Evolutionary dynamics on a social multiplex network

Let us consider a multiplex network of M layers, α = {1, …, M}, and N nodes, i = {1, …, N}, which is a set of M networks G_α = (V, E_α). The set of nodes V is the same for each layer, whereas the set of links E changes according to the layer.

Each layer corresponds to a different kind of social interaction between nodes of the multiplex network and it may exhibit a different topology of connections. Modeling a social network as a multiplex network represents a more realistic approach, since in most real-world social networks, different and not mutually exclusive relationships can be considered between the same individuals (e.g. friends, relatives, colleagues, etc). Thus, the framework of social multiplex network allows us to analyse the complex dynamic patterns involving these individuals, shedding light on how the different types of interactions at various layers impact on the realistic dynamics of human behaviors.

We start from the key assumption that each entity or node in the multiplex network is an individual with a different gut metabolic profile, which in turn reflects its specific diet (see Metabolic model). Each network G_α is described by its adjacency matrix , where each element is given by: , if nodes i and j are connected at layer α, or otherwise.

The idea is to define clusters of individuals based on the concept of “gut similarity”, a metabolic similarity measure related to their metabolic profiles and indicated by θ_ij, calculated as an Euclidean distance between metabolic profiles. Since the metabolic profile is linked to diet, nodes with similar dietary habits will belong to the same cluster. Nodes are the same in every layer of the multiplex network, so they will belong to the same cluster in all the layers of the multiplex network. In terms of homophily, we consider an overall measure that includes both the concepts of centrality and homophily in the multiplex structure [30]. Moreover, we weigh the coupling between layers in the multiplex exploiting the communicability function defined in [30], which quantifies the number of possible routes that two nodes have to communicate with each other. In order to analyze the emergent behaviors in the clusters and in the whole multiplex, we consider the mathematical framework of EGT on multiplex networks [30, 31]. In particular, we choose the Snowdrift Game (SG), a pairwise social dilemma [59, 60] where nodes can select one of the two strategies, cooperation or defection. The symmetric game can be described with the payoff matrix in Table 2.

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Table 2. Payoff matrix of the Snowdrift Game.

In this social dilemma, a task has to be done and b is the benefit of accomplishing the task, while c is the cost for doing the task (where and b > c). We can observe that if neither of the two players puts an effort to cooperate, the task would not be accomplished with a negative effect for both players.

https://doi.org/10.1371/journal.pcbi.1006714.t002

In this model of conflict, a task has to be done and defectors benefit (b) from cooperators without paying a cost (c) for doing the task, but without cooperators the task would not be accomplished. Thus, despite mutual cooperation yields both an individual and total benefit higher than that of mutual defection, pure defection is the favored strategy when the other player cooperates, which occurs at the cost of the overall group payoff. It is clear how both strategies are best replies to each other, which leads to a “coexistence game” [61]. In our model, the choice of cooperating, namely following a healthy lifestyle and diet and sharing dietary habits with neighborhood, can derive from the direct and indirect reciprocity mechanisms [62]. Indeed, good eating habits may positively influence the overall behavior of the group a node interacts with, by stimulating the other nodes to be more proactive and producing, as a consequence, an amplifying effect. Instead, the selfish choice of defecting, leveraging the cooperativeness of the other neighboring nodes, without paying any cost of cooperating, can produce a major payoff in the short time, but it constitutes a major risk if also the other players will decide to defect.

One of the main reasons behind the choice of an Iterated Snowdrift Game (ISG) in this scenario is linked with its more realistic nature compared to other social dilemmas [59]. Indeed, it corresponds to more frequently observed natural situations where cooperators contribute to a public good or common task, represented in our model by good dietary habits, that is exploitable by cheaters but also provides immediate direct benefit to the cooperator [59]. In evolutionary games, payoffs determine reproductive fitness, and the target is to detect the Evolutionarily Stable Strategy (ESS). This can be formalized using replicator dynamics [30, 31] which, in the case of SG, admits a mixed stable equilibrium, where both cooperation and defection coexist. The pairwise nature of the game is translated to a population scale by making the nodes play with each other, and accumulating payoffs obtained from each interaction, over the rounds of the game. Thus, we consider an ISG where, after each round of the game, the strategies of the nodes are updated so that those nodes with less payoff are inclined to imitate the strategy of the fittest individuals. We focus on memory-1 game [63], so that choices at each round depend only on the previous round. In [63] the authors have proved that, giving only a finite memory of previous play, the payoff obtained is exactly the same as if we would consider a player with a longer memory.

