Competition for fluctuating resources reproduces statistics of species abundance over time across wideranging microbiotas
Abstract
Across diverse microbiotas, species abundances vary in time with distinctive statistical behaviors that appear to generalize across hosts, but the origins and implications of these patterns remain unclear. Here, we show that many of these macroecological patterns can be quantitatively recapitulated by a simple class of consumerresource models, in which the metabolic capabilities of different species are randomly drawn from a common statistical distribution. Our model parametrizes the consumerresource properties of a community using only a small number of global parameters, including the total number of resources, typical resource fluctuations over time, and the average overlap in resourceconsumption profiles across species. We show that variation in these macroscopic parameters strongly affects the time series statistics generated by the model, and we identify specific sets of global parameters that can recapitulate macroecological patterns across wideranging microbiotas, including the human gut, saliva, and vagina, as well as mouse gut and rice, without needing to specify microscopic details of resource consumption. These findings suggest that resource competition may be a dominant driver of community dynamics. Our work unifies numerous time series patterns under a simple model, and provides an accessible framework to infer macroscopic parameters of effective resource competition from longitudinal studies of microbial communities.
Editor's evaluation
This paper introduces an elegant mathematical and ecological framework to model the fluctuations of microbial abundances in microbiomes along time series. The modeling approach considers consumerresource properties and is regulated by few parameters. Applied to timeseries microbiome data the model suggests the existence of recurrent patterns of microbial dynamics that are quite dependent on resource competition.
https://doi.org/10.7554/eLife.75168.sa0Introduction
Microbial communities are ubiquitous across our planet, and strongly affect host and environmental health (Sekirov et al., 2010; Tkacz and Poole, 2015). Predictive models of microbial community dynamics would accelerate efforts to engineer microbial communities for societal benefits. A promising class of models is consumerresource (CR) models, wherein species growth is determined by the consumption of environmental resources (Chesson, 1990). CR models capture a core set of interactions among members of a community based on their competition for nutrients, and have demonstrated the capacity to recapitulate important properties of microbial communities such as diversity and stability (Niehaus et al., 2019; Posfai et al., 2017; Tikhonov and Monasson, 2017). However, while model parameters such as resource consumption rates are beginning to be uncovered in the context of in vitro experiments (Goldford et al., 2018; Hart et al., 2019; Liao et al., 2020), it remains challenging to determine all parameters for a community of native complexity from the bottomup. A more accessible approach to parametrize CR models and to understand the features that drive communitylevel properties is needed.
To interrogate the dynamics of in vivo microbiotas, a common, topdown strategy is longitudinal sampling followed by 16S amplicon or metagenomic sequencing, thereby generating a relative abundance time series. Analyses of longitudinal data have shown that species abundances fluctuate around stable, hostspecific values in healthy humans (Caporaso et al., 2011; David et al., 2014; Faith et al., 2013). Recently, it was discovered that such time series exhibit distinctive statistical signatures, sometimes referred to as macroecological dynamics, that can reflect the properties of the community and its environment (Descheemaeker and de Buyl, 2020; Grilli, 2020; Ji et al., 2020; Shoemaker et al., 2017). For example, in human and mouse gut microbiotas, the temporal variance of different species scales as a power of their mean abundance (‘Taylor’s law’, Taylor, 1961) and deviations from this trend can highlight species that are transient invaders (Ji et al., 2020). Time series modeling can also provide insights into the underlying ecological processes. For example, the relative contributions of intrinsic versus environmental processes can be distinguished using autoregressive models whose output values depend linearly on values at previous times and external noise (Gibbons et al., 2017). Time series can also be correlated to environmental metadata such as diet to generate hypotheses about how environmental perturbations affect community composition (David et al., 2014), and to identify environmental drivers of transitions between distinct ecological states (Levy et al., 2020).
A growing body of work has shown that time series generated by simple mathematical models can exhibit statistics similar to experimental data sets, reinforcing the utility of such models for providing information about community dynamics even when many microscopic details are unknown. Some statistics can be recapitulated by phenomenological models, such as a noninteracting, constrained random walk in abundances (Grilli, 2020), while others can be described by a generalized LotkaVolterra (gLV) model with colored noise (Descheemaeker and de Buyl, 2020) or by ecological models describing the birth, immigration, and death of species (Azaele et al., 2006). However, the origins of and relationships among time series statistics have yet to be explained. Here, we sought to address this question using CR models, and simultaneously to use time series statistics as an accessible approach for parametrizing CR models.
Since the network of resource consumption in a community will typically depend on thousands of underlying parameters, directly measuring all parameters is intrinsically challenging. We sought to overcome this combinatorial complexity by adopting an indirect, coarsegrained approach, in which resources describe effective groupings of metabolites or niches, and model parameters are randomly drawn from a common statistical ensemble. We show that this simple formulation generates statistics that quantitatively match those observed in experimental time series across wideranging microbiotas without needing to specify the exact parameters of resource competition, allowing us to infer the global properties of resource competition that can recapitulate experimentally observed time series statistics. We further show that our effective CR model captures the behavior of a broader class of ecological interactions, and can guide the development and analysis of other models and their time series statistics. Our work thus provides an accessible connection between complex microbiotas and the effective resource competition that could underlie their dynamics, with broad applications for engineering communities relevant to human health and to agriculture.
Results
A coarsegrained CR model under fluctuating environments
To determine the nature of time series statistics generated by resource competition, we considered a minimal CR model in which $N$ consumers compete for $M$ resources via growth dynamics described by
Here, ${X}_{i}$ denotes the abundance of consumer i, ${Y}_{j}$ the amount of resource $j$, and ${R}_{ij}$ the consumption rate of resource $j$ by consumer i. The resources in this model are defined at a coarsegrained level, such that individual resources represent effective groups of metabolites or niches. We assumed that the resource consumption rates ${R}_{ij}$ were independent of the external environment and constant over time, thereby specifying the intrinsic ecological properties of the community with a collection of $N\times M$ microscopic parameters. To simplify this vast parameter space, we conjectured that the macroecological features of our experimental time series might be captured by typical profiles of resource consumption drawn from a statistical ensemble. This is a crucial simplification: while these randomly drawn values will never match the specific resource consumption rates of a given microbiota, previous work suggests that they can often recapitulate the largescale behavior of sufficiently diverse communities (Cui et al., 2021). This simplification allows us to test whether particular ensembles of resource consumption rates can reproduce the time series statistics we observe. Specifically, we considered an ensemble in which each ${R}_{ij}$ was randomly selected from a uniform distribution between 0 and ${R}_{\text{max}}$. To model the sparsity of resource competition within the community, each ${R}_{ij}$ was set to zero with probability $S$ (Figure 1A). This ensemble approach allows us to represent arbitrarily large communities with just two global parameters, $S$ and ${R}_{\text{max}}$.
We simulated the dynamics in Equation 1 using a serial dilution scheme (Erez et al., 2020) to mimic the punctuated turnover of gut microbiotas due to multiple feedings and defecations between sampling times. During a sampling interval $T$, each dilution cycle was seeded with an initial amount of each resource, ${Y}_{j,0}\left(T\right)$, and Equation 1 was simulated until all resources were depleted ($d{Y}_{j}/dt=0$ for all $j$). The community was then diluted by a factor $D$ and resources were replenished to their initial amounts ${Y}_{j,0}\left(T\right)$ (Figure 1B). To mimic the effects of a reservoir of species that could potentially compete for the resources (Ng et al., 2019), we initialized the first dilution cycle of each sampling interval by assuming that $N$ consumers were present at equal abundance. Additional dilution cycles were then performed until an approximate ecological steady state was reached (Figure 1B, Materials and methods). Consumer abundances at sampling time $T$ were defined by this approximate ecological steady state. For the relevant parameter regimes we considered, this approximate steady state was reached within a reasonable number of generations (5–6 dilutions or ~40 generations for $D=200$). Although the precise details of microbiota turnover are largely unknown in humans, our modeling results were robust to the precise value of $D$ and threshold for ecological steady state (Figure 1—figure supplement 1). Similarly, our results did not depend on the precise composition of the reservoir (Figure 1—figure supplement 2), although they did depend on its existence and relative size (Figure 1—figure supplement 2).
Under the assumptions of this model, any temporal variation in consumer abundances must arise through external fluctuations in the initial resource levels ${Y}_{j,0}\left(T\right)$, which might come, for example, from dietary fluctuations. To model these fluctuations, we assumed that the initial resource levels undergo a biased random walk around their average values $\stackrel{}{{Y}_{j}}$:
where ${\xi}_{j}\left(T\right)$ is a normally distributed random variable with zero mean and unit variance, $\sigma $ determines the magnitude of resource fluctuations, and $k$ is the strength of a restoring force that ensures the same resource environment on average over time (Figure 1A). The absolute value enforces ${Y}_{j,0}$ to be positive. If $k=0$, there is no restoring force and hence ${Y}_{j,0}\left(T\right)$ performs an unbiased random walk; if $k=1$, ${Y}_{j,0}\left(T\right)$ fluctuates about its set point $\overline{{Y}_{j}}$ independent of its value at the previous sampling time. For all $k>0$, the model exhibits longterm stability without drift. As above, we used an ensemble approach to model the set points $\overline{{Y}_{j}}$, assuming that each $\overline{{Y}_{j}}$ was independently drawn from a uniform distribution between $0$ and ${Y}_{\text{max}}$. These assumptions yield a Markov chain of fluctuating resource amounts ${Y}_{j,0}\left(T\right)$ and their corresponding consumer relative abundances ${x}_{i}\left(T\right)={X}_{i}\left(T\right)/\sum _{n}{X}_{n}\left(T\right)$ (Figure 1C).
