## Highlights

- •Following nutritional upshifts, bacteria rapidly increase their growth rate
- •A square root relation exists between the pre-shift and post-shift growth rates
- •The square root relation quantifies ribosomal spare capacity
- •Spare capacity speeds response to change and avoids metabolic overshoots

## Summary

## Graphical Abstract

## Keywords

## Introduction

*R*(

Mori, M., Marinari, E., and De Martino, A. (2017a). A yield-cost tradeoff governs Escherichia coli’s decision between fermentation and respiration in carbon-limited growth. arXiv, arXiv:1703.00748v1, https://arxiv.org/abs/1703.00748.

## Results

### Optimal Resource Allocation Model for the Immediate Growth Rate after Upshifts

*C*sector) and a ribosomal sector (denoted as the

*R*sector). The catabolic sector, which includes carbon transporters and catabolic enzymes, is responsible for carbon uptake and conversion into intracellular substrates (denoted by

*x*). The ribosomal sector, which includes ribosomes and translational machinery, converts these substrates into biomass. A third sector (the

*Q*sector; Supplemental Experimental Procedures S1) includes all proteins which under limiting carbon conditions do not change with growth rate (

*Q*, the experimentally observed tradeoff between making $R$ and $C$ sector proteins is summarized by $R+C=1$ (

$g\left(x\right)$ and $R\left(x\right)$ are normalized between zero and one such that ${\mu}_{sat}$ is the growth rate when $x$ is saturating. $g\left(x\right)$, the average elongation rate, is an increasing function of $x$, describing ribosome utilization (

- Li S.H.-J.
- Li Z.
- Park J.O.
- King C.G.
- Rabinowitz J.D.
- Wingreen N.S.
- Gitai Z.

*bioRxiv.*2018; https://doi.org/10.1101/224204

Nutrients are imported and catabolized by the $C$ sector at a rate $\beta h\left(x\right)$, where $\beta $ represents nutrient availability. $h\left(x\right)$, the import rate, is a decreasing function of $x$ that describes inhibition of the transporters by intracellular substrates (

The function $f\left(x\right)$ describes transcriptional control that determines the fraction of the total biomass production rate $\mu $ that goes to the $R$ sector (

*C*and

*R*sectors, $h\left(x\right)$ and $g\left(x\right)$, define the possible set of steady states (i.e., the curve in Figure 1C), $f\left(x\right)$ determines the chosen steady state among this set. Importantly,

Intuitively, this optimal regulation function $f\left(x\right)$ determines the best trade-off between $R$ and $C$ by balancing the relative advantage of investing in each of these sectors according to the logarithmic sensitivities of their activity curves ${h}^{\text{'}}/h$ and ${g}^{\text{'}}/g$ (

_{2}/k

_{1}represents the cellular saturation level (Figure 1D; Supplemental Experimental Procedures S4). Full saturation, in which ribosomes and transporters work close to saturation in a wide range of substrate levels, is captured by κ≪1 (Figure 1D, left). Extreme sub-saturation, in which ribosomes and transporters work far from their full capacity in most conditions, means κ≫1 (Figure 1D, right). Intermediate sub-saturation, in which the halfway coefficients of ribosomes and transporters are equal, is captured by κ = 1 (Figure 1D, middle).

*C*sector and measured the resulting growth rate (black dots in Figure 1C). These experiments indicated that the saturation halfway points are approximately equal, ${k}_{1}={k}_{2}$, and hence κ∼1.

At full saturation, κ≪1, the expression reduces to $\tilde{{\mu}_{1}}=\tilde{{\mu}_{0}}$, because when ribosomes are saturated before the shift, growth rate cannot immediately increase after the upshift. At extreme sub-saturation, κ≫1, the model gives a linear relation between the normalized pre-shift and post-shift growth rates with an intercept of ½ (Figure 2B).