We simulate the evolutionary microscopic dynamics using the standard Monte Carlo simulation procedure, composed of elementary steps, where each step gives a chance for every player to change its strategy once on average. The first step includes the distribution of competing strategies, which is an elementary step entails randomly selecting a player and one of its neighbors, calculating the payoffs of both players, and finally attempting a strategy adoption. First, a randomly selected player i acquires its payoff P_i by playing the game with all its neighbors on the layer α. Next, player i randomly chooses one neighbor j on the layer β, who then also acquires its payoff P_j on the layer β in the same way as previously did player i. Thus, a player j imposes its strategy S_j onto player i with a probability determined by the Fermi function, defined as follows [30]: (1) Therefore, a player i adopts the strategy S_j of another player j in function of the payoff difference P_i − P_j, the two similarity measures (δ_ij and θ_ij) and the factors η_i and ψ_i, related to multiplexity and unconscious bias, respectively. In particular, we consider the homophily measure, δ_ij, related to social interactions, and the gut similarity measure, θ_ij, related to gut metabolic profiles. Moreover, we include two statistical estimators η_i and ψ_i in the multiplex network, where η_i is a factor related to communicability measure in the multiplex network [30], while ψ_i represents the balance of micro-affirmations (MA) and micro-inequities (MI) of each node in the multiplex (see Micro-affirmations and micro-inequities). The selection intensity K quantifies the uncertainty in the strategy adoption process [30]. The homophily measure δ_ij represents the homophily difference between two players, so that if this values is high, a player i is more likely to imitate the strategy of j at each round.

The scaling factor η_i allows including the dependency of the strategy adopted by the player i on the strategies of related players from the other layers [30], so it depends on the communicability function between layers [30, 64]: (2) where the numerator is the sum of the communicability functions calculated between the node i on the layer α and all the neighboring nodes j on the layer β, adopting the same strategy as player i. The denominator represents the sum of the communicability functions calculated between the node i on the layer α and all the neighboring nodes j belonging to the layer β. Therefore, the ratio quantifies the influence, in terms of communicability, on the strategy adoption of the player i on the layer α, due to the strategies adopted by the counterpart node and its neighbors on the layer β. In particular, the more are players on the layer β with a high communicability with the node i and adopting the same strategy as player i, the more likely i will adopt the same strategy in the next round. On the other hand, if there are nodes on the layer β with a high communicability, but adopting a different strategy, the player i will most likely tend to change its strategy. Thus, this ratio depends on the communicability function and it may result in a bias regarding the strategy adoption of the player i in the next round of the game.

The communicability function has been introduced in order to shed light on the importance of the coupling between layers in exploring the evolution of behaviors in the multiplex structure. To this aim we have exploited the communicability function defined in [64], which quantifies the number of possible routes that two nodes have to communicate with each other. Starting from the definition of the communicability function between two nodes i and j in the multiplex provided in [64], we consider the communicability matrix G, where each element is the matrix representing the communicability between every pair of nodes belonging to two different layers α and β, of the multiplex network. It is defined as in [30] and [G_αβ]_ij represents the communicability between the node i in the layer α and the node j in the layer β.

Micro-affirmations and micro-inequities

In order to deal with unconscious bias, in our model we introduce a statistical estimator ψ_i, impacting on Eq 1. This bias, which is the result of quick judgments or evaluations of people and situations, is influenced by our individual experiences or environment in which we are merged, occurs automatically and may impact on our choices [28]. “Micro-inequities” (MI) are one of the manifestations of this bias, and they are defined as “apparently small events which are often ephemeral and hard-to-prove, events which are covert, often unintentional, frequently unrecognized by the perpetrator, which occur wherever people are perceived to be different” [28]. In order to address it, Rowe defined the concept of micro-affirmations (MA) as “apparently small acts, which are often ephemeral and hard-to-see, events that are public and private, often unconscious but very effective, which occur wherever people wish to help others to succeed” [28]. Micro-affirmations therefore act as a countermeasure to micro-inequities, so that if individuals are constantly proactive about rewarding and professing the achievements of others, people will be less inclined to micro-inequities and it will trigger a reciprocity effect, producing a psychological reward.