The statistical properties of these time series are primarily determined by five global parameters: the total number of consumers in the reservoir $N$, the number of resources in the environment $M$, the sparsity $S$ of the resource consumption matrix, and the resource fluctuation parameters $\sigma $ and $k$. The absolute magnitudes of ${R}_{\text{max}}$ and ${Y}_{\text{max}}$ are not important for our purposes since they do not affect the predictions of consumer relative abundances at ecological steady state. We extracted $N$ from experimental data as the number of consumers that were present for at least one sampling time point, leaving only four free global parameters.
Previous studies have suggested that the family level is an appropriate coarse graining of metabolic capabilities (Goldford et al., 2018; Louca et al., 2016; Tian et al., 2020), thus we assumed, unless otherwise specified, that each consumer grouping i within our model represents a taxonomic family, and combined abundances of empirical operational taxonomic units (OTUs) or amplicon sequencing variants (ASVs; Callahan et al., 2016) at the family level for analyses (Materials and methods). Given the typical limits of detection of 16S amplicon sequencing data sets, we only examined time series statistics for taxa with relative abundance >10^{–4} at any given time point. Experimental and simulated data were processed equivalently to enable consistent comparisons of their time series statistics.
As expected, we found that random realizations of our model (i.e., different resource consumption matrices drawn from the same ensemble) generated similar time series statistics, whose typical behavior strongly varied with the global parameters of the model. In particular, only small subsets of the parameters led to time series statistics that agreed with experiments, as we show below. An example simulation using the macroscopic parameters $\left(N,M,S,\sigma ,k\right)=\left(50,30,0.1,0.2,0.8\right)$ is shown in Figure 1. This set of parameters produced relative abundance time series with highly similar statistical behaviors as in experiments involving daily sampling of human stool (Figure 1D). Given this agreement, we next systematically analyzed the time series statistics generated by our model across the macroscopic parameter space and compared against experimental behaviors to estimate model parameters for wideranging microbiotas.
Model reproduces the statistics of human gut microbiota time series
To test whether our model can recapitulate major features of experimental time series, we first focused on a data set of daily sampling of the gut microbiota from a human subject (Caporaso et al., 2011; Figure 2). These data were previously shown (Ji et al., 2020) to exhibit several distinctive statistical behaviors: (1) the variance ${\sigma}_{{x}_{i}}^{2}$ of family i over the sampling period scaled as a power law with its mean $\u2329{x}_{i}\u232a$ (Figure 2B and F); (2) the log_{10}(abundance change) ${\mathrm{\Delta}l}_{i}\left(T\right)={\mathrm{log}}_{10}\left({x}_{i}\left(T+1\right)/{x}_{i}\left(T\right)\right)$ , pooled over all families and across all sampling times, was well fit by an exponential distribution with standard deviation ${\sigma}_{\mathrm{\Delta}l}$ (Figure 2B and G); and (3) the distributions of residence times ${t}_{\text{res}}$ and return times ${t}_{\text{ret}}$ (the durations of sustained presence and absence, respectively) pooled over all families were well fit by power laws with an exponential cutoff (Figure 2D and K). Through an exhaustive search of parameter space, we identified a specific combination of parameters that could reproduce all of these behaviors within our simple CR model (Figure 2F, G and K).
In addition, several other important statistics were reproduced without any additional fitting: (1) the distribution of richness $\alpha \left(T\right)$ , the number of consumers present at sampling time $T$ (Figure 2A and E); (2) the distribution of the restoring slopes ${s}_{i}$ of the linear regression of $\mathrm{\Delta}{l}_{i}\left(T\right)$ against ${l}_{i}\left(T\right)\equiv {\mathrm{log}}_{10}\left({x}_{i}\left(T\right)\right)$ across all $T$ (Figure 2C and H); (3) the distribution of prevalences ${p}_{i}$ , the fraction of sampling times for which family i is present (Figure 2A1); (4) the relationship between ${p}_{i}$ and $\u2329{x}_{i}\u232a$ (Figure 2J); and (5) the rank distribution of mean abundances $\u2329{x}_{i}\u232a$ (Figure 2L).
Therefore, our model was able to simultaneously capture at least eight statistical behaviors in a microbiota time series with only four parameters, each of which may represent biologically relevant features of the community.
To determine whether our model can be used to analyze time series statistics at other taxonomic levels, we analyzed the same data set (Caporaso et al., 2011) at finer (genus) and coarser (class) taxonomic levels, both of which exhibited qualitatively similar statistical behaviors as the family level. Our modeling framework was able to quantitatively recapitulate almost all statistics at both levels (Figure 2—figure supplements 1 and 2). A notable exception is that the Bacteroides genus dominated the observed rank abundance distribution at the genus level, while our CR model predicted a more even distribution (Figure 2—figure supplement 2). Nevertheless, the relative abundances among the remaining genera were still well captured by the model predictions (Figure 2—figure supplement 2). These results demonstrate that our model and its applications can be generalized across taxonomic levels.
Systematic characterization of the effects of CR dynamics on time series statistics
Since our model can reproduce the observed statistics in gut microbiota time series, we sought to determine how these statistics would respond to changes in model parameters, and thus how experimental measurements constrain the ensemble parameters across various data sets. To do so, we simulated our model across all relevant regions of parameter space. $S$ and $k$ were varied across their entire ranges, and $M$ and $\sigma $ were varied across relevant regions outside of which the model clearly disagreed with the observed data. For each set of parameters, each time series statistic was averaged across random instances of ${R}_{ij}$ and ${Y}_{j,0}\left(T\right)$ drawn from the same statistical ensemble. For each statistic $z$, its global susceptibility $C(z,w)$ to parameter $w$ was calculated as the change in $z$ when $w$ is varied, averaged over all other parameters and normalized by the standard deviation of $z$ across the entire parameter space. Due to the normalization, $C\left(z,w\right)$ varies approximately between –3 and 3, where a magnitude close to 3 indicates that almost all the variance of $z$ is due to changing $w$.
By clustering and ranking susceptibilities, we identified four statistics with $\leftC\left(z,w\right)\right>2$ that were largely determined by one of each of the four model parameters (Figure 3, Figure 3—figure supplement 1): mean richness $\u2329\alpha \u232a$, the power law exponent $\beta $ of ${\sigma}_{{x}_{i}}^{2}$ versus $\u2329{x}_{i}\u232a$, the standard deviation in log_{10}(abundance change) ${\sigma}_{\mathrm{\Delta}l}$, and the mean restoring slope $\u2329s\u232a$ were almost exclusively susceptible to variations in $M$, $S$, $\sigma $, and $k$, respectively. Similar results were also obtained for local versions of the susceptibility, in which individual parameters were varied around the best fit values for the human gut microbiota in Figure 2 (Figure 3figure supplement 2). These susceptibilities broadly illustrate how various time series statistics are affected by coarsegrained parameters of resource competition; we further investigate some specific examples in the next section.
The exclusive susceptibilities of these four statistics suggest that they can serve as informative metrics for estimating model parameters. Therefore, we estimated model parameters by minimizing the sum of errors between model predictions and experimental measurements of these four statistics, and obtained estimation bounds by determining parameter variations that would increase model error by 5% of the mean error across all parameter space. As we will show, the resulting bounds are small relative to the differences among distinct microbiotas, indicating that meaningful conclusions can be drawn from the best fit values of the ensemble level parameters of resource competition. In summary, the four model parameters were fit to four summary statistics: mean richness $\u2329\alpha \u232a$, variancemean scaling exponent $\beta $, standard deviation of abundance change ${\sigma}_{\mathrm{\Delta}l}$, and mean restoring slope $\u2329s\u232a$ (Figure 2E–H, respectively). The shapes of their corresponding distributions and scalings, as well as at least four other statistics (Figure 2I–L), are all parameterfree predictions.
Origins of distinctive statistical behaviors in species abundance time series
To understand the mechanisms that underlie the susceptibilities of various time series statistics to model parameters, we investigated their origins within our model, focusing on how they constrain the parameters.
The average richness $\u2329\alpha \u232a$ is a fundamental descriptor of community diversity. Within our model, $\u2329\alpha \u232a$ is largely determined by and increases with increasing resource number $M$ ($C\left(\alpha ,N/M\right)=2.6$), as expected for CR dynamics. The sparsity of resource use $S$ impacts the power law exponent $\beta $ between ${\sigma}_{{x}_{i}}^{2}$ and $\u2329{x}_{i}\u232a$ ($C\left(\beta ,S\right)=2.0$). Together, $\alpha $ and $\beta $ constrain the parameters of resource competition $M$ and $S$.
