scholarly journals The statistical mechanics of a polygenic character under stabilizing selection, mutation and drift

2010 ◽  
Vol 8 (58) ◽  
pp. 720-739 ◽  
Author(s):  
Harold P. de Vladar ◽  
Nick H. Barton
Keyword(s):  
Population Genetics ◽  
Statistical Mechanics ◽  
Generating Function ◽  
Genetic Drift ◽  
Infinite Series ◽  
Genetic Variance ◽  
Directional Selection ◽  
Entropy Measure ◽  
Stabilizing Selection ◽  
Approximation Procedure

By exploiting an analogy between population genetics and statistical mechanics, we study the evolution of a polygenic trait under stabilizing selection, mutation and genetic drift. This requires us to track only four macroscopic variables, instead of the distribution of all the allele frequencies that influence the trait. These macroscopic variables are the expectations of: the trait mean and its square, the genetic variance, and of a measure of heterozygosity, and are derived from a generating function that is in turn derived by maximizing an entropy measure. These four macroscopics are enough to accurately describe the dynamics of the trait mean and of its genetic variance (and in principle of any other quantity). Unlike previous approaches that were based on an infinite series of moments or cumulants, which had to be truncated arbitrarily, our calculations provide a well-defined approximation procedure. We apply the framework to abrupt and gradual changes in the optimum, as well as to changes in the strength of stabilizing selection. Our approximations are surprisingly accurate, even for systems with as few as five loci. We find that when the effects of drift are included, the expected genetic variance is hardly altered by directional selection, even though it fluctuates in any particular instance. We also find hysteresis, showing that even after averaging over the microscopic variables, the macroscopic trajectories retain a memory of the underlying genetic states.

scholarly journals Quantitative genetic variability maintained by mutation-stabilizing selection balance: sampling variation and response to subsequent directional selection

1989 ◽  
Vol 54 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Peter D. Keightley ◽  
William G. Hill
Keyword(s):  
Genetic Variance ◽  
Directional Selection ◽  
Quantitative Character ◽  
Stabilizing Selection ◽  
Effective Population ◽  
Selection Responses ◽  
Quantitative Genetic ◽  
Before And After ◽  
Cage Population ◽  
Selection Of

SummaryA model of genetic variation of a quantitative character subject to the simultaneous effects of mutation, selection and drift is investigated. Predictions are obtained for the variance of the genetic variance among independent lines at equilibrium with stabilizing selection. These indicate that the coefficient of variation of the genetic variance among lines is relatively insensitive to the strength of stabilizing selection on the character. The effects on the genetic variance of a change of mode of selection from stabilizing to directional selection are investigated. This is intended to model directional selection of a character in a sample of individuals from a natural or long-established cage population. The pattern of change of variance from directional selection is strongly influenced by the strengths of selection at individual loci in relation to effective population size before and after the change of regime. Patterns of change of variance and selection responses from Monte Carlo simulation are compared to selection responses observed in experiments. These indicate that changes in variance with directional selection are not very different from those due to drift alone in the experiments, and do not necessarily give information on the presence of stabilizing selection or its strength.

Isolation by distance in a quantitative trait.

Genetics ◽  
1991 ◽  
Vol 128 (2) ◽  
pp. 443-452 ◽  
Author(s):  
R Lande
Keyword(s):  
Genetic Drift ◽  
Genetic Variance ◽  
Two Dimensions ◽  
Quantitative Character ◽  
Stabilizing Selection ◽  
Neighborhood Size ◽  
Phenotypic Variance ◽  
Two Dimensional ◽  
Random Genetic Drift ◽  
Dispersal Function

Abstract Random genetic drift in a quantitative character is modeled for a population with a continuous spatial distribution in an infinite habitat of one or two dimensions. The analysis extends Wright's concept of neighborhood size to spatially autocorrelated sampling variation in the expected phenotype at different locations. Weak stabilizing selection is assumed to operate toward the same optimum phenotype in every locality, and the distribution of dispersal distances from parent to offspring is a (radially) symmetric function. The equilibrium pattern of geographic variation in the expected local phenotype depends on the neighborhood size, the genetic variance within neighborhoods, and the strength of selection, but is nearly independent of the form of the dispersal function. With all else equal, geographic variance is smaller in a two-dimensional habitat than in one dimension, and the covariance between expected local phenotypes decreases more rapidly with the distance separating them in two dimensions than in one. The equilibrium geographic variance is less than the phenotypic variance within localities, unless the neighborhood size is small and selection is extremely weak, especially in two dimensions. Nevertheless, dispersal of geographic variance created by random genetic drift is an important mechanism maintaining genetic variance within local populations. For a Gaussian dispersal function it is shown that, even with a small neighborhood size, a population in a two-dimensional habitat can maintain within neighborhoods most of the genetic variance that would occur in an infinite panmictic population.

scholarly journals On the distribution of the mean and variance of a quantitative trait under mutation-selection-drift balance.