It is important to observe how the unconscious bias, expressed in terms of MI and MA and quantified by the factor ψ_i (see Eq 3), is linked with several environmental (cultural, geographical, social, etc.) factors, metabolic needs and emotional states [21], influencing our food choices and the resulting gut microbial composition [65, 66]. ψ_i is defined as follows: (3) where MA_ji and MI_ji are respectively the micro-affirmations and micro-inequities obtained at each round of the game by the node i from all the interactions with its neighboring nodes j. At each round, the factor ψ_i is calculated in game-theoretical terms, by taking into account all the payoffs obtained by the node i in the previous round of the game. In particular, if a node gets a “positive” payoff (see the first column of the payoff matrix, Table 2), then it will get a MA, otherwise a MI. Thus, ψ_i produces a bias in the choices and, in particular if MI_ji ≥ MA_ji, then a player will have a higher tendency to change its strategy in the next round. It may reflect a sort of ‘delusion’ or ‘sadness’ of the node i, related to the payoff obtained in the previous round after having played with its neighboring nodes j. Otherwise, if MI_ji ≤ MA_ji, it means that node i has experienced positive affirmations in its interactions (“positive” payoff) and it will likely tend to keep the same choice adopted in the previous round. Analogously to η_i [30], the two limit values, and , are chosen in order to avoid the “frozen states”. ψ_i acts as a key bias towards a change in strategy when MI_ji ≫ MA_ji, which means a strong imbalance of MI over MA, while if MA_ji ≫ MI_ji, a node tends to keep the same strategy of the previous round. At the beginning we assume that MI_ji and MA_ji are initialized with a value equal to zero, which creates an initial condition where nodes are on average satisfied of their neighborhood and diet, given that there have not been interactions and hence they have not yet experienced any MA or MI in the network.

Feedback mechanism

Our work is aimed at modeling and quantifying the mutual influences between gut microbiota and the social multiplexity of networked individuals in the “gut-human behavior axis”. To this end, we introduce a control mechanism, acting from dietary behaviors influenced by social interactions to gut microbiota and, backwards, from gut to social behaviors. Although the diet-induced dynamics of the gut microbial communities has a relatively high responsiveness within 24 hours [9], diets and bacterial composition of gut microbiota are determined by long-term dietary habits [10]. Thus, we evaluate the evolutionary dynamics in a relatively long time-span, including a certain amount of rounds of the game (see Results).

To measure the impact of these long-term dietary behaviors on diets and human gut microbiota, we introduce a statistical estimator γ_i, defined as follows: (4) where NC is the number of cooperations in the previous t − 1 rounds before feedback, while N_r is the number of rounds before feedback, which happens at time step t. γ_i, called “happiness index”, reflects the attitude of each individual in the network, that is how much a node has been “happy” (or cooperative) in the previous rounds. Therefore, γ_i will act as a modulator, influencing gut metabolic profiles and modifying a fraction of diet components (nutrients, etc.), and ultimately producing a different gut profile. γ_i ranges in [0, 1], where γ_i = 0 reflects a lack of cooperativeness while γ_i = 1 means that a node has been fully cooperative with a proactive attitude towards its community. γ_i quantifies the impact on gut bacteria deriving from human behaviors. The updated gut metabolic model for each individual will then be used to update the distance among individuals for the next round, therefore driving the re-clustering of the multiplex network based on the gut similarity of the updated gut profiles (see Metabolic model for details on how a different diet composition is achieved and used to run te gut model, finally updating the gut flux profile of each individual).

As well as producing a re-clustering of the nodes in the multiplex network, γ_i can produce a remarkable MA or MI, that is respectively a positive or negative bias from gut. This bias, weighted by another statistical estimator ϵ_i, called “gut bias”, is defined as the ratio between the number of MA or MI obtained in all the rounds before the feedback mechanism and the number of rounds in the same temporal window. We have a dual definition of ϵ_i, depending on the sign of the bias from gut, so that in the case of a positive bias, it is defined as: (5) whereas in the case of a negative bias it is defined as: (6)

In terms of evolutionary dynamics, the “gut bias” amplifies the number of MA or MI, impacting on ψ_i and, in turn, on the strategy adoption process (see Eq 1). In particular, we define a threshold γ_th so that, if γ_i ≥ γ_th, it means that the node has experienced a positive feedback, resulting in: MA = MA + ϵ_i ⋅ MA, while if γ_i < γ_th, the node has experienced a negative feedback and there is an amplifying effect of MI as follows: MI = MI + ϵ_i ⋅ MI. The re-clustering of the network, obtained at each control mechanism, modifies the spatial distribution of the network. It is interesting to observe the patterns of clustering depending on the evolutionary dynamics of nodes behaviors, due to the described feedback mechanism.

