Complementary resource preferences spontaneously emerge in diauxic microbial communities

A model of diauxic community assembly

Community models studying diauxie should mimic serial dilution cultures instead of chemostats, in order to make their predictions both experimentally and ecologically relevant. Experimentally, microbial community assembly assays frequently utilize serial dilution cultures. Ecologically, diauxic growth is best suited to a “feast and famine” lifestyle, which a serial dilution culture mimics^30,31,32. Therefore, throughout this manuscript, we model the assembly of a microbial community undergoing a sequence of growth-dilution cycles (see Fig. 1a). Community assembly occurs gradually through the addition of microbial species from a diverse species pool one at a time. Each species in the pool consumes resources diauxically, i.e., one at a time according to its resource preference.

We begin by illustrating the growth of a single species (labeled α) grown in an environment with four resources (Fig. 1a–c). The species first grows on its most preferred resource (R₁) with a growth rate g_α1 until time T₁, when this resource gets exhausted. After a lag period τ, the species switches to growing on its next preferred resource (R₃) with growth rate g_α3 until time T₃, when this resource also gets exhausted. This process of diauxic growth by sequential utilization of resources continues until either all resources are depleted, or the cycle ends at time T. At this point, a fraction 1/D of the medium containing the species is transferred to a fresh medium replete with resources. This corresponds to the dilution of species abundances by a factor D, mimicking serial dilution experiments in the laboratory.

After several transfers, species dynamics converge to a steady state, where each species starts a cycle with the same initial abundance as the previous cycle. At this point, we add a small population of a new invader species, chosen randomly from the species pool, to the steady-state community (Fig. 1d, e). (Hence, we assume that species invasions are rare enough such that communities always reach a steady state before the next invasion.) The invader may differ from the resident species in both resource preference order and growth rates on each resource (Fig. 1a). Once introduced, the invader may grow and establish itself in the community in a new steady state (Fig. 1d, e), or it may fail, returning the community to its previous steady state.

The growth rates and preference orders completely characterize a species, while the set of resource depletion times (T₁, T₂, etc.) characterize the current state of the abiotic environment. As we will later show, these resource depletion times are important observables in a community, since they determine the success or failure of an invader.

A realistic example of a community captured by our model is the human gut microbiome, specifically the assembly of primary consumers (e.g., Bacteroides species) on the polysaccharides (e.g., starch, cellulose, and mucin) that they consume. Here, there is a significant overlap between the metabolic capabilities of the microbes, but they nevertheless coexist. These species often consume polysaccharides diauxically, and engage in resource competition. Moreover, several of these species have different resource preferences, which others have hypothesized help them coexist^26,33.

Throughout this paper, we neglect diauxic lag times (τ = 0) for simplicity. We will later show that adding lag times only quantitatively strengthens our main results (see “Discussion” and Fig. 5). We also assume that the supplied resource concentrations are sufficiently large, enabling species to always grow exponentially at their resource-specific growth rates. Further, we assume a balanced supply of resources, i.e., that resources are supplied in equal concentrations (see “Discussion” and Supplementary Text for results in an unbalanced resource supply).

We simulated the assembly of 1000 communities, each being colonized from a pool of ~10,000 species (see “Methods”). Species could utilize all 4 supplied resources diauxically. Each species had a random resource preference order and different growth rates on each resource, which were picked randomly from a rectified normal distribution (with mean 0.25 and standard deviation 0.05). We assumed that the growth rate distributions for each of the 4 resources were the same, such that no resource was consistently better than the other. This is a simplifying assumption, but it nevertheless captures a variety of experimental observations showing remarkable growth rate variability of different microbial species on the same carbon sources^34,35,36. Community assembly proceeded via introduction of species one at a time, in a random order, until each species had attempted to invade exactly once.

Emergent properties of diauxic community assembly

To study the emergent properties of communities of diauxic species, we followed the assembly process from a species pool via invasion of species one at a time. We used the number of invasion attempts to track time; communities matured over successive invasions. We found that the assembly process became slower over time—successful invasions became rarer as the community matured (Fig. 2a inset). Throughout the assembly process, we recorded four key properties of communities: total resource depletion time, species diversity, complementarity of the community, and prevalence of anomalous species (defined below).

