Trade-offs are thought to arise from inevitable, biophysical limitations that prevent organisms from optimizing multiple traits simultaneously. A leading explanation for biodiversity maintenance is a theory that if the shape, or geometry, of a trade-off is right, then multiple species can coexist. Testing this theory, however, is difficult as trait data is usually too noisy to discern shape, or trade-offs necessary for the theory are not observed in vivo. To address this, we infer geometry directly from the biophysical mechanisms that cause trade-offs, deriving the geometry of two by studying nutrient uptake and metabolic properties common to all living cells. To test for their presence in vivo we isolated Escherichia coli mutants that vary in a nutrient transporter, LamB, and found evidence for both trade-offs. Consistent with data, population genetics models incorporating the trade-offs successfully predict the co-maintenance of three distinct genetic lineages, demonstrating that trade-off geometry can be deduced from fundamental principles of living cells and used to predict stable genetic polymorphisms.
Trade-offs between life-history traits are seen as a key part of the processes that maintain the rich biodiversity observed in microbes1, plants2, insects3 and rain forests4. Due to its importance to biodiversity theory, a search for trade-off data has become the subject of a burgeoning experimental literature using microbial populations in the laboratory1,5,6,7,8,9,10,11,12. However, despite the significance of trade-off geometry to the theory and the long-postulated hypothesis that trade-offs arise from life-history constraints, we know of no successful derivation of trade-off geometry in living systems from fundamental physical or biochemical principles. Resolving this absence is critical if we are to successfuly predict conditions that maintain diverse microbial communities. Here we present a mechanistic theory, supported by data, on the geometry of two trade-offs among microbes: one is a feeding trade-off, one is a growth trade-off. These trade-offs have been studied before but their shapes have remained elusive, yet here we find agreement between theory and data showing that one trade-off is convex and the other is sigmoidal and both are implicated in diversity maintenance.
The first trade-off we study is the metabolic rate–yield trade-off (the RYTO or ‘waste from haste’ trade-off13,14) whose shape we found by culturing Escherichia coli at different growth rates. The second trade-off shape we deduce is that of the nutrient uptake rate-affinity trade-off (the RATO14,15) by measuring the growth kinetics of a library of mutant E. coli that varied in a sugar transporter, LamB. Given these two geometries, prior theory predicts that there could be nutrient conditions where divergent consumer strategies would coexist in a microbial population16. To test this, we used bespoke theoretical population genetics models to determine environmental conditions that would support a multiplicity of nutrient transporter genotypes in E. coli. However, this theory successfully predicted the frequency dynamics of three genetic lineages in a laboratory microcosm only when it incorporated two trade-offs, not one. We therefore establish that trade-offs are a consequence of the physical necessities of life and moreover genetic polymorphisms and biodiversity patterns can be predicted from the resulting trade-off geometries.
Many mechanisms have been proposed for the RYTO in bacteria, both physiological17 and metabolic13, and a quadratic shape is thought to describe it13. Attempts to find the RYTO in vivo, however, have been mixed. A negative correlation between growth rate and yield has been observed but without the quadratic form12. Moreover, positive rate–yield correlations, the antithesis of a rate–yield trade-off, have been observed in other bacterial species9 and, curiously, this relationship is also predicted to be constrained by a quadratic geometry10. We can reconcile these apparent contradictions by following a suggestion9 that r–k relationships can be manipulated by varying the carbon richness of the environment in which bacteria grow (for clarity, we use k for carrying capacity where others use K). We hypothesize that r and k are indeed positively correlated and that the shape of the RYTO can be extracted from r–k data sets.
To study the RYTO, we cultured E. coli B (REL606) in Davis minimal (DM) medium supplemented with a range of sugar (maltotriose) concentrations. We determined their r (exponential growth rate) and k (maximal population size) values by fitting the mathematical logistic model to growth data, where x(t) denotes bacterial population density at time t (Supplementary Discussion, section C). At low sugar concentrations, ranging from 1 to 8 μg ml−1 the experiment recapitulates a prior data set9, whereby a positive, linear correlation between r and k is observed (Fig. 1a). However, when observations from higher maltotriose concentrations are added to the data, ranging from 1 to 125 μg ml−1, the r–k relationship becomes nonlinear because r saturates. This is expected because growth rate is limited, whereas k increases proportionally to the maltotriose supplied (Fig. 1a, Supplementary Fig. 15). Although r and k adaptation are classically viewed as conflicting life-history strategies18 here they are positively correlated. Nevertheless, this is still consistent with a RYTO, as we now explain.