Metabolic model

The happiness index γ_i is used to update the diet of each individual during each feedback round. The impact of the new diet on the gut microbiota is assessed using a GSMM. GSMMs can be used for generating context-specific or personalized gut microbiota models through nutrient composition or data integration at various omic levels, and have recently been augmented with three-dimensional metabolite and protein structure data [67–70]. The model used to represent the gut consists of a human GSMM (Recon 2) and reconstructions for eleven bacterial species representing commensal, probiotic and pathogenic associations which can occur between the human host and gut microbiome [71]. Flux balance analysis (FBA) is then used to run the updated model. FBA is characterized by the definition of precise linear constraints used to cumulatively reduce the solution space of the computed states of a metabolic model. After selecting a cellular objective, the model is finally simulated through linear programming [72].

For a genome-scale metabolic model representing a single organism or community, biomass production is the most commonly considered objective for optimization as it can serve as a proxy for growth rate. To improve this estimation, in recent years procedures have been developed for checking biomass weight, generating accurate stoichiometric coefficients for the biomass objective function and enforcing time-averaged equality of growth rates in microbial communities [73–75]. For this combination of models we use the COBRA Toolbox in MATLAB [76] to perform FBA whilst optimizing a linear combination of multiple biomass objectives, which represents hierarchical maximization of biomass production by each member of the microbial community (see Table 3).

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Table 3. Ordered list of biomass reactions chosen as objectives for multi-level linear optimization in FBA for the different species considered in this study.

https://doi.org/10.1371/journal.pcbi.1006714.t003

In our modeling framework, a series of 100 diets with varying compositions of carbohydrate, fat and protein intake was created to simulate the effect of various diets on metabolic flux through the optimized pathway(s) or reaction(s). With the exception of a few more extreme diets, percentages of macro-nutrients within these diets were generated using a random number generator. Subsequently, uptake rates were calculated by multiplying percentages of each macro-nutrient by a basal uptake rate B, thus accounting for differences in molecular size by scaling uptakes to the number of carbon atoms in each nutrient [71].

Following Heinken and Thiele [71], 53 metabolites belonging to three classes of macronutrients—carbohydrates, fats and proteins (C, F, and P) were associated with a basal uptake rate B calculated from a data survey conducted by the USDA [77]. These basal uptake rates were multiplied by the specific percentage of each macronutrient in an individual’s diet. With the exception of the Western diet (94.36% sugars, 5.64% fiber), carbohydrate composition for all diets was divided into 50% sugars and 50% fiber. Therefore, basal uptake rates for fats and proteins were multiplied by 2 to re-scale them in accordance with the carbohydrate uptake. All of the diets contain the same number of metabolites, but uptake rates were varied by adjusting lower bounds for the exchange reactions of each metabolite with respect to the type of diet simulated. For example, the calculation of metabolite uptake rates (A_u) for a diet consisting of 34% carbohydrate, 44% fat and 22% protein would be calculated as follows: (7)

Series of uptake rates were calculated separately for each group of nutrients (i.e. C_u, F_u and P_u) before being concatenated within a single vector of uptake rates (A_u). Once uptake rates for all metabolites were calculated, these values were set as lower bounds of exchange reactions in the model using indexes corresponding to each reaction. In this way, FBA returns a complete flux distribution for each diet, elucidating beneficial, detrimental or neutral effects of dietary changes on human health.

When this modeling approach is combined with social relationships using a hierarchical clustering based on gut profiles, the complex interplay between dietary habits and social relationships can be analyzed by examining interactions between nodes representing metabolic profiles and social ties. In modeling diets, we must take into account some perturbations of diets as people do not adhere to dietary regimes, corresponding to the most typical diets (e.g. Western diet, high-fat, high-carbohydrate, high-protein), but most often they change some dietary components producing diet modifications or perturbations, that will result in a change in the microbiota [10].

We jointly optimize the growth of all species (including the host) by setting biomass reactions for each species as objectives using a multi-level linear optimization program based on GEMsplice [78] which seeks trade-offs between multiple objectives. The order of the chosen biomass objectives was defined as: BT, FP, EC, LP, LL, ST, HP, KP, STy, ECs, ECe and HS [79] (see Table 3). The growth of commensal and probiotic species were considered to be more important as they are thought to produce beneficial metabolites for the host as opposed to toxic or harmful compounds produced by the pathogenic species. In accordance with [71], a minimal growth rate was enforced to maintain growth of all species by constraining the lower bounds for the chosen biomass objectives to 0.01. To set this, we define v_biomass as all of the biomass reactions within the full flux vector v (i.e. those reactions identified by the objectives c₁, …, c₁₂ and detailed in Table 3). For each diet, fluxes through the biomass and ATP synthase reaction for each species were recorded. In each distribution, fluxes for the biomass and ATP synthase reaction are used to calculate new values for γ_i, which are averaged over all previous rounds of EGT. Formally, we defined the twelve-level linear program as follows: (8) where Λ is the stoichiometric matrix of the model, x_j represents the concentration of each metabolite M_j.

