## SUMMARY

Cell size control emerges from a regulated balance between the rates of cell growth and division. In bacteria, simple quantitative laws connect cellular growth rate to ribosome abundance. However, it remains poorly understood how translation regulates bacterial cell size and shapes under growth perturbations. Here we develop a whole-cell model for growth dynamics in rod-shaped bacteria that links ribosomal abundance with cell geometry, division control, and the extracellular environment. Our study reveals that cell shape maintenance under nutrient perturbations requires a balanced trade-off between ribosomes and division protein synthesis. Deviations from this trade-off relationship is predicted under translational perturbations, leading to distinct modes of cell morphological changes, in agreement with single-cell experimental data on *Escherichia coli.* Furthermore, by calibrating our model with experimental data, we predict how combinations of nutrient-, translational- and shape perturbations can be chosen to optimize bacterial growth fitness and drug resistance.

## INTRODUCTION

Cell size maintenance is essential for regulating cell physiology, function and fitness [1]. Maintaining a characteristic cell size necessitates an intricate balance between cell growth and division rates. How this balance is achieved in different growth conditions remains an outstanding question. It has been known for over six decades that bacteria modulate their size in response to changes in nutrient conditions. Quantifying the cell size and growth rates of *Salmonella enterica* grown in different nutrient media, Schaechter and colleagues discovered the *nutrient growth law* – bacterial cell size increases exponentially with the population growth rate [2]. High-throughput single-cell studies in recent years have confirmed this result for evolutionary divergent *Escherichia coli* and *Bacillus subtilis* [3–6], suggesting common strategies for bacterial cell size control. However, single-cell data show deviations from the nutrient growth law in experiments altering cellular proteomics [4, 7], leaving open the connection between cell size, growth rate and protein synthesis.

At the single-cell level, cell size homeostasis is achieved via the *adder* mechanism, whereby cells add a constant volume between consecutive division events, irrespective of the cell size at birth [8–11]. As a result of this strategy, cells deviating from the average homeostatic size quickly converge to the average size within a few generations [5, 12–15]. This strategy for cell size homeostasis is followed by a wide range of bacterial species including *E. coli*, *B. subtilis*, *C. crescentus* and *P. aeruginosa* [11, 12, 14–16], but does not provide a molecular-level understanding of the mechanism for size control [17]. Recent studies [14, 18–20] have linked the adder model of cell size homeostasis to the accumulation of a threshold amount of division proteins (FtsZ) to trigger cell division. However, it remains unknown how the synthesis of division proteins is altered by nutrients or translational perturbations in order to regulate cell size.

A key component in understanding cell size regulation is the interdependence between growth rate and the macromolecular composition of the cell. The nutritional content of the growth medium sets the specific growth rate [2, 21], which in turn regulates the macromolecular composition of cells [22, 23]. For exponentially growing *E.coli* cells, RNA and ribosome abundance increases linearly with increasing the specific growth rate [7, 24–28]. This implies an upregulation in translation leading to increased protein production for growth [7, 29, 30] and cell size inflation. While this model is in agreement with experimental observations for cell size increase with increasing nutrient concentrations, it fails to explain cell size changes under translation inhibition [4, 7]. In particular, it remains unclear whether translation inhibition would lead to an increase in cell size such that there is a positive correlation between cell size and ribosome abundance, or a decrease in cell size with growth rate reduction. Both these behaviors are observed in experiments [4]. To explain how translation and nutrient quality regulates cell morphologies, we develop a whole-cell coarse-grained theory that links ribosomes with cell geometry, division control and the extracellular environment.

