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# A universal trade-off between growth and lag in fluctuating environments

## Abstract

The rate of cell growth is crucial for bacterial fitness and drives the allocation of bacterial resources, affecting, for example, the expression levels of proteins dedicated to metabolism and biosynthesis1,2. It is unclear, however, what ultimately determines growth rates in different environmental conditions. Moreover, increasing evidence suggests that other objectives are also important3,4,5,6,7, such as the rate of physiological adaptation to changing environments8,9. A common challenge for cells is that these objectives cannot be independently optimized, and maximizing one often reduces another. Many such trade-offs have indeed been hypothesized on the basis of qualitative correlative studies8,9,10,11. Here we report a trade-off between steady-state growth rate and physiological adaptability in Escherichia coli, observed when a growing culture is abruptly shifted from a preferred carbon source such as glucose to fermentation products such as acetate. These metabolic transitions, common for enteric bacteria, are often accompanied by multi-hour lags before growth resumes. Metabolomic analysis reveals that long lags result from the depletion of key metabolites that follows the sudden reversal in the central carbon flux owing to the imposed nutrient shifts. A model of sequential flux limitation not only explains the observed trade-off between growth and adaptability, but also allows quantitative predictions regarding the universal occurrence of such tradeoffs, based on the opposing enzyme requirements of glycolysis versus gluconeogenesis. We validate these predictions experimentally for many different nutrient shifts in E. coli, as well as for other respiro-fermentative microorganisms, including Bacillus subtilis and Saccharomyces cerevisiae.

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## Data availability

Lag times are provided in Supplementary Tables 2, 3. All other data are found in downloadable Excel files for each figure. Data for Fig. 3a were taken from ref. 3 and are deposited with the paper on the Molecular Systems Biology website. Source data are provided with this paper.

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## Acknowledgements

We thank A. Murray for helpful comments and suggestions, and V. Patsalo for technical support and development of the proteomics method. M.B. acknowledges a SystemsX.ch fellowship. T. Honda acknowledges a Japan Student Services Organization (JASSO) long-term graduate fellowship award. Work in the Hwa laboratory is supported by the National Institutes of Health (NIH) through grant R01GM109069 and by the Simons Foundation through grant 330378. J.R.W. acknowledges NIH support through grant R01GM118850.

## Author information

Authors

### Contributions

M.B., T. Hwa and U.S. designed the study. Experiments were performed by M.B., T. Honda, M.H., Y.-F.C., E.L., A.M., H.O., B.R.T., J.M.S. and C.S., and all authors contributed to the analysis of experimental data. Specifically, lag times for E. coli were measured by M.B. and T. Honda. Metabolite measurements and analysis were performed by M.B. and M.H. Proteomics measurements were performed by T. Honda and H.O. Proteomics data analysis was performed by T. Honda, D.C., J.M.S and J.R.W. Genetic constructs were made by M.B., Y.-F.C and H.O. Lag times for S. cerevisiae and B. subtilis were measured by Y.-F.C. Growth rates for B. thetaiotaomicron were measured by B.R.T. Experiments for single-cell lag phases from microfluidics and plates were performed and analysed by E.L., A.M. and C.S. M.B., D.C., T. Hwa and U.S. developed the model. M.B., J.P., T. Hwa and U.S. wrote the paper and the Supplementary Information.

### Corresponding authors

Correspondence to Markus Basan, Terence Hwa or Uwe Sauer.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Jeff Gore, Christopher Marx, Arjan de Visser and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Growth curves for shifts.

a, Growth curves following shifts from different glycolytic carbons to acetate by filtration. Long lag phases can consist of several hours without detectable biomass production. There are large variations in the duration of lag phases following shifts from different carbon sources. The duration of the lag phase correlates with the preshift growth rate (Fig. 1): fast growth before the shift results in very long lag times. bd, Comparisons of lag times following filtration shifts and in diauxie experiments (which involve no shift, but rather growth on medium containing two sugars, with one sugar running out). b, Shift from 1.7 mM glucose to 60 mM acetate. Here the diauxie medium contained glucose plus acetate. c, Shift from 1.7 mM glucose to 30 mM succinate. d, Shift from 1.7 mM glucose to 40 mM pyruvate. Lag times resulting from filtration shifts and from classical diauxie experiments are mostly comparable. In c, the presence of pyruvate in the medium in addition to glucose adversely affected the growth rate, resulting in a shorter lag time in the diauxie shift, consistent with our general observation of the growth-rate dependence of lag times.