Upon obtaining the first set of γ_i values for every node (characterized by a specific diet) in the network, the initial percentage of each macronutrient (carbohydrate, fat and protein) ±(1 − γ_i) ⋅ 10 is used to calculate lower and upper limits, between which the new fraction of each dietary component is selected for that node using a random number generator for the next feedback, ensuring that the sum of all three percentages is equal to 100%. For example, if a given node in the first feedback round has a carbohydrate percentage of 50 and the γ_i value for this diet characterizing the node i is 0.0800586, we calculate: (1 − γ) ⋅ 10 = (1 − 0.0800586) ⋅ 10 = 9.19941. This means that in the next feedback round, the lower carbohydrate limit will be equal to: 50 − 9.19941 = 40.80059, and the upper carbohydrate limit will be: 50 + 9.19941 = 59.19941. Therefore, for this particular node i, the carbohydrate percentage in next feedback round is set to a randomly drawn integer between 40.80059 and 59.19941, e.g. 54%. The calculation is repeated using the same γ_i for adjusting fat and protein content in this node i, and using a different γ_i value for each node i in all four feedback rounds.

The gut community model used for this study comprises multiple bacterial species within the human host. There are also particular gut communities, that is those formed by patients with celiac disease [80], autistic individuals [81] or people with irritable bowel syndrome [82]. In [83], an inter-organ communication during metabolic disorders was identified, and in fact almost every sick organ guides the formation of a community. Although some studies consider a larger number of bacterial species, in [14] the number of species considered is similar. Here the cause of community formation is the disease, but it may also be the therapy administration (e.g. metformin for type 2 diabetes). We can observe how specific diseases, geographical locations or even cultural choices (e.g. vegan diets) link the community and gut in a stronger way than other situations.

Following [35, 84], the metabolic gut flux profiles of each individual are then used to define distances among individuals. This metabolic similarity, represented by θ_ij and calculated as an Euclidean distance between metabolic profiles is then used to define clusters of individuals.

Correlation analysis

To gain insight into how the “happiness index” is influenced by particular pathways, we here investigate the correlation of γ_i with absolute average flux rates across subsystems in the model. We observe correlations between average flux in each pathway and happiness index in each node, or diet. Our table of results for the correlation analysis is provided in S1 Table. Overall, the positive correlations which stand out as being significantly higher are folate metabolism and purine/pyrimidine synthesis in the pathogen S. typhimurium, during the second feedback round (> 0.5). These pathways are linked in several ways, e.g. the folate cofactor tetrafolate is involved in the synthesis of purines [89].

Both these pathways are also responsible for producing fundamental biomass precursors, which is significant as biomass is the objective maximized for all species in our model during FBA. It is well established that purines and pyrimidines form the building blocks for the nucleotides which comprise DNA, but they are also vital for the synthesis of important coenzymes, such as ATP and NAD. Folate is required for important cellular processes such as DNA replication, repair and methylation, as well as playing a vital role in the metabolism of many amino acids. Maternal diet, such as periconceptional folic acid supplementation, has proven to impact DNA methylation in offspring and can affect their neurodevelopment and health outcomes [90]. In humans, folate cannot be synthesized de-novo and must therefore be obtained from the diet or gut microbiota. In this regard, the increase in dietary folate intake and introduction of folate-producing probiotic bacteria in the form of live microbial supplements was previously discussed [91].

In accordance with protein-rich diets yielding a high γ_i value, it can be seen that the metabolism of various amino acids are among the top positively and negatively correlated subsystems with γ_i in all four feedback rounds. In the first and second feedback rounds, the synthesis and recycling of murein in E.coli has a significant role to play, as murein is a polysaccharide which provides structural integrity to bacterial cell walls whilst allowing the exchange of metabolites [92, 93].

As a key intermediate in human cellular respiration, acetyl co-enzyme A is involved in pyruvate oxidation, as well as cholesterol synthesis and fatty acid oxidation during lipid catabolism, where it acts as an acyl carrier [94]. Fatty acids are an essential component of both human and bacterial cell membranes and can be synthesized via glycolysis or amino acid deamination. Human cholesterol metabolism (in the third flux iteration) has a moderately high negative correlation; this seems to support the idea that diets high in cholesterol are likely to lead to poor health.

Code availability

The developed methodology in terms of metabolic modeling, that is the multi-objective optimization of gut community metabolic model, is available to the scientific community at the following repository: https://github.com/svijayakumar32/multi-objective-optimization-of-gut-community-metabolic-model.