Generalizing game-changing species across microbial communities

Tractable systems

Tractable systems are models typically of known and reduced complexity that can be operationalized and reproduced over relatively short periods of time. To formalize our study using tractable systems, we consider a (regional) pool ({mathcal{R}} =left{1,2,cdots ,Sright}) of (S) resident species and one non-resident species denoted by “(I)” (Fig. 1A). We denote the resident community by ({mathcal{M}} subseteq {mathcal{R}}), which is the species collection that coexists obtained by assembling all resident species simultaneously. Additionally, let ({ {mathcal{M}} }_{p}) denote the perturbed community formed by the species collection that coexists when assembling all residents and the non-resident species simultaneously assuming the possibility of multiple introductions. Note that this mechanism corresponds to a top-down assembly process.³³ Then, the non-resident species is classified as a game-changing species if it changes the number of resident species that coexist: (|{ {mathcal{M}} }_{p}{I}|,ne, | {mathcal{M}} |). Figure 1A illustrates the concept of a game-changing species in a hypothetical microbial community of (S=2) resident species. Here, one possible context for the resident species is that one species excludes the other, e.g., ({mathcal{M}} ={1}) (Fig. 1B). In this case, the non-resident species is a game-changing species if it promotes the establishment of the other resident (Fig. 1C). Note that we do not consider a change when eliminating the current resident species. The other context for the resident species is that both coexist ({mathcal{M}} ={1,2}) (Fig. 1D). In this case, a non-resident species is a game-changing species if it suppresses the establishment of at least one of the residents, e.g., ({ {mathcal{M}} }_{p}={1,I}) (Fig. 1E). Note that a game-changing species can be either colonizer or transient depending on the dynamics.

We study the generalization of game-changing species under controlled conditions using in vitro experimental soil communities and under changing conditions using in vivo gut microbial communities (see SI for details about these experimental systems). Note that in vitro experiments usually create ad hoc conditions for species by putting them outside of their natural changing habitats, assuring that species survive in monocultures, and by forming interspecific interactions that may not occur otherwise. Instead, in vivo experiments are performed within living systems, resembling much closer the natural habitat of species and their interspecific interactions. Both types of systems are of reduced complexity, allowing the monitoring and reproducibility of experiments. The studied in vitro soil communities are formed by experimental trials of eight interacting heterotrophic soil-dwelling microbes:³⁰ Enterobacter aerogenes, Pseudomonas aurantiaca, Pseudomonas chlororaphis, Pseudomonas citronellolis, Pseudomonas fluorescens, Pseudomonas putida, Pseudomonas veronii, and Serratia marcescens. These experiments were performed by co-inoculating species at different growth–dilution cycles into fresh media. Each species was cultured in isolation. All experiments were carried out in duplicate. The studied in vivo gut communities are formed by experimental trails of five interacting microbes commonly found in the fruit fly Drosophila melanogoster gut microbiota:³⁴ Lactobacillus plantarum, Lactobacillus brevis, Acetobacter pasteurianus, Acetobacter tropicalis, and Acetobacter orientalis. These experiments were performed by co-inoculating species through frequent ingestion in different flies. All experiments were replicated at least 45 times. These two data sets are, to our knowledge, the closest and best described systems of two- and three-species communities currently available describing species coexistence (not just presence/absence records) under two contrasting environmental conditions.

Focusing on in vitro communities, we studied all 28 pairs and 56 trios formed by the eight soil species.³⁰ This provided 168 cases, where it is possible to investigate the expected result (({ {mathcal{M}} }_{p})) of assembling a non-resident species together with a resident community (21 cases for each of the 8 studied microbes). The overall competition time was chosen such that species extinctions would have sufficient time to occur, while new mutants would typically not have time to arise and spread. Similarly for the in vivo communities, we studied all 10 pairs and 10 trios formed by the 5 gut species, which provided 30 cases equivalent to the soil experiments.³⁴ Because species extinctions in in vivo communities are harder to establish, we classified as an expected extinction to any species whose relative abundance was less than 10% in at least 71% of all (47–49) replicates, which corresponds to less than 1% of cases under a binomial distribution with (p=0.5) (slightly different thresholds produce qualitatively similar results). Each of these 168 and 30 cases for soil and gut communities, respectively, represents a given resident community (({mathcal{M}})) formed by a pool of two resident species (({mathcal{R}} ={1,2}), ({mathcal{M}} subseteq {mathcal{R}})) where the target for a non-resident species ((I)) can be either to promote or to suppress the establishment of resident species (({mathcal{R}} {cup }left{Iright}), ({mathcal{M}}_{p} setminus {I},ne, {mathcal{M}})). Non-resident species that are expected to survive in ({ {mathcal{M}} }_{p}) are classified as colonizers; otherwise they are classified as transients.

