Phase separation and coexistence in spatial coordination games between microbes

Abstract

Dense, microbial communities are shaped by local interactions between cells. Both the nature of interactions, spanning antagonistic to cooperative, and the strength of interactions vary between and across microbial species and strains. These local interactions can influence the emergence and maintenance of microbial diversity. However, it remains challenging to link features of local interactions with spatially mediated coexistence dynamics given the significant variation in the microscopic mechanisms involved in cell-to-cell feedback. Here, we explore how microbial interactions over a broad range of ecological contexts spanning antagonism to cooperation can enable coexistence as spatially explicit domains emerge. To do so, we introduce and analyze a family of stochastic coordination games, where individuals do better when playing (i.e., interacting) with individuals of the same type than when playing with individuals of a different type. Using this game-theoretic framework, we show that the population dynamics for coordination games is governed by a double-well shaped interaction potential. We find that in a spatial setting this double-well potential induces phase separation, facilitating coexistence. Moreover, we show that for microbes engaged in symmetric coordination games, phase separation takes on a universal scaling that follows 'Model A' coarsening, consistent with prior experimental observations for Vibrio cholerae mutual killers. Finally, we derive a PDE equivalent of the spatial stochastic game, confirming both the double-well nature of spatial coordination games and the universality of phase separation. Altogether, this work extends prior findings on the link between microbial interactions and population structure and suggests generic mechanisms embedded in local interactions that can enable coexistence.

FIG. S1:

Mapping from two-player games to inter- and intra-specific interactions. The sign (+/−) of payoff values correspond to the positive (+) and negative (−) interactions. There are two layers, the outer-layer represents the intra-specific interactions and the inner-layer represents the inter-specific interactions. Some intra-/inter-specific ecological interactions cannot be mapped to any coordination game (gray blocks), e.g., an example of anti-coordination game with $\text{sign}\{A\}=[-,+;-,+]$. Some pairs of games are symmetric (blocks with same color), i.e., one can be transformed to another via swapping player 1 and player 2. Hence, there are six distinct inter-/intra-specific ecological interactions that are mapped to coordination games as shown in Fig. 2 in main text.

FIG. S2:

Workflow of the image processing and analysis. a) Black-white image of a phase separated lattice, and b) the power spectrum of the Fourier transformed image in (a), displayed with logarithmic scaling. c) Magnified cutout of the center of the Fourier transformed image (b) illustrating the radial integration for calculating the structure factor $S(q)$. Intensities of pixels between concentric circles of $q$ and $q+dq$ (red) are integrated and normalized by the number of pixels in this ring. This procedure is performed for the full range of $q$-values to provide the structure factor $S(q)$ as a function of distance $q$ from the center of the Fourier transformed image. Note that the radial axis is displayed with a logarithmic scale, and the image center corresponds to $q=0$.

FIG. S3:

Spatial coordination games with a low degree of asymmetry in the payoff matrix. (a) Time lapse images of a two player coordination game with payoff matrix $A=[10,-5;-5.5,10]$. The images show the spatial configuration (lattice size $L=256$) at $t=0$, 2, 4, 6, 8, 10 with dt = 0.05. White corresponds to player 1 and black is player 2. Starting from a well mixed initial condition with an initial fraction of 0.49 for player 1 (unstable equilibrium of corresponding replicator equation), the two players separate into domains which grow over time. (b) Structural analysis of simulated spatial coordination games that correspond to asymmetric potentials with a small deviation ($\eta =0.5$). For both panels (left and middle), there are 6 different colors, each corresponding to one type of interaction class in Fig. 2. We choose one example payoff matrix from each class, and every data point is averaged over 50 realizations. The payoff matrices are [10, 2; 4.5, 7] (red circle), [10, −3; 4.5, 2] (green square), [10, −7; 4.5, −2] (blue triangle), [2, −1; −3.5, 4] (cyan triangle), [2, −7; −3.5, −2] (magenta triangle) and [−5, −7; −10.5, −2] (yellow triangle). The relationship between $q_m$ and $t$ is summarized in the left panel; all games closely follow a universal $t^{-1∕2}$ trend. Similarly, $S(q_m)$ curves collapse when $S(q_m)$ is plotted versus $q_m$, all the games undergo the same coarsening process, see middle panel. The plot of the potential function on the right panel shows the low degree of asymmetry.