### Extended Data Fig. 2 Lag-time/growth-rate relations.

af, The inverse of lag times following a shift to the indicated sugars (a, to acetate; b, to pyruvate; c, to succinate; d, to fumarate; e, to lactate; f, to malate) is plotted as a function of the preshift growth rate in glycolytic conditions. The preshift growth rate was modulated using different carbon sources (circles) and through lactose-uptake titration (squares). Solid lines show nonlinear least-squares fits (Matlab lsqcurvefit function) of lag times as a function of preshift growth rates according to the relation given by equation (1). Most lag phases agree very well with equation (1); only a few shifts, with short lag times (low growth rates), deviate somewhat from this relation. This is partly the result of plotting inverse lag times, which amplifies relatively small experimental variations in lag times for short lag phases. These fits allow us to estimate 95% confidence intervals for model parameters (Matlab nlparci function), most importantly for the critical growth rates λ0. For acetate, λC = (1.10 ± 0.01) h−1, α = 0.78 ± 0.10, n = 17; for pyruvate, λC = (1.12 ± 0.03) h−1, α = 0.33 ± 0.07, n = 17; for succinate, λC = (1.13 ± 0.04) h−1, α = 0.33 ± 0.10, n = 14; for fumarate, λC = (1.08 ± 0.02) h−1, α = 0.23 ± 0.07, n = 5; for lactate, λC = (1.09 ± 0.05) h−1, α = 0.22 ± 0.15, n = 5; for malate, λC = (1.17 ± 0.09) h−1, α = 0.22 ± 0.11, n = 5. g, Lag times as a function of steady-state growth rates in the postshift medium for different preshift media. Coloured solid lines show linear regressions of the corresponding coloured data points. Carbon sources that allow slower growth rates tend to result in longer lag phases when they are the postshift carbon sources. This intuitive correlation has previously been characterized13.

### Extended Data Fig. 3 Single-cell behaviour during a glucose-to-acetate shift in microfluidics.

a, Diagram showing the microfluidic device (mother machine) in which bacterial cells are grown. The cells are loaded in narrow trenches (inset), where they are diffusively fed from the medium flowing through the feeding lane. As cells grow out of the trenches, they are washed away by the medium flow. We focused solely on the cells at the closed end of each trench, also called ‘mother cells’, as they are kept for the duration of the experiment. b, Diagram outlining the experimental protocol. Cells were recovered from the mother machine using glucose medium, and then connected to a flask with culture growing in the same medium43. The medium was switched as for batch cultures, and the flow was restarted towards the mother machine. Cells continued growing for a short time after filtration both in batch and in the mother machine, presumably because of residual glucose in the system; therefore the experiment resembles a diauxic shift. c, Instantaneous single-cell growth rates determined from cell length. Length traces from individual cells were used to compute instantaneous growth rates; the blue points and blue shaded area represent population averages and standard deviations. The orange trace is the instantaneous growth rate trace of an example cell. d, Single-cell lag-time distribution. The lag time is defined as the time delay in growth after the switch compared with instantaneous growth at the maximum postshift growth rate. Instantaneous growth-rate traces were used to compute single-cell lag times (Methods). The red dashed line shows the mean of the lag-time distribution of the tracked cells. Cells tracked in the mother machine introduce a bias towards long lag times, because growing cells are washed away instead of being amplified, as happens in batch culture. Therefore, we also calculated the expected batch lag time (2.69 h; grey dashed line), taking into account cell growth (Methods). e, The postshift growth curve (grey) of the batch culture connected to the microfluidic chamber was used to determine the batch lag time (4.14 h). The maximum growth rate along the growth curve corresponds to the approximately linear part of the log(OD(t)) (grey dotted line), for a growth rate of 0.51 h−1. The red dotted line shows the time derivative of log(OD(t)). The quantitative agreement between the microfluidics and the batch is not perfect. Nevertheless, the single-cell distribution of lag times shows that the response of individual cells after the shift is unimodal, and that the lag time is not governed by a small subpopulation of cells that grows immediately on acetate, as expected in ref. 12. We see no reason why this cell population should not be present in the microfluidics if it were present in the batch. Our data also showed no evidence for the prediction12 that most cells would never recover and grow after the shift. However, because the cells were grown in a microfluidic chip, our experiment cannot definitively rule out the possibility that the recovery of growth observed here is due to differences in the conditions. To determine whether such a nongrowing population exists in the batch culture, we performed another experiment (Extended Data Fig. 4). n = 681 cells. We carried out the growth-curve experiment once, with two independent lanes (one with YFE44, one with the wild-type strain); the plots are relative to results obtained from the flask inoculated with YFE44.