Assume, for the moment, that c is a constant representing the biomass yield of E. coli per maltotriose, when cultured with maltotriose as the sole carbon source at supply concentration S0 μg ml−1. We assume Monod growth kinetics, so the bacterial population density B changes as and maltotriose reduces in concentration as , where Vmax is the maximal rate of uptake of maltotriose and Km is the maltotriose half saturation constant. The initial condition S(0)=S0 applies here, B(0) represents a bacterial inoculate whose size is assumed negligibly small.
A conservation law holds in this model, B(t)/c+S(t)=B(0)/c+S0, so that when S≈0 at carrying capacity, k=B≈c·S0 thus k is proportional to the maltoriose supplied, if we neglect B(0). If we approximate exponential growth using the Monod model, so r≈cVmaxS0/(Km+S0), then k/r≈(Km+S0)/Vmax and so k and k/r should both exhibit linear dependence on S0, hence k and r should be positively correlated. This is consistent with our observations (for k versus S0: R2>0.99, P<10−8; for k/r versus S0: R2≈0.93, P<10−4; Supplementary Fig. 15). However, observations also indicate that yield, meaning c or k/S0, is not constant because it decreases with increasing maltotriose supply (Fig. 1b, F-test versus constant model, F=17.3, P<0.00014).
We therefore need a more accurate description of the variable-yield data in Fig. 1c. So, we relax our assumption that yield is a constant independent of the carbon supplied. Maltotriose import and metabolism is carried out by the genes of the mal regulon19. This set is induced by malT, which itself is expressed when maltodextrins like maltotriose are present. The uptake of maltodextrins is facilitated by the LamB maltoporin in combination with the periplasmic maltodextrin-binding protein MalE and the ATP-driven inner-membrane transport system MalGFK2. Once in the cytoplasm, maltotriose is degraded by MalZ into maltose and glucose, MalQ then produces two glucoses for each available maltose. Given this, a theoretical form for the rate and yield of maltose metabolism can be derived (Supplementary Discussion, section B) by representing glycolysis and the tricarboxylic acid (TCA) cycle as a branched pathway20 (Fig. 1a). Using ATP yield as a proxy for biomass yield, this model predicts a relationship between carbon supply, S0, and cell yield. Writing c(S0) to represent this relationship, it has the following repression-activation form
Here chi represents the highest yield attainable, achieved at the lowest sugar concentrations, whereas clo is the yield attained when sugar is abundant, p is a phenotype that controls the rate of decrease in yield with increasing sugar supply.
The positive r–k correlation predicted by the constant and variable-yield assumptions are consistent with r–k data (Fig. 1b) with agreement across 2 orders of magnitude of maltotriose supply in both cases. However, only the decreasing yield model correctly captures the yield data across the full range of maltotriose concentrations (Fig. 1c, R2>0.99, P≪10−15 from above F-test). The resulting five-parameter, within-strain geometric RYTO model
has a sigmoidal shape that compares favourably with data (Fig. 1c). Note that equation (2) has only four independent parameters, Vmax merely converts cell yield into a rate.
The logistic equation contains no explicit term for nutrient availability so, we deployed the following improvement to that model that better captures different bacterial growth phases and which can also represent N-strain, frequency- and density-dependent competition for maltotriose (where N≥1):
Here Bi is density of the i-th strain, growth rate is Gi(S)=ci·Ui(S), ci is cell yield and Ui(S) is nutrient uptake rate (at nutrient concentration S):
The variable ni represents a subpopulation of the inoculate that may be in a non-vegetative state and the associated νi-dependent terms allow the model to capture the transition from lag to exponential phase. When N=1 equation (3) is consistent with the RYTO data for E. coli strain REL606 (Supplementary Fig. 14) and when N>1, it can predict the outcome of multi-strain competition experiments (Supplementary Discussion, section C).
Pursuing evidence of a between-strain RYTO, we generated a library of E. coli mutants that vary in growth rate because of mutations that interfere with maltotriose uptake. E. coli uses LamB to transport maltotriose across the outer membrane, however, it may also be taken up by OmpF, albeit over an order of magnitude more slowly21,22. LamB is also a receptor for the virus λ, which provides a convent way to generate LamB mutations. Previous work has shown that when E. coli and λ are co-cultured, E. coli rapidly evolves resistance through an array of lamB mutations23,24, some of which alter the diffusion of maltodextrins across the outer membrane by altering the pore dimensions25. We therefore co-cultured 100 separate populations in batch for 3 days and isolated 2 bacteria per population. The large number of populations and low sampling intensity within each population helped maximize the number of unique lamB mutants isolated, the short time period ensured only a single mutation occurred in the lamB gene and nowhere else in the genome. Altogether, we harvested 50 novel E. coli strains with SNPs, insertions and deletions in the lamB gene, the library also includes frameshift mutations that render LamB inoperable (Supplementary Table 1).
