Fifty years of Theoretical Population Biology

A matrix-analytical sampling formula for time-homogeneous coalescent processes under the infinite sites mutation model

In this paper we develop a general framework for calculating the probability of a genetic sample under a time-homogeneous coalescent process and the infinite sites mutation model. The evolutionary model that we consider can be characterized as a two-step procedure: A coalescent process that describes the ancestral relatedness of the samples and a sprinkling of mutations in separate sites on the ancestral tree according to a Poisson process. The coalescent process is defined using multivariate phase-type theory. The requirements are a rate matrix that determines the transition rates between the ancestral states, an initial state probability vector, and a reward matrix that informs about the characteristics of the ancestral states. For example, the reward matrix could contain information about the number of singleton, doubleton or higher-order lineages in the ancestral states. We analyze the probability generating function for the evolutionary model as a function of the initial state probability vector, the transition rate matrix, the reward matrix, and the mutation rate. The matrix-analytical expression of the probability generating function allows us to develop a general method for calculating the probability of a population genetic data set. We demonstrate that the method is computationally attractive for a small number of mutations and provide a simple and easy-to-implement algorithm for determining the probability of a sample from the evolutionary model. The method is computationally stable and only involves a single inverse matrix operation, matrix multiplications and matrix additions. We provide comprehensive understanding of the procedure by detailed calculations and discussions of several elementary examples. These examples include different sample representations (labeled samples and the site frequency spectrum) and different demographic and genetic models (the structured coalescent and the Beta-coalescent). We apply the sampling formula to calculate probabilities of spectra for the Kingman coalescent and the Beta-coalescent. Even for a small number of samples and mutations we find that the probabilities for spectra vary in huge orders of magnitudes. We compare the probabilities of the spectra to the values of Tajima's D-statistics, and find that the D-statistic is a poor predictor for the probability of a spectrum. Finally, we investigate how the probabilities of the spectra vary with the parametrization of the Beta-coalescent.

Phase-type distributions in mathematical population genetics: An emerging framework

A phase-type distribution is the time to absorption in a continuous- or discrete-time Markov chain. Phase-type distributions can be used as a general framework to calculate key properties of the standard coalescent model and many of its extensions. Here, the 'phases' in the phase-type distribution correspond to states in the ancestral process. For example, the time to the most recent common ancestor and the total branch length are phase-type distributed. Furthermore, the site frequency spectrum follows a multivariate discrete phase-type distribution and the joint distribution of total branch lengths in the two-locus coalescent-with-recombination model is multivariate phase-type distributed. In general, phase-type distributions provide a powerful mathematical framework for coalescent theory because they are analytically tractable using matrix manipulations. The purpose of this review is to explain the phase-type theory and demonstrate how the theory can be applied to derive basic properties of coalescent models. These properties can then be used to obtain insight into the ancestral process, or they can be applied for statistical inference. In particular, we show the relation between classical first-step analysis of coalescent models and phase-type calculations. We also show how reward transformations in phase-type theory lead to easy calculation of covariances and correlation coefficients between e.g. tree height, tree length, external branch length, and internal branch length. Furthermore, we discuss how these quantities can be used for statistical inference based on estimating equations. Providing an alternative to previous work based on the Laplace transform, we derive likelihoods for small-size coalescent trees based on phase-type theory. Overall, our main aim is to demonstrate that phase-type distributions provide a convenient general set of tools to understand aspects of coalescent models that are otherwise difficult to derive. Throughout the review, we emphasize the versatility of the phase-type framework, which is also illustrated by our accompanying R-code. All our analyses and figures can be reproduced from code available on GitHub.

Allen Orr and the genetics of adaptation

Over most of the 20th century, evolutionary biologists predominantly subscribed to a strong form of "micro-mutationism," in which adaptive phenotypic divergence arises from allele frequency changes at many loci, each with a small effect on the phenotype. To be sure, there were well-known examples of large-effect alleles contributing to adaptation, yet such cases were generally regarded as atypical and unrepresentative of evolutionary change in general. In 1998, Allen Orr published a landmark theoretical paper in Evolution, which showed that both small- and large-effect mutations are likely to contribute to "adaptive walks" of a population to an optimum. Coupled with a growing set of empirical examples of large-effect alleles contributing to divergence (e.g., from QTL studies), Orr's paper provided a mathematical formalism that converted many evolutionary biologists from micro-mutationism to a more pluralistic perspective on the genetic basis of evolutionary change. We revisit the theoretical insights emerging from Orr's paper within the historical context leading up to 1998, and track the influence of this paper on the field of evolutionary biology through an examination of its citations over the last two decades and an analysis of the extensive body of theoretical and empirical research that Orr's pioneering paper inspired.

Skin deep: The decoupling of genetic admixture levels from phenotypes that differed between source populations

OBJECTIVES

In genetic admixture processes, source groups for an admixed population possess distinct patterns of genotype and phenotype at the onset of admixture. Particularly in the context of recent and ongoing admixture, such differences are sometimes taken to serve as markers of ancestry for individuals-that is, phenotypes initially associated with the ancestral background in one source population are assumed to continue to reflect ancestry in that population. Such phenotypes might possess ongoing significance in social categorizations of individuals, owing in part to perceived continuing correlations with ancestry. However, genotypes or phenotypes initially associated with ancestry in one specific source population have been seen to decouple from overall admixture levels, so that they no longer serve as proxies for genetic ancestry. Here, we aim to develop an understanding of the joint dynamics of admixture levels and phenotype distributions in an admixed population.

METHODS

We devise a mechanistic model, consisting of an admixture model, a quantitative trait model, and a mating model. We analyze the behavior of the mechanistic model in relation to the model parameters.

RESULTS

We find that it is possible for the decoupling of genetic ancestry and phenotype to proceed quickly, and that it occurs faster if the phenotype is driven by fewer loci. Positive assortative mating attenuates the process of dissociation relative to a scenario in which mating is random with respect to genetic admixture and with respect to phenotype.

CONCLUSIONS

The mechanistic framework suggests that in an admixed population, a trait that initially differed between source populations might serve as a reliable proxy for ancestry for only a short time, especially if the trait is determined by few loci. It follows that a social categorization based on such a trait is increasingly uninformative about genetic ancestry and about other traits that differed between source populations at the onset of admixture.

Learning enables adaptation in cooperation for multi-player stochastic games

Interactions among individuals in natural populations often occur in a dynamically changing environment. Understanding the role of environmental variation in population dynamics has long been a central topic in theoretical ecology and population biology. However, the key question of how individuals, in the middle of challenging social dilemmas (e.g. the 'tragedy of the commons'), modulate their behaviours to adapt to the fluctuation of the environment has not yet been addressed satisfactorily. Using evolutionary game theory, we develop a framework of stochastic games that incorporates the adaptive mechanism of reinforcement learning to investigate whether cooperative behaviours can evolve in the ever-changing group interaction environment. When the action choices of players are just slightly influenced by past reinforcements, we construct an analytical condition to determine whether cooperation can be favoured over defection. Intuitively, this condition reveals why and how the environment can mediate cooperative dilemmas. Under our model architecture, we also compare this learning mechanism with two non-learning decision rules, and we find that learning significantly improves the propensity for cooperation in weak social dilemmas, and, in sharp contrast, hinders cooperation in strong social dilemmas. Our results suggest that in complex social-ecological dilemmas, learning enables the adaptation of individuals to varying environments.