Evolutionary Multiplayer Games

Evolutionary game theory has become one of the most diverse and far reaching theories in biology. Applications of this theory range from cell dynamics to social evolution. However, many applications make it clear that inherent non-linearities of natural systems need to be taken into account. One way of introducing such non-linearities into evolutionary games is by the inclusion of multiple players. An example is of social dilemmas, where group benefits could e.g.\ increase less than linear with the number of cooperators. Such multiplayer games can be introduced in all the fields where evolutionary game theory is already well established. However, the inclusion of non-linearities can help to advance the analysis of systems which are known to be complex, e.g. in the case of non-Mendelian inheritance. We review the diachronic theory and applications of multiplayer evolutionary games and present the current state of the field. Our aim is a summary of the theoretical results from well-mixed populations in infinite as well as finite populations. We also discuss examples from three fields where the theory has been successfully applied, ecology, social sciences and population genetics. In closing, we probe certain future directions which can be explored using the complexity of multiplayer games while preserving the promise of simplicity of evolutionary games.Comment: 14 pages, 2 figures, review pape

Mutualism and evolutionary multiplayer games: revisiting the Red King

Coevolution of two species is typically thought to favour the evolution of faster evolutionary rates helping a species keep ahead in the Red Queen race, where `it takes all the running you can do to stay where you are'. In contrast, if species are in a mutualistic relationship, it was proposed that the Red King effect may act, where it can be beneficial to evolve slower than the mutualistic species. The Red King hypothesis proposes that the species which evolves slower can gain a larger share of the benefits. However, the interactions between the two species may involve multiple individuals. To analyse such a situation, we resort to evolutionary multiplayer games. Even in situations where evolving slower is beneficial in a two-player setting, faster evolution may be favoured in a multiplayer setting. The underlying features of multiplayer games can be crucial for the distribution of benefits. They also suggest a link between the evolution of the rate of evolution and group size

Chaotic provinces in the kingdom of the Red Queen

The interplay between parasites and their hosts is found in all kinds of species and plays an important role in understanding the principles of evolution and coevolution. Usually, the different genotypes of hosts and parasites oscillate in their abundances. The well-established theory of oscillatory Red Queen dynamics proposes an ongoing change in frequencies of the different types within each species. So far, it is unclear in which way Red Queen dynamics persists with more than two types of hosts and parasites. In our analysis, an arbitrary number of types within two species are examined in a deterministic framework with constant or changing population size. This general framework allows for analytical solutions for internal fixed points and their stability. For more than two species, apparently chaotic dynamics has been reported. Here we show that even for two species, once more than two types are considered per species, irregular dynamics in their frequencies can be observed in the long run. The nature of the dynamics depends strongly on the initial configuration of the system; the usual regular Red Queen oscillations are only observed in some parts of the parameter region

Detection of suspicious interactions of spiking covariates in methylation data

BACKGROUND: In methylation analyses like epigenome-wide association studies, a high amount of biomarkers is tested for an association between the measured continuous outcome and different covariates. In the case of a continuous covariate like smoking pack years (SPY), a measure of lifetime exposure to tobacco toxins, a spike at zero can occur. Hence, all non-smokers are generating a peak at zero, while the smoking patients are distributed over the other SPY values. Additionally, the spike might also occur on the right side of the covariate distribution, if a category "heavy smoker" is designed. Here, we will focus on methylation data with a spike at the left or the right of the distribution of a continuous covariate. After the methylation data is generated, analysis is usually performed by preprocessing, quality control, and determination of differentially methylated sites, often performed in pipeline fashion. Hence, the data is processed in a string of methods, which are available in one software package. The pipelines can distinguish between categorical covariates, i.e. for group comparisons or continuous covariates, i.e. for linear regression. The differential methylation analysis is often done internally by a linear regression without checking its inherent assumptions. A spike in the continuous covariate is ignored and can cause biased results. RESULTS: We have reanalysed five data sets, four freely available from ArrayExpress, including methylation data and smoking habits reported by smoking pack years. Therefore, we generated an algorithm to check for the occurrences of suspicious interactions between the values associated with the spike position and the non-spike positions of the covariate. Our algorithm helps to decide if a suspicious interaction can be found and further investigations should be carried out. This is mostly important, because the information on the differentially methylated sites will be used for post-hoc analyses like pathway analyses. CONCLUSIONS: We help to check for the validation of the linear regression assumptions in a methylation analysis pipeline. These assumptions should also be considered for machine learning approaches. In addition, we are able to detect outliers in the continuous covariate. Therefore, more statistical robust results should be produced in methylation analysis using our algorithm as a preprocessing step

The pace of evolution across fitness valleys

How fast does a population evolve from one fitness peak to another? We study the dynamics of evolving, asexually reproducing populations in which a certain number of mutations jointly confer a fitness advantage. We consider the time until a population has evolved from one fitness peak to another one with a higher fitness. The order of mutations can either be fixed or random. If the order of mutations is fixed, then the population follows a metaphorical ridge, a single path. If the order of mutations is arbitrary, then there are many ways to evolve to the higher fitness state. We address the time required for fixation in such scenarios and study how it is affected by the order of mutations, the population size, the fitness values and the mutation rate