### Extended Data Fig. 4 Single-cell behaviour during a glucose-to-acetate shift through time-lapse microscopy of batch culture.

a, Diagram showing the experimental protocol (Methods section on‘Batch microscopy’). After the medium shift from glucose to acetate, the culture was split into two identical six-well glass-bottom plates. One was briefly centrifuged and placed into an incubator on a microscope for time-lapse microscopy, and phase-contrast images were recorded. The other plate was placed in a shaker incubator as a control, and OD600 was monitored manually. b, Growth curves from two biological repeats (circles and squares), obtained by monitoring OD600 from the control six-well plate after the media switch. The calculated lag time is 295 min, virtually identical to the batch-culture lag time that we characterized in Fig. 1, indicating that the environment of the six-well plate is almost identical to that of the batch culture as far as the lag time is concerned. c, Normalized single-cell-area traces (two biological repeats) from the other plate (n = 1,761 traces). We use cell area as a metric for biomass growth. Light blue traces show the 1,500 cells that crossed an arbitrary 10% threshold for increase in area within our observation time (Methods). Red traces indicate the 261 cells that did not cross this threshold before they became unobservable, either because they detached from the glass or because they were were flooded by other cells. d, Histogram showing the distribution of time required for individual cells to increase their area by 10%. e, The percentages of cells that grew in cell area (y-axis) by at least the amount shown on the x-axis, relative to their initial size, are plotted. All of these data show that most cells recover after an initial lag phase, eventually growing on acetate. Despite the relatively short observation window of 5–6 h (beyond which the plate became too crowded by dividing cells to allow imaging of individual cells), which is roughly equal to the batch lag time, most cells exhibit substantial growth (e). A 10% increase in cell area is easily detectable, and roughly 85% of cells crossed this threshold. The cells that did cross this threshold grew continuously throughout the observation period, exhibiting a single-cell growth curve and a lag time (c, d) that was similar to the batch lag time. Thus, no more than roughly 15% of cells were completely growth arrested after the shift to actetate, even during this limited observation window. Therefore, in the lag phases studied here, the dormant subpopulations proposed previously12 had a negligible role in determining lag times. (As an example, even if the roughly 15% of growth-arrested cells never grew again, they would contribute only approximately 21 min to the total lag time of 295 min.).