FIG. S4:

Spatial coordination games with a high degree of asymmetry in the payoff matrix. (a) Time lapse images of a two player coordination game with payoff matrix $A=[10,-5;-8,10]$. The images show the spatial configuration (lattice size $L=256$) at $t=0$, 2, 4, 6, 8, 10 with dt = 0.05. White corresponds to player 1 and black is player 2. Starting from a well mixed initial condition with an initial fraction of 0.45 for player 1 (unstable equilibrium of corresponding replicator equation), the two players separate into domains which grow over time. (b) Structural analysis of simulated spatial coordination games that correspond to asymmetric potentials with a large deviation ($\eta =3$). For both panels (left and middle), there are 6 different colors, each corresponding to one type of interaction class in Fig. 2. We choose one example payoff matrix from each class of interaction, and every data point is averaged over 50 realizations. The payoff matrices are [10, 2; 2, 7] (red circle), [10, −3; 2, 2] (green square), [10, −7; 2, −2] (blue triangle), [2, −1; −6, 4] (cyan triangle), [2, −7; −6, −2] (magenta triangle) and [−5, −7; −13, −2] (yellow triangle). The relationship between $q_m$ and $t$ is summarized in the left panel; the games deviate from a universal $t^{-1∕2}$ trend. Similarly, $S(q_m)$ curves do not collapse when $S(q_m)$ is plotted versus $q_m$, see middle panel. The plot of the potential function on the right panel shows the high degree of asymmetry.

FIG. 1:

Stochastic Games. a) To characterize a two-species system, we define intra-specific interactions when a focal player with strategy 1 (blue species) or strategy 2 (orange species) play against their kin, quantified by payoffs $a_{11}$ and $a_{22}$, respectively, and inter-specific interactions when a player with strategy 1 plays against a player with strategy 2 (payoff $a_{12}$), and vice versa (payoff $a_{21}$). b) The interaction potential $V(n)$ is shown as a function of the frequency $n$ of player 1 for three representative coordination games: equally competitive players (competitiveness $C=(a_{11}-a_{21})∕(a_{22}-a_{12})=1$), and for games where player 1 is dominant ($C>1$) or inferior ($C<1$). c) Spatial coordination game on a two-dimensional lattice with focal player $f$ with strategy 1 and next neighbor interactions defined by payoff values $a_{ij}$.

FIG. 2:

Partitioning games into inter- and intra-specific interactions. Each box corresponds to one class of interaction scenarios, and in analogy to Fig. 1a, the sign (+/−) correspond to payoff values $a_{ij}$, characterizing positive (+) and negative (−) interactions. For example, the mutual killing system (cyan box) has positive intra-specific interactions, and negative inter-specific interactions. Coordination games exist within the classes that are outlined with a colored box, but not in classes displayed in gray only.

FIG. 3:

Time lapse visualizations of a two player coordination game with payoff matrix $A=[10,-5;-5,10]$. The images show the spatial configuration (lattice size $L=256$) at $t=0$, 2, 4, 6, 8, 10 with dt = 0.05. White corresponds to player 1 and black is player 2. Starting from well mixed initial condition, initial fraction of player 1 is 0.5, the two players separate into domains whose characteristic length scales (patch size) grow over time.

FIG. 4:

Structural analysis of 6 simulated, symmetric spatial coordination games. In both panels, each curve corresponds to one interaction class in Fig. 2. We chose one example payoff matrix from each class of ecological interactions, every data point is averaged over 50 realizations. The payoff matrices are [10, 2; 5, 7] (red circle), [10, −3; 5, 2] (green square), [2, −1; −3, 4] (cyan triangle), [10, −7; 5, −2] (blue triangle), [2, −7; −3, −2] (magenta triangle) and [−5, −7; −10, −2] (yellow triangle). The relationship between $q_m$ and $t$ follow a universal $t^{-1∕2}$ trend. Similarly, $S(q_m)$ curves collapse when $S(q_m)$ is plotted versus $q_m$ (right), hence, all the games are undergoing the same coarsening transition.