We estimated uptake rate, yield and therefore growth rate for each strain by culturing them separately in DM media supplemented with 10, 25 and 250 μg ml−1 maltotriose by fitting equation (3) to that data (Supplementary Discussion, section C). This consistently produced similar values for yield between strains and environments and significant evidence for a negative rate–yield correlation between strains was observed but only at a maltotriose concentration of 250 μg ml−1, not at 25 μg ml−1 (P<0.00064 and P>0.16, respectively; see Fig. 2a).
The same data set points to a second possible trade-off: a correlation between Km and Vmax (Fig. 3a, R2≈0.58, F≈70.5, P<10−10). This prediction is consistent with prior single transporter models for small-molecule uptake, which show that changes to a transporter’s affinity for the molecule will produce concomitant changes in its maximal translocation rate26,27,28. This was demonstrated by distorting the performance of transmembrane channels by creating potential energy wells within a synthetic pore using holographic tweezers. These experiments showed that an optimal transporter configuration must balance a high translocation probability and ‘not too long’ a blockage of the channel27; this is the physical basis of the RATO.
Different theories are possible for RATO geometries depending on the assumptions they use. A linear relationship between Km and Vmax can be derived27 (as is done in equation (6) of the latter reference), whereby Km=pVmax+q for parameters (p, q). This result is consistent with a theoretical enzymatic model based on a transporter embedded within a single membrane (Supplementary Discussion, section B), so this would appear to be the description of the trade-off we seek. However, affinity for a carbon source, a, is usually defined by and so strains with different Vmax and Km phenotypes could have a common maltotriose affinity, if q=0. However, RATO data observed between the E. coli library strains is too noisy to eliminate linearity or to resolve shape (Fig. 3a). To remedy this, we used a related Saccharomyces cerevisiae data set where glucose transporters had been genetically manipulated26. As this data exhibits significant nonlinearity (see Fig. 3b for the relative likelihoods, RLlinear, of linear and nonlinear data fits where RLlinear<0.013), the prior argument supporting a linear Km–Vmax relationship will thus have to be modified.
On reflection, small-molecule uptake in some microbes is a two-step process: entry into the cytoplasm of Gram-negative bacteria and yeast requires the passage through two diffusive barriers, not one. Modifying the above single membrane model to account for this, Km depends nonlinearly on Vmax in the resulting theory (Supplementary Discussion, section B):
where (p, q, r, s) are parameters. A quadratic nonlinearity here is expected because if transport through one diffusive barrier leads to a linear Km–Vmax relationship, two barriers positioned in serial separated by a periplasm should produce a quadratic-like function, which equation (5) is. A power-law fit to empirical (Vmax, Km) data of the form is consistent with equation (5) because b≈1.86±0.23 (mean±estimated s.e.m., degree of freedom=8, Fig. 3b). We note that the role played by MalE in E. coli is neglected as part of the derivation of equation (5) (in Supplementary Discussion, section B), but if we were to assume that MalE–maltotriose binding is so rapid that no free maltotriose is found in the periplasm, then the form of equation (5) will not change.
To test whether the RATO could promote the maintenance of bacterial diversity, we isolated a subset of our library that exhibited a RATO. Three strains, labelled 9a (a predicted high-affinity strain; an AAC deletion at 871 bp in lamB), 19a (a frameshift mutant with nonfunctional LamB, T insertion at 610 bp) and 62b (predicted low-affinity strain; A→G at 1,211 bp) were identified with different maltotriose transport properties (Fig. 3b), one-way analysis of variance, F≈28.2, P<7 × 10−13). The three-strain version of equation (3) predicts they should coexist in a fluctuating resource environment, such as a batch culture, where bacteria are inoculated into fresh flasks each day with high initial resources that are depleted throughout a 24-h cycle as the bacteria grow (Fig. 4a).
We cultured 9a, 19a and 62b in DM with 300 μg ml−1 of maltotriose for 18 days in such a batch culture microcosm and found that, consistent with predictions, all three genotypes persisted. However, the model is not quantitative (Fig. 4b): it fails to predict the relative frequencies at which each genotype was maintained. Most notably, the microcosm was dominated by the high-affinity strain, 9a, whereas the model predicts that the low-affinity strain, 62b, should prevail (Fig. 4a). This shows that consumer lifestyle and the between-strain RATO alone are not sufficient to explain the dynamics we observe.