Our theoretical framework combines a mechanistic model of cell shape and division with an extended ribosomal resource allocation model, allowing us to quantitatively predict cell size changes under nutrient shifts and translational perturbations. We use ribosome abundance as the one of the key regulatory variable as approximately 85% of cellular RNA encodes for rRNA that is folded in ribosomes [31, 32]. We also assume that all the nutrients transported from the extracellular medium into the cell are used in the production of ribosomes and other proteins. This is because over 80% of cell’s energy requirements for biomass is spent on proteins and rRNA synthesis [33]. Using this framework, we uncover a model for balanced allocation of ribosomal resources towards cell growth and division. We find that a balanced trade-off between the rates of cell growth and division proteins synthesis sets bacterial size under nutrient shifts. As a result, in rich media, cells produce division proteins slower than the rate of cell elongation leading to larger cell sizes.

We then extend our framework to predict cell morphologies under translation inhibition across different nutrient media. Our model predicts three different types of cell morphological response unifying past experimental observations [4, 7, 31]. First, cells deprived of nutrients allocate more ribosomes towards growth which results in an increase in volume. Second, cells grown in rich nutrient media favor resource allocation towards division and thus a decrease in volume is observed under translation inhibition. Under optimal growth conditions cells preserve the balance between growth and division protein synthesis, such that cell size is invariant under translational perturbations. We show that cell size changes are intimately coupled to the regulation of cell surface-to-volume ratio – a fitness metric that controls nutrient and antibiotic influx rates. We therefore investigate the relationship between cell shape, nutrient quality and bacterial growth rate under translational perturbations. We predict that round cells are most resistant to translation-inhibitory antibiotics, and that drug resistance increases with increasing nutrient quality. Thus, induced filamentation could have a negative impact on bacterial growth fitness [34], whereas cell rounding could promote bacterial resistance to ribosome-targeting antibiotics.

## RESULTS

### Cell size control emerges from nutrient-dependent trade-off between rates of cellular growth and division protein synthesis

To understand how bacterial cell size changes with the nutrient specific growth rate, we develop a model for the allocation of ribosomal resources towards cell growth and division protein synthesis. During each cell cycle, cells elongate exponentially in volume (*V*) at a rate *κ*. At steady-state, *κ* depends linearly on the ribosomal mass fraction *r* (≈ RNA/protein ratio), such that
where *κ*(*r*) = *κ _{t}*(

*r*−

*r*

_{min}) [31]. Here,

*κ*can be interpreted as the translational capacity of the cell, which correlates with the speed of translational elongation [35], and

_{t}*r*

_{min}is the minimum mass fraction of ribosomes needed for growth (Fig. 1C inset). The value for

*r*

_{min}is obtained from the intercept of

*κ*as a function of

*r*from experimental data [4, 31].

We combine this model for growth with a model for the control of cell division (Fig. 1A). The division proteins, *X*, are synthesised at a rate proportional to the cell volume:
where *k _{p}*(

*r*) is the rate of synthesis of division proteins that is assumed to be a function of the ribo-some mass fraction. Cell division is triggered when a threshold copy number of division proteins,

*X*

_{0}, is accumulated at the mid-plane of the cell (Fig. 1B). While various proteins could be potential candidates for division initiation [36–39], a recent study identifies FtsZ as the key initiator protein that assembles a ring-like structure in the mid-cell region to trigger septation [19]. We therefore suggest that

*X*represents FtsZ copy number, and assume that its turnover rate in the ring-bound state is much faster than its rate of synthesis [40]. As a result, all the newly synthesised FtsZ in the cytoplasm are assumed to be recruited in the ring.

Solving Eq. (1) and Eq. (2), we obtain: *X*_{0} = Δ*V k _{p}/κ*, where Δ

*V*is the added volume per generation. As

*X*

_{0},

*k*and

_{p}*κ*are constant for a given growth medium, cells add a constant volume Δ

*V*in each growth generation, consistent with the phenomenological adder model. Furthermore, for symmetrically dividing bacterium, the average newborn cell volume, ⟨

*V*⟩, asymptotes to Δ

*V*[12]. Therefore, average cell volume in a given growth medium is given by the ratio of growth rate to the rate of division protein synthesis: where

*κ*=

_{p}*k*

_{p}/X_{0}, is the normalized rate of division protein synthesis. Thus, cell volume can be modulated by perturbations in translation, as both