The effect of $S$ on $\beta $ can be partially understood by considering limiting behaviors as follows. When sparsity is high ($S\approx 1$), there is little competition and each consumer consumes almost distinct sets of resources from other consumers. In the limit in which each consumer utilizes a single unique resource, ${\sigma}_{{x}_{i}}^{2}$ is determined by the noise in resource level, which has a $\beta =2$ scaling according to Equation 2. In the limit of large $M$ and high sparsity, the variation in the number of resources consumed by each consumer can be large relative to the mean, and both ${\sigma}_{{x}_{i}}^{2}$ and $\u2329{x}_{i}\u232a$ scale with the number of resources consumed, hence $\beta =1$. Simulations of a nocompetition model in which consumers consume distinct sets of resources confirmed the scalings in these limits (Figure 3—figure supplement 3). By contrast, when sparsity is low ($S\approx 0$), each consumer utilizes almost all resources and hence variation in the number of resources consumed is small relative to the mean. Despite the obvious presence of competition in our CR model, we nevertheless attempted to understand the low sparsity limit by extrapolating the nocompetition model above to a case in which all consumers consume distinct sets of the same number of resources. For large number of resources, these simulations predicted that $\beta \approx 1.5$ (Figure 3—figure supplement 3), as did our CR model for $S=0.1$ (Figure 3—figure supplement 3). These findings suggest that the effect of $S$ on $\beta $ can be partially attributed to differences in the number of resources consumed.
The distribution of $\mathrm{\Delta}l$ describes the nature of abundance changes. As expected, the width of the distribution is largely determined by and increases with increasing $\sigma $ ($C\left({\sigma}_{\mathrm{\Delta}l},\sigma \right)=2.6$). For the gut microbiota data set in Figure 2, the shape of the distribution was well fit by an exponential. Within our model, the shape of the distribution aggregated across all consumers is determined by $N/M$ and the sparsity $S$, emerging from the mixture of each consumer’s individual distribution (Figure 3—figure supplement 4). When $N/M<1$ and the sparsity $S$ is low, individual distributions of $\mathrm{\Delta}l$ are well fit by normal distributions, and pool together to generate another normal distribution. When $N/M<1$ and sparsity $S$ is high, individual distributions remain normal, but can pool together to generate a nonnormal distribution that is well fit by an exponential (see also Allen et al., 2001). By contrast, when $N/M>1$, individual distributions can be well fit by an exponential and can pool together to approximate another exponential. Simulations of the nocompetition model considered above led to individual and aggregate distributions that were normal in all cases, indicating that in our model resource competition is responsible for generating the nonnormal distributions of $\mathrm{\Delta}l$ (Figure 3—figure supplement 3). Although it is challenging to discern the shape of individual distributions in most experimental data sets given the limited numbers of samples, the shape of the aggregate distribution of $\mathrm{\Delta}l$ informs the parameters of resource competition $M$ and $S$. In particular, an exponential distribution of $\mathrm{\Delta}l$ suggests either strong resource competition in the form of $N>M$ or substantial niche differentiation in the form of high $S$. Other statistics such as $\beta $ can help to distinguish between these two regimes.
The distribution of restoring slopes ${s}_{i}$ describes the tendency with which consumers revert to their mean abundances following fluctuations. As expected, the mean $\u2329s\u232a$ is almost completely determined by $k$, which describes the autocorrelation in resource levels ($\u2329s\u232a\approx k$ and $C\left(\u2329s\u232a,k\right)=3.0$). Together, the distributions of $\mathrm{\Delta}l$ and ${s}_{i}$ constrain the parameters of external fluctuations $\sigma $ and $k$.
Within our model, resource fluctuations can lead to the temporary ‘extinction’ of certain species when they drop below the detectability threshold of 10^{–4}. The distributions of residence and return times, ${t}_{\text{res}}$ and ${t}_{\text{ret}}$, reflect the probabilities of extinction as well as correlations between sampling times. For all parameter sets explored, these distributions can be well fit by power laws, with an exponential cutoff to account for finite sampling (Ji et al., 2020). As expected, the power law slopes ${\nu}_{\text{res}}$ and ${\nu}_{\text{ret}}$ decrease (become more negative) with increasing $\sigma $ or $k$ (Figure 3—figure supplement 1), since increasing external noise or decreasing correlations in time increases the probability of fluctuating between existence and extinction for each consumer. By contrast, ${\nu}_{\text{res}}$ and ${\nu}_{\text{ret}}$ change in opposite directions in response to variation in $M$ (Figure 3—figure supplement 1). Increasing $M$ leads to a larger number of highly prevalent consumers, thereby increasing the mean and broadening the distribution of ${t}_{\text{res}}$ and decreasing the mean and narrowing the distribution of ${t}_{\text{ret}}$. Since the four ensemble level parameters are already fixed by other statistics, the distributions of ${t}_{\text{res}}$, ${t}_{\text{ret}}$, and ${p}_{i}$ are parameterfree predictions of our model. In other words, a macroscopic characterization of the effective resource competition and resource fluctuations is sufficient to predict the statistics of ‘extinction’ dynamics, as well as the abundance rank distribution and the relationship between consumer abundance and prevalence.
Since the distributions of $\mathrm{\Delta}l$, ${t}_{\text{res}}$, and ${t}_{\text{ret}}$ are dependent on correlations between sampling times, it was initially puzzling that their distributions in some data sets remained similar after shuffling sampling times, raising questions as to what extent these statistics hold information about the underlying intrinsic dynamics (Tchourine et al., 2021; Wang and Liu, 2021a). Our results assist in reconciling the apparent conundrum, since within our model richness $\u2329\alpha \u232a$ and Taylor’s law exponent $\beta $ do not depend on correlations between sampling times and are also the statistics that are most informative about the intrinsic parameters $M$ and $S$ (Figure 3). As a result, the shuffled time series were also well fit by our model and yielded best fit values that were identical to those produced by the actual time series except with $k=1$, as expected due to the absence of correlation across sampling times (Figure 3—figure supplement 5). Thus, our results suggest that while external fluctuations in resource levels may be responsible for generating species abundance variations, the intrinsic properties of resource competition can determine the resulting scaling exponents of many statistical behaviors.
Taken together, our analyses demonstrate the complex relationships among time series statistics and highlight their unification within our model using only a small number of global parameters, whose values are strongly constrained by macroecological patterns.
CR model guides the identification of other models that can reproduce time series statistics
We have shown that many time series statistics can be recapitulated by a simple model that does not require knowing many detailed features of real microbiota (Figure 2). The success of this approach implies that these macroecological fluctuations must be independent of at least some model details, which suggests that there may be other ecological models that could also recapitulate the same data (Figure 4, Figure 4—figure supplements 1–5). The relationships between ecological models are generally poorly characterized. To explore these possibilities, we sought to compare our calibrated CR models against several common alternatives.
First, we aimed to determine the extent to which the simulated statistics depend on the assumptions of our CR model. Our parametrization of the consumption rates introduces a correlation between the maximum growth rate of a consumer and the number of resources it consumes. To remove this correlation, we normalized the sum of consumption rates ${\sum}_{j}{R}_{ij}$ for consumer i to a fixed capacity ${\stackrel{~}{R}}_{i}$ that was randomly drawn from the original growth rates ${\sum}_{j}{R}_{ij}{Y}_{j,0}$ (Good et al., 2018; Posfai et al., 2017; Tikhonov and Monasson, 2017). This modification preserves the variation in consumer fitness while implementing a metabolic tradeoff. The resulting time series statistics were essentially unaffected, also recapitulating experimental data (Figure 4—figure supplement 1).
Moreover, the CR dynamics in Equation 1 do not consider other biologically plausible scenarios such as saturation kinetics (Momeni et al., 2017; Niehaus et al., 2019). To probe the robustness of the results of our model to the dynamical assumptions, we implemented saturation kinetics with all other details kept the same (Materials and methods). When this model was simulated with the best fit parameters of the original model, the resulting dynamics were less variable across sampling times than without saturation kinetics, since the saturated regime is unaffected by small changes in resource levels (Figure 4—figure supplement 2). Nonetheless, experimental statistics were again reproduced once the strength of environmental fluctuations $\sigma $ was increased appropriately (Figure 4—figure supplement 2). This suggests that our results are robust to assumptions regarding metabolic tradeoffs and saturation kinetics.
We next considered a noninteracting null model in which consumer abundances were drawn from independent normal distributions whose means and variances were fitted directly from the data. Even with a large number of free parameters, this null model was unable to capture some of the time series statistics reproduced by our CR model, including Taylor’s law as well as the distributions of richness and restoring slopes (Figure 4—figure supplement 3). We reasoned that the discrepancies between experimental data and the null model could be due to the lack of interspecies interactions. To test this hypothesis, we examined the pairwise correlations between the consumer abundances across sampling times. The measured distribution of pairwise correlations is much broader than the prediction of the noninteracting model, which is sharply peaked about zero as expected (Figure 4). By contrast, the distribution of correlations predicted by our CR model without any additional fitting was in much closer agreement with the experimental data (Figure 4). These findings imply that interspecies interactions are required to capture important details of community dynamics.