Intuitively, this square-root law results from the following situation: in poor conditions, both ribosomal content

*R*and saturation level

*g*are a small value $\mathit{\epsilon}$ and the growth rate is ${\mu}_{0}\sim {\mathit{\epsilon}}^{2}$, whereas soon after the upshift, ribosomes are still $\mathit{\epsilon}$ but saturation is high due to the presence of nutrient, resulting in ${\mu}_{1}\sim \mathit{\epsilon}$. In Supplemental Experimental Procedures S5, we relax the assumption that

*h*(

*x*) and

*g*(

*x*) are MM-like and derive a similar square-root law for general

*h*(

*x*),

*g*(

*x*) functions (Supplemental Experimental Procedures S5).

where the growth rate far after the shift is denoted ${\mu}_{post}$ and $\tilde{{\mu}_{post}}={\mu}_{post}/{\mu}_{sat}$. In the case of large upshifts to saturating carbon, ${\mu}_{post}={\mu}_{sat}$, and the formula reduces to the square-root formula of Equation 6 (Supplemental Experimental Procedures S6).

*U*can be defined as the fold change in growth rate after the shift, $U=\frac{{\mu}_{1}}{{\mu}_{0}}$. This definition, together with Equation 6, leads to a spare capacity of

as shown in (Figure 2C). Spare capacity is smallest (

*U*= 1) when cells are close to their saturating growth rate. Spare capacity increases the poorer the medium (the smaller ${\mu}_{0}/{\mu}_{sat}$). For example, in a medium that allows only 10% of the growth rate on saturating carbon, ${\mu}_{0}/{\mu}_{sat}$ = 0.1, the cells grow $U=\sqrt{10}\phantom{\rule{0.25em}{0ex}}\sim 3$ times faster when shifted to saturating carbon.

### Experimental Tests for Nutritional Upshifts Support the Square-Root Formula

*E. coli*MG1655 cells were shifted from exponential growth in a poor medium to saturating glucose medium. We used two experimental systems: (1) a chemostat, in which slowly dividing cultures in glucose-limited medium (0.02%) were shifted to growth in 0.2% glucose (Figures 3A and S1), and (2) batch culture in a multi-well robotic assay at several temperatures (25°C, 30°C, and 37°C), in which cultures growing exponentially on different carbon sources (acetate, sorbitol, rhamnose, or pyruvate) were shifted to 0.4% glucose medium (Figures 3B, S2, and S3). Because we are interested in biomass growth rate, we measured the optical density (OD) of the cells at a temporal resolution of 0.5 min in the chemostat and 3.6 min in the batch culture, with an error of 4%–10% in growth rate between biological repeats.

*E. coli*15T¯, on a different bacterial species,

*S. typhimurium*(

*Saccharomyces cerevisiae*(

^{−1}), species, and measurement methods (Table S1).

^{−26}).

### Ribosome Spare Capacity Prevents Large Substrate Overshoot and Is Beneficial in Frequently Changing Environments

*Δrrn*) to wild-type strains in chemostat and batch culture. The

*Δrrn*strains outcompeted wild-type strains in a chemostat, but not in batch culture conditions. We interpret these findings in the light of the present model: the

*Δrrn*strains have fewer ribosomes (

*Δrrn*stains do worse after an upshift (shift from overnight to fresh batch culture), due to the predicted benefits of sub-saturation in the wild-type strain.

(see Supplemental Experimental Procedures S6 for the range of validity of this formula and for a more general formula), where ${\mu}_{post}$ is the steady-state growth rate in the post-shift medium. Because downshifts are experimentally harder to explore, we defer a test of this prediction to future studies.

## Discussion

*E. coli*, Salmonella, and yeast. The growth law is also a quantitative measure of the spare capacity of cells for growth. We suggest that sub-saturation of ribosomes can be beneficial in frequently changing environments, because it prevents large overshoots in metabolic intermediates and allows rapid initial increase in growth rate following an upshift.