### Extended Data Fig. 5 Absolute and relative concentrations of key metabolites in the shift from glucose to acetate.

a, Intracellular concentrations of F6P in the three biological repeats (circles, squares and triangles) of the shift from glucose to acetate presented in Fig. 2. The dashed line represents the steady-state level of F6P for growth on acetate. The F6P concentration is low compared with the Michaelis constants of key enzymes Pgi and TktA, which catalyse the first reactions from F6P in the synthesis of E4P and R5P, essential precursors for biomass production. b, Intracellular concentrations of PEP during the lag phase that follows a shift from glucose to acetate and from mannose to acetate. Steady-state (s.s.) concentrations are also shown. PEP acts as a key repressor of glycolytic flux by inhibiting Pfk51. The PEP concentration remained low throughout lag phase, even by comparison with the steady-state concentration on glucose, when Pfk is very active. c, Time courses of FBP and PEP concentrations throughout lag phase during a shift from glucose to acetate. We normalized the concentrations by their steady-state concentration (dashed line) during exponential growth on acetate. During the lag phase, FBP drops from its steady-state level for growth on glucose, which is more than 100-fold higher than its steady-state level on acetate (normalized to 1). PEP remains at very low concentrations and slowly builds up, together with FBP, 1.5 h after the shift. In our model, we attribute this slow build-up to the need for protein synthesis to increase levels of gluconeogenic enzymes.

### Extended Data Fig. 6 Proteomics-based characterization of lag-phase dynamics.

af, Gluconeogenic enzymes. Relative levels of gluconeogenic enzymes at different times during lag phase following a shift from glucose to acetate (ace t0, immediately after shift; ace t6, exiting lag phase, 6 h after shift) and glucose to pyruvate (pyr t0, immediately after shift; pyr t1, exiting lag phase, 1 h after shift) and in different steady-state conditions on glucose (glu), pyruvate (pyr), acetate (ace) and mannose (man). a, Isocitrate lyase (AceA); b, malate synthase (AceB); c, fructose-1,6-bisphosphatase (Fbp); d, malate dehydrogenase (MaeB); e, phosphoenolpyruvate carboxykinase (Pck); f, PEP synthase (Pps). gj, Glycolytic enzymes. Relative levels of irreversible glycolytic enzymes at different times during lag phase following a shift from glucose to acetate and from glucose to pyruvate, as well as in different steady-state conditions, as for af. g, 6-Phosphofructokinase I (PfkA); h, 6-phosphofructokinase II (PfkB); i, PEP carboxylase (Ppc); j, pyruvate kinase I (PykF). Black dots indicate weighted median values. These were obtained from multiple measurements and weighted by the confidence of a sample’s quality, as derived from a support vector model (Methods) set up to classify samples into ‘high’ or ‘low’ quality, based on a training set of several thousand hand-classified samples15. The weights’ range is [0,1] and can be found as a separate attribute (‘svmPred’) for each sample in the accompanying source file. Grey dots indicate individual measurements; the size of each dot indicates the associated confidence (the larger the dot, the higher the confidence that a measurement is of high quality). Dot sizes were defined using the ‘MarkerSize’ attribute of the ‘plot’ function in Matlab. Specifically, a dot size was calculated as the confidence value of a measurement (the ‘svmPred’ attribute) multiplied by 11 (which allowed clearer plotting and ease of visual inspection). If the product of this multiplication for a certain measurement was below a certain minimum value (in our case, 1.8), we set the dot size to this minimum (below that value, the dot would not be visible with the naked eye).