Resource depletion time

In each community, resources disappear at specific times and in a well-defined order (Fig. 1c). The total resource depletion time measures how quickly the community consumes all supplied resources. In this way, the total resource depletion time characterizes the overall speed at which a community consumes resources. The total resource depletion time decreased as communities assembled (Fig. 2a, solid line). The rate and degree of this decrease depend on the mean and variance of the growth rate distribution and the number of invasion attempts. In addition, the variability in depletion times between communities reduced over community assembly (Fig. 2a, gray lines; coefficient of variation reduces by 47%, see Fig. S1). Thus the assembly process selects for communities that collectively consume resources quickly.

Species diversity

The species diversity was quantified as the number of species coexisting in the steady-state community. In the model, like in other consumer-resource models, the number of coexisting species at steady state is limited by the number of resources, 4 (Fig. 2b, dashed line)^10,37. This is a natural consequence of competition for resources in our model (see Supplementary Text, section F for a derivation). Notably, species with the same resource preferences can coexist in the model, as long as the number of species is less than the number of resources (e.g., pairs of E. coli strains can coexist in media with glucose and xylose, see below). We found that the average community diversity increased over time, but the rate slowed as the community matured (Fig. 2b; note the logarithmic x-axis scale). Communities displayed significant variability in the trajectories of increasing diversity (Fig. 2b, gray lines). We discuss the slow increase of diversity, and observed variability, in the next section.

Top choice complementarity

The top choice complementarity of a community measured the overlap in the top choice resource of each of the species residing in the community. We defined the top choice complementarity of a community as the number of unique top choice resources among community residents, divided by the number of residents. Thus the top choice complementarity varied between 1, in a maximally complementary community where each resident species had a unique top choice resource (Fig. 2c, right), and 1 divided by the number of coexisting species in the community, where all residents chose the same resource as the top choice (Fig. 2c, left). During community assembly, top choice complementarity stayed close to the maximum value throughout the assembly process (Fig. 2c, blue). This observation was in sharp contrast to the prediction from a null model for the complementarity (Fig. 2c, red). We obtained the null prediction by measuring the complementarity of a group of randomly chosen species from the species pool (group size being the number of coexisting species in the community). This null prediction decreased during the assembly process, due to the increasing community diversity, unlike the top choice complementarity which remained close to the maximum value. We also recorded the complementarity in the second, third, and fourth choice resource of the assembled community (defined similarly to the top choice complementarity). The complementarity of all other choices agreed with the null prediction (Fig. 2c). Together, these observations suggest that communities of coexisting diauxic species exhibit high complementarity on the top-choice resources, in a manner reminiscent of niche partitioning in consumer-resource models.

Prevalence of anomalous species

Intuition gleaned from experiments with E. coli dictates that microbes often grow fastest on their top choice resource (glucose for E. coli)^18,20. However, exceptions to this trend also exist, such as Bacteroides species in the human gut that often prefer polysaccharides that they grow slower on^22,26,38. Based on this intuition, we defined anomalous microbes as microbes that do not grow fastest on their top choice resource. To investigate which resource preferences might give microbes a competitive advantage during community assembly, we tracked the fraction of anomalous resident species during community assembly. Despite the majority (75%) of species in the pool being anomalous (since growth rates and preferences were randomly picked; see “Methods”), anomalous species were absent in mature communities. The fraction of anomalous resident species decreased rapidly during assembly (Fig. 2d). Thus, anomalous resource preferences are strongly selected against during community assembly. Further investigation revealed a reduced selection pressure against anomalous species if either resource supply was severely imbalanced (i.e., the imbalance has to be comparable to the dilution factor, D = 100), or if the dilution factor was small (see Figs. S4 and S5; also see Supplementary Text, sections C and H). However, microbes with anomalous resource preferences were eventually outcompeted in all conditions.

Top choice resources chiefly drive emergent assembly patterns

To understand what factors drove the maintenance of top choice complementarity—despite the steady increase in species diversity, expected to reduce complementarity—we focused on growth on top choice resources. We hypothesized that the reason for the much higher than expected top choice complementarity was the following: diauxic species derived most of their growth, and spent most of their time growing on their top choice resources. Co-utilizing microbes, instead, grow on multiple resources simultaneously, spending roughly equal time on each utilized resource.