*U*for growth as a function of the growth rate ${\mu}_{0}$ and the saturating growth rate ${\mu}_{sat}$, namely $U=\sqrt{\frac{{\mu}_{sat}}{{\mu}_{0}}}$. Spare capacity is larger the poorer the carbon source in the medium (the smaller $\frac{{\mu}_{0}}{{\mu}_{sat}}$).

*rrn*deletion strains (

## Experimental Procedures

### Multi-well Batch Culture Experiments

*E. coli*wild-type strain, MG1655. Cells were grown overnight in M9 minimal medium (42 mM Na2HPO4, 22 mM KH2PO4, 8.5 mM NaCl, 18.5 mM NH4Cl, 2 mM MgSO4, and 0.1 mM CaCl) supplemented with 0.4% glucose (w/v) and 0.05% casamino acids at 37°C. Using a robotic liquid handler (FreedomEvo; Tecan), 96-well plates were prepared with 150 μL of M9 minimal medium (without casamino acids) with the indicated carbon sources (0.2%). The wells were inoculated with bacteria at a 1:500 dilution from the overnight culture. Wells were covered with 100 μL of mineral oil (Sigma) to prevent evaporation, a step which we previously found not to significantly affect growth (

### Analysis of Robotic Batch Culture Experiments

*e*. We computed separately the pre-shift growth curve and the post-shift growth curve, without mixing points before the shift with points after the shift. For time points next to the shift point, we used an asymmetric window of less than 15 points. For the computation of the pre-shift growth rate ${\mu}_{0}$, we used the control experiment growth curves. For each condition, we defined the exponential phase as the range in which growth rate was fairly constant (SD < 0.05), over a window of 3–6 hr, and we averaged over the growth rate in this window. For the computation of the post-shift growth rate ${\mu}_{1}$, we used the upshift experiments. The growth rate temporarily stabilized on a value smaller than the growth rate in glucose after ∼15–50 min, depending on the temperature. We computed this value by averaging the growth rate over a window of stability: 20–60 min (37°C), 40–70 min (30°C), or 55–85 min (25°C) after the shift. The saturating growth rate ${\mu}_{sat}$ was computed from a batch culture in 0.4% glucose in each temperature by taking the logarithmic derivative of the OD at mid-exponential growth, in a window of 2–5 hr (37°C), 2–4 hr (30°C), or 3–8 hr (25°C; Figure S4A). Errors for ${\mu}_{0}$, ${\mu}_{1}$, and $\phantom{\rule{0.25em}{0ex}}{\mu}_{sat}$ are SE of 3 day-day repeats. For all points in Figures 3C and 3D, we computed the error in $F\left({\mu}_{0},\phantom{\rule{0.25em}{0ex}}{\mu}_{sat}\right)=\phantom{\rule{0.25em}{0ex}}{\mu}_{0}/{\mu}_{sat}$ from the errors in ${\mu}_{0},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\mu}_{sat}$ using the formula $\Delta F\left({\mu}_{0},\phantom{\rule{0.25em}{0ex}}{\mu}_{sat}\right)=\sqrt{{\left(\Delta {\mu}_{0}\frac{dF}{d{\mu}_{0}}\right)}^{2}+{\left(\Delta \phantom{\rule{0.25em}{0ex}}{\mu}_{sat}\frac{dF}{d{\mu}_{sat}}\right)}^{2}}$ and similarly for ${\mu}_{1}/{\mu}_{sat}$, $\phantom{\rule{0.25em}{0ex}}{\mu}_{1}/{\mu}_{0}$, and $\sqrt{{\mu}_{0}{\mu}_{sat}}$.

