Continuous model for the rock-scissors-paper game between bacteriocin producing bacteria

Modelling population dynamics in a unicellular social organism community using a minimal model and evolutionary game theory

Most unicellular organisms live in communities and express different phenotypes. Many efforts have been made to study the population dynamics of such complex communities of cells, coexisting as well-coordinated units. Minimal models based on ordinary differential equations are powerful tools that can help us understand complex phenomena. They represent an appropriate compromise between complexity and tractability; they allow a profound and comprehensive analysis, which is still easy to understand. Evolutionary game theory is another powerful tool that can help us understand the costs and benefits of the decision a particular cell of a unicellular social organism takes when faced with the challenges of the biotic and abiotic environment. This work is a binocular view at the population dynamics of such a community through the objectives of minimal modelling and evolutionary game theory. We test the behaviour of the community of a unicellular social organism at three levels of antibiotic stress. Even in the absence of the antibiotic, spikes in the fraction of resistant cells can be observed indicating the importance of bet hedging. At moderate level of antibiotic stress, we witness cyclic dynamics reminiscent of the renowned rock–paper–scissors game. At a very high level, the resistant type of strategy is the most favourable.

Trends in mathematical modeling of host-pathogen interactions

Pathogenic microorganisms entail enormous problems for humans, livestock, and crop plants. A better understanding of the different infection strategies of the pathogens enables us to derive optimal treatments to mitigate infectious diseases or develop vaccinations preventing the occurrence of infections altogether. In this review, we highlight the current trends in mathematical modeling approaches and related methods used for understanding host-pathogen interactions. Since these interactions can be described on vastly different temporal and spatial scales as well as abstraction levels, a variety of computational and mathematical approaches are presented. Particular emphasis is placed on dynamic optimization, game theory, and spatial modeling, as they are attracting more and more interest in systems biology. Furthermore, these approaches are often combined to illuminate the complexities of the interactions between pathogens and their host. We also discuss the phenomena of molecular mimicry and crypsis as well as the interplay between defense and counter defense. As a conclusion, we provide an overview of method characteristics to assist non-experts in their decision for modeling approaches and interdisciplinary understanding.

Toxin-mediated competition in weakly motile bacteria

Many bacterial species produce toxins that inhibit their competitors. We model this phenomenon by extending classic two-species Lotka-Volterra competition in one spatial dimension to incorporate toxin production by one species. Considering solutions comprising two adjacent single-species colonies, we show how the toxin inhibits the susceptible species near the interface between the two colonies. Moreover, a sufficiently effective toxin inhibits the susceptible species to such a degree that an 'inhibition zone' is formed separating the two colonies. In the special case of truly non-motile bacteria, i.e. with zero bacterial diffusivity, we derive analytical expressions describing the bacterial distributions and size of the inhibition zone. In the more general case of weakly motile bacteria, i.e. small bacterial diffusivity, these two-colony solutions become travelling waves. We employ numerical methods to show that the wavespeed is dependent upon both interspecific competition and toxin strength; precisely which colony expands at the expense of the other depends upon the choice of parameter values. In particular, a sufficiently effective toxin allows the producer to expand at the expense of the susceptible, with a wavespeed magnitude that is bounded above as the toxin strength increases. This asymptotic wavespeed is independent of interspecific competition and due to the formation of the inhibition zone; when the colonies are thus separated, there is no longer direct competition between the two species and the producer can invade effectively unimpeded by its competitor. We note that the minimum toxin strength required to produce an inhibition zone increases rapidly with increasing bacterial diffusivity, suggesting that even moderately motile bacteria must produce very strong toxins if they are to benefit in this way.

Polymorphism in rapidly changing cyclic environment

Selection in a time-periodic environment is modeled via the continuous-time two-player replicator dynamics, which for symmetric payoffs reduces to the Fisher equation of mathematical genetics. For a sufficiently rapid and cyclic (fine-grained) environment, the time-averaged population frequencies are shown to obey a replicator dynamics with a nonlinear fitness that is induced by environmental changes. The nonlinear terms in the fitness emerge due to populations tracking their time-dependent environment. These terms can induce a stable polymorphism, though they do not spoil the polymorphism that exists already without them. In this sense polymorphic populations are more robust with respect to their time-dependent environments. The overall fitness of the problem is still given by its time-averaged value, but the emergence of polymorphism during genetic selection can be accompanied by decreasing mean fitness of the population. The impact of the uncovered polymorphism scenario on the models of diversity is exemplified via the rock-paper-scissors dynamics, and also via the prisoner's dilemma in a time-periodic environment.