*κ*and

*κ*are functions of the ribosomal mass fraction. A key proposition of our model is that there is a tradeoff between ribosomes allocated for synthesizing growth and division proteins such that: where can be interpreted as the rate of production of FtsZ per ribosomes, and is the ribosome mass fraction when growth rate is maximum. By combining the expressions for growth rate and division proteins synthesis rate, we find: such that average cell size increases with increasing ribosome abundance. We fit the expression in Eq. (5) to experimental data [4] in order to determine the parameters and (Fig. 1C). We can then express the average volume as a function of nutrient specific growth rate, recapitulating Schaecter et al.’s

_{p}*nutrient growth law*[2] that cell size increases monotonically with increasing growth rate (Fig. 1D):

Notably, our result deviates from the phenomenological model of exponential dependence between cell size and growth rate, and predicts a maximum growth rate, , when all ribosomal resources are allocated towards growth.

To understand the mechanistic origin of the ribosomal tradeoff between growth and division protein synthesis (Eq. 4), we consider a model for allocation of ribosomal resources, extending the framework of Scott et al [31]. The total protein content of the cell can be decomposed into four classes (Fig. 1E): ribosome-affiliated proteins (R, mass fraction *ϕ _{R}*), house-keeping proteins not affected by translation (Q, mass fraction

*ϕ*), division proteins (X, mass fraction

_{Q}*ϕ*), and the rest non-ribosomal proteins (P, mass fraction

_{X}*ϕ*). The mass fractions are constrained by the equation: . For different combinations of the nutritional and translation capacities of the cell, efficient resource allocation requires that the abundance of P- and R-class proteins be adjusted so that the rate of nutrient influx by P matches the rate of protein synthesis achievable by R: , where

_{P}*κ*is the nutritional capacity of the cell. This results in the following relation between the mass fractions of ribosomes and division proteins:

_{n}We assume that the rate of production of division proteins, *κ _{p}*, is proportional to

*ϕ*:

_{X}*κ*=

_{p}*λϕ*, and

_{X}*r*=

*ϕ*, where

_{R}/ρ*ρ*and

*λ*are constant conversion factors. This results in Eq. (4), where we identify , and . Growth rate

*κ*decreases with increased allocation of resources towards division proteins

*ϕ*(Fig. 1E-F):

_{X}With all the model parameters inferred from experimental data (Methods, Table 1), we can plot the dependency of *κ _{p}* on

*κ*(Fig. 1F), showing the negative correlation between the division protein synthesis rate and the volumetric growth rate. Cells growing in poor nutrient medium allocate a smaller fraction of ribosomes towards growth, resulting in smaller size on average. However, cells growing in rich nutrients inflate their size by allocating a larger fraction of ribosomes towards growth (Fig. 1E).

### Translation inhibition breaks balanced allocation of ribosomal resources

In a given nutrient medium, *κ/κ _{p}* is maintained at a constant value, indicating a balance between growth and division protein synthesis. If

*κ/κ*remains invariant under translation inhibition, we expect cell size to remain unchanged, as previously suggested by Basan

_{p}*et al.*[7]. However, experimental data [4] show that cell size could either increase, decrease or remain unchanged when

*E. coli*cells are subjected to varying concentrations of Chloramphenicol – a ribosome-targeting antibiotic. We therefore hypothesize that translation inhibition breaks balanced allocation of ribosomal resources towards growth and division proteins, by differentially reducing the rates

*κ*and

*κ*.