### Chemostat Experiments

*E. coli*MG1655 was used to inoculate the parallel bioreactors, each containing 150 mL (1:50; OD600 ∼0.05) M9 media supplemented with vitamin B1 and 0.02% glucose. The cultures were initially grown at 37°C in batch mode to early log phase and then shifted to growth in chemostat mode using the same media as above for feeding, with dilution rates of 0.28, 0.17, 0.09, and 0.06 hr

^{−1}(corresponding to doubling times of 2.5 hr, 4 hr, 8 hr, and 12 hr) and the following controlled parameters: 200 rpm and 0.66 VVM (volume of air/volume of media/min). When the oxygen in the reactors became limiting (DO = <20% [dissolved oxygen]), a feedback cascade of mixing and aeration was engaged (300–800 rpm and 0.66–2.0 VVM, respectively). When the OD of all the cultures stabilized, the cultures were spiked with 1.5 mL of 20% glucose (final concentration 0.2%) and further grown for at least 4 hr in chemostat mode with the above media containing 0.2% glucose, while monitoring and logging their DO, OD, and pH.

### Analysis of Chemostat Experiments

### Analysis of Published Upshift Experiments

*S. typhimurium*cells were transferred from glycerol-minimal medium to broth. Growth rate was measured by radioactive amino acid incorporation. Maaløe and Kjeldgaard report the steady-state growth rate values in the pre-shift medium and in the post-shift medium, as well as the growth rate value in the first 15–25 min after the shift. We take these values to be ${\mu}_{0},\phantom{\rule{0.25em}{0ex}}{\mu}_{sat}$, and ${\mu}_{1}$, respectively. Note that Maaløe and Kjeldgaard define growth rate in base 2. We divide the reported values by

*ln(2)*to transfer them to base

*e*. No error estimate for these values was supplied.

*E. coli*15T¯ cells from M9 medium with fumarate, succinate, aspartate, or glyoxylate to glucose 0.4%. They measured OD at a resolution of ∼5 min. We extracted numerical data from the figures using the grabit package of MATLAB. We computed the post-shift growth rate ${\mu}_{1}$ by taking the logarithmic derivative of the OD in a window of 6 points (∼30 min) after the shift, except for fumarate, for which we used a window of 8 points to avoid a large fitting error. Error bars are the fit 95% confidence intervals. For the pre-shift values ${\mu}_{0}$, we use the authors report for the doubling times in the indicated carbon sources (no error estimate is supplied). For ${\mu}_{sat}$, we use the steady-state growth rate value in the post-shift medium. This value is in agreement with the doubling time reported by the authors. Errors were computed from the fit confidence interval.

*S. cerevisiae*) from galactose to glucose 0.4%. Cells were grown and imaged in a flow cell, and their volume was measured at a time resolution of 5 min. We computed the growth curve by taking the logarithmic derivative of the OD in a window of 5 time points (25 min). We computed the pre-shift growth rate ${\text{\mu}}_{0}$ by averaging on a window of 1.5 hr before the shift. The error was taken to be the SE of the growth rate values in this window. The post-shift growth rate ${\text{\mu}}_{1}$ was computed using a window of 25 min after the shift. Error bars represent 95% confidence intervals. The growth rate on saturating carbon ${\text{\mu}}_{\text{sat}}$ was computed using a doubling time of 86 min, which is achieved in growth on 2% glucose (data kindly supplied by Metzl-Raz et al.).

### Transformation of Data Points from Towbin et al

*E. coli*mutant strain, which cannot endogenously produce cAMP (a strain deleted for the enzymes

*cyaA*and

*cpdA*that synthesize and degrade cAMP). cAMP activates the transcription factor cAMP receptor protein (CRP), which controls the expression of many carbon catabolic enzymes (

^{∗}, defined as the promoter activity of a CRP reporter divided by the promoter activity of a σ70 reporter) were measured. We transformed the CRP

^{∗}values into ribosomal sector values R by: R = 1 − CRP

^{∗}/C

_{m}, where C

_{m}is the maximal CRP

^{∗}= 1.1 as found in

## Acknowledgments

### Author Contributions

### Declaration of Interests

## Supplemental Information

- Document S1. Supplemental Experimental Procedures, Figures S1–S5, and Tables S1 and S2

- Data S1. Summary of Growth Rate Values for All Upshift Experiments Described in this Paper, Related to Figures 3 and S5 and Table S1

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