Genetics ◽  
1994 ◽  
Vol 138 (3) ◽  
pp. 901-912 ◽  
Author(s):  
R Bürger ◽  
R Lande
Keyword(s):  
Genetic Drift ◽  
Quantitative Trait ◽  
Quantitative Traits ◽  
Genetic Variance ◽  
Stochastic Simulations ◽  
Stabilizing Selection ◽  
Phenotypic Evolution ◽  
Theoretical Predictions ◽  
Mean And Variance ◽  
The Mean

Abstract The distributions of the mean phenotype and of the genetic variance of a polygenic trait under a balance between mutation, stabilizing selection and genetic drift are investigated. This is done by stochastic simulations in which each individual and each gene are represented. The results are compared with theoretical predictions. Some aspects of the existing theories for the evolution of quantitative traits are discussed. The maintenance of genetic variance and the average dynamics of phenotypic evolution in finite populations (with Ne < 1000) are generally simpler than those suggested by some recent deterministic theories for infinite populations.

scholarly journals THE SELECTIVE VALUE OF ALLELES UNDERLYING POLYGENIC TRAITS

Genetics ◽  
1984 ◽  
Vol 108 (4) ◽  
pp. 1021-1033
Author(s):  
Michael Lynch
Keyword(s):  
Molecular Evolution ◽  
Genetic Drift ◽  
Genetic Variance ◽  
Neutral Theory ◽  
Mechanistic Explanation ◽  
Neutral Evolution ◽  
Stabilizing Selection ◽  
Average Effect ◽  
Absolute Value ◽  
The Mean

ABSTRACT To define the genetic and ecological circumstances that are conductive to evolution via genetic drift at the allelic level, the selection coefficient for a constituent allele of arbitrary effect is derived for a polygenic character exposed to stabilizing selection. Under virtually all possible conditions, alleles within the class for which the absolute value of the average effect is <10-2 phenotypic standard deviations are neutral with respect to each other. In addition, when the mean phenotype is at the optimum and the genetic variance is in selection-drift-mutation equilibrium, a considerable amount of neutral evolution is expected in the class of alleles with intermediate effects on the phenotype. These results help clarify how molecular evolution via genetic drift may occur at a locus despite intense selection and provide a potential mechanistic explanation for the neutral theory of molecular evolution.

scholarly journals Polygenic adaptation after a sudden change in environment

2019 ◽  
Author(s):  
Laura K. Hayward ◽  
Guy Sella
Keyword(s):  
Genetic Variance ◽  
Sudden Change ◽  
Directional Selection ◽  
Stabilizing Selection ◽  
Effective Population ◽  
Short Term ◽  
Exponential Process ◽  
Optimal Value ◽  
Polygenic Adaptation ◽  
The Mean

AbstractPolygenic adaptation in response to selection on quantitative traits is thought to be ubiquitous in humans and other species, yet this mode of adaptation remains poorly understood. We investigate the dynamics of this process, assuming that a sudden change in environment shifts the optimal value of a highly polygenic quantitative trait. We find that when the shift is not too large relative to the genetic variance in the trait and this variance arises from segregating loci with small to moderate effect sizes (defined in terms of the selection acting on them before the shift), the mean phenotype’s approach to the new optimum is well approximated by a rapid exponential process first described by Lande (1976). In contrast, when the shift is larger or large effect loci contribute substantially to genetic variance, the initially rapid approach is succeeded by a much slower one. In either case, the underlying changes to allele frequencies exhibit different behaviors short and long-term. Over the short term, strong directional selection on the trait introduces small differences between the frequencies of minor alleles whose effects are aligned with the shift in optimum versus those with effects in the opposite direction. The phenotypic effects of these differences are dominated by contributions from alleles with moderate and large effects, and cumulatively, these effects push the mean phenotype close to the new optimum. Over the longer term, weak directional selection on the trait can amplify the expected frequency differences between opposite alleles; however, since the mean phenotype is close to the new optimum, alleles are mainly affected by stabilizing selection on the trait. Consequently, the frequency differences between opposite alleles translate into small differences in their probabilities of fixation, and the short-term phenotypic contributions of large effect alleles are largely supplanted by contributions of fixed, moderate ones. This process takes on the order of ~4Ne generations (where Ne is the effective population size), after which the steady state architecture of genetic variation around the new optimum is restored.