_{p}Under translation inhibition, bacteria produce more ribosomes to compensate for the inactive ribosomes that are bound by antibiotics [31]. By measuring bacterial growth rates and ribosome mass fractions for increasing concentrations of Chloramphenicol, Scott *et al.* [31] obtained:
where *κ _{n}* is the nutritional capacity that depends on nutrient quality, and

*r*

_{max}is the maximum ribosome fraction that cells can produce under translation inhibition, weakly correlating with the nutrient specific growth rate [31]. By combining Eq. (10) with Eq. (4), we find: where can be interpreted as the excess ribosomal mass fraction allocated to division protein synthesis under translation inhibition. Unlike nutrient perturbations, we find that

*κ*and

_{p}*κ*are positively correlated under translation inhibition (Fig. 2A), such that they both decrease with increasing antibiotic concentration. Eq. (11) can be combined with Eq. (3) to determine how cell volume changes as a function of growth rate under translation inhibition:

Interestingly, the above expression predicts three distinct behaviors (Fig. 2B):

We determine the parameters *δr* and *κ _{n}* for each growth medium, by fitting Eq. (11) to the experimental data for cell growth rate and volume under Chloramphenicol perturbations [4] (Fig. 2C). We find that

*δr*< 0 in poor media,

*δr*> 0 in rich growth media, whereas

*δr*≈ 0 for cells growing with medium growth rates (Fig. 2D). We interpret the above result as cells allocating excess ribosomes for growth in poor media, whereas in rich media cells tend to allocate more ribosomal resources for division protein synthesis. The predicted volume curves for each growth conditions quantitatively capture the trend in the experimental data without further fitting (Fig. 2E).

### Cells actively regulate shapes to adapt to translational perturbations

Under translation inhibition, decrease in cell volume in rich media is indicative of a higher surface-to-volume ratio that may drive an increased nutrient influx. Conversely, in poor media, increase in cell volume may be indicative of a lower surface-to-volume ratio that in turn would reduce antibiotic influx (Fig. 2F). Therefore the surface-to-volume ratio of the cell may play a crucial role in controlling cellular adaptive response to perturbations in the growth medium. To test this hypothesis, we construct a model coupling cell growth and geometry to nutrient and antibiotic transport.

#### Nutrient dynamics

The dynamics of nutrient concentration inside the cell, [*n*], is given by:
where *J _{n}* = [

*n*

_{ext}]

*P*

_{in}

*A/V*is the nutrient influx, [

*n*

_{ext}] is the nutrient concentration in the extra-cellular medium,

*P*

_{in}is the cell envelope permeability and

*κ*is the rate at which ribosomes are produced from the nutrients. The model for nutrient transport across the cell membrane is consistent with the one proposed in [41] if we assume that the number of metabolic proteins (transporters) scales with the surface area of the cell. The interplay between nutrients and ribosome synthesis is schematically represented in Fig. 3A. The intracellular concentration of nutrients determines the specific growth rate as:

_{r}*κ*

_{specific}=

*κ*

_{0}[

*n*]

*/*([

*n*] +

*n**) [42], where

*κ*

_{0}is the maximum growth rate characteristic of the medium, and

*n** is the value of [

*n*] when

*κ*

_{specific}

*/κ*

_{0}= 0.5. When the nutrients inside the cell reach saturation, i.e. d[

*n*]/

*dt*= 0,

*κ*=

*κ*

_{specific}.

#### Antibiotic dynamics

The action of ribosome-targeting antibiotics is illustrated using the diagram in Fig. 3A, which consists of two key components: the flux of antibiotics *J _{a}* entering the cell, and the binding of antibiotics to the active pool of ribosomes,

*r*. The dynamics are described by the following set of equations, extending the model of Elf

_{a}*et al.*[43] and Greulich

*et al.*[44], where

*a*

_{ex}is the extracellular antibiotic concentration,

*a*

_{in}is the intracellular concentration of the antibiotic,

*r*is the concentration of the active pool of ribosomes in the cell,

_{a}*r*is concentration of the pool of ribosomes bound by the antibiotics, and

_{b}*s*is the rate of synthesis of ribosomes. Unlike previous models [43, 44], here we account for the dependence of

*J*on cell shape as: where

_{a}*P*

_{in}and

*P*

_{out}are the cell envelope permeabilities in the inward and outward directions, respectively. The ribosome-antibiotic interactions are defined by:

