Session Q42: Focus Session: Stochastic Population Dynamics II - Games and Spatial Dynamics

Bacterial Games

Microbial laboratory communities have become model systems for studying the complex interplay between evolutionary selection forces, stochastic fluctuations, and spatial organization. Two fundamental questions that challenge our understanding of evolution and ecology are the origin of cooperation and biodiversity. Both are ubiquitous phenomena yet conspicuously difficult to explain since the fitness of an individual or the whole community depends in an intricate way on a plethora of factors, such as spatial distribution and mobility of individuals, secretion and detection of signaling molecules, toxin secretion leading to inter-strain competition and changes in environmental conditions. We discuss two possible solutions to these questions employing concepts from evolutionary game theory, nonlinear dynamics, and the theory of stochastic processes. Our work provides insights into some minimal requirements for the evolution of cooperation and biodiversity in simple microbial communities. It further makes predictions to be tested by new microbial experiments.   

Evolutionary dynamics of range expansions with curved fronts and inflationary directed percolation

We compare the evolutionary dynamics of populations expanding into a new territory with flat and curved fronts. When actively reproducing individuals confined to a thin, uniform population front experience deleterious mutations, the evolutionary dynamics fall into the directed percolation (DP) universality class. At the DP phase transition, the selective advantage of the fit individuals balances the deleterious mutation rate. Curvature in the front changes the dynamics: Sectors of the population become causally disconnected after a time $t_* = R_0/v$, where $R_0$ is the initial radius of the population and $v$ is the radial front propagation speed. The reproducing population size increases, creating an inflationary effect that prevents the loss of fit individuals due to sector boundary diffusion and sector interactions. We develop a generalization of the Domany-Kinzel model on amorphous, isotropic lattices to simulate radial expansions. We find scaling functions characterizing the effects of inflation at criticality. We also discuss analytic results for two-point correlation functions and survival probabilities in the two limiting cases of no mutations (compact DP)and no selection.   

Competition and cooperation in one-dimensional stepping stone models

Mutualism and cooperation are major biological forces sustaining ecosystems and enabling complex evolutionary adaptations. Although spatial degrees of freedom and number fluctuations often significantly affect evolutionary dynamics, their effects on mutualism are not fully understood. We show that, even when mutualism confers a distinct selective advantage, it persists only in populations with high density and frequent migrations. When these parameters are reduced, number fluctuations lead to the local extinctions of one of the species, segregating the species in space and decreasing the size of regions where cooperation occurs. The segregated and mutualistic states are separated by a second order nonequilibrium phase transition. Generically, this transition is in the universality class of directed percolation (DP), but the phase diagram is strongly influenced by an exceptional symmetric directed percolation (DP2) transition. This influence is manifested in a strong increase in the resilience to number fluctuations of symmetric mutualism, when organisms benefit equally from interacting.   

Range expansion of mutualists

The expansion of a species into new territory is often strongly influenced by the presence of other species. This effect is particularly striking for the case of mutualistic species that enhance each other's proliferation. Examples range from major events in evolutionary history, such as the spread and diversification of flowering plants due to their mutualism with pollen-dispersing insects, to modern examples like the surface colonisation of multi-species microbial biofilms. Here, we investigate the spread of cross-feeding strains of the budding yeast \textit{Saccharomyces cerevisiae} on an agar surface as a model system for expanding mutualists. Depending on the degree of mutualism, the two strains form distinctive spatial patterns during their range expansion. This change in spatial patterns can be understood as a phase transition within a stepping stone model generalized to two mutualistic species.   