While our CR model assumes pairwise interactions between consumers and resources, the effective interactions between consumers are not necessarily pairwise. To explore whether these higherorder contributions are necessary for recapitulating the data, we considered models explicitly based on pairwise interspecies interactions, which despite differences compared with CR models (Momeni et al., 2017) can also reproduce some properties of experimental time series (Descheemaeker and de Buyl, 2020; Wang and Liu, 2021b). To further explore the properties of models focused on pairwise interactions, we investigated gLV models in which $N$ taxa grow and interact via
where ${X}_{i}$ denotes the relative abundance of taxon i, ${r}_{i}$ its growth rate, and ${A}_{ij}$ its interaction coefficient with taxon $j$. $\mathrm{\Gamma}\left(t\right)=\sum _{i}{r}_{i}{X}_{i}+\sum _{i,j}{A}_{ij}{X}_{i}{X}_{j}$ is a normalizing term that ensures that the relative abundances always sum to one (Joseph et al., 2020). Since this classical model is generally unstable for randomly drawn interaction coefficients (May, 1972), we sought to focus on particular instances of the gLV model that were closest to our original CR model. This conversion between models was achieved by converting the consumption rates ${R}_{ij}$ and resource levels ${Y}_{j,0}$ at each sampling time $T$ to the growth rates ${r}_{i}$ and interaction coefficients ${A}_{ij}$ that characterize the dynamics when consumption rates are similar to the mean value (Materials and methods). This conversion results in negative, symmetric ${A}_{ij}$ whose magnitudes depend on the niche overlap between the interacting taxa (Good et al., 2018). Moreover, fluctuations in ${Y}_{j,0}$ result in corresponding fluctuations in both ${r}_{i}$ and ${A}_{ij}$ across $T$. These CRconverted gLV models generated time series statistics that reproduced the experimental data to a similar extent as the original CR model (Figure 4—figure supplement 4). In light of this correspondence, we asked whether more general ensembles of pairwise interaction could also reproduce the experimental data. We randomly selected ${r}_{i}$ and ${A}_{ij}$ values from normal distributions with means and variances equal to those in the CRconverted gLV models while enforcing symmetric and negative interactions. The resulting gLV models yielded a poor fit to the data (Figure 4—figure supplement 5). Together, these results suggest that while pairwise interactions between taxa are likely sufficient to recapitulate the experimental data, their parameters must be drawn from particular ensembles that can be more simply described in the CR framework.
These examples reinforce that only a particular subset of models can recapitulate the data, and therefore, that the underlying community properties are highly constrained by macroecological dynamics. Moreover, our calibrated CR model can guide the parametrization of other models that can satisfy those constraints, while also identifying model features that are necessary for recapitulating data.
Time series statistics distinguish wideranging microbiotas
Having developed a simple method to estimate parameters of our CR model that recapitulate time series statistics, we applied this method to data sets involving wideranging microbial communities. Although the various communities considered are drastically different in many aspects, we hypothesized that our CR model framework could still be applied to identify the statistical ensembles that can describe their macroecological dynamics. In addition to microbiotas from the human and mouse gut (Caporaso et al., 2011; Carmody et al., 2015; David et al., 2014), we examined communities from the human vagina (Song et al., 2020), human saliva (David et al., 2014), and in and around rice roots (Edwards et al., 2018). The time series statistics of these microbiotas varied broadly (Figure 5A). Nevertheless, our model successfully reproduced the experimental statistics across all communities (Figure 5—figure supplements 1–6), suggesting that simple CR models can capture many of the macroscopic features of these microbiotas.
The best fit parameters suggest that the effective resource competition dynamics occur in distinct regimes across microbiotas (Figure 5B). Human gut microbiotas were best described by $N>M$, suggesting that there are more species in the reservoir than resources in the environment, by contrast to mouse gut microbiotas that were best described by $N<M$. In terms of resource niche overlaps, human gut microbiotas were best fit with sparsity $S<0.3$, while mouse gut microbiotas were best fit with $S>0.3$, suggesting that on average, pairs of bacterial families are more metabolically distinct in the mouse versus the human gut.
Unlike gut microbiotas, a human saliva microbiota yielded best fit parameters $N\approx M$ and $S\approx 0.8$, suggesting that this community has access to abundant resources and that each effective resource is competed for by a small fraction of the extant bacterial families. All vaginal microbiotas were best fit with $S<0.1$, suggesting intense resource competition.
Like vaginal microbiotas, microbial communities residing in the bulk soil around rice roots and in the associated rhizoplane and rhizosphere were well described by $S<0.1$. By contrast, the community in the associated endosphere was best described by $S\approx 0.6$, suggesting that resource competition is less fierce within plant roots than around them.
In addition, inferences about the nature of environmental fluctuations can be made from the best fit values of $\sigma $ and $k$ (Figure 5B). Apart from the two vaginal microbiota data sets, the best fit values of $\sigma $ ranged from 0.1 to 0.3, indicating that changes in resource levels smaller than this magnitude will generate abundance changes that look like typical fluctuations. The best fit values of $k$ varied between 0.5 and 1 across data sets, suggesting that the dynamics of microbial communities occur faster than or comparable to the typical sampling frequency of longitudinal studies. While it is unclear whether the internal time scales are faster than the sampling frequency for all of these communities, simulation results were robust to the dilution factor and threshold change defining ecological steady state (Figure 1—figure supplement 1), two main factors that affect the relationship between the internal and sampling time scales.
Inferences about intrinsic parameters of resource competition and external parameters of environmental fluctuations were also consistent with expectations for in vitro passaging of complex communities derived from humanized mice (ArandaDíaz et al., 2022). The resulting time series statistics were best fit by the smallest value of $\sigma $ among the data sets studied, indicating that the in vitro environment has relatively low noise across sampling times (as expected); the nonzero $\sigma $ presumably arises from technical variations that result in effective noise in resource levels. The best fit value of $M$ was larger than the reservoir size $N$, suggesting that there are many distinct resources in the complex medium used for passaging and consistent with the ability of more diverse inocula to support more diverse in vitro communities (ArandaDíaz et al., 2022). The consistency of these results further supports the utility of our model.
Taken together, our model infers ensemblelevel parameters of resource competition and external parameters of environmental fluctuations for several widely studied microbial communities that can inform future mechanistic studies.
Discussion
Here, we presented a coarsegrained CR model that generates species abundance time series from fluctuating environmental resources. We demonstrated that this model reproduces several statistical behaviors (Figure 2) and elucidated how these observations constrain the parameters of resource competition within the model (Figure 3). Moreover, we successfully fitted the model to wideranging microbiotas, which allowed us to draw inferences about the parameters of their effective resource competition. In sum, our work provides an existence proof that a CR model can recapitulate experimentally observed time series statistics in microbiotas from diverse environments.
An important feature of our model is that it does not need to specify the individual resource uptake rates of different taxa, which could be too numerous and complex to be tractable. Instead, our model reproduces many statistical behaviors with a small number of global parameters that describe the distributions of resource uptake rates. To what extent these macroscopic parameters can be interpreted mechanistically is an interesting open question that could be explored in future work. Although by no means exhaustive, our framework nevertheless addresses several pertinent questions regarding construction of useful models of microbiota dynamics. The success of our CR model in reproducing experimental time series statistics is consistent with bioinformaticsguided analyses of complex communities demonstrating that metabolic capability is a major determinant of community composition (Louca et al., 2016; Tian et al., 2020). Our results also suggest that the contributions of a reservoir of species or other forms of species reintroduction are important for the dynamics of wideranging microbiotas. Within our model, the lack of species reintroduction renders poor consumers unable to recover to meaningful abundance within a sampling time even when resource fluctuations are in their favor, thereby distorting time series statistics. The existence of a reservoir is consistent with previous experimental work in mice (Ng et al., 2019), but further work is required to investigate how species reintroduction occurs in other systems. Similarly, further experimental work is required to ascertain the amount of growth and change that occurs during sampling time scales, and further theoretical work is required to infer such internal time scales from microbiota time series.
In terms of intrinsic metabolic properties, our results provide a baseline expectation for the effective number of resources or available niches in the wideranging systems examined here, and to what extent they are competed for by extant consumers. In terms of environmental properties, our results provide a baseline expectation to help distinguish between typical fluctuations and large perturbations in resources. These expectations may aid in the engineering of complex microbiotas.
In general, our work demonstrates that it is feasible to reproduce time series statistics using CR models of microbiota dynamics, thereby generating mechanistic hypotheses for further investigation. Our CR model and fitting procedure can also be used to aid the parametrization of other models such as LotkaVolterra models (Figure 4—figure supplements 1–5), comparisons among which can reveal the model details that are required to recapitulate experimental data. In the future, more detailed hypotheses can be generated by investigating how time series statistics are affected by modifications to baseline CR dynamics, such as the incorporation of metabolic crossfeeding (Goldford et al., 2018; Li et al., 2020) or physical interactions such as type VI killing (Verster et al., 2017), functional differentiation from genomic analysis (Arkin et al., 2018; Machado et al., 2021; Pollak et al., 2021), and physical variables such as pH (ArandaDíaz et al., 2020; Ratzke and Gore, 2018), temperature (Lax et al., 2020), and osmolality (Cesar et al., 2020). In addition, recent studies have shown that evolution can substantially affect the dynamics of human gut microbiotas (Garud et al., 2019; Yaffe and Relman, 2020; Zhao et al., 2019). It will therefore be illuminating to incorporate evolutionary dynamics into CR models under fluctuating environments (Good et al., 2018). Such extended models can then be applied to probe the underlying mechanisms in microbiotas for which frequent sampling and deeper understanding could be translated to urgent applications, including those in marine environments, wastewater treatment plants, and the guts of insect pests and livestock.