Empirical contexts

To investigate the role of context dependency in the game-changing capacity of microbial species, we study the extent to which a given species can be classified as a game-changer regardless of the resident community it interacts with or if it is the resident community that provides the opportunity for a non-resident species to be a game-changer. Specifically, for each species, we calculate the fraction of times such a species changes the resident community conditioned on the type (whether it is a colonizer or a transient) and target (whether promoting or suppressing). Then, we calculate the probability ((p) value) of observing a fraction greater than or equal to the observed fraction under the given type/target (using a one-sided binomial test with mean value given by the empirical frequency within each type/target). High (p) values (e.g., (> 0.05)) would be indicative of the importance of context-dependency and not of the intrinsic capacity of species.

Next, we quantify the average effect of empirical contexts shaping the game-changing capacity of non-resident species. Specifically, we measure the type’s average effect on changing the community using ({E}_{Y}=P(C=1|Y=1)-P(C=1|Y=0)). Here, (C=1) if the non-resident species was a game-changer ((C=0) if it was not), and where (Y=1) if the species was a colonizer ((Y=0) if the species was transient). The non-parametric quantity (P(C|Y)) corresponds to the frequency of observing (C) given (Y). Thus, ({E}_{Y} ,> , 0) (resp. ({E}_{Y} ,<, 0)) indicates that a game-changing species is more likely to be a colonizer (resp. transient). Similarly, we measure the target’s average effect on changing the community using ({E}_{T}=P(C=1|T=1)-P(C=1|T=0)), where (T=1) if the target was to suppress ((T=0) if the target was to promote) and (P(C|T)) corresponds to the frequency of observing (C) given (T). Thus, ({E}_{T} ,> , 0) (resp. (,{E}_{T} ,<, 0)) indicates that a game-changing species is more likely to suppress (resp. promote) the establishing of resident species. High effects (and statistically different from what would be expected by chance using a G({}^{2})-test) would be indicative of the impact of the empirical contexts on the game-changing capacity of species.

Tractable theoretical systems: mutual invasibility theory

We use tractable theoretical systems to establish sufficiently operationalizable algorithms that can move us away from context-specific cases to regularities shaping the capacity of game-changing species. The first premise follows a heuristic assembly rule based on mutual invasibility theory.³⁰ Mutual invasibility theory has been a widely-adopted tractable premise in ecology and evolution,^29,35 This theory states that in a multispecies community, species that all coexist with each other in sub-communities will survive, whereas species that are excluded by any of the surviving species will go extinct³⁰ (Fig. 2A–D). Despite its strict assembly requirements, this heuristic rule has proved relatively successful in predicting the outcome of surviving species in the studied in vitro soil communities.³⁰ Thus, to operationalize this assembly rule, we introduce a binary variable (({{V}})) that when anticipating the promotion (resp. suppression) of resident species becomes ({{V}}=1) (resp. ({{V}}=0)) if and only if all the resident species survive when paired with any other species; otherwise ({{V}}=0) (resp. ({{V}}=1)). Then, we measure the expected effect of mutual invasibility on the capacity of game-changing species while keeping all other factors constant at whatever value they would have obtained under a non-invasibility case—known as the direct natural effect:³⁶ ({{rm{NE}}}_{{{V}}}={Sigma }_{{{T}},{{Y}}}[P(C=1|{{V}}=1,,T,Y)-P(C=1|{{V}}=0,T,Y)]P(T,Y|{{V}}=0)). Positive effects (and statistically different from what would be expected by chance using a G({}^{2})-test) would be indicative of the usefulness of this heuristic rule as a general context for identifying game-changing species.

Tractable theoretical systems: structuralist theory

While mutual invasibility theory has provided key insights regarding population dynamics,^29,30,35 it has been shown that it cannot be directly generalized to multispecies communities.^37,38,39 Hence, as an alternative potential generalization, we introduce a second heuristic rule based on structuralist theory,^32,40,41 Across many areas of biology, the structuralist view has provided a systematic and probabilistic platform for understanding the diversity that we observe in nature,^31,42,43 In ecology, structuralist theory assumes that the probability of observing a community is based on the match between the internal constraints established by species interactions (treated as physico-chemical rules of design) within a community and the changing external conditions (treated as unknown conditions).^32,44,45 This other premise has also been shown to be as successful as mutual invasibility in predicting the outcome of surviving species in the studied in vitro soil communities,³² but it has not been tested for its generality.