To test this hypothesis, we first simulated the growth of a single diauxic species in monoculture using our model. We found that indeed, the species derived the overwhelming majority of its growth (measured in generations of growth) and spent most of its time growing on its top choice resource (54%, Fig. 3a, b, left). For a simpler case, where a single species had the same growth rate g while growing on two resources (both supplied at the same concentration), and preferring resource R₁ over R₂, we derived the ratio of time spent growing on the top choice resource R₁ (T₁) versus the second choice R₂ (T₂ − T₁). We obtained the following approximate expression for a large dilution factor D (see Supplementary Text, section A):

which explains that the fraction of time spent growing on the top choice resource increases with the dilution factor.

Strikingly, the fraction of time spent growing on the top choice resource became even larger if the species grown in monoculture (Fig. 3b, top row) were instead part of a diverse community (i.e., in co-culture with 3 other species, top choice share 70% versus 54% in monoculture, Fig. 3b, bottom row and top row, respectively). This is because of the following reason. In our model, while a species consumes and grows on all available resources in monoculture, in co-culture, it may not have the opportunity to consume all the resources it can grow on because other species might deplete them first. This further skews growth in favor of the top choice resource. Such a phenomenon only occurs in diauxic species, not co-utilizing species (Supplementary Text, section I).

Invader success

Interestingly, once we understood that the top choice chiefly drove species growth, we could explain the other emergent patterns in diauxic communities. Importantly, the success of an invader depended on the growth rate on their top choice resource. As community assembly proceeded, the top choice growth rate of successful invaders increased consistently (Fig. 3c, blue line), while their growth rates on all other choices remained constant and close to the average growth rate (Fig. 3c, green lines). Selection on the top choice growth rate in diauxic communities is in striking contrast with co-utilizing communities, which we found select for the average growth rate across all resources instead (Supplementary Text, section I). Further, an invader whose top choice resource coincided with the last depleted resource in the community had the highest probability of invasion success (Fig. 3d). Invaders whose top choice resource was not depleted last had lesser time to grow on it, and thus a lower rate of invasion success. By depleting the last resource faster, invaders reduced the total resource depletion time in the community, thus explaining the trend observed in Fig. 2a. In addition, after a successful invasion, the community’s steady state could have a different resource depletion order.

Complementarity and diversity

Successful invasions could be classified into one of two types based on the “invaded resource”, i.e., the invader’s top choice. If the invaded resource was not the top choice of any other resident community member, we called it an invasion of an “unoccupied” resource (Fig. 3e; in our simulations, 33% of cases). If the invaded resource was instead already the top choice of at least one resident, we called it an invasion of an “occupied” resource (Fig. 3e; 67% of cases). Both types of successful invasions had different effects on species diversity, but interestingly, both maintained complementarity (on the top choice, as in Fig. 2c). invasions of unoccupied resources usually increased community diversity by 1 (62% of cases), and were less likely to result in the extinction of one or more other species (38% of cases). This is because, in that case, the invader did not have to compete with other residents for its top choice resource. For communities with a complementarity <1, unoccupied invasions increased complementarity (Fig. 3e). For communities with complementarity 1, they maintained complementarity (Fig. 3e).

In contrast, an invasion of an occupied resource usually brought the invader in conflict with at least one resident with the same top choice (Fig. 3f, cartoon). To be successful, an invader typically had to have a better growth rate than the resident with a matching top choice. A successful invasion of this kind thus typically maintained species diversity (by replacing one resident with the invader; Fig. 3f) and sometimes, could decrease it (by knocking out more species; Fig. 3f). It is because of the prevalence of invasions of this kind (94% of invasions)—which often did not change or could even decrease diversity—that overall species diversity increased slowly in the community. However, because the outcome of all such invasions was driven primarily by the top choice resources, replacements by successful invaders maintained top choice complementarity (Fig. 3f). Thus, even though successful invasions could impact species diversity in different and often contrasting ways, they maintained species complementarity on their top choices. Moreover, because all other choices had much smaller effects on species’ growth, the emergence of complementarity was restricted to only the top choice.