Materials and methods
Simulations of a CR model with fluctuating resource amounts
Request a detailed protocolUnder a serial dilution scheme, an ecological steady state is reached when the dynamics in subsequent passages are identical, which is the case when all consumers are either extinct or have a growth ratio (the ratio of a consumer’s final and initial abundances within one passage) equal to the dilution factor $D$. Due to the slow path to extinction of some consumers, reaching an exact ecological steady state can require hundreds of passages, presumably more than realistically occurs between sampling times in the data sets examined here. Thus, we assumed instead that between sampling times the system only approximately reaches an ecological steady state, defined as the growth ratios of all species changing by less than a threshold between subsequent passages that was defined as a fraction of $D$. Throughout this study, $D$ was set to 200 and the steady state threshold was 5%, under which a steady state was approximately reached in about 5 dilutions (Figure 1B). In this manner, our model assigns a welldefined state of consumer abundances to each resource environment while ensuring that only a reasonable amount of change occurs between sampling times. Note that in human gut microbiotas, abundances can change by more than 1000fold between daily samplings (Figure 2B), indicating that at least 10 generations can occur between sampling times. The precise value of $D$ did not affect time series statistics, and steadystate thresholds between 1% and 10% generated similar time series statistics (Figure 1—figure supplement 1). We therefore expect our results to be robust to the values of these two parameters. Simulations were carried out in Matlab, and all code is freely available online in Matlab and Python at https://bitbucket.org/kchuanglab/consumerresourcemodelformicrobiotafluctuations/.
CR model with saturation kinetics
Request a detailed protocolSaturation kinetics were implemented into the CR dynamics of Equation 1 as
where ${Y}_{s}$ denotes the saturation constant. For simplicity, ${Y}_{s}$ was assumed to be equal for all resources, and set to an intermediate value of ${Y}_{s}=\u2329{Y}_{j,0}\u232a/3$ such that both saturated and linear kinetics could affect community dynamics. Other model details are the same as the original CR model.
LotkaVolterra models
Request a detailed protocolThe gLV model in Equation 3 was parametrized in two ways. The first parametrization, which we refer to as CRconverted gLV models, was motivated by the successful recapitulation of experimental time series statistics with our CR model. The CR model can be rewritten as a gLV model when resource consumption rates are similar to the mean value (Good et al., 2018). Under this assumption, the mapping is ${r}_{i}=2{\sum}_{j}{R}_{ij}{Y}_{j,0}$ and ${A}_{ij}=\frac{1}{{R}_{\text{max}}}{\sum}_{k}{R}_{ik}{R}_{jk}{Y}_{k,0}$ . The converted interaction coefficients are negative and symmetric, and their magnitudes depend on the niche overlap between the interacting taxa. Since the resource levels ${Y}_{j,0}$ are involved in this parametrization, fluctuations in ${Y}_{j,0}$ across sampling times $T$ translate into fluctuations in ${r}_{i}$ and ${A}_{ij}$ .
In the second parametrization, ${r}_{i}$ and ${A}_{ij}$ were randomly drawn from normal distributions with means and variances equal to those in the CRconverted gLV model. ${A}_{ij}$ were forced to be negative and symmetric.
The gLV models were initialized with equal relative abundances for all taxa, and simulated for a fixed amount of time such that a similar range of relative abundances was generated as in the CR model at approximate ecological steady state.
Analysis of 16S amplicon sequencing data
Request a detailed protocolRaw 16S sequencing data from David et al., 2014; Song et al., 2020, were downloaded from the European Nucleotide Archive and the Sequence Read Archive, respectively, and ASVs were extracted using DADA2 (Callahan et al., 2016) with default parameters. OTUs or ASVs from other studies were downloaded and analyzed in their available form. All code for data processing is available in the repository listed above.
Data availability
The current manuscript is a computational study, so no data have been generated for this manuscript. Modelling code is uploaded at https://bitbucket.org/kchuanglab/consumerresourcemodelformicrobiotafluctuations/.

EBIID ERP006059. Host lifestyle affects human microbiota on daily timescales.

NCBI BioProjectID PRJNA637322. Daily Vaginal Microbiota Fluctuations Associated with Natural Hormonal Cycle, Contraceptives, Diet, and Exercise.
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Decision letter

Nicola SegataReviewing Editor; University of Trento, Italy

Wendy S GarrettSenior Editor; Harvard T.H. Chan School of Public Health, United States

Sean GibbonsReviewer
In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.
Decision letter after peer review:
[Editors’ note: the authors submitted for reconsideration following the decision after peer review. What follows is the decision letter after the first round of review.]
Thank you for submitting the paper "Competition for fluctuating resources reproduces statistics of species abundance over time across wideranging microbiotas" for consideration by eLife. Your article has been reviewed by 2 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and a Senior Editor. The following individuals involved in review of your submission have agreed to reveal their identity: Sean Gibbons (Reviewer #1).
Comments to the Authors:
We are very sorry to share that, after extensive consultation with the reviewers, we have decided that this work will not be considered further for publication by eLife. Both reviewers and the editor think that the mathemathical model proposed is of potential great relevance for the field, but despite the elegant formulation and the interesting results fo some of the analyses, quite a significant amout of additional work would be needed to address most of the reviewers' points and be considered for publication in eLife (see below). We are sorry to convey this negative decision, as we addressing the points of the reviewers most likely goes beyond the usual effort for a revision at eLife. We are, however, open to considering a substantially revised manuscript in the future.
Reviewer #1:
The authors propose a simple consumerresource (CR) model, where the dynamics of microbial communities are governed by fluctuations in external resources and by competition for these resources between taxa. The model is elegant in its simplicity, while also being biologically intuitive and subtly clever in its implementation. The authors show how the model accurately predicts many of the macroecological patterns found in microbiome time series. Unlike many papers I've read that focus on macroecological patterns (with some exceptions), the authors do a great job connecting model parameters to measured properties of microbial ecosystems and show how these parameterizations of realworld ecosystems can provide potential mechanistic insights into the ecology of the system. I really enjoyed reading this manuscript. The writing is clear, as are the formalisms and the analyses. The model provided many expected results, but also revealed some surprising insights. This is a promising approach for generating novel hypotheses for how microbial ecosystems behave. Overall, I think this is a valuable contribution. My only caveat is that many different mechanistic models can be constructed to explain a given phenomenon  so I suggest the authors remain somewhat humble about whether or not 'fluctuating resources' are the major drivers of these complex dynamics. They might be! The fact that such a simple model makes so many predictions is promising. But in the end, this is just one possible model among many.
Major Strengths/Weaknesses:
1) I like the simplicity of the CR model. More than this, I like the subtlety with which you handled community dynamics. Many prior studies have erroneously treated microbiome time series as if they directly represent growth curves of all the taxa in the system (e.g. fitting LV models to human gut time series). Your method simulates serial dilution and growth of microbial taxa over several cycles to approximate a steadystate community composition for each sample time point. This fits with my biological intuition.
2) One minor weakness in the data processing was that the most resolved taxonomic level that was analyzed was the family level. Why not start with genuslevel? Genuslevel annotations can usually be estimated from 16S reads. Another question that I had was whether or not the model assumes absolute or relative abundances? I'm guessing absolute, in which case, I found the rarefaction and renormalization of the counts to frequencies to be a slight concern. I'd suggest the authors perform a centered logratio (CLR) transform (or some other form of isometric logratio transform) on the nonrarified count data, and only remove lowfrequency taxa after the transformation. I doubt this will substantially impact the results, but this is considered best practice.
3) The 'origins of distinctive statistical behaviors…' section is really great. The authors do a great job mapping their model parameters to features that can be estimated directly from the empirical time series (i.e. αdiv and the βslope constrain N, M, and S, while δl and si constrain σ and k). However, I'm not sure I understood your explanation for why lowsparsity leads to a steeper Taylor's Law slope, and how this is essentially equivalent to a competitionfree mode. Naively, I'd expect competition to be greater at low sparsity, due to multiple species consuming the same sets of resources.
4) The noninteracting null model is an appropriate null. However, the authors should be humble about whether or not their competition model is capturing the mechanisms driving community dynamics. For example, direct microbemicrobe killing (antimicrobials or type VI secretion systems) is not captured. Host antimicrobials and immunesystem interactions aren't captured. Diet is implicitly captured with the nutrient fluctuations. That being said, I think the model is still reasonable and the insights should be fairly robust  the environmental fluctuations in the model probably capture a lot of this systemscale variance (in a statistical mechanics kind of way  the averaging together of a lot of different factors giving rise to a predictable statistical outcome).
5) There seem to be two assumptions regarding time in your model. First, I think you need to be operating within a stationary/stable system (i.e. where there's no longterm drift), correct? I think that's fine but wanted to clarify. The second assumption is that you're sampling from a steadystate endpoint of fast internal growth dynamics within the system. I think this is an excellent assumption in the human or mouse gut, but you might want to think about the timescales of sampling and microbial growth in the various systems you are sampling. If you are sampling within the timescales of the faster dynamics (e.g. possible for in vitro systems…maybe in the vaginal system?), how would this impact your results? You mention that your k values were between 0.5 and 1.0, suggesting that internal dynamics were faster than sampling timescales. Due to the ecological steadystate assumption of your modeling, would it be possible for your parameters to tell you that dynamics were slower than sampling timescales?
Overall, I think the authors achieve their aims and that their conclusions are supported by their results. This is an elegant and useful modeling framework that should have a sizable impact on the field and provide potential mechanistic insight into existing and future longitudinal microbiome data sets. I found many of the model predictions to be intuitive, and a few to be surprising, which is always a good sweet spot. I'd like to commend the authors on writing a nice manuscript that clearly communicates their results with a set of beautiful and easytoread figures.