The Neutral Divergence of Quantitative Traits

2018 ◽  
Author(s):  
Bruce Walsh ◽  
Michael Lynch
Keyword(s):  
Gene Expression ◽  
Genetic Drift ◽  
Joint Action ◽  
Quantitative Traits ◽  
Directional Selection ◽  
Stabilizing Selection ◽  
Data Sets ◽  
Diverse Range ◽  
Over Time

The joint action of genetic drift and mutation results in the divergence of trait means over time. This chapter examines the expected amount of divergence, which forms the basis for a number of tests on whether an observed pattern is either too large relative to drift (suggesting directional selection) or two small (suggesting stabilizing selection). It then applies these results to examine tests for selection over a very diverse range of data sets, ranging from a stratophenetic series of fossils to divergence in gene expression over time. It also examines a number of trait-augmented marked-based tests (such as using the QTLs or GWAS hits for a trait) for departures from neutrality.

scholarly journals Polygenic Adaptation in a Population of Finite Size

Entropy ◽  
2020 ◽  
Vol 22 (8) ◽  
pp. 907
Author(s):  
Wolfgang Stephan ◽  
Sona John
Keyword(s):  
Genetic Drift ◽  
Evolutionary Biology ◽  
Genetic Variance ◽  
Finite Size ◽  
Stabilizing Selection ◽  
Rapid Phase ◽  
Exponential Process ◽  
Polygenic Adaptation ◽  
The Mean ◽  
High Equilibrium

Polygenic adaptation in response to selection on quantitative traits has become an important topic in evolutionary biology. Here we review the recent literature on models of polygenic adaptation. In particular, we focus on a model that includes mutation and both directional and stabilizing selection on a highly polygenic trait in a population of finite size (thus experiencing random genetic drift). Assuming that a sudden environmental shift of the fitness optimum occurs while the population is in a stochastic equilibrium, we analyze the adaptation of the trait to the new optimum. When the shift is not too large relative to the equilibrium genetic variance and this variance is determined by loci with mostly small effects, the approach of the mean phenotype to the optimum can be approximated by a rapid exponential process (whose rate is proportional to the genetic variance). During this rapid phase the underlying changes to allele frequencies, however, may depend strongly on genetic drift. While trait-increasing alleles with intermediate equilibrium frequencies are dominated by selection and contribute positively to changes of the trait mean (i.e., are aligned with the direction of the optimum shift), alleles with low or high equilibrium frequencies show more of a random dynamics, which is expected when drift is dominating. A strong effect of drift is also predicted for population size bottlenecks. Our simulations show that the presence of a bottleneck results in a larger deviation of the population mean of the trait from the fitness optimum, which suggests that more loci experience the influence of drift.

scholarly journals Adaptive landscapes, genetic distance and the evolution of quantitative characters

1987 ◽  
Vol 49 (2) ◽  
pp. 157-173 ◽  
Author(s):  
N. H. Barton ◽  
Michael Turelli
Keyword(s):  
Genetic Distance ◽  
Covariance Matrix ◽  
Genetic Drift ◽  
Genetic Variance ◽  
Adaptive Landscape ◽  
Stabilizing Selection ◽  
Mathematical Framework ◽  
Adaptive Landscapes ◽  
House Of Cards ◽  
Genotype Frequencies

SummaryThe maintenance of polygenic variability by a balance between mutation and stabilizing selection has been analysed using two approximations: the ‘Gaussian’ and the ‘house of cards’. These lead to qualitatively different relationships between the equilibrium genetic variance and the parameters describing selection and mutation. Here we generalize these approximations to describe the dynamics of genetic means and variances under arbitrary patterns of selection and mutation. We incorporate genetic drift into the same mathematical framework.The effects of frequency-independent selection and genetic drift can be determined from the gradient of log mean fitness and a covariance matrix that depends on genotype frequencies. These equations describe an ‘adaptive landscape’, with a natural metric of genetic distance set by the covariance matrix. From this representation we can change coordinates to derive equations describing the dynamics of an additive polygenic character in terms of the moments (means, variances, …) of allelic effects at individual loci. Only under certain simplifying conditions, such as those derived from the Gaussian and house-of-cards approximations, do these general recursions lead to tractable equations for the first few phenotypic moments. The alternative approximations differ in the constraints they impose on the distributions of allelic effects at individual loci. The Gaussian-based prediction that evolution of the phenotypic mean does not change the genetic variance is shown to be a consequence of the assumption that the allelic distributions are never skewed. We present both analytical and numerical results delimiting the parameter values consistent with our approximations.

Population Genetics: Consanguinity, Genetic Drift

1997 ◽  
pp. 549-582 ◽  
Author(s):  
Friedrich Vogel ◽  
Arno G. Motulsky
Keyword(s):  
Population Genetics ◽  
Genetic Drift

Applications of Supercomputers in Population Genetics

2015 ◽  
pp. 176-200
Author(s):  
Gerard G. Dumancas
Keyword(s):  
Population Genetics ◽  
Natural Selection ◽  
Genetic Drift ◽  
Critical Role ◽  
Allele Frequency Distribution ◽  
Modern Approach ◽  
Mathematical Discipline ◽  
Modern Evolutionary Synthesis ◽  
Changes Over Time ◽  
Over Time

Population genetics is the study of the frequency and interaction of alleles and genes in population and how this allele frequency distribution changes over time as a result of evolutionary processes such as natural selection, genetic drift, and mutation. This field has become essential in the foundation of modern evolutionary synthesis. Traditionally regarded as a highly mathematical discipline, its modern approach comprises more than the theoretical, lab, and fieldwork. Supercomputers play a critical role in the success of this field and are discussed in this chapter.

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