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A model of diauxic community assemblyCommunity 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 mimics30,31,32. 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.Fig. 1: Model of community assembly with diauxie and serial dilution.a Tables of growth rates and resource preferences of two species α (red) and β (yellow), each capable of consuming all four available resources, R1 to R4. The resource preference sets the sequence in which a microbial species utilizes resources, and the corresponding rates gXi indicate the growth rate while consuming each resource (see “Methods”). b Diauxic growth curve of species α during one serial dilution cycle, which has 4 phases of growth on each individual resource, with rates gα1, gα3, gα2, and gα4, respectively (with a brief lag period between two phases). At the end of each dilution cycle, we dilute the population by a factor D = 100, and supply fresh resources (see “Methods”). c Resource depletion curves corresponding to (b), where each resource is represented by a different color. R1 is exhausted at time T1; then species α consumes R3 which runs out at T3, which is followed by exhaustion of R2 at T2, and so on. d Schematic of serial dilution experiment. During community assembly, new species are added one by one from a species pool. After each successful invasion, the system undergoes several growth-dilution cycles until it reaches a steady state. e Population dynamics corresponding to the assembly process in (d). Panels (b) and (c) correspond to a small section of this process (highlighted in gray), where the community dynamics consist only of species α (red) reaching a steady state.Full size imageWe 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 gα1 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 gα3 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 coexist26,33.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 sources34,35,36. 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 assemblyTo 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).Fig. 2: Emergent properties of diauxic microbial communities.In all plots, solid bold lines represent the average over 958 individual community assembly simulations, while gray lines correspond to 100 randomly chosen community assembly simulations. a Total resource depletion time during community assembly (the time taken by the community to deplete all available resources). (Inset) Number of successful invasions during community assembly. b Total species diversity during community assembly (number of surviving species at steady state). c Resource utilization complementarity during community assembly. For each time point, the nth choice complementarity was calculated as a number of unique resources among the n-th preferred choices of all species in the community, divided by the number of unique resources in the environment. For a certain community, the null expectation (complementarity without selection) was defined by the complementarity of a random set of species from the pool that has the same diversity of that community. Colored lines show the average trend of complementarity on each preferred resource choice: top (light blue), second (cyan), third (deep green), and fourth (light green). The red dash-dotted line shows the average trend of null expectation. The gray dash-dotted line at the top corresponds to the perfect complementarity, which is 1. d Frequency of species with anomalous resource preferences during community assembly. The gray dash-dotted line is the expectation of fraction of anomalous species (75%) in the pools.Full size imageResource depletion timeIn 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 diversityThe 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 complementarityThe 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 speciesIntuition 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 on22,26,38. 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 patternsTo 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):$$frac{{T}_{1}}{{T}_{2}-{T}_{1}}=frac{,{{mbox{log}}}(D/2)}{{{mbox{log}}},(2)},$$
(1)
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 successInterestingly, 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 diversitySuccessful 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 {T}_{2},$$
(2)
where gα1 and gα2 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 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. 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).Fig. 3: Top choice resources chiefly drive community diversity and complementarity.a (top) Table showing the preferences of a diauxic microbial species (purple) for 4 resources, R1 to R4. (bottom) Plots showing the depletion of the 4 resources by the purple species during one serial dilution cycle, when grown alone in our model. b (top) Bar plots showing the time taken by the purple species in (a) to grow on each of the 4 resources. Percentages on each bar represent the fraction of time spent growing on each resource. (bottom) Bar plots showing the number of generations grown, or the number of doublings by the species when growing on each resource. In both cases, the plots on the left show the quantities when the purple species is in monoculture (growing alone), and those on the right show them when the purple species is in a community with 3 other species. c Mean growth rates of successful invaders during community assembly. The blue line corresponds to the invader’s top choice, while the other colors correspond to all other choices. The horizontal dashed line shows the mean growth rate of the species pool. Each quantity represents a moving average from 958 independent community assembly simulations. Error bars represent s.e.m. d Fraction of the successful invasions as a function of the order in which the invader’s top choice resource is depleted, 1 indicating cases where the invader prefers the earliest depleted resource, and 4 where it prefers the last depleted resource. Each bar represents the mean of such a fraction over 958 independent community assembly simulations, and error bars represent s.e.m. e, f Effect of invasions of community diversity and complementarity, based on whether the invader’s top choice was (e) unoccupied or (f) occupied. Cartoons show the typical effect of an invasion. Pie charts show the fraction of invasions that increase, decrease or maintain a community’s species diversity (middle) and complementarity (right). On unoccupied resources, diversity typically increases (62%), but sometimes stays constant (32%) or decreases (6%). On occupied resources, diversity typically stays the same (68%), but sometimes decreases (26%) and rarely increases (6%). In almost all cases complementarity either stays maintained or increases ( >95%), and very rarely decreases ( More

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