Formally, the structuralist framework assumes that the per-capita growth rate of an (i)th species can be approximated by a general phenomenological function ({f}_{i}({N}_{1},cdots ,{N}_{S},{N}_{I};{boldsymbol{theta}})), i.e.,

Above, ({N}_{i}) represents the abundance (or biomass) of species (i). The functions ({f}_{i}) encode the internal constraints of the community dynamics.⁴⁶ The vector parameter ({mathbf{theta}}) encodes the external (unknown) conditions acting on the community, which can change according to some probability distribution (p({mathbf{theta}})). For a particular value ({mathbf{theta}}={{mathbf{theta}}}^{ast }), a species collection ({mathscr{Z}}subseteq {mathcal{R}} {cup }{I}) is said feasible (potentially observable) for Eq. (1) if there exists equilibrium abundances ({N}_{i}^{ast } ,> , 0) for all species (iin {mathscr{Z}}) and ({N}_{i}^{ast }=0) for (i,notin, {mathscr{Z}}) (i.e., ({f}_{i}({N}_{1}^{ast },cdots ,{N}_{S}^{ast },{N}_{I}^{ast };{{mathbf{theta}}}^{ast })=0) for all (i)).⁴⁰ Then, we can use Eq. (1) to push-forward (p({mathbf{theta}})) and estimate the probability that a randomly chosen species (i) is feasible with ((iin {mathcal{R}} {cup }{I})) and without ((iin {mathcal{R}})) the non-resident species under isotropic changing conditions, respectively. In this form, the effect of a non-resident species (I) on a resident community can be characterized by the expected maximum impact on its feasibility, i.e., ({Delta }_{{{F}}}=p(i| {mathcal{R}} {cup }{I})-p(i| {mathcal{R}} )) (Fig. 2E–H).

To make this framework tractable, we leverage on the mathematical properties of the linear Lotka–Volterra (LV) system⁴⁷ with the per-capita growth rate ({f}_{i}({N}_{1},cdots ,{N}_{S},{N}_{I};{mathbf{theta}})=mathop{sum}limits_{jin {mathcal{R}} {cup }{I}}{a}_{ij}{N}_{j}+{theta}_{i}) for (iin {mathcal{R}} {cup }{I}). While the linear LV system can be interpreted under many different assumptions,⁴⁷ we follow its most general interpretation as a first-order approximation to Eq. (1).⁴⁶ In this system, the time-invariant community structure consists of the intraspecific and interspecific species interactions ({bf{A}}=({a}_{ij})in {{mathbb{R}}}^{(S+1)times (S+1)}), and the external factors ({mathbf{theta}}=({theta }_{1},cdots ,{theta }_{S},{theta }_{I})in {{mathbb{R}}}^{S+1}) consist of density-independent intrinsic per-capita growth rates of all species. We assume that (p({boldsymbol{theta }})) is uniform over the positive parameter space (conforming with ergodicity in dynamical systems³²) and find analytically the external conditions compatible with the feasibility of a randomly chosen species within a given community ({bf{A}}), i.e., (p(i|{bf{A}})) (see SI). This framework is robust to changes in the system dynamics since (p(i|{bf{A}})) is identical for all systems that are topologically equivalent to the linear LV system^32,48 and a lower bound for systems with higher-order terms.⁴¹ Note also that while higher-order interactions may impact the dynamics of microbial communities^49,50, their incorporation into ecological models as higher-order polynomials rend intractable and super-sensitive systems (no closed-form solutions can be found in terms of radicals)^41,51,52,53.

To quantify the contribution to feasibility (({Delta }_{{rm{F}}})) of a non-resident species under the structuralist framework defined above, we infer both the resident interaction matrix ({bf{A}}) and the perturbed interaction matrix ({{bf{A}}}_{p}) using only information from experimental monocultures and pairwise cocultures. Interaction matrices were inferred by fitting the linear LV system using the different repetitions of the observed survival data (see SI and Fig. S1). To make the structuralist framework comparable with the mutual invasibility framework, we introduce a heuristic rule based on structuralist theory formalized in a binary variable (({{F}})) that when anticipating the promotion (resp. suppression) of resident species becomes ({{F}}=1) if ({Delta }_{{{F}}} ,> , 0); otherwise ({{F}}=0) if ({Delta }_{{{F}}} ,<, 0). Then, we measure the expected effect of structuralist theory on the capacity of game-changing species while keeping all other factors constant at whatever value they would have obtained under an opposite scenario:³⁶ ({{rm{NE}}}_{{{F}}}=mathop{sum}limits_{{{T}},{{Y}}}[{{P}}({{C}}=1|{{F}}=1,{{T}},{{Y}})-{{P}}({{C}}=1|{{F}}=0,{{T}},{{Y}})]{{P}}({{T}},{{Y}}|{{F}}=0)). Positive effects (and statistically different from what would be expected by chance using a G({}^{2})-test) would be indicative of the usefulness of this heuristic rule as a general context for identifying game-changing species.

Source: Ecology - nature.com