Reviewer #2:
This paper discusses a consumerresource model, where microbial families are considered consumers and their nutrients are resources. The model is used to simulate microbial abundances over time: batch feeding events allow populations to grow, dilutions in between feeding events reduce populations. Coefficients of the model, such as the number of resources and the rates at which each family can consume them, are fit to data from different microbiomes by comparing summary statistics of simulated and observed time series. Different microbiome time series, e.g. from mice or humans, have different summary statistics. The model can be optimized to simulate time series with summary statistics similar to each of those from different microbiome data sets.
The model is very simple, allowing the reader to easily understand what is going on. This is a strength of the manuscript. The overlap in resources consumed between consumers in this model is revealed as a crucial parameter because it exhibits the most interesting changes when fitting different microbiome data sets. However, in the model there is no trade off between the rate at which a species may consume a resource and the number of resources it can consume. Therefore, the more different nutrients a species can consume, the fitter it will be. It may be interesting to reevaluate the major results when this assumption is changed.
A weakness of the paper is that it overstates the implications of the theoretical findings. Simulated timelines from the presented model can generate summary statistics that look like those in real data sets. This will also be possible with other models, even simpler ones or more complex ones. The article ought to include a more critical discussion and validation with simpler (e.g. pairwise interaction) or more complex (e.g. saturating growth kinetics) models.
The article is also poorly referenced, e.g. Niehaus et al. 2019 develop a resource driven model for microbial populations (doi.org/10.1038/s4146701910062x), and Momeni et al. 2017 discussed the importance of resource mediated interactions (doi.org/10.7554/eLife.25051).
Finally, the article is not very carefully put together. I received two figures labeled as "Figure 1". The methods appear unfinished.
I recommend reducing the amount of fluff terms throughout the manuscript. For example, the sentence from the abstract:
"Our coarsegrained model parametrizes the intrinsic consumerresource properties of a community using a small number of macroscopic parameters, including the total number of resources, typical resource fluctuations over time, and the average overlap in resourceconsumption profiles across species"
would read fine without the illdefined filler words:
"Our model parametrizes the consumerresource properties of a community using parameters that include the total number of resources, resource fluctuations over time, and the average overlap in resourceconsumption profiles across species."
In my opinion, simplicity and clarity strengthen theoretical papers, increasing their impact.
[Editors’ note: further revisions were suggested prior to acceptance, as described below.]
Thank you for submitting your article "Competition for fluctuating resources reproduces statistics of species abundance over time across wideranging microbiotas" for consideration by eLife. Your article has been reviewed by 2 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Wendy Garrett as the Senior Editor. The following individual involved in review of your submission has agreed to reveal their identity: Sean Gibbons (Reviewer #1).
The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.
Essential revisions:
The paper has improved with the revision and it meets the standard for publication in eLife. However, the paper is rather technical and in some parts there is the risk of misinterpretation or overestimating/overinterpreting the potential of the model. The authors should better highlight the intrinsic limitations and strong assumptions of the model throughout the paper, starting – for example – from the abstract. It is not a problem of the model or the data per se, but it is rather the way it is communicated considering that the large majority of the readership will have different backgrounds and cannot necessarily understand the limitations directly. Thus, we would like to see a revised manuscript addressing these specific issues as soon as possible.
Reviewer #1:
The authors have done a commendable job responding to the reviewer comments. The additional analyses and model simulations have greatly strengthened their work. The authors have provided their code in a more accessible format. And, they have made the suggested improvements in how they discuss their results. I have no further concerns or comments.
Reviewer #2:
My main concern remains: a simulation of timeseries is presented that has summary statistics as observed in data. Upon revision, based on my comment that this is not special to the model presented, another model is used; this also reproduces summary statistics similar to those from data. This is not a broad impact result and will, with the current narrative, be easily misunderstood by a nonspecialist readership.
In my opinion, such timeseries summary statistics offer little insight and have limited biological meaning. Thus, my original opinion has not shifted much.
https://doi.org/10.7554/eLife.75168.sa1Author response
[Editors’ note: the authors resubmitted a revised version of the paper for consideration. What follows is the authors’ response to the first round of review.]
Reviewer #1:
The authors propose a simple consumerresource (CR) model, where the dynamics of microbial communities are governed by fluctuations in external resources and by competition for these resources between taxa. The model is elegant in its simplicity, while also being biologically intuitive and subtly clever in its implementation. The authors show how the model accurately predicts many of the macroecological patterns found in microbiome time series. Unlike many papers I've read that focus on macroecological patterns (with some exceptions), the authors do a great job connecting model parameters to measured properties of microbial ecosystems and show how these parameterizations of realworld ecosystems can provide potential mechanistic insights into the ecology of the system. I really enjoyed reading this manuscript. The writing is clear, as are the formalisms and the analyses. The model provided many expected results, but also revealed some surprising insights. This is a promising approach for generating novel hypotheses for how microbial ecosystems behave.
We thank the reviewer for a careful reading of our manuscript and appreciate the reviewer’s support!
Overall, I think this is a valuable contribution. My only caveat is that many different mechanistic models can be constructed to explain a given phenomenon  so I suggest the authors remain somewhat humble about whether or not 'fluctuating resources' are the major drivers of these complex dynamics. They might be! The fact that such a simple model makes so many predictions is promising. But in the end, this is just one possible model among many.
We agree with the reviewer that the core of our work is an existence proof, which does not rule out the possibility that other models can also capture experimental data. We have edited the text throughout to better reflect this point.
Moreover, to further explore other models, we have added extensive new simulations of (1) a consumerresource model with metabolic tradeoffs, (2) a consumerresource model with saturation kinetics, and (3) generalized LotkaVolterra (gLV) models involving pairwise interactions. It is challenging to exhaustively analyze any particular modeling framework due to the high dimensionality of parameter space, particularly for gLV models that have many more interaction parameters than our macroscopically parametrized consumerresource (CR) model. To overcome this obstacle, we exploited the fact that our CR model establishes the existence of a simple model that can recapitulate the statistics in microbiota time series, and analyzed the behavior of other models near the parameter space occupied by our successful model. While this approach cannot rule out the existence of other parameter regimes that recapitulate timeseries statistics for other models, we show that it can nevertheless shed light on the features of other models, such as interspecies interactions in LV models, that are necessary to explain the observed statistics. Our approach also highlights some of the challenges that other models may face in describing experimental data. We describe the results for each of modeling framework below in response to reviewer #2, who had similar concerns. We have also revised the text and added several supplemental figures (Figure S10, S11, S13, S14) to incorporate these analyses.
Major Strengths/Weaknesses:
(1) I like the simplicity of the CR model. More than this, I like the subtlety with which you handled community dynamics. Many prior studies have erroneously treated microbiome time series as if they directly represent growth curves of all the taxa in the system (e.g. fitting LV models to human gut time series). Your method simulates serial dilution and growth of microbial taxa over several cycles to approximate a steadystate community composition for each sample time point. This fits with my biological intuition.
Thank you! We are glad that the reviewer found the model to be biologically intuitive.
(2) One weakness in the data processing was that the most resolved taxonomic level that was analyzed was the family level. Why not start with genuslevel? Genuslevel annotations can usually be estimated from 16S reads.
We apologize for the confusion; genuslevel annotations can indeed be obtained for the data sets analyzed, but we focused on the family level because it has been suggested to be an appropriate coarsegraining of metabolic capabilities. For instance, prior work found that for diverse soil communities grown in simple medium, family level abundances converged despite substantial variability within families (Goldford et al. Science 2018). We therefore reasoned that the family level would be a natural coarsegraining resolution for comparison to a CR model.
Nonetheless, our model and analysis can straightforwardly be applied to other taxonomic levels, and in the original submission, we included an analysis at the class level that was qualitatively similar as the family level (Figure S3 of this revision). Comprehensively simulating systems with hundreds of taxa (e.g. the hundreds to thousands of species found in some gut and soil microbiotas) would require extensive computation time, not to mention the likelihood of metabolic correlations between species in the same genus. To balance these points with the reviewer’s question, in our revision we included an analysis of the gut microbiota timeseries data set from Caporaso et al. (as in Figure 2) at the genus level. This analysis (Figure 2—figure supplement 2) showed that our CR model could again largely recapitulate all experimental statistics at the genus level. The only statistic that showed a discrepancy was the dominance of the Bacteroides genus that disrupted the rank distribution of mean abundances, and recalculation of relative abundances without the Bacteroides restored close agreement with model predictions, providing further support that our model can be used at various taxonomic levels.
Another question that I had was whether or not the model assumes absolute or relative abundances? I'm guessing absolute, in which case, I found the rarefaction and renormalization of the counts to frequencies to be a slight concern. I'd suggest the authors perform a centered logratio (CLR) transform (or some other form of isometric logratio transform) on the nonrarified count data, and only remove lowfrequency taxa after the transformation. I doubt this will substantially impact the results, but this is considered best practice.
We thank the reviewer for bringing to attention the centered logratio transform and its uses in analyzing compositional data. First, we reaffirm that in all our analyses, experimental data and numerical simulations were processed and analyzed equivalently, ensuring the validity of the analyses. Our model was formulated using absolute abundances, which were then always normalized to sum to one before comparing to compositional (relative abundance) experimental data. In fact, since absolute abundances within the model were never analyzed directly, the total resource levels can be normalized to sum to one a priori without affecting the results.