*f*(

*r*

_{a}, r_{b}, a_{in}) = −

*k*

_{on}

*a*

_{in}(

*r*−

_{a}*r*

_{min})+

*k*

_{off}

*r*, where

_{b}*k*

_{on}is the rate of binding of antibiotics to ribosomes, and

*k*

_{off}is the rate of unbinding. These rate constants for Chloramphenicol are known from literature [43, 44]. Furthermore, cells produce more ribosomes to compensate for the inactive ribosomes bound by antibiotics [31]. This is captured by the source term:

*s*=

*κ*(

*r*

_{max}−

*κ*Δ

*r*(1

*/κ*

_{specific}−

*κ*Δ

_{t}/*r*)), where Δ

*r*=

*r*

_{max}−

*r*

_{min}.

*Cell shape dynamics.*Having described the dynamics of cell volume (Eq. 1), division control (Eq. 2), nutrient and antibiotic transport (Eq. 13–14), we need to additionally account for cell surface area synthesis to predict cell shape changes. We assume that rate of synthesis cell surface area is proportional to cell volume [45]: where

*β*is the rate of surface area production, which depends on cell shape, growth rate and division protein synthesis rate. Solving Eq. (1) and Eq. (16), one obtains

*A/V*=

*β/κ*, at steady-state [45]. In recent work [20] we found that

*E. coli*cells obey the relation:

*A*=

*νV*

^{2/3}, under nutrient and translational perturbations, where

*ν*is a geometric factor related to the cell aspect ratio

*η*as: . Therefore, sur4face a12rea production rate varies non-monotonically with growth rate as:

*β*=

*νκ*(

*κ/κ*)

_{p}^{−1/3}(Fig. 3B).

Taken together, our model accounts for the key functions of ribosomes in controlling cell growth rate (*κ*), rate of production of division proteins (*κ _{p}*), and the rate of surface area synthesis (

*β*). Under translation inhibition, both

*κ*and

*κ*decreases as shown in Fig. 2A and C. Surface area production rate is also impacted by translation inhibition, as shown in Fig. 3B, albeit in a different manner from the growth rate. Differential reduction of

_{p}*κ*and

*β*under translation inhibition is in-dicative of changes in steady-state cell surface-to-volume ratio (∝

*β/κ*). To test this quantitatively, we simulated the coupled equations (Methods) for single-cell growth (Eq. 1, 2 and 16), nutrient (Eq. 13) and ribosome-antibiotic dynamics (Eq. 14) under stresses induced by ribosome-targeting antibiotics (Fig. 3C). In response to a step pulse of antibiotic in a rich nutrient medium at

*t*= 0 h, the concentration of antibiotic inside the cell and the influx increase rapidly. This in turn reduce cell elongation rate as a result of antibiotic binding to ribosomes, and leads to longer interdivision times, a decreased (increased) average birth volume and a concomitant increase (decrease) in surface-to-volume ratio for cells growing in rich (poor) nutrients. These results confirm our hypotheses that in poor media, cells reduce their surface-to-volume ratio to inhibit antibiotic influx, while in rich media cells increase their surface-to-volume ratio to import more nutrients.

While all the model parameters can be calibrated from available experimental data (Table 1), the relative magnitude of the permeabilities, *P*_{in}*/P*_{out} remains undetermined. To this end, we fit our model to the experimental growth-inhibition curves [4] in differents nutrient conditions (Fig. 3D), treating *P*_{in}*/P*_{out} as a fitting parameter. Interestingly, we find that *P*_{in}*/P*_{out} is nutrient-dependent and decreases with increasing specific growth rate (Fig. 3E). Including nutrient-dependent regulation of membrane permeability, our model predictions capture the experimental data for the decrease in division protein synthesis rate under Chloramphenicol inhibition (Fig. 3F) and the changes in cell volume (Fig. 3G). Consistent with our hypothesis and experimental data, we find that cell surface-to-volume increases in rich nutrient media (synthetic rich in experiments), suggestive of increased nutrient influx (Fig. 3H). Conversely, in poor nutrient medium (glycerol in experiments), cell surface-to-volume reduces with increasing drug dosage, suggesting that cells are countering the influx of antibiotics if sufficient nutrients are not available.

