The equilibrium allele frequency distribution for a population with reproductive skew
Ricky Der, Joshua B. Plotkin
(Submitted on 20 Jun 2013)

We study the population genetics of two neutral alleles under reversible mutation in the \Lambda-processes, a population model that features a skewed offspring distribution. We describe the shape of the equilibrium allele frequency distribution as a function of the model parameters. We show that the mutation rates can be uniquely identified from the equilibrium distribution, but that the form of the offspring distribution itself cannot be uniquely identified. We also introduce an infinite-sites version of the \Lambda-process, and we use it to study how reproductive skew influences standing genetic diversity in a population. We derive asymptotic formulae for the expected number of segregating sizes as a function of sample size. We find that the Wright-Fisher model minimizes the equilibrium genetic diversity, for a given mutation rate and variance effective population size, compared to all other \Lambda-processes.

Analysis and rejection sampling of Wright-Fisher diffusion bridges
Joshua G. Schraiber, Robert C. Griffiths, Steven N. Evans
(Submitted on 14 Jun 2013)

We investigate the properties of a Wright-Fisher diffusion process started from frequency x at time 0 and conditioned to be at frequency y at time T. Such a process is called a bridge. Bridges arise naturally in the analysis of selection acting on standing variation and in the inference of selection from allele frequency time series. We establish a number of results about the distribution of neutral Wright-Fisher bridges and develop a novel rejection sampling scheme for bridges under selection that we use to study their behavior.

The Moran model with selection: Fixation probabilities, ancestral lines, and an alternative particle representation
Sandra Kluth, Ellen Baake
(Submitted on 12 Jun 2013)

We reconsider the Moran model in continuous time with population size $N$, two types, and selection. We introduce a new particle representation, which we call labelled Moran model, and which has the same empirical type distribution as the original Moran model, provided the initial values are chosen appropriately. In the new model, individuals are labelled $1,2, \dots, N$; neutral resampling events may take place between arbitrary labels, whereas selective events only occur in the direction of increasing labels. With the help of elementary methods only, we do not only recover fixation probabilities, but obtain detailed insight into the number and nature of the selective events that play a role in the fixation process forward in time.

Incentive Processes in Finite Populations
Marc Harper, Dashiell Fryer
(Submitted on 11 Jun 2013)

We define the incentive process, a natural generalization of the Moran process incorporating evolutionary updating mechanisms corresponding to well-known evolutionary dynamics, such as the logit, projection, and best-reply dynamics. Fixation probabilities and internal stable states are given for a variety of incentives, including new closed-forms, as well as results relating fixation probabilities for members of two one-parameter families of incentive processes. We show that the behaviors of the incentive process can deviate significantly from the analogous properties of deterministic evolutionary dynamics in some ways but are similar in others. For example, while the fixation probabilities change, their ratio remains constant.

Evolutionary accessibility of modular fitness landscapes
Benjamin Schmiegelt, Joachim Krug
(Submitted on 8 Jun 2013)

A fitness landscape is a mapping from the space of genetic sequences, which is modeled here as a binary hypercube of dimension $L$, to the real numbers. We consider random models of fitness landscapes, where fitness values are assigned according to some probabilistic rule, and study the statistical properties of pathways to the global fitness maximum along which fitness increases monotonically. Such paths are important for evolution because they are the only ones that are accessible to an adapting population when mutations occur at a low rate. The focus of this work is on the block model introduced by A.S. Perelson and C.A. Macken [Proc. Natl. Acad. Sci. USA 92:9657 (1995)] where the genome is decomposed into disjoint sets of loci (`modules’) that contribute independently to fitness, and fitness values within blocks are assigned at random. We show that the number of accessible paths can be written as a product of the path numbers within the blocks, which provides a detailed analytic description of the path statistics. The block model can be viewed as a special case of Kauffman’s NK-model, and we compare the analytic results to simulations of the NK-model with different genetic architectures. We find that the mean number of accessible paths in the different versions of the model are quite similar, but the distribution of the path number is qualitatively different in the block model due to its multiplicative structure. A similar statement applies to the number of local fitness maxima in the NK-models, which has been studied extensively in previous works. The overall evolutionary accessibility of the landscape, as quantified by the probability to find at least one accessible path to the global maximum, is dramatically lowered by the modular structure.

Density behavior of spatial birth-and-death stochastic evolution of mutating genotypes under selection rates
Dmitri Finkelshtein, Yuri Kondratiev, Oleksandr Kutoviy, Stanislav Molchanov, Elena Zhizhina
(Submitted on 5 Jun 2013)

We consider birth-and-death stochastic evolution of genotypes with different lengths. The genotypes might mutate that provides a stochastic changing of lengthes by a free diffusion law. The birth and death rates are length dependent which corresponds to a selection effect. We study an asymptotic behavior of a density for an infinite collection of genotypes. The cases of space homogeneous and space heterogeneous densities are considered.

Coalescence, genetic diversity and adaptation in sexual populations
Richard A. Neher, Taylor A. Kessinger, Boris I. Shraiman
(Submitted on 5 Jun 2013)

In diverse sexual populations, selection operates neither on the whole genome — which is repeatedly taken apart and reassembled by recombination — nor on individual alleles which are tightly linked to the chromosomal neighborhood. Those tightly linked alleles affect each others dynamics which reduces the efficiency of selection and distorts patterns of genetic diversity. Inference of evolutionary history from diversity shaped by linked selection requires an understanding of these patterns. Here, we reexamine this problem in the light of recent progress in coalescent theory of rapidly adapting asexual populations. We present a simple but powerful scaling analysis identifying the unit of selection as the genomic “linkage block” with characteristic length \xi_b, which is determined in a self-consistent manner by the condition that the rate of recombination within the block is comparable to the fitness differences between different alleles of the block. We find that an asexual model with strength of selection tuned to that of the linkage block provides an excellent description of genetic diversity and the site frequency spectra when compared to computer simulations of population dynamics. This correspondence holds for the entire spectrum of strength of selection. When fitness differentials arise from the collective contribution of numerous weakly selected polymorphisms, the rate of adaptation increases as the square root of the recombination rate. Linkage block approximation thus provides a simple but powerful tool for understanding interference and collective behavior of dense weakly selected loci.

Interference limits resolution of selection pressures from linked neutral diversity
Benjamin H. Good, Aleksandra M. Walczak, Richard A. Neher, Michael M. Desai
(Submitted on 5 Jun 2013)

Pervasive natural selection can strongly influence observed patterns of genetic variation, but these effects remain poorly understood when multiple selected variants segregate in nearby regions of the genome. Classical population genetics fails to account for interference between linked mutations, which grows increasingly severe as the density of selected polymorphisms increases. Here, we describe a simple limit that emerges when interference is common, in which the fitness effects of individual mutations play a relatively minor role. Instead, molecular evolution is determined by the variance in fitness within the population, defined over an effectively asexual segment of the genome (a “linkage block”). We exploit this insensitivity in a new “coarse-grained” coalescent framework, which approximates the effects of many weakly selected mutations with a smaller number of strongly selected mutations with the same variance in fitness. This approximation generates accurate and efficient predictions for the genetic diversity that cannot be summarized by a simple reduction in effective population size. However, these results suggest a fundamental limit on our ability to resolve individual selection pressures from contemporary sequence data alone, since a wide range of parameters yield nearly identical patterns of sequence variability.