Moreover, given the typical limits of detection of 16S amplicon sequencing data sets, we only examined time series statistics for taxa with relative abundance >10^{4} at any given time point. Previously, we renormalized the relative abundances after removing taxa below the detection threshold. The results without renormalization were unchanged (Author response image 1) , which is intuitive because taxa below the detection threshold comprise only a tiny fraction of the community.
Finally, the denominator in the CLR is the geometric mean, which cannot naturally handle cases with zero reads. It is therefore not a natural metric to describe the timeseries statistics of lowabundance taxa that are fluctuating above and below the limit of detection. On the other hand, relative abundances naturally incorporate cases with zero reads. Since we processed and analyzed experimental and simulated data equivalently, statistics such as the residence and return times can be compared consistently across samples and between data and simulations.
We have revised the text to reflect these points.
(3) The 'origins of distinctive statistical behaviors…' section is really great. The authors do a great job mapping their model parameters to features that can be estimated directly from the empirical time series (i.e. αdiv and the βslope constrain N, M, and S, while δl and si constrain σ and k). However, I'm not sure I understood your explanation for why lowsparsity leads to a steeper Taylor's Law slope, and how this is essentially equivalent to a competitionfree mode. Naively, I'd expect competition to be greater at low sparsity, due to multiple species consuming the same sets of resources.
We would like to stress that our explanation for the value of the slope in Taylor’s law is partial and does not account for all possible effects, and have edited the text to reflect this point. Our reasoning was that since our model has no tradeoffs or metabolic constraints between the number of resources consumed and a consumer’s total consumption rate (see e.g. Posfai et al. PRL 2017, Good et al. PNAS 2018, and response to reviewer #2), the number of resources consumed necessarily strongly affects consumer abundance. This effect dominates at high sparsity, in which consumers typically consume distinct sets of resources and the number of resources consumed is relatively variable, as depicted in Figure S7A. Indeed, the Taylor’s law slope predicted by this nocompetition model (Figure S7A) matched closely with simulations of our model at high sparsity (Figure S7B).
Despite obvious competition in the actual CR model when sparsity is low, we nevertheless attempted to understand the lowsparsity limit by extrapolating the nocompetition model to the limiting case of zero sparsity, in which the number of resources consumed is the same for all consumers. Surprisingly, when the mean number of resources consumed is large, the predictions from the nocompetition model matched qualitatively with simulations of the actual CR model, suggesting that reduction in the variance of the number of resources consumed partially explains the dependence of Taylor’s law on sparsity. We emphasize that we do not claim that zero sparsity is equivalent to a competitionfree model, and we have revised the text to explain this point more clearly.
(4) The noninteracting null model is an appropriate null. However, the authors should be humble about whether or not their competition model is capturing the mechanisms driving community dynamics. For example, direct microbemicrobe killing (antimicrobials or type VI secretion systems) is not captured. Host antimicrobials and immunesystem interactions aren't captured. Diet is implicitly captured with the nutrient fluctuations. That being said, I think the model is still reasonable and the insights should be fairly robust  the environmental fluctuations in the model probably capture a lot of this systemscale variance (in a statistical mechanics kind of way  the averaging together of a lot of different factors giving rise to a predictable statistical outcome).
We agree with the reviewer that there exist other mechanisms that might affect community dynamics. We have revised the text to emphasize that our model is an existence proof and does not rule out that other mechanisms might drive community dynamics.
In addition, we tested the robustness of the macroscopic parameters of our model by considering variants incorporating saturation kinetics and metabolic tradeoffs (see response to reviewer #2). Despite these modifications, the bestfit parameters for the original model still reproduced data under mild assumptions, suggesting that the insights obtained are robust to model details to a reasonable extent. We have revised the text to include this discussion.
(5) There seem to be two assumptions regarding time in your model. First, I think you need to be operating within a stationary/stable system (i.e. where there's no longterm drift), correct? I think that's fine but wanted to clarify. The second assumption is that you're sampling from a steadystate endpoint of fast internal growth dynamics within the system. I think this is an excellent assumption in the human or mouse gut, but you might want to think about the timescales of sampling and microbial growth in the various systems you are sampling. If you are sampling within the timescales of the faster dynamics (e.g. possible for in vitro systems…maybe in the vaginal system?), how would this impact your results? You mention that your k values were between 0.5 and 1.0, suggesting that internal dynamics were faster than sampling timescales. Due to the ecological steadystate assumption of your modeling, would it be possible for your parameters to tell you that dynamics were slower than sampling timescales?
Yes, we assume longterm stability without drift, which we now clarify in the text.
We were indeed motivated by gut microbiotas when we assumed that the internal time scales between samplings were faster, and we agree that this assumption may not be the case for all systems analyzed here. In particular, the in vitro system was sampled every log_{!} 200 ≈ 7.6 generations, and hence may not have reached ecological steady state between samplings. Our model nevertheless produced a good fit (Figure S20), indicating that model results were robust to a relatively broad range of internal time scales (see Figure S1 and the discussion below).
More systematically, three factors control the relationship between the internal and sampling time scales: the dilution factor, the threshold change for ecological steady state, and the reservoir composition. The dilution factor and threshold affect the number of generations between samplings, while reservoir composition affects the correlation between sampling times. Simulation results were not substantially affected for a relatively broad range of dilution factors and thresholds (Figure S1). On the other hand, if the reservoir inherits a substantial fraction of its composition from the previous sampling time, simulation results can be affected (Figure S2). These results show that the relationship between the internal and sampling time scales can affect timeseries statistics in complex ways. It remains an interesting open question to infer internal time scales from microbiota time series, and our work provides a strong starting point to do so. We have revised the text to incorporate this discussion.
Overall, I think the authors achieve their aims and that their conclusions are supported by their results. This is an elegant and useful modeling framework that should have a sizable impact on the field and provide potential mechanistic insight into existing and future longitudinal microbiome data sets. I found many of the model predictions to be intuitive, and a few to be surprising, which is always a good sweet spot. I'd like to commend the authors on writing a nice manuscript that clearly communicates their results with a set of beautiful and easytoread figures.
We thank the reviewer for their kind words and helpful review.
Reviewer #2:
This paper discusses a consumerresource model, where microbial families are considered consumers and their nutrients are resources. The model is used to simulate microbial abundances over time: batch feeding events allow populations to grow, dilutions in between feeding events reduce populations. Coefficients of the model, such as the number of resources and the rates at which each family can consume them, are fit to data from different microbiomes by comparing summary statistics of simulated and observed time series. Different microbiome time series, e.g. from mice or humans, have different summary statistics. The model can be optimized to simulate time series with summary statistics similar to each of those from different microbiome data sets.
The model is very simple, allowing the reader to easily understand what is going on. This is a strength of the manuscript. The overlap in resources consumed between consumers in this model is revealed as a crucial parameter because it exhibits the most interesting changes when fitting different microbiome data sets.
We thank the reviewer for their careful reading of our manuscript, and appreciate the reviewer’s support for the strength of our work.
However, in the model there is no trade off between the rate at which a species may consume a resource and the number of resources it can consume. Therefore, the more different nutrients a species can consume, the fitter it will be. It may be interesting to reevaluate the major results when this assumption is changed.
We appreciate the reviewer’s point and note indeed that metabolic tradeoffs have been investigated previously in other contexts (e.g. Posfai et al. PRL 2017, Tikhonov and Monasson PRL 2017, Good et al. PNAS 2018). To explore how metabolic tradeoffs affect timeseries statistics, we simulated our original model with the constraint that the sum of consumption rates $\sum _{j}{R}_{\text{ij}}$ for consumer і is normalized to a fixed capacity $\stackrel{~}{R}}_{i$. We further assumed $\stackrel{~}{R}}_{i$ to be randomly drawn from the original growth rates $\sum _{j}{R}_{\text{ij}}{Y}_{j,0}$ (consumption rate times resource level), in effect preserving the variation in consumer fitness while removing its correlation to the number of resources consumed. Interestingly, this model largely reproduced all the timeseries statistics using the same bestfit parameters of the original model, indicating that the correlation between fitness and the number of resources in the original model is not required to reproduce experimental data (Figure S10 of the revision). We have revised the text to incorporate this finding.
A weakness of the paper is that it overstates the implications of the theoretical findings. Simulated timelines from the presented model can generate summary statistics that look like those in real data sets. This will also be possible with other models, even simpler ones or more complex ones. The article ought to include a more critical discussion and validation with simpler (e.g. pairwise interaction) or more complex (e.g. saturating growth kinetics) models.
We appreciate the reviewer’s point and have now taken more care not to overstate the implications of our findings. We now emphasize in the text that the core of our work is an existence proof of a model that can recapitulate the statistics of experimental time series, and that our work does not rule out the possibility that other models can also capture experimental statistics.
Moreover, we have endeavored to directly address the reviewer’s points about pairwise interactions in the form of generalized Lotka Volterra (gLV) models and the original CR model with saturating kinetics, as described below and in the text.