A geometric approach to understand microbial communities undergoing serial dilution

To further understand what shaped the observed emergent community properties, we developed a geometric approach to visualize, analyze, and understand community assembly during serial dilution. The approach is inspired by previous work by Tilman^39,40, where he developed geometric methods to analyze continuously diluted (chemostat-like) communities. The geometric method is easiest to visualize for a community in a two-resource environment, and so we will restrict ourselves to this scenario in this text.

The key insight to developing a geometric approach for serially diluted communities is the following: each resource environment can be characterized by a set of steady-state resource depletion times, T_i in our model. At steady state, a species starts consecutive growth cycles at the same abundance i.e., its abundance grows by a factor equal to the dilution factor D every growth cycle. The set of resource depletion times that allows a species to grow exactly by a factor D defines a set of curves in the space of T_i. We term these curves zero net growth isoclines (ZNGIs) following Tilman and others^{39,40,41,42,43}.

Figure 4a illustrates the ZNGI for a species α growing on two resources, R₁ and R₂), preferring R₂ over R₁. The isoclines are composed of two separate lines in the two triangular regions: T₁ < T₂ (Fig. 4a, yellow) and T₁ > T₂ (Fig. 4a, green) defined by the following equations:

where g_α1 and g_α2 are the species α’s growth rates on R₁ and R₂, respectively. The two triangular regions separated by the diagonal define two complementary scenarios: when T₁ < T₂, R₁ is depleted first and the species grows exclusively on its preferred resource R₂; when T₁ > T₂, R₁ is depleted second and the species grows on R₁ after R₂ is depleted.

For a given set of initial resource and species concentrations, community dynamics must converge to a steady state lying on the ZNGI of the surviving species (e.g., the bold purple point in Fig. 4a). This point defines the resource depletion times by the resident species at steady state. Changing the resource supply or dilution factor moves this point along the ZNGI.

The ZNGI of a species also separates the resource environment space into two regions: a region inside the ZNGI (towards the origin) where that species grows by a factor <D, and a region outside the ZNGI (away from the origin) where the species grows by a factor >D. An invader is successful if it is able to grow by a factor ≥D in the community it invades. Geometrically, the invader’s ZNGI must be closer to the origin than the resource environment corresponding to the invaded community (Fig. 4b). In this way, our geometric approach allows easy visualization of invasion criteria.

We can also visualize invasion outcomes. A successful invasion of a single-species community leads to either displacement of the resident or coexistence between the invader and resident. For example, in Fig. 4b, because the ZNGI of the invader (blue) lies fully inside the ZNGI of the resident (purple), the invader displaces the resident. This is because the invader reduces the resource depletion times in the environment to a point where the resident can no longer survive, driving it extinct (bold blue point in Fig. 4b). In contrast, in Fig. 4c, the ZNGI of the invader (orange) intersects with the new resident (blue), in a manner that leads to coexistence between both species (albeit at a new set of resource depletion times, i.e., their intersection point in Fig. 4c). In general, whether two species will coexist depends on various factors, such as the supplied resource concentrations, but whenever two species coexist, they will do so at the intersection of their ZNGIs (Supplementary Text, section A). As a corollary, two species whose ZNGIs do not intersect cannot coexist. Notably, the orange and blue species in Fig. 4c coexist stably with each other; a short perturbation to the resource supply is quickly compensated by species growth, and the resource depletion times returned to the coexistence point (see Supplementary Text, section B for details).

The geometric approach provides an alternative explanation to why species with complementary top choices are more likely to coexist than species with the same top choice (Fig. 2c). The ZNGIs of species sharing the same top choice are unlikely to intersect with each other (e.g., the blue and purple species in Fig. 4b). This is because of two reasons: (1) their segments in the yellow region are parallel to each other since both species prefer the same resource (R₂), and (2) for the slanted segments in the green region to intersect, the blue species would need a higher growth rate on R₁ than the purple species. This is as likely as the outcome of a coin toss, since both growth rates derive from the same distribution. Thus, an invasion of an occupied resource often leads to displacement of the resident, not coexistence (Fig. 4b, d) and no change in community diversity, while an invasion of an unoccupied resource often leads to coexistence (Fig. 4c) and an increase in community diversity (Fig. 3e).

