Evolutionary shift dynamics on a cycle

Population genetics in microchannels

Many microbial populations proliferate in small channels. In such environments, reproducing cells organize in parallel lanes. Reproducing cells shift these lanes, potentially expelling other cells from the channel. In this paper, we combine theory and experiments to understand how these dynamics affects the diversity of a microbial population. We theoretically predict that genetic diversity is quickly lost along lanes of cells. Our experiments confirm that a population of proliferating Escherichia coli in a microchannel organizes into lanes of genetically identical cells within a few generations. Our findings elucidate the effect of lane formation on populations evolution, with potential applications ranging from microbial ecology in soil to dynamics of epithelial tissues in higher organisms.

Spatial constraints, such as rigid barriers, affect the dynamics of cell populations, potentially altering the course of natural evolution. In this paper, we investigate the population genetics of Escherichia coli proliferating in microchannels with open ends. Our analysis is based on a population model, in which reproducing cells shift entire lanes of cells toward the open ends of the channel. The model predicts that diversity is lost very rapidly within lanes but at a much slower pace among lanes. As a consequence, two mixed, neutral E. coli strains competing in a microchannel must organize into an ordered regular stripe pattern in the course of a few generations. These predictions are in quantitative agreement with our experiments. We also demonstrate that random mutations appearing in the middle of the channel are much more likely to reach fixation than those occurring elsewhere. Our results illustrate fundamental mechanisms of microbial evolution in spatially confined space.

Cooperative success in epithelial public goods games

Cancer cells obtain mutations which rely on the production of diffusible growth factors to confer a fitness benefit. These mutations can be considered cooperative, and studied as public goods games within the framework of evolutionary game theory. The population structure, benefit function and update rule all influence the evolutionary success of cooperators. We model the evolution of cooperation in epithelial cells using the Voronoi tessellation model. Unlike traditional evolutionary graph theory, this allows us to implement global updating, for which birth and death events are spatially decoupled. We compare, for a sigmoid benefit function, the conditions for cooperation to be favoured and/or beneficial for well-mixed and structured populations. We find that when population structure is combined with global updating, cooperation is more successful than if there were local updating or the population were well-mixed. Interestingly, the qualitative behaviour for the well-mixed population and the Voronoi tessellation model is remarkably similar, but the latter case requires significantly lower incentives to ensure cooperation.

Evolution of cooperation in an epithelium

Cooperation is prevalent in nature, not only in the context of social interactions within the animal kingdom but also on the cellular level. In cancer, for example, tumour cells can cooperate by producing growth factors. The evolution of cooperation has traditionally been studied for well-mixed populations under the framework of evolutionary game theory, and more recently for structured populations using evolutionary graph theory (EGT). The population structures arising due to cellular arrangement in tissues, however, are dynamic and thus cannot be accurately represented by either of these frameworks. In this work, we compare the conditions for cooperative success in an epithelium modelled using EGT, to those in a mechanical model of an epithelium—the Voronoi tessellation (VT) model. Crucially, in this latter model, cells are able to move, and birth and death are not spatially coupled. We calculate fixation probabilities in the VT model through simulation and an approximate analytic technique and show that this leads to stronger promotion of cooperation in comparison with the EGT model.

Computation and Simulation of Evolutionary Game Dynamics in Finite Populations

The study of evolutionary dynamics increasingly relies on computational methods, as more and more cases outside the range of analytical tractability are explored. The computational methods for simulation and numerical approximation of the relevant quantities are diverging without being compared for accuracy and performance. We thoroughly investigate these algorithms in order to propose a reliable standard. For expositional clarity we focus on symmetric 2 × 2 games leading to one-dimensional processes, noting that extensions can be straightforward and lessons will often carry over to more complex cases. We provide time-complexity analysis and systematically compare three families of methods to compute fixation probabilities, fixation times and long-term stationary distributions for the popular Moran process. We provide efficient implementations that substantially improve wall times over naive or immediate implementations. Implications are also discussed for the Wright-Fisher process, as well as structured populations and multiple types.

Effects of motion in structured populations

In evolutionary processes, population structure has a substantial effect on natural selection. Here, we analyse how motion of individuals affects constant selection in structured populations. Motion is relevant because it leads to changes in the distribution of types as mutations march towards fixation or extinction. We describe motion as the swapping of individuals on graphs, and more generally as the shuffling of individuals between reproductive updates. Beginning with a one-dimensional graph, the cycle, we prove that motion suppresses natural selection for death-birth (DB) updating or for any process that combines birth-death (BD) and DB updating. If the rule is purely BD updating, no change in fixation probability appears in the presence of motion. We further investigate how motion affects evolution on the square lattice and weighted graphs. In the case of weighted graphs, we find that motion can be either an amplifier or a suppressor of natural selection. In some cases, whether it is one or the other can be a function of the relative reproductive rate, indicating that motion is a subtle and complex attribute of evolving populations. As a first step towards understanding less restricted types of motion in evolutionary graph theory, we consider a similar rule on dynamic graphs induced by a spatial flow and find qualitatively similar results, indicating that continuous motion also suppresses natural selection.

Hamilton's rule

This paper reviews and addresses a variety of issues relating to inclusive fitness. The main question is: are there limits to the generality of inclusive fitness, and if so, what are the perimeters of the domain within which inclusive fitness works? This question is addressed using two well-known tools from evolutionary theory: the replicator dynamics, and adaptive dynamics. Both are combined with population structure. How generally Hamilton's rule applies depends on how costs and benefits are defined. We therefore consider costs and benefits following from Karlin and Matessi's (1983) "counterfactual method", and costs and benefits as defined by the "regression method" (Gardner et al., 2011). With the latter definition of costs and benefits, Hamilton's rule always indicates the direction of selection correctly, and with the former it does not. How these two definitions can meaningfully be interpreted is also discussed. We also consider cases where the qualitative claim that relatedness fosters cooperation holds, even if Hamilton's rule as a quantitative prediction does not. We furthermore find out what the relation is between Hamilton's rule and Fisher's Fundamental Theorem of Natural Selection. We also consider cancellation effects - which is the most important deepening of our understanding of when altruism is selected for. Finally we also explore the remarkable (im)possibilities for empirical testing with either definition of costs and benefits in Hamilton's rule.

