Multiple stable states in microbial communities explained by the stable marriage problem

Abstract

Experimental studies of microbial communities routinely reveal that they have multiple stable states. While each of these states is generally resilient, certain perturbations such as antibiotics, probiotics, and diet shifts, result in transitions to other states. Can we reliably both predict such stable states as well as direct and control transitions between them? Here we present a new conceptual model-inspired by the stable marriage problem in game theory and economics-in which microbial communities naturally exhibit multiple stable states, each state with a different species' abundance profile. Our model's core ingredient is that microbes utilize nutrients one at a time while competing with each other. Using only two ranked tables, one with microbes' nutrient preferences and one with their competitive abilities, we can determine all possible stable states as well as predict inter-state transitions, triggered by the removal or addition of a specific nutrient or microbe. Further, using an example of seven Bacteroides species common to the human gut utilizing nine polysaccharides, we predict that mutual complementarity in nutrient preferences enables these species to coexist at high abundances.

Fig. 1

Ranked interaction tables encode microbes’ nutrient preferences and competitive abilities. Two ranked tables with each microbe’s preferences towards nutrients (a) and their competitive abilities with respect to each particular nutrient (b) fully define our model. We illustrate them using two microbial species, M₁ and M₂, represented correspondingly by dark and light circles, and three nutrients, N₁, N₂, and N₃. Both species can use all three nutrients. a Microbial nutrient preferences: the dark species prefers nutrient N₁ the most (rank 1 in the table above), N₂ next (rank 2), and N₃ the least (rank 3), while the yellow species prefers nutrients in the order: N₃ > N₁ > N₂. b Microbial competitive abilities: the dark species (rank 1) can displace the light species (rank 2) in a competition for utilizing the nutrient N₂, but will be displaced by the light species when competing for nutrients N₁ and N₃

Fig. 2

Community restructuring following external perturbations. Two ranked tables of microbes’ nutrient utilization preferences and competitive abilities are shown on top of each panel. Colored circles represent different microbial species M₁, M₂, M₃. The size of each circle corresponds to the rank of a nutrients microbe currently utilizes—bigger sizes correspond to better ranks and thus larger populations. Different nutrients are labeled N₁, N₂, N₃. Oblique dashed lines indicate transient states for microbial competition. a The introduction of a new probiotic microbe, dark species (M₃), causes grey (M₁) and light (M₂) species to enter into a competition with the invader. The dynamics of the stable marriage model results in a community restructuring to the state B, such that the grey (M₁) and light (M₂) species shift their currently utilized nutrients to their second choices. The invading dark species (M₃) fails to establish itself and disappears from the system. b A transient addition of a prebiotic nutrient, N₃, restructures the community from state B back to state A, in which each microbe once again uses its most preferred nutrient

Fig. 3

Multiple stable states and the network of transitions between them. Two ranked tables of microbes’ nutrient utilization preferences and competitive abilities are shown on the left. a The list of all stable states (labeled S₁ through S₅) in the model. In each stable state, every microbe (colored circles with tails; sizes indicative of how preferred the consumed nutrient in a state is) exclusively consumes one nutrient (labeled N₁ through N₇). b The “microbe-optimality” of stable states S₁ S₅ (lower is better for microbes) quantified by the rank of the consumed nutrient averaged over all microbes. Microbe-optimality can be improved by transiently removing microbes and deteriorated by transiently removing nutrients. c, d The stable states are connected via “restructuring networks”. The community in the model gradually restructures from S₁ towards S₅ by transient nutrient removal (for details, see the Results section) and from S₅ back towards S₁ by transient microbe removal. In this restructuring network, a pair of stable states is connected by a directed link, if the community can transition between these states via transient removal of just one nutrient (removed nutrient and directionality are shown in c) or of a single microbe (removed microbe and directionality are shown in d). e Average number of stable states for communities with different numbers of microbes (M, x-axis) and nutrients (N, y-axis) and randomized interaction tables. (Inset, top) For (M, N) = (50, 50), we show the distribution of the number of steady states (in orange) for 1000 random interaction tables. The distribution has a pronounced peak and an exponential tail

Fig. 4

Complementary polysaccharide prioritization allows robust coexistence in gut Bacteroides species. a The polysaccharide utilization network of Bacteroides species in the human gut (data taken from ref. [45]). The character labels represent nine different polysaccharides: S starch, M mucin, G galactan, P pectin, A arabinogalactan, HC hemicellulose, C cellulose, H hyaluronan, CS chondroitin sulfate (CS)—known to be frequently present in human diets (legend in the box on the left), whereas the colored circles represent seven different Bacteroides species routinely found in human gut microbiome: Bacteriodes fragilis, B. ovatus, B.vulgatus, B.caccae, B. cellulosilyticus, B. thetaiotaomicron, Parabacteroides distasonis. Undirected links between microbes and polysaccharides indicate a species’ ability to metabolize that polysaccharide. b Examples of microbial nutrient preferences (the most preferred nutrient of each of the microbes) are sorted into three categories: complementary (top) where microbes’ top preferred nutrients (#1) are all distinct from each other; random (middle) preferences where all ranked lists are randomly generated; and maximal conflict (bottom) which represents the maximum intersection between the sets of top (#1) and second (#2) preferred nutrients of different microbes. c For 1000 randomly sampled microbial preferences from each category, we simulated the stable marriage model to compute the expected per species microbial abundances (Methods section: Studying complementarity through different ranked interaction tables) for each case as box plots. The box plots quantify the distribution of average microbial abundance assumed to be inversely proportional to the rank of utilized nutrient. The average abundance is the largest in the case of complementary nutrient choices. All differences between distributions of abundances in each category are highly statistically significant according to the Kolmogorov–Smirnov test with a P-value threshold of 0.01