To understand why theory and experiments disagreed, we reasoned that the RYTO could play a role because it could lead fast-uptake genotypes to have low efficiencies which would impact growth rates. To justify this we followed the path of maltotriose through the cell which is converted to glucose and an excess of glucose produced at high maltotriose concentrations will lead glycolytic NADH to repress citrate synthase and, therefore, the TCA cycle29. As total ATP yield from the TCA cycle is greater per glucose supplied than from glycolysis, this repression should strengthen the within-strain RYTO we observed (Fig. 1). We therefore included a glucose-dependent, maltotriose-to-biomass conversion efficiency into the model to mimic the RYTO (Supplementary Discussion, section D). Only then, with two trade-offs, do we have a theory consistent with the 18-day mean frequency dynamics of the strains that also exhibits the long-term coexistence of all three (Fig. 4c).
To test for stable coexistence, we inoculated the three strains into a fresh microcosm at relative frequencies predicted by the trained dual trade-off model after running it to equilibrium. After so doing, we found no evidence of a deviation from the predicted strain frequencies when propagating them for further 7 days of seasonal co-culture (Fig. 4c and Supplementary Discussion, section D). The eventual deviation that we observed of the microcosm from the theoretical prediction is only to be expected as the coexisting lineages likely gain further mutations that are not encoded within the model (Supplementary Discussion, section D).
Trade-off shape is believed to be a key to understanding species diversity in global ecosystems and our analyses concur. Our theoretical models and empirical analyses elucidate the shape of two, the RATO and RYTO. According to our theory, both must be invoked simultaneously to explain the stable maintenance of genetic diversity in an E. coli transporter gene, lamB, when mutants are co-cultured within a seasonal resource dynamic. The RATO, with its quadratic shape (Fig. 3b), is the result of a biophysical constraint that arises because the transport of extracellular nutrients into the cell is optimized. As different transporter structures are optimal for different extracellular nutrient concentrations, a continuum of potential sugar uptake niches is created (Fig. 4a) that here maintains three genetic lineages. However, nutrient consumption alone cannot account for strain frequencies and differential efficiencies resulting from downstream metabolism due to the RYTO, with its sigmoidal shape (Fig. 1c), ensure that efficient and not greedy consumers prevail.
Trade-offs have long been postulated as a mechanism supporting genetic diversity in ecosystems and it has become a fundamental problem in ecology to find them in living systems30. Importantly, we show they are predictable: because trade-offs are the result of physical processes within the cell, their shapes can be derived mathematically and corroborated against empirical data. Consistent with trade-off theories of biodiversity, we also show that polymorphisms in specific feeding genes are predictable. However, just one trade-off was not enough, we need to account for the simultaneous presence of two to correctly predict allele frequencies in a bacterial microcosm. Therefore, theories of multiple simultaneous trade-offs31 are likely needed to capture the diversity observed in natural ecosystems.
lamB mutant library generation
We co-cultured E. coli strain B(REL606), from the laboratory of Richard Lenski, Michigan State University, Michigan, USA, with the obligatorily lytic λ strain cI26, from the laboratory of Donald Court, National Cancer Institute, Maryland, USA. When co-cultured with cI26, REL606 experiences strong pressure to evolve resistance because E. coli B strains lack generalized phage defenses such as mucoid cell formation, restriction modification or CRISPR adaptive immunity32,33. This pressure is magnified by the lytic phage’s increased virulence as compared with its lysogenic relatives34. About 100 flasks were initiated with 103 bacterial cells and even fewer phage (~102 particles). We initiated the study with small populations to increase the likelihood that mutations for defense would arise de novo, which improved our chances of isolating unique lamB mutations. Culturing occurred in 50 ml Erlenmeyer flasks, filled with 10 ml of modified DM35 (125 μg ml−1 maltotriose instead of glucose and 1 μg ml−1 of magnesium sulphate), incubated for 24 h, at 37 °C and shaken at 120 r.p.m. Bacteria were pre-conditioned for a full 24-h cycle before being co-cultured with phage. After the first 24 h of incubation in the mixed cultures, a random 100 μl sample of each flask was transferred to a fresh flask and the bacteria and phage were allowed to grow again. This cycle was repeated twice and bacteria were sampled after the third day of growth. We ended the experiment at this early time-point to ensure that the bacteria only had enough time to evolve a single mutation for defense. In addition, in previous experiments we observed the greatest genotypic diversity of bacteria on the third day. E. coli evolve resistance in this environment through mutations in many loci, however, preliminary experiments revealed that lamB is the most common locus for λ resistance mutations. For this study, we focused just on genotypes that had mutations in lamB.