Cellular cooperation with shift updating and repulsion

Population structure can facilitate evolution of cooperation. In a structured population, cooperators can form clusters which resist exploitation by defectors. Recently, it was observed that a shift update rule is an extremely strong amplifier of cooperation in a one dimensional spatial model. For the shift update rule, an individual is chosen for reproduction proportional to fecundity; the offspring is placed next to the parent; a random individual dies. Subsequently, the population is rearranged (shifted) until all individual cells are again evenly spaced out. For large population size and a one dimensional population structure, the shift update rule favors cooperation for any benefit-to-cost ratio greater than one. But every attempt to generalize shift updating to higher dimensions while maintaining its strong effect has failed. The reason is that in two dimensions the clusters are fragmented by the movements caused by rearranging the cells. Here we introduce the natural phenomenon of a repulsive force between cells of different types. After a birth and death event, the cells are being rearranged minimizing the overall energy expenditure. If the repulsive force is sufficiently high, shift becomes a strong promoter of cooperation in two dimensions.

Evolution of staying together in the context of diffusible public goods

We study the coevolution of staying together and cooperation. Staying together means that replicating units do not separate after reproduction, but remain in proximity. For example, following cell division the two daughter cells may not fully separate but stay attached to each other. Repeated cell division thereby can lead to a simple multi-cellular complex. We assume that cooperators generate a diffusible public good, which can be absorbed by any cell in the system. The production of the public good entails a cost, while the absorption leads to a benefit. Defectors produce no public good. Defectors have a selective advantage unless a mechanism for evolution of cooperation is at work. Here we explore the idea that the public good produced by a cooperating cell is absorbed by cells of the same complex with a probability that depends on the size of the complex. Larger complexes are better at absorbing the public goods produced by their own individuals. We derive analytical conditions for the evolution of staying together, thereby studying the coevolution of clustering and cooperation. If cooperators and defectors differ in their intrinsic efficiency to absorb the public good, then we find multiple stable equilibria and the possibility for coexistence between cooperators and defectors. Finally we study the implications of disadvantages that might arise if complexes become too large.

Adaptive dynamics with interaction structure

Evolutionary dynamics depend critically on a population's interaction structure-the pattern of which individuals interact with which others, depending on the state of the population and the environment. Previous research has shown, for example, that cooperative behaviors disfavored in well-mixed populations can be favored when interactions occur only between spatial neighbors or group members. Combining the adaptive dynamics approach with recent advances in evolutionary game theory, we here introduce a general mathematical framework for analyzing the long-term evolution of continuous game strategies for a broad class of evolutionary models, encompassing many varieties of interaction structure. Our main result, the canonical equation of adaptive dynamics with interaction structure, characterizes expected evolutionary trajectories resulting from any such model, thereby generalizing a central tool of adaptive dynamics theory. Interestingly, the effects of different interaction structures and update rules on evolutionary trajectories are fully captured by just two real numbers associated with each model, which are independent of the considered game. The first, a structure coefficient, quantifies the effects on selection pressures and thus on the shapes of expected evolutionary trajectories. The second, an effective population size, quantifies the effects on selection responses and thus on the expected rates of adaptation. Applying our results to two social dilemmas, we show how the range of evolutionarily stable cooperative behaviors systematically varies with a model's structure coefficient.

Global migration can lead to stronger spatial selection than local migration

The outcome of evolutionary processes depends on population structure. It is well known that mobility plays an important role in affecting evolutionary dynamics in group structured populations. But it is largely unknown whether global or local migration leads to stronger spatial selection and would therefore favor to a larger extent the evolution of cooperation. To address this issue, we quantify the impacts of these two migration patterns on the evolutionary competition of two strategies in a finite island model. Global migration means that individuals can migrate from any one island to any other island. Local migration means that individuals can only migrate between islands that are nearest neighbors; we study a simple geometry where islands are arranged on a one-dimensional, regular cycle. We derive general results for weak selection and large population size. Our key parameters are: the number of islands, the migration rate and the mutation rate. Surprisingly, our comparative analysis reveals that global migration can lead to stronger spatial selection than local migration for a wide range of parameter conditions. Our work provides useful insights into understanding how different mobility patterns affect evolutionary processes.

Evolution of emotions on networks leads to the evolution of cooperation in social dilemmas

We show that the resolution of social dilemmas in random graphs and scale-free networks is facilitated by imitating not the strategy of better-performing players but, rather, their emotions. We assume sympathy and envy to be the two emotions that determine the strategy of each player in any given interaction, and we define them as the probabilities of cooperating with players having a lower and a higher payoff, respectively. Starting with a population where all possible combinations of the two emotions are available, the evolutionary process leads to a spontaneous fixation to a single emotional profile that is eventually adopted by all players. However, this emotional profile depends not only on the payoffs but also on the heterogeneity of the interaction network. Homogeneous networks, such as lattices and regular random graphs, lead to fixations that are characterized by high sympathy and high envy, while heterogeneous networks lead to low or modest sympathy but also low envy. Our results thus suggest that public emotions and the propensity to cooperate at large depend, and are in fact determined by, the properties of the interaction network.