Nutrient-dependent regulation membrane permeability to antibiotics (Fig. 3E) can be a result of different metabolic pathways. It has been observed that *E.coli* cells have different metabolic pathways for nutrients depending on the growth conditions [46]. Furthermore, if the cells are subjected to a nutrient downshift, the proteome reallocates such that a larger fraction of proteins is allocated to the sector responsible for carbon catabolism which in turn reduces the available proteome fraction for other sectors [47, 48]. The transition from one metabolic mechanism to another can be justified using a proteome allocation model as suggested by Basan *et al.* [48] and Mori *et al.* [47], or by increasing the glucose uptake rates. The drop in cell envelope permeability that we observe around *κ* = 0.6 h^{−1} (Fig. 3E) matches the maximum growth rate that *E.coli* cells can achieve while staying below the critical limit on energy dissipation [49].

### Cell surface area production promotes bacterial growth inhibition by ribosome-targeting antibitiocs

Our theory predicts that bacterial growth response to translation-inhibitory antibiotics is governed by nutrient-dependent cell shape changes (Fig. 2–3). To systematically study how bacterial growth inhibition depends on cell shape and nutrient quality, we simultaneously perturbed cell shape and ribosomal translation in varying growth media using our computational model. These simulations can be realised experimentally by simultaneously applying two antibiotics - one that changes cell shapes (e.g. by targeting the cell wall), while the other affects the translational machinery by inhibiting ribosomal activity. The resultant effect can be suppressive, antagonistic, or synergistic depending on what the combined effect of the two drugs is with respect to the individual effect of each [50, 51].

In simulations we simultaneously applied a surface area *modifier* and chloramphenicol to a cell growing at steady-state. To achieve rounder cells, the *modifier* is a surface area synthesis inhibitor that decreases the surface production rate *β*, by decreasing the cell’s geometric factor *ν* (= *A/V*^{2/3}), which in turn reduces cellular aspect ratio. By contrast, long filamentous cells are obtained when a surface area promoter is added (increasing *ν*), leading to higher aspect ratio cells.

We investigated the response of growth rate to increasing Chloramphenicol concentrations for cells with varying aspect ratios – ranging from *η* = 1 for coccoidal cells to *η* = 10 for filamentous cells (Fig. 4A). The response of *κ* to the concentration of the applied antibiotic can be characterized by a Hill function of the form [52] (Fig. 4A):
where IC_{50} is the half-inhibitory concentration of the antibiotic, and the Hill coefficient *n* quantifies the dose-sensitivity of the growth rate to relative changes in drug concentration. We take IC_{50} as a measure of drug *resistance* [52].

For a range of aspect ratios and nutrient conditions, we fitted the growth inhibition curves to the Hill function in Eq. (17), and obtained the values for IC_{50} (Fig. 4B) and the dose-sensitivity *n* (Fig. 4C). Our model predicts that IC_{50} (resistance) increases with decreasing aspect ratio in rich-nutrient medium, while being less sensitive to changes in cell aspect ratio in poor-nutrient medium (Fig. 4B). Dose-sensitivity to changes in drug concentration increases with decreasing aspect ratio and increasing nutrient quality (Fig. 4C), such that dose-sensitivity is positively correlated with drug resistance (Fig. 4D). These results indicate that cellular response to translation inhibitory antibiotics is sensitive to both the nutrient quality as well as cell shape. We find that round coccoidal cells are most drug-resistant, while filamentous cells are least resistant (Fig. 4E). Furthermore, depending on nutrient-quality, cellular morphological response to translation inhibitory drugs is different. While cells increase their surface-to-volume ratio to import more nutrients in nutrient-poor medium, cells prefer to reduce their surface-to-volume ratio in rich-nutrient medium to inhibit antibiotic influx (Fig. 2–3). These findings predict that bacterial growth inhibition can be maximized by simultaneously inhibiting ribosomal translation and promoting surface area production in nutrient-poor media.