Generalized LotkaVolterra models: In the gLV model, Ν taxa grow and interact via $\frac{{\text{dX}}_{i}}{\text{dt}}={X}_{i}\left({r}_{i}+\sum _{j=i}^{N}{A}_{\text{ij}}{X}_{j}\right)$ where ✕_{"} denotes the abundance of taxon і, r_{"} its growth rate, and A_{ij} its interaction coefficient with taxon j. Since this classical model is generally unstable for randomly drawn interaction coefficients (May Nature 1972), we focused on instances of the gLV model near the parameter space corresponding to the dynamics of our CR model. This conversion between models was achieved by converting the consumption rates R_{ij} and resource levels Y_{j,0} at each sampling time T to the growth rates r_{"} and interaction coefficients A_{"#} that characterize the dynamics when consumption rates are similar to the mean value (Good et al. PNAS 2018). Under this assumption, the mapping is $r}_{i}=2\sum _{j}{R}_{\text{ij}}{Y}_{j,o$ and $A}_{\text{ij}}=\frac{1}{{R}_{max}}\sum _{k}{R}_{\text{ik}}{R}_{\text{jk}}{Y}_{k,0$ , which we refer to as CRconverted cLV models. The converted interaction coefficients are negative and symmetric, and their magnitudes depend on the niche overlap between the interacting taxa. Since the resource levels Y_{j,}_{0} are involved in this parameterization, fluctuations in Y_{#,%} across sampling times T translate into fluctuations in r_{i} and A_{ij}.
Finally, since the data being examined is compositional, the gLV model was amended to describe only the dynamics of relative abundances (Joseph et al. PLoS Comp Biol 2020) by including a normalizing term Γ(t),
$\frac{{\text{dX}}_{i}}{\text{dt}}={X}_{i}\left({r}_{i}+\sum _{j=i}^{N}{A}_{\text{ij}}{X}_{j}\mathrm{\Gamma}(t)\right)$where ✕_{i} now describes the relative abundance of taxon і and Γ(t) = ∑_{i} r_{i}✕_{i} + ∑_{I,j} A_{ij}✕_{i}✕_{j.} Relative abundances were initialized with equal values across all taxa, and the normalizing term Γ(t) enforces ∑_{i} ✕_{j} = 1. The cLV models were simulated for a fixed amount of time such that a similar range of relative abundances is generated as in the CR model at approximate ecological steady state. These CRconverted compositional LotkaVolterra models generated time series statistics that reproduced the experimental data to a similar extent as the original CR model (Figure S13A). Similarly, the CRconverted cLV models better predicted the experimentally observed distribution of pairwise correlations compared with the noninteracting null model (Figure S13B).
Finally, we asked whether more general ensembles of r_{i} and A_{ij} could also reproduce the experimental data. We randomly selected r_{i} and A_{ij} from normal distributions with means and variances equal to those in the CRconverted cLV models while enforcing symmetric and negative interactions. The resulting cLV models yielded a poor fit to the data (Figure S14). Together, these results suggest that pairwise interactions between taxa are likely sufficient to recapitulate the experimental data, although their parameters must be drawn from particular statistical ensembles.
Saturatingkinetics model: The CR model with saturating kinetics is the same as the original CR model, except that the dynamical equations are now $\frac{d{X}_{i}}{\text{dt}}={X}_{i}\left(\sum _{j=1}^{M}{R}_{\text{ij}}\frac{{Y}_{j}}{{Y}_{s}+{Y}_{j}}\right)$$\frac{d{Y}_{j}}{\text{dt}}=\frac{{Y}_{j}}{{Y}_{s}+{Y}_{j}}\left(\sum _{i=1}^{N}{R}_{\text{ij}}{X}_{i}\right)$ where $Y}_{s$ denotes the saturation constant. For simplicity, $Y}_{s$ was assumed to be equal for all resources, and set to an intermediate value of ${Y}_{s}=\u27e8{Y}_{j,0}\u27e9/3$ such that both saturated and linear kinetics could affect community dynamics. When this model was simulated with the bestfit parameters of the original model, the resulting dynamics were much less variable across sampling times than without saturating kinetics (Figure S11). Intuitively, this result is because the saturated regime is unaffected by small changes in resource levels.
Indeed, experimental statistics were again reproduced after the strength of environmental fluctuations σ was increased.
The above results demonstrate how our approach can reveal the features of other models that are necessary (or not) to explain experimental data. We have added the above figures and discussions to the text, expanding the context for our CR model through comparisons with these other models.
The article is also poorly referenced, e.g. Niehaus et al. 2019 develop a resource driven model for microbial populations (doi.org/10.1038/s4146701910062x), and Momeni et al. 2017 discussed the importance of resource mediated interactions (doi.org/10.7554/eLife.25051).
We agree that these papers are important to include and apologize for our oversight. They are now cited appropriately.
Finally, the article is not very carefully put together. I received two figures labeled as "Figure 1". The methods appear unfinished.
We thank the reviewer for their attention to detail. We have carefully combed through the manuscript to ensure that all figures, citations, and references are correct. The Methods were in fact complete, but we have expanded the section for clarity and to reflect the addition of several models.
I recommend reducing the amount of fluff terms throughout the manuscript. For example, the sentence from the abstract:
"Our coarsegrained model parametrizes the intrinsic consumerresource properties of a community using a small number of macroscopic parameters, including the total number of resources, typical resource fluctuations over time, and the average overlap in resourceconsumption profiles across species"
would read fine without the illdefined filler words:
"Our model parametrizes the consumerresource properties of a community using parameters that include the total number of resources, resource fluctuations over time, and the average overlap in resourceconsumption profiles across species."
In my opinion, simplicity and clarity strengthen theoretical papers, increasing their impact.
We appreciate the reviewer’s point and have edited the text for conciseness throughout. We have edited that particular sentence as suggested but left in the adjective “coarsegrained” as we feel it is important to convey to the reader that the “resources” do not necessarily correspond to individual metabolites.
[Editors’ note: what follows is the authors’ response to the second round of review.]
Essential revisions:
The paper has improved with the revision and it meets the standard for publication in eLife. However, the paper is rather technical and in some parts there is the risk of misinterpretation or overestimating/overinterpreting the potential of the model. The authors should better highlight the intrinsic limitations and strong assumptions of the model throughout the paper, starting – for example – from the abstract. It is not a problem of the model or the data per se, but it is rather the way it is communicated considering that the large majority of the readership will have different backgrounds and cannot necessarily understand the limitations directly. Thus, we would like to see a revised manuscript addressing these specific issues as soon as possible.
Reviewer #2:
My main concern remains: a simulation of timeseries is presented that has summary statistics as observed in data. Upon revision, based on my comment that this is not special to the model presented, another model is used; this also reproduces summary statistics similar to those from data. This is not a broad impact result and will, with the current narrative, be easily misunderstood by a nonspecialist readership.
Our key finding in response to the reviewer’s previous comment was the following: Although the generalized LotkaVolterra (gLV) model can also reproduce experimental statistics, its parametrization was guided by the CR model. The guidance provided by the consumerresource (CR) model was crucial because null parametrizations of GLV models that one might use, e.g., normally distributed interaction coefficients, did not reproduce experimental statistics. These results were shown in Figure S13 and S14. In other words, without having first identified the CR models that reproduce data, it would be highly unlikely to find the mathematically related gLV models that also reproduce data. This finding therefore strengthens the implications of our modeling framework, which can aid the investigation of other ecological models.
We apologize for not stating our findings more clearly and have revised the text throughout for clarity and to avoid misinterpretation. The extensive changes can be found highlighted in the “highlighted” version of the manuscript file.
In my opinion, such timeseries summary statistics offer little insight and have limited biological meaning. Thus, my original opinion has not shifted much.
A growing body of work has begun to show that time series statistics can be a useful window into the difficulttoaccess inner workings of complex microbiotas. We highlight some important results from this body of work and clarified their implications in the introduction. In particular only subsets of models can reproduce experimental statistics, implying that these time series statistics are informative of the underlying dynamics. Our work extends the variety of existing insights garnered from time series statistics and offers a baseline parametrization of CR models for complex microbiotas. Thus, we believe that these results have broad applications, as we elaborate on in the discussion.
https://doi.org/10.7554/eLife.75168.sa2Article and author information
Author details
Funding
National Institutes of Health (F32 GM14385901)
 PoYi Ho
National Institutes of Health (R01 AI147023)
 Kerwyn Casey Huang
National Institutes of Health (NIH RM1 GM135102)
 Kerwyn Casey Huang
Alfred P. Sloan Foundation (FG202115708)
 Benjamin H Good
National Science Foundation (EF2125383)
 Kerwyn Casey Huang
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We thank members of the Huang lab and Lisa Maier, Rui Fang, Jie Lin, and Felix Wong for helpful discussions. We thank Stephanie Song and Nicholas Chia for sharing metadata. This work was funded by a Stanford School of Medicine Dean’s Postdoctoral Fellowship (to PH), NIH F32 GM14385901 (to PH), an Alfred P Sloan Research Fellowship FG202115708 (to BHG), a Stanford Terman Fellowship (to BHG), NSF grant EF2125383 (to KCH), NIH Award R01 AI147023 (to KCH), and NIH Award RM1 GM135102 (to KCH). KCH and BHG are Chan Zuckerberg Biohub Investigators.
Senior Editor
 Wendy S Garrett, Harvard T.H. Chan School of Public Health, United States
Reviewing Editor
 Nicola Segata, University of Trento, Italy
Reviewer
 Sean Gibbons
Publication history
 Preprint posted: May 14, 2021 (view preprint)
 Received: November 1, 2021
 Accepted: March 24, 2022
 Version of Record published: April 11, 2022 (version 1)
Copyright
© 2022, Ho et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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