Pervasive Selection for Cooperative Cross-Feeding in Bacterial Communities

Bacterial communities are taxonomically highly diverse, yet the mechanisms that maintain this diversity remain poorly understood. We hypothesized that an obligate and mutual exchange of metabolites, as is very common among bacterial cells, could stabilize different genotypes within microbial communities. To test this, we developed a cellular automaton to model interactions among six empirically characterized genotypes that differ in their ability and propensity to produce amino acids. By systematically varying intrinsic (i.e. benefit-to-cost ratio) and extrinsic parameters (i.e. metabolite diffusion level, environmental amino acid availability), we show that obligate cross-feeding of essential metabolites is selected for under a broad range of conditions. In spatially structured environments, positive assortment among cross-feeders resulted in the formation of cooperative clusters, which limited exploitation by non-producing auxotrophs, yet allowed them to persist at the clusters' periphery. Strikingly, cross-feeding helped to maintain genotypic diversity within populations, while amino acid supplementation to the environment decoupled obligate interactions and favored auxotrophic cells that saved amino acid production costs over metabolically autonomous prototrophs. Together, our results suggest that spatially structured environments and limited nutrient availabilities should facilitate the evolution of metabolic interactions, which can help to maintain genotypic diversity within natural microbial populations.

Evolutionary game theory: cells as players

In two papers we review game theory applications in biology below the level of cognitive living beings. It can be seen that evolution and natural selection replace the rationality of the actors appropriately. Even in these micro worlds, competing situations and cooperative relationships can be found and modeled by evolutionary game theory. Also those units of the lowest levels of life show different strategies for different environmental situations or different partners. We give a wide overview of evolutionary game theory applications to microscopic units. In this first review situations on the cellular level are tackled. In particular metabolic problems are discussed, such as ATP-producing pathways, secretion of public goods and cross-feeding. Further topics are cyclic competition among more than two partners, intra- and inter-cellular signalling, the struggle between pathogens and the immune system, and the interactions of cancer cells. Moreover, we introduce the theoretical basics to encourage scientists to investigate problems in cell biology and molecular biology by evolutionary game theory.

Cyclic dominance in evolutionary games: a review

Rock is wrapped by paper, paper is cut by scissors and scissors are crushed by rock. This simple game is popular among children and adults to decide on trivial disputes that have no obvious winner, but cyclic dominance is also at the heart of predator-prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms and competition in microbial populations. Cyclical interactions also emerge spontaneously in evolutionary games entailing volunteering, reward, punishment, and in fact are common when the competing strategies are three or more, regardless of the particularities of the game. Here, we review recent advances on the rock-paper-scissors (RPS) and related evolutionary games, focusing, in particular, on pattern formation, the impact of mobility and the spontaneous emergence of cyclic dominance. We also review mean-field and zero-dimensional RPS models and the application of the complex Ginzburg-Landau equation, and we highlight the importance and usefulness of statistical physics for the successful study of large-scale ecological systems. Directions for future research, related, for example, to dynamical effects of coevolutionary rules and invasion reversals owing to multi-point interactions, are also outlined.

Evolution and stability of altruist strategies in microbial games

When microbes compete for limited resources, they often engage in chemical warfare using bacterial toxins. This competition can be understood in terms of evolutionary game theory (EGT). We study the predictions of EGT for the bacterial "suicide bomber" game in terms of the phase portraits of population dynamics, for parameter combinations that cover all interesting games for two-players, and seven of the 38 possible phase portraits of the three-player game. We compare these predictions to simulations of these competitions in finite well-mixed populations, but also allowing for probabilistic rather than pure strategies, as well as Darwinian adaptation over tens of thousands of generations. We find that Darwinian evolution of probabilistic strategies stabilizes games of the rock-paper-scissors type that emerge for parameters describing realistic bacterial populations, and point to ways in which the population fixed point can be selected by changing those parameters.

