigate how surface-to-volume ratio is regulated at the single cell level we analysed the morphologies of E. coli cells grown in the mother machine [6] (Fig. 1D). For five different growth media, mean volume and surface area of newborn cells also follow the relationship S = 2πV 2/3, suggesting that a fixed aspect ratio is maintained on average. Deviation from the mean trend is a consequence of intergenerational or cell-to-cell variabilities in length and width (Figure 1—figure supplement 1C-D). Importantly, the probability distribution of aspect ratio is independent of the growth media (Fig. 1E), implying that cellular aspect ratio is independent of cell size as well as growth rate (Fig. 1F). We further analysed cell shape data for seven additional rod-shaped and one coccoid bacteria (Fig. 1G). Surprisingly, all rod-like cells follow the same universal surface-to-volume scaling, while the coccoid S. aureus maintains a much lower aspect ratio η = 1.38 ± 0.18 [21]. This suggests that aspect ratio homeostasis likely emerges from a mechanism that is common to diverse bacterial species.
To elucidate 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 (Fig. 2A). E. coli and other rod-like bacteria maintain a constant width during their cell cycle while elongating exponentially in length L [6]: dL/dt = 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 [18, 25, 26]. While many molecular candidates have been suggested as initiators of division [27], a recent study [28] 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 (Appendix). First, a cytoplasmic component with abundance Pc grows in proportion to cell size (∝ L), as ribosome content increases with cell size [30]. Second, a ring-bound component, Pr, is assembled from the cytoplasmic pool at a constant rate. At the start of the division cycle, Pc = P∗ (a constant), Pr = 0, and the cell divides when Pr reaches a threshold amount, P0, required for the completion of ring assembly. A key ingredient of our model is that P0 scales linearly with the cell circumference πw, preserving the density of FtsZ in the ring. This is consistent with experimental findings that the total FtsZ scales with the cell width [31]. Accumulation of division proteins, P = Pc + Pr − P∗, follows the equation: dP/dt = kPL, where kP is the production rate of division proteins, with P = 0 at the start of the division cycle and P = P0 at cell division (Fig. 2A). As a result, during one division cycle cells grow by adding a length ∆L = P0k/kP, which equals the homeostatic length of newborn cells. Furthermore, recent experiments suggest that the amount of FtsZ synthesised per unit cell length, dP/dL, is constant [28]. 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 [29, 32]. Indeed single cell E. coli data [6] show that ∆L/w is constant on average and independent of growth conditions (Fig. 2B, Figure 2—figure supplement 1A). Thus cell width is a good predictor for added cell length.
To predict cell-shape dynamics under perturbations to growth conditions we simulated our single-cell model (Fig. 2A, Appendix) with an additional equation for cell width that we derived using the model [9]: dS/dt = βV, where β is the rate of surface area synthesis relative to volume (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 (Figure 2—figure supplement 1C-E), changes in cell width result in a transient increase in aspect ratio (η = L/w) during nutrient downshift, or a transient decrease in η during nutrient upshift (Fig. 2C). 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 (∝ 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 [6] (Figure 2—figure supplement 1F). In our model, cell shape changes are controlled by two parameters: the ratio k/kP that determines cell aspect ratio, and k/β that controls cell width (Fig. 2D). Nutrient upshift or downshift only changes the ratio k/β while keeping the steady-state aspect ratio (∝ k/kP) constant.
We further used our model to predict drastic shape changes, leading to deviations from the home-ostatic aspect ratio, when cells are perturbed by FtsZ knockdown, MreB depletion, and antibiotic treatments that induce non steady state filamentation (Fig. 2E). First, FtsZ depletion results in long cells while the width stays approximately constant [29]. We modelled FtsZ knockdown by decreasing kP and simulations quantitatively agree with experimental data. Second, MreB depletion increases the cell width and slightly decreases cell length while keeping growth rate constant [29]. We modelled MreB knockdown by decreasing β as expected for disruption in cell wall synthesis machinery, while simultaneously increasing kP. This increase in kP is consistent with a prior finding that in MreB mutant cells of various sizes, the total FtsZ scales with the cell width [31]. Third, transient long filamentous cells resulted from exposure to high dosages of cell-wall targeting antibiotics that prevent cell division, or DNA-targeting antibiotics that induce filamentation via SOS response [12]. 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 kP, while slightly decreasing surface synthesis rate β. For DNA targeting antibiotics, FtsZ is directly sequestered during SOS response resulting in delayed ring formation and septation [33]. Surprisingly all filamentous cells have a similar aspect ratio of 11.0 ± 1.4, represented by a single curve in the S-V plane (Fig. 2E).