### Extended Data Fig. 7 Sequential flux limitation model and trade-off between growth and lag.

a, Intuitively, in our model, lag phases emerge because the gluconeogenic flux, JGNG (blue arrow), limits the synthesis of proteins, which include gluconeogenic enzymes (green arrow). Therefore, the production rate of limiting gluconeogenesis is proportional to the gluconeogenic flux: $$\frac{d}{dt}{\varphi }_{\text{GNG},\mathrm{lower}}\propto {J}_{\text{GNG}}$$, in which $${\varphi }_{\mathrm{GNG},\mathrm{lower}}$$ denotes the abundance of lower gluconeogenic enzymes. JGNG in turn depends on limiting metabolite concentrations. b, To understand the dynamic scaling of these metabolite concentrations, based on the biochemistry of the pathway, we describe gluconeogenesis by a coarse-grained model comprising two irreversible steps (upper and lower gluconeogenesis), connected by reversible reactions. Upper gluconeogenesis does not appear to be limited by its enzyme (Fbp), whose abundance changed only moderately throughout the lag phase and across growth conditions (Extended Data Fig. 6 and proteomics data in ref. 3). We thus assume the flux through upper gluconeogenesis (top blue arrows) to be limited by the concentration of its substrate, FBP, thus $${J}_{\mathrm{GNG}}\propto [\text{FBP}]$$. The FBP concentration is connected to the output of lower gluconeogenesis, PEP, by the relation $$[FBP]\propto {[PEP]}^{2}$$, owing to the stoichiometry of the reversible reactions (grey arrows). The enzymes of lower gluconeogenesis do appear to be limiting, given previous proteomics data3 (Fig. 3a and Extended Data Fig. 6). We assume that the lag phase is dominated by a quasistationary period, where transcriptional regulation can be considered constant. The abundances of gluconeogenic enzymes are assumed to change in proportion to each other, characterized by ϕGNG,lower. The latter assumption is plausible, as the expression of gluconeogenic enzymes is primarily controlled by a common transcription factor Cra. Indeed we note that for different preshift (steady-state) conditions, the abundances of different gluconeogenic enzymes are proportional to each other, as they show the same linear growth-rate dependence (Fig. 3a). The flux through lower gluconeogenesis (bottom blue arrow), which is proportional to [PEP], is then governed by ϕGNG,lower. Thus, $$[\text{PEP}]\propto {\varphi }_{\text{GNG},\mathrm{lower}}$$, resulting in $${J}_{\text{GNG}}\propto {\varphi }_{\text{GNG},\mathrm{lower}}^{2}$$. c, During fast glycolytic growth (top), glycolytic enzymes are highly abundant (thick red arrows), whereas gluconeogenic enzymes are scarce (thin green arrows). The enzyme composition therefore strongly favours glycolysis, resulting in severe depletion of carbon-based metabolites (blue circles) after a shift to gluconeogenic conditions, and hence a long lag phase. For slow glycolytic growth (bottom), the ratio of glycolytic and gluconeogenic enzymes is much more balanced (red and green arrows of similar thickness), resulting in an improved carbon supply to gluconeogenesis after shift and hence a shorter lag. The thick blue and pink arrows illustrate influx from uptake of glycolytic and gluconeogenic substrates, respectively. The thin blue and pink arrows illustrate flux branching off from central carbon metabolism to provide biomass building blocks.

### Extended Data Fig. 8 Preshift overexpression of glycolytic enzymes.

ad, Lag times following shifts from glucose to: a, acetate; b, pyruvate; c, malate; d, succinate. The graphs compare the effects of preshift overexpression of the glycolytic enzymes PykF (strain NQ1543) and Pfk (strain NQ1544) with a control enzyme, ArgA (strain NQ1545). Each protein was overexpressed from the same plasmid (pNT3) using the tac promoter. Horizontal lines and error bars indicate means and standard deviation (n = 4). Lag times more than doubled as a result of preshift overexpression of Pfk or PykF. Thus, the residual activity of glycolytic enzymes is important in lag phase, despite the allosteric regulation of these glycolytic enzymes. Consistent with this picture, the concentration of PEP—a key regulatory metabolite and repressor of glycolytic flux—remained low throughout lag phase, even compared with steady-state levels on glycolytic carbons (Extended Data Fig. 5).