Bacterial isolation and storage
To isolate bacteria we streaked a sample on Luria Bertani (LB) agar plates36, incubated them for 24 h at 37 °C, and arbitrarily picked two colonies. Next, we re-plated each colony two more times serially to remove all phage. Finally, we grew each colony in liquid LB overnight and preserved two 1 ml samples in 15% glycerol and froze it at −80 °C. We tested whether each isolate had evolved resistance to λ by spot assay37, where 2 μl of the concentrated phage stock is dried on top of a matrix of 4 ml of soft agar (LB agar with only 4% w/v agar) and 0.5 ml of the LB overnight culture. The matrix is suspended on top of a typical LB agar Petri dish. Each plate was incubated at 37 °C for 24 h and then scored the next day. A strain was considered resistant if no clearing occurred under the spot where the phage lysate was applied. We sequenced each of the resistant genotypes to determine if resistance had evolved through mutations in lamB.
Sequencing was performed with an automated ABI sequencer maintained at Michigan State University Research Technology Support Facility. PCR-amplified fragments that were purified with GFX columns and were used as templates. Fragments containing lamB were amplified with primer sequences 5′-TTCCCGGTAATGTGGAGATGC-3′ and 5′-AATGTTTGCCGGGACGCTGTA-3′, placed 1,398 bases up and 504 bases downstream of the lamB gene, respectively.
Each genotype was revived by inoculating a few microliters of frozen culture into a tube with 4 ml of liquid LB. Cultures were grown at 37 °C and shaken at 160 r.p.m. After 12 h, 10 μl of each culture was transferred to a fresh tube with modified DM250 and grown for 12 h at 37 °C and shaken at 160 r.p.m. This acclimation step to DM250 was repeated again because preliminary studies showed that growth was much more consistent when two sequential acclimation steps were performed. Finally, 2 μl of each culture (106 cells) was inoculated into a well with 200 μl of modified DM. Wells were randomized in a 96-well plate, however, cultures were never inoculated into the outer-rim to avoid edge effects caused by increased rates of evaporation. Next, the cells were incubated at 37 °C and the optical densities (at 420 nm wave length) of each well was measured once every 5 min for 12 h using a VersaMax automated plate reader (Molecular Devices). Each genotype was grown in modified DM supplemented with either 25 μg ml−1 of maltotriose or 250 μg ml−1, in four separate replicates. Optical densities (OD) was converted into cell densities by comparing counts of colony-forming units to ODs at a number of different cell concentrations. Growth curves were created identically for the ancestor, REL606, except at many more concentrations of maltotriose (1, 2, 4, 8, 16, 32, 64 and 128 μg ml−1). These data were used to measure the RYTO.
Coexistence experiments were run under the same conditions as the growth dynamics were measured, except maltotriose concentrations were increased to 333 μg ml−1, plates were continuously shaken at 220 r.p.m. and the experiments were run for 18 days by transferring a random sample of 2% of the culture after 24 h of growth to a well with fresh medium. Coexistence was measured by tracking co-occurring populations of three genotypes—9a, 19a and 62b. To track the populations, we used the ara genetic marker. Each of the three lamB mutant genotypes were unable to metabolize L-arabinose (ara-) because of a single nucleotide substitution in araA that caused a G-to-D amino acid change at position 92. We selected for spontaneous reversions of this mutation by growing ~109 cells of each strain on minimal arabinose plates for 48 h at 37 °C (ref. 35). Reversions were verified by targeted sequencing of araA using the same protocol for lamB, except the following primers were used for PCR amplification: 5′-CCGATACGCTCATGGGCTTGTTTA-3′ and 5′-CTGCCCAGGCCGTTGCGACTCTAT-3′. The ara genetic marker was chosen because previous work has shown that it does not confer a cost35. We cultured each ara+ genotype with the ara− version of the other two lamB mutants. The density of the focal genotype was measured by counting colonies on minimal arabinose plates. Total bacterial population size was also measured by counting colonies on LB plates. For each ara+ genotype, we constructed four replicate populations making 12 coexistence experiments in total.
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We thank Chris Marx, Tom Ferenci, Christina Burch and other members of a NESCent working group ‘Mathematical Models, Microbes and Evolutionary Diversification’ for helpful discussions. We thank Richard Lenski and Donald Court for supplying bacteria and phage.
The authors declare no competing financial interests.
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Meyer, J., Gudelj, I. & Beardmore, R. Biophysical mechanisms that maintain biodiversity through trade-offs. Nat Commun 6, 6278 (2015). https://doi.org/10.1038/ncomms7278
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