## DISCUSSION

We develop a whole-cell coarse-grained model for bacterial growth dynamics that connects intra-cellular control of translation, with cell shape, division control and extracellular environment. This provides a promising theoretical framework that quantitatively captures available experimental data for bacterial cell size and shape dynamics under nutrient and translational perturbations. Out study reveals that during nutrient shifts, the ribosomal resources are optimally allocated to maintain a balanced trade-off between the rates of cell growth and division protein synthesis. In rich nutrient media, more ribosomes are used for growth than division protein synthesis, leading to cell size inflation with increasing nutrient quality. Conversely in nutrient-poor media, cells allocate more ribosomal resources for division protein synthesis than growth, leading to a reduction in average cell size. This principle underlies the molecular basis for the celebrated *nutrient growth law* [2, 4], and can be interpreted as an optimization principle for cellular economy. Based on this principle, the resources allocated to a particular proteomic sector are inversely proportional to the efficiency of that sector [41]. In nutrient-rich media, cells invest more ribosomal resources to growth in order to compensate for a lower translational capacity. The latter can arise from an increased dilution rate of ribosomes under fast growth conditions, lowering the efficiency of protein synthesis. In nutrient-poor media, cells have a lower nutritional capacity that they compensate by allocating more resources to metabolism and division protein synthesis.

Comparing our theory to experimental data, we uncover several feedback pathways between cell shape, growth rate, protein synthesis and extracellular transport that were previously unknown (Fig. 4F). In particular, we predict that under translation inhibition, cells break the balanced trade-off between ribosomes and division protein synthesis, leading to cell size inflation, reduction or size invariance, in a nutrient-dependent manner. Our model predictions are in quantitative agreement with experimental data on *E. coli* cells subjected to Chloramphenicol perturbations across various nutrient conditions [4]. If cells are grown in nutrient-rich media, the excess ribosomes produced under translation inhibition are allocated towards division, leading to smaller cell sizes and higher surface-to-volume ratios. This is in agreement with Chloramphenicol treated *E.coli* cells grown in synthetic rich medium. Conversely, in nutrient-poor media cells allocate excess ribosomes towards growth, leading to cell size inflation and lower surface-to-volume ratios, in agreement with *E.coli* cell data in glycerol medium. This suggests that cells shape changes in response to translation-inhibitory antibiotics may confer certain fitness advantages under stress. In nutrient-rich media it is more favorable for cells to reduce their surface-to-volume in order to minimize antibiotic influx. Whereas in nutrient-poor media, cells adapt to import more nutrients by increasing their surface-to-volume ratios.

To quantitatively test the role of cell shape and nutrient quality on bacterial growth inhibition under antibiotic stress, we simulated bacterial growth under simultaneous perturbation of surface area production and translation inhibition in varying nutrient media. From growth-inhibition curves we measured bacterial response to antibiotics by quantifying resistance (half-inhibitory concentration of the drug) and dose-sensitivity to increasing concentration of the drug. Our study reveals that that round-shaped cells are fitter and more drug-resistant than higher aspect ratio filamentous cells, and that dose-sensitivity increases with increasing nutrient quality. These results can be tested experimentally by measuring bacterial growth rates in response to simultaneous application of cell-wall targeting and ribosome-targeting antibiotics, in different nutrient concentrations. Interestingly, we predict that bacterial growth-inhibition can be maximized by simultaneously inhibiting ribosomal translation and promoting surface area production in nutrient-poor media.

## AUTHOR CONTRIBUTIONS

SB and DS designed research. DS, NO and SB developed theory. DS performed simulations and analysed data. DS and SB wrote the paper.