### Extended Data Fig. 9 Improved growth of Cra-knockout E. coli, and trade-offs for other microbes.

a, Growth rates of the Cra knockout (Δcra) on glycolytic carbon sources: growth rates on the slow glycolytic sources (fructose and mannose) are markedly improved compared with the wild type (WT). The Cra knockout expresses very low levels of most gluconeogenic enzymes, and glycolytic enzymes are derepressed; hence it cannot grow on most gluconeogenic carbon sources. bd, Growth–adaptation tradeoff in wild-type yeast and B. subtilis. We grew two different wild-type yeast strains (YPS163 and YPS128) and a B. subtilis strain at different preshift growth rates (λpre) on different media, before shifting them to acetate (b, c) and fumarate (d) minimal media. After the shift, culture density (OD600) was monitored as a function of time. Data points indicate means; error bars show standard deviations from three biological replicates. The lag time of the growth curves increases with increasing preshift growth, suggesting a trade-off similar to that characterized for E. coli (Fig. 1). e, Growth comparison for E. coli and B. thetaiotaomicron, an obligatory anaerobe. The growth rate of E. coli NCM3722 on a number of common carbon substrates from the ‘top’ of central carbon metabolism (glycolysis and pentosephosphate pathways) exhibit a range of values, from 0.9 h−1 down to 0.5 h−1 (black bars). The growth rates of B. thetaiotaomicron (B. theta) on the same substrates in anaerobic conditions (red bars) are all within 10% of each other. For comparison, we also show the growth rates of NCM3722 on the same substrates in anaerobic conditions (blue bars). These show a similar pattern of variation as the aerobic growth rates, with the fast ones comparable to that of B. thetaiotaomicron (roughly 0.6 h−1) and the slow ones about one-fifth of the fast ones. Saturating amounts of substrates (15 mM) were used, except in the case of E. coli on mannose (40 mM).

### Extended Data Fig. 10 Optimal growth rate as a function of the expected substrate abundance in an environment.

a, Cells initially grow exponentially by a factor N (reflecting the expected carbon abundance) over time Tgrowth at growth rate λ. When carbon runs out, the cells enter lag phase, chartacterized by the lag time, Tlag. Cells then again grow exponentially; in the example here, they use the fermentation product acetate at growth rate λace. b, The optimal strategy for the cell minimizes the total time before postshift exponential growth (resulting in the same cell number, but resuming growth the fastest after the lag phase). The total time before postshift growth resumes is the sum of the growth time, Tgrowth = log(N)/λ, and the lag time, given by equation (1), Tlag = 1/[α(λ0 – λ)], both of which are influenced by the growth rate λ. The optimal growth rate, λ*, minimizes this total time, and is obtained from: $${\lambda }^{* }={\lambda }_{0}\frac{\sqrt{\alpha \,\mathrm{ln}(N)}}{1+\sqrt{\alpha \,\mathrm{ln}(N)}}$$ c, For strain NCM3722, the optimal growth rate, λ*, given by this equation, is plotted against the expected carbon abundance, given by N. The value of α was determined from the fit in Fig. 1d to the majority of glycolytic carbon sources (black line). For realistic carbon abundances, the range of optimal growth rates spans precisely the relatively narrow range of growth rates on naturally occurring carbon sources observed for the wild-type E. coli strain NCM3722 (ref. 2): for example, glucose, 0.95 h−1; mannitol, 0.90 h−1; maltose, 0.79 h−1; glycerol, 0.70 h−1; galactose, 0.59 h−1; mannose, 0.49 h−1. The optimal growth rate drops substantially below 0.5 h−1 only when the expected preshift carbon abundance allows for less than a single doubling, N < 2, and surpasses 1.0 h−1 at enormous, unrealistically high carbon abundances, N > 1012, explaining the absence of naturally occurring carbon sources that result in such growth rates.

## Supplementary information

### Supplementary Information

This file contains Supplementary Notes 1 and 2, Supplementary Tables S1-S4 and Supplementary References.

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Basan, M., Honda, T., Christodoulou, D. et al. A universal trade-off between growth and lag in fluctuating environments. Nature 584, 470–474 (2020). https://doi.org/10.1038/s41586-020-2505-4

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