The conserved surface-to-volume scaling in diverse bacterial species, S ∼ V 2/3, is a direct consequence of aspect-ratio homeostasis at the single-cell level. We present a regulatory model where aspect-ratio control is the consequence of a constant ratio between the rate of cell elongation (k) and division protein accumulation (kP). Deviation from the homeostatic aspect ratio is a consequence of altered k/kP, as observed in filamentous cells or MreB depleted cells. 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 (Fig. 1C). Under translation inhibition by ribosome-targeting antibiotics, bacterial cells increase their volume while preserving aspect ratio [3, 9]. 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 [34, 35]. The highest foraging speed has been observed for aspect ratios in the range 4-6 [34], suggesting that the maintenance of an optimal aspect ratio may have evolutionary benefits for cell swarmers.
Author Contributions
NO and SB conceived and designed research. NO and DS performed research. NO and SB wrote the paper.
Acknowledgements
We thank Suckjoon Jun lab (UCSD) for providing single cell shape data for E. coli, and Javier López-Garrido, Guillaume Charras, and Deb Sankar Banerjee for useful comments. We gratefully acknowledge funding from EPSRC grant EP/R029822/1 (SB & NO), Royal Society Tata University Research Fellowship URF/R1/180187 (SB), Royal Society grant RGF/EA/181044 (SB & DS), and UCL Department of Physics & Astronomy (DS).
Cell shape analysis
Bacterial cell surface area and volume are obtained directly from previous publications where these values were reported [3, 9, 14], or they are calculated assuming a sphero-cylindrical cell geometry using reported values for population-averaged cell length and width [12, 13, 17–20, 22, 23, 29]. For a spherocylinder of pole-to-pole length L and width w, the surface area is S = wLπ, and volume is given by . In the case of S. aureus, surface area and volume are computed assuming prolate spheroidal shape using reported population averaged values of cell major axis, c, and minor axis a [21]. Surface area of a prolate spheroid is , and volume is .
Initiator model for cell division
We considered a two-component model for FtsZ dynamics [28] - a cytoplasmic component with abundance Pc, and a Z-ring bound component with abundance Pr. Production of cytoplasmic and ring-bound FtsZ are given by: , , where kP is the constant production rate of cytoplasmic FtsZ, kb is the rate of binding of cytoplasmic FtsZ to the Z-ring, and kd is the rate of disassembly of Z-ring bound FtsZ. At the start of the cell cycle, Pc = P∗ (a constant), Pr = 0, and the cell divides when the number of division proteins at the cell surface, Pr, reaches the threshold amount P0 ∝ πw. Net accumulation of division proteins, P = Pc + Pr − P∗, then follows the simple equation: , where P = 0 at the start of the cell cycle. Assuming kb » kd all the newly synthesized division proteins are recruited to the Z-ring such that cell division occurs when P = P0 (Fig. 2A). Upon division P is reset to 0. In the limit of FtsZ knockdown, septal ring formation occurs may be prevent, which can be realised in the limit that kb is much smaller than the cell elongation rate.
Cell growth simulations
We simulated the single-cell model using coupled equations for the dynamics of cell length (L), division protein (P) production, and cell width (w) (Fig. 2A) [18, 26]. In simulations, when P reaches the threshold, P0, the mother cell divides into two daughter cells whose lengths are 0.5 ± δ fractions of the mother cell. Parameter δ is picked from Gaussian distribution (µ = 0, σ = 0.05). For nutrient shift simulations we simulated 105 asynchronous cells growing with k = 0.75 h−1 (Fig. 2C). In Equation 3, parameter β = 4k/w is obtained from the fit to experimental data for 4k/w vs k (Figure 2—figure supplement 1B) [3]. At time 0 h we change k corresponding to nutrient upshift (k = 1.25, 2 h−1) or nutrient downshift (k = 0.75, 0.25 h−1). We calculated population average of length and width (Fig. 2C, middle), and population average of aspect ratio of newborn cells (Fig. 2C, right). Aspect ratio of newborn cells are binned in time and the bin average is calculated for a temporal bin size of 10 min. Examples of single cell traces during the nutrient shift are shown in Figure 2—figure supplement 1C-E. FtsZ depletion experiment [29] was simulated for w = 1 µm while kP was reduced to 40 % of its initial value. Best fit for MreB depletion experiment [29] was obtained for η ≈ 2.7, by simulating reduction in division protein production rate, kP, and by varying β so that width spans range from 0.9 to 1.8 µm. The best fit for long filamentous cells (resulting from DNA or cell-wall targeting antibiotics) was obtained for η ≈ 11.0. Filamentation was simulated by decreasing kP and β so that w spans the range from 0.9 to 1.4 µm as experimentally observed [12].