Cooperation, cheating, and collapse in microbial populations

Natural populations can suffer catastrophic collapse in response to small changes in environmental conditions, and recovery after such a collapse can be exceedingly difficult. We have used laboratory yeast populations to study proposed early warning signals of impending extinction. Yeast cooperatively breakdown the sugar sucrose, meaning that there is a minimum number of cells required to sustain the population. We have demonstrated experimentally that the fluctuations in the population size increase in magnitude and become slower as the population approaches collapse. The cooperative nature of yeast growth on sucrose suggests that the population may be susceptible to cheater cells, which do not contribute to the public good and instead merely take advantage of the cooperative cells. We have confirmed this possibility experimentally by using a cheater yeast strain that lacks the gene encoding the cooperative behavior [1]. However, recent results in the lab demonstrate that the presence of a bacterial competitor may drive cooperation within the yeast population.\\[4pt] [1] Gore et al, \textit{Nature} \textbf{459}, 253 -- 256 (2009)   

Evolution of cooperation in microbial biofilms - A stochastic model for the growth and survival of bacterial mats

Cooperative behavior is essential for microbial biofilms. The structure and composition of a biofilm change over time and thereby influence the evolution of cooperation within the system. In turn, the level of cooperation affects the growth dynamics of the biofilm. Here, we investigate this coupling for an experimentally well-defined situation in which mutants of the Pseudomonas fluorescens strain form a mat at the liquid-air interface by the production of an extra-cellular matrix [1]. We model the occurrence of cooperation in this bacterial population by taking into account the formation of the mat. The presence of cooperators enhances the growth of the mat, but at the same time cheaters can infiltrate the population and put the viability of the mat at risk. We find that the survival time of the mat crucially depends on its initial dynamics which is subject to demographic fluctuations [2]. More generally, our work provides conceptual insights into the requirements and mechanisms for the evolution of cooperation.\\ $[1]$ P. Rainey et al., Nature 425, 72 (2003).\\ $[2]$ A. Melbinger et al., PRL 105, 178101 (2010).   

Bacterial Cheating Limits the Evolution of Antibiotic Resistance

The emergence of antibiotic resistance in bacteria is a significant health concern. Bacteria can gain resistance to the antibiotic ampicillin by acquiring a plasmid carrying the gene beta-lactamase, which inactivates the antibiotic. This inactivation may represent a cooperative behavior, as the entire bacterial population benefits from removal of the antibiotic. The presence of a cooperative mechanism of resistance suggests that a cheater strain - which does not contribute to breaking down the antibiotic - may be able to take advantage of resistant cells. We find experimentally that a ``sensitive'' bacterial strain lacking the plasmid conferring resistance can invade a population of resistant bacteria, even in antibiotic concentrations that should kill the sensitive strain. We use a simple model in conjunction with difference equations to explain the observed population dynamics as a function of cell density and antibiotic concentration. Our experimental difference equations resemble the logistic map, raising the possibility of oscillations or even chaotic dynamics.   

Does fast migration imply well-mixing?

A popular assumption in population dynamics is that the population is well mixed, i.e., a spatial character of the interactions between the individuals can be neglected. A common justification of this assumption is that the rate of migration between the local possibly well-mixed population-dynamics domains is much larger than the rates of interactions within a domain. We consider a system of local well-mixed domains of varying carrying capacity. In the limit of infinite migration rate we calculate the stationary probability distribution of the total population and find that generally it is not equivalent to the stationary probability distribution of a single well-mixed domain with a large carrying capacity. This proves that fast migration does not generally justify the well-mixed population assumption.   

Turing patterns and a stochastic individual-based model for predator-prey systems

Reaction-diffusion theory has played a very important role in the study of pattern formations in biology. However, a group of individuals is described by a single state variable representing population density in reaction-diffusion models and interaction between individuals can be included only phenomenologically. Recently, we have seamlessly combined individual-based models with elements of reaction-diffusion theory. To include animal migration in the scheme, we have adopted a relationship between the diffusion and the random numbers generated according to a two-dimensional bivariate normal distribution. Thus, we have observed the transition of population patterns from an extinction mode, a stable mode, or an oscillatory mode to the chaotic mode as the population growth rate increases. We show our phase diagram of predator-prey systems and discuss the microscopic mechanism for the stable lattice formation in detail.