Finally, we can also explain the relative scarcity of anomalous species (Fig. 2d). The ZNGI of an anomalous species is defined by an acute angle between the two line segments that comprise it (Supplementary Text, section C and Fig. S11), in contrast with an obtuse angle for a non-anomalous species (e.g., Fig. 4a). Briefly, it is possible for an anomalous species to displace a species with the same resource preference, but nevertheless be displaced by that same species once community diversity increases. This scenario is described in detail in the Supplementary Text, section C. Taken together, our geometric approach helps explain all the emergent properties of diauxic communities described in the text.

Diauxic lags promote complementarity, coexistence, and multistability

In our model, we have so far assumed the absence of diauxic lags to simplify our calculations and simulations (τ = 0). Diauxic lags represent a period of no growth after exhausting a resource during which microbes re-wire their metabolic machinery for utilization of the next resource^27,30,44,45. To test how adding diauxic lags would affect our results, we considered two variants of our model, first with fixed lag times, and second with variable lag times in an experimentally relevant scenario.

In the first variant of our model with lags, we added a fixed diauxic lag uniformly for all species and resources, and repeated our community assembly simulations. We found that adding diauxic lags promotes species diversity as well as complementarity, thus strengthening our results (Fig. 5; Supplementary Text, section H). As microbes spend more time switching from one resource to another, they wait longer to grow on less preferred resources. Increasing lag times thus skews their growth even more towards their top choice resource. Recent experimental work has shown that there is a trade-off between diauxic lag times and microbial growth rates, such that microbes that grow faster tend to experience longer lags³⁰. Adding such a growth-lag trade-off to our model promotes diversity and complementarity even further (Figs. S3 and S6). Our geometric approach explains why a growth-lag trade-off promotes coexistence. Namely, there is a greater chance for the ZNGIs of two microbes exhibiting such a trade-off to intersect with each other and lead to stable coexistence (see Supplementary Text, section H).

In the second variant, we applied our model to predict the outcomes of pairwise co-cultures of 13 laboratory E. coli strains. We knew the growth parameters, such as growth rates and yields, for these strains from experimental measurements (data from Barthe et al.⁴⁶). These strains represented a much less diverse pool than the one we had assumed so far, and thus all had the same preference order, preferring glucose over xylose (Fig. 6b). While the diauxic lag times were measured in a population shifting from glucose to xylose for the first time, they were shown to be sensitive to experimental conditions and characterized by long memory effects, and thus could potentially vary across multiple dilution cycles. Further, the initial (regrowth) times of these strains relevant for the serial dilution experiments were not measured. Therefore, we sampled both the diauxic and initial lag times from uniform distributions in the reported experimentally relevant ranges (Fig. 6c; see “Methods”). For a randomly chosen set of lag times, our simulations yielded three interesting observations. First, even for these closely related strains, there was a wide variety of qualitative outcomes (Fig. 6d), ranging from competitive exclusion (95%), coexistence (4%), and in rare cases, bistability (where both species mutually excluded each other; 1%). Notably, these results show that bistability is another possible outcome for diauxic species (see Supplementary Text, section G for a more detailed discussion). In our simulations, bistability resulted from a difference in biomass yields for glucose and xylose across strains. Using geometric theory developed in our study, we found that bistability (and even multistability) can also result from other factors, such as the presence of diauxic lags, and anomalous species (Supplementary Text, section D and H). Second, strains with the same preference order could coexist (Fig. 6e), showing that competition between strains did not only occur on the top choice, but on multiple resources. A strain that grows slower on its top choice (i.e., an anomalous species) could still coexist if it grew faster on its second choice, or if it had a shorter lag (Fig. S22). Third, the qualitative co-culture outcomes were relatively robust to varying the lag (both initial and diauxic) times, suggesting that the uncertainty in these parameters was unlikely to affect most qualitative outcomes (Fig. 6e). Multiple simulations where we randomly sampled the lag times in an experimentally relevant range showed that most outcomes (e.g., coexistence) were robust to uncertainty in the lag times. Together, these results show that the presence of lags strengthens our results, ultimately allowing us to make testable predictions for laboratory strains.

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