## DECLARATION OF INTERESTS

The authors declare no competing interests.

## METHODS

### Cell growth simulations

To investigate the dynamic response of cell shape and growth to applied antibiotic and nutrient shifts, we simulated single-cell growth over multiple generations. We first initiated cells at different stages in their cell cycle, and upon division followed the daughter cells over a number of generations until steady-state is reached. During each cell generation *i*, we evolved the following seven coupled differential equations for cell volume *V _{i}*, division protein abundance

*X*, surface area

_{i}*A*, nutrient concentration inside the cell [

_{i}*n*], antibiotic concentration inside the cell , active ribosomes , and inactive or antibiotic-bound ribosomes, .

_{i}In the above equations, we have (dropping ‘*i*’ for simplicity):
*J _{n}*(

*A, V,*[

*n*]) = [

*n*

_{ext}]

*P*

_{in}([

*n*])

*A/V*, where [

*n*

_{ext}] is the extracellular nutrient concentration, and

*P*

_{in}([

*n*]) is the nutrient-dependent inward permeability (Fig. 3E)

For each cell cycle *i*, Eq. (18)-(24) are evolved for *t* ≤ *τ _{i}*, where

*τ*is the interdivision time for the

_{i}*i*

^{th}generation. Division is triggered when

*X*>

_{i}*X*

_{0}, with

*X*

_{0}a constant. Upon division, we set:

*V*

_{i}_{+1}(0) =

*D*(

_{R}V_{i}*τ*),

_{i}*X*

_{i}_{+1}(0) = 0,

*A*

_{i}_{+1}(0) =

*D*(

_{R}A_{i}*τ*), [

_{i}*n*

_{i}_{+1}](0) = [

*n*](

_{i}*τ*), , where

_{i}*D*is a Gaussian random variable with mean 0.5 and standard deviation 0.05. We initialize the nutrient concentration inside the cell ([

_{R}*n*]) close to zero, and calibrate the extracellular nutrient concentration [

_{i}*n*

_{ext}] to reach the growth rate of the medium we choose to simulate. Over time [

*n*] reaches the steady-state value

_{i}*n**

*κ*

_{specific}

*/*(

*κ*

_{0}−

*κ*

_{specific}), such that

*κ*=

*κ*

_{specific}. We run simulations for additional 5h after the nutrient concentration reaches steady-state, to record the average values of cell volume, area, and ribosome concentration. Antibiotic perturbation is applied after 10h from the start of the simulations and continued for another 20h, when we compute the average values for the various cellular variables.

### Model parameters

We extracted the parameters *κ _{t}* and

*r*

_{min}by fitting the equation

*κ*=

*κ*(

_{t}*r*−

*r*

_{min}) to the data for growth rate vs RNA/protein ratio [4] (Table 1). Using our theoretical model, we obtained the expression for cell volume

*V*as a function of

*r*(Eq. (5)), which we fitted to experimental data [4], in order to to extract the parameters , and (Table 1). For cells under Chloramphenicol stress, the nutrient-dependent parameters

*κ*and

_{n}*δr*were obtained by fitting Eq. (10) and Eq. (11) to the experimental dataset for each nutrient condition (Table 1). From experimental data [4], we estimated the division protein production rate as

*κ*=

_{p}*κ/V*. To determine the permeability of the cell envelope to nutrient and antibiotic transport, we fitted the growth inhibition curves resulting from our simulations to the growth inhibition curves from the data in [4], using a method of least-squares. We find that

*P*

_{in}

*/P*

_{out}is a function of nutrient quality, and used that as an input to our model simulations. Tables 1 and 2 list a complete set of parameter values used in our model simulations.

## ACKNOWLEDGMENTS

We thank Suckjoon Jun lab (UCSD) for providing single cell shape data for *E. coli*, and Guillaume Charras for many useful discussions. SB acknowledges funding from EPSRC grant EP/R029822/1, Royal Society grants URF/R1/180187 and RGF/EA/181044.