Le Chatelier's principle in replicator dynamics

The Le Chatelier principle states that physical equilibria are not only stable, but they also resist external perturbations via short-time negative-feedback mechanisms: a perturbation induces processes tending to diminish its results. The principle has deep roots, e.g., in thermodynamics it is closely related to the second law and the positivity of the entropy production. Here we study the applicability of the Le Chatelier principle to evolutionary game theory, i.e., to perturbations of a Nash equilibrium within the replicator dynamics. We show that the principle can be reformulated as a majorization relation. This defines a stability notion that generalizes the concept of evolutionary stability. We determine criteria for a Nash equilibrium to satisfy the Le Chatelier principle and relate them to mutualistic interactions (game-theoretical anticoordination) showing in which sense mutualistic replicators can be more stable than (say) competing ones. There are globally stable Nash equilibria, where the Le Chatelier principle is violated even locally: in contrast to the thermodynamic equilibrium a Nash equilibrium can amplify small perturbations, though both types of equilibria satisfy the detailed balance condition.

Competitive interactions in Escherichia coli populations: the role of bacteriocins

Explaining the coexistence of competing species is a major challenge in community ecology. In bacterial systems, competition is often driven by the production of bacteriocins, which are narrow-spectrum proteinaceous toxins that serve to kill closely related species, providing the producer better access to limited resources. Bacteriocin producers have been shown to competitively exclude sensitive, nonproducing strains. However, the dynamics between bacteriocin producers, each lethal to its competitor, are largely unknown. In this study, we used in vitro, in vivo and in silico models to study competitive interactions between bacteriocin producers. Two Escherichia coli strains were generated, each carrying a DNA-degrading bacteriocin (colicins E2 and E7). Using reporter-gene assays, we showed that each DNase bacteriocin is not only lethal to its opponent but, at lower doses, can also induce the expression of its opponent's toxin. In a well-mixed habitat, the E2 producer outcompeted its adversary; however, in structured environments (on plates or in mice colons), the two producers coexisted in a spatially 'frozen' pattern. Coexistence occurred when the producers were initiated with a clumped spatial distribution. This suggests that a 'clump' of each producer can block invasion of the other producer. Agent-based simulation of bacteriocin-mediated competition further showed that mutual exclusion in a structured environment is a relatively robust result. These models imply that colicin-mediated colicin induction enables producers to successfully compete and defend their niche against invaders. This suggests that localized interactions between producers of DNA-degrading toxins can lead to stable coexistence of heterogeneously distributed strains within the bacterial community and to the maintenance of diversity.

The dual role of bacteriocins as anti- and probiotics

Bacteria employed in probiotic applications help to maintain or restore a host's natural microbial floral. The ability of probiotic bacteria to successfully outcompete undesired species is often due to, or enhanced by, the production of potent antimicrobial toxins. The most commonly encountered of these are bacteriocins, a large and functionally diverse family of antimicrobials found in all major lineages of Bacteria. Recent studies reveal that these proteinaceous toxins play a critical role in mediating competitive dynamics between bacterial strains and closely related species. The potential use of bacteriocin-producing strains as probiotic and bioprotective agents has recently received increased attention. This review will report on recent efforts involving the use of such strains, with a particular focus on emerging probiotic therapies for humans, livestock, and aquaculture.

Use of game-theoretical methods in biochemistry and biophysics

Evolutionary game theory can be considered as an extension of the theory of evolutionary optimisation in that two or more organisms (or more generally, units of replication) tend to optimise their properties in an interdependent way. Thus, the outcome of the strategy adopted by one species (e.g., as a result of mutation and selection) depends on the strategy adopted by the other species. In this review, the use of evolutionary game theory for analysing biochemical and biophysical systems is discussed. The presentation is illustrated by a number of instructive examples such as the competition between microorganisms using different metabolic pathways for adenosine triphosphate production, the secretion of extracellular enzymes, the growth of trees and photosynthesis. These examples show that, due to conflicts of interest, the global optimum (in the sense of being the best solution for the whole system) is not always obtained. For example, some yeast species use metabolic pathways that waste nutrients, and in a dense tree canopy, trees grow taller than would be optimal for biomass productivity. From the viewpoint of game theory, the examples considered can be described by the Prisoner's Dilemma, snowdrift game, Tragedy of the Commons and rock-scissors-paper game.