Change and maintenance of variation in quantitative traits in the context of the Price equation

A scaling law of multilevel evolution: how the balance between within- and among-collective evolution is determined

Numerous living systems are hierarchically organised, whereby replicating components are grouped into reproducing collectives-e.g., organelles are grouped into cells, and cells are grouped into multicellular organisms. In such systems, evolution can operate at two levels: evolution among collectives, which tends to promote selfless cooperation among components within collectives (called altruism), and evolution within collectives, which tends to promote cheating among components within collectives. The balance between within- and among-collective evolution thus exerts profound impacts on the fitness of these systems. Here, we investigate how this balance depends on the size of a collective (denoted by N) and the mutation rate of components (m) through mathematical analyses and computer simulations of multiple population genetics models. We first confirm a previous result that increasing N or m accelerates within-collective evolution relative to among-collective evolution, thus promoting the evolution of cheating. Moreover, we show that when within- and among-collective evolution exactly balance each other out, the following scaling relation generally holds: Nmα is a constant, where scaling exponent α depends on multiple parameters, such as the strength of selection and whether altruism is a binary or quantitative trait. This relation indicates that although N and m have quantitatively distinct impacts on the balance between within- and among-collective evolution, their impacts become identical if m is scaled with a proper exponent. Our results thus provide a novel insight into conditions under which cheating or altruism evolves in hierarchically-organised replicating systems.

In defence of doing sums in genetics

There has been a long history of the use of mathematics in genetics, ranging from the use of statistics to analyse genetic data to genetic models of evolutionary processes. Contemporary research into the genomic basis of disease and complex traits exemplifies the importance of statistical methods in genetics. Some examples of the development and application of population genetic models are described, which are intended to highlight the utility of such models for understanding variation and evolution in natural populations. The effects of selection on variability at sites linked to the targets of selection illustrate how fruitful interactions between theory and data can be.

An Alternative to the Breeder’s and Lande’s Equations

The breeder’s equation is a cornerstone of quantitative genetics, widely used in evolutionary modeling. Noting the mean phenotype in parental, selected parents, and the progeny by E(Z₀), E(Z_W), and E(Z₁), this equation relates response to selection R = E(Z₁) − E(Z₀) to the selection differential S = E(Z_W) − E(Z₀) through a simple proportionality relation R = h²S, where the heritability coefficient h² is a simple function of genotype and environment factors variance. The validity of this relation relies strongly on the normal (Gaussian) distribution of the parent genotype, which is an unobservable quantity and cannot be ascertained. In contrast, we show here that if the fitness (or selection) function is Gaussian with mean μ, an alternative, exact linear equation of the form R′ = j²S′ can be derived, regardless of the parental genotype distribution. Here R′ = E(Z₁) − μ and S′ = E(Z_W) − μ stand for the mean phenotypic lag with respect to the mean of the fitness function in the offspring and selected populations. The proportionality coefficient j² is a simple function of selection function and environment factors variance, but does not contain the genotype variance. To demonstrate this, we derive the exact functional relation between the mean phenotype in the selected and the offspring population and deduce all cases that lead to a linear relation between them. These results generalize naturally to the concept of G matrix and the multivariate Lande’s equation Δz¯=GP−1S. The linearity coefficient of the alternative equation are not changed by Gaussian selection.

Current applications of models of genetic effects with interactions across the genome

Models of genetic effects integrate the action of genes, regulatory regions and interactions among alleles across the genome. Such theoretical frameworks are critical for applied studies in at least two ways. First, discovering genetic networks with specific effects underlying traits in populations requires the development of models that implement those effects as parameters-adjusting the implementation of epistasis parameters in genetic models has for instance been a requirement for properly testing for epistasis in gene-mapping studies. Second, studying the properties and implications of models of genetic effects that involve complex genetic networks has proven to be valuable, whether those networks have been revealed for particular organisms or inferred to be of interest from theoretical works and simulations. Here I review the current state of development and recent applications of models of genetic effects. I focus on general models aiming to depict complex genotype-to-phenotype maps and on applications of them to networks of interacting loci.

Genetic analysis of environmental variation

Environmental variation (VE) in a quantitative trait - variation in phenotype that cannot be explained by genetic variation or identifiable genetic differences - can be regarded as being under some degree of genetic control. Such variation may be either between repeated expressions of the same trait within individuals (e.g. for bilateral traits), in the phenotype of different individuals, where variation within families may differ, or in both components. We consider alternative models for defining the distribution of phenotypes to include a component due to heterogeneity of VE. We review evidence for the presence of genetic variation in VE and estimates of its magnitude. Typically the heritability of VE is under 10%, but its genetic coefficient of variation is typically 20% or more. We consider experimental designs appropriate for estimating genetic variance in VE and review alternative methods of estimation. We consider the effects of stabilizing and directional selection on VE and review both the forces that might be maintaining levels of VE and heritability found in populations. We also evaluate the opportunities for reducing VE in breeding programmes. Although empirical and theoretical studies have increased our understanding of genetic control of environmental variance, many issues remain unresolved.

Replication and explorations of high-order epistasis using a large advanced intercross line pedigree

Dissection of the genetic architecture of complex traits persists as a major challenge in biology; despite considerable efforts, much remains unclear including the role and importance of genetic interactions. This study provides empirical evidence for a strong and persistent contribution of both second- and third-order epistatic interactions to long-term selection response for body weight in two divergently selected chicken lines. We earlier reported a network of interacting loci with large effects on body weight in an F(2) intercross between these high- and low-body weight lines. Here, most pair-wise interactions in the network are replicated in an independent eight-generation advanced intercross line (AIL). The original report showed an important contribution of capacitating epistasis to growth, meaning that the genotype at a hub in the network releases the effects of one or several peripheral loci. After fine-mapping of the loci in the AIL, we show that these interactions were persistent over time. The replication of five of six originally reported epistatic loci, as well as the capacitating epistasis, provides strong empirical evidence that the originally observed epistasis is of biological importance and is a contributor in the genetic architecture of this population. The stability of genetic interaction mechanisms over time indicates a non-transient role of epistasis on phenotypic change. Third-order epistasis was for the first time examined in this study and was shown to make an important contribution to growth, which suggests that the genetic architecture of growth is more complex than can be explained by two-locus interactions only. Our results illustrate the importance of designing studies that facilitate exploration of epistasis in populations for obtaining a comprehensive understanding of the genetics underlying a complex trait.

Dynamics of gene duplication and transposons in microbial genomes following a sudden environmental change

A variety of genome transformations can occur as a microbial population adapts to a large environmental change. In particular, genomic surveys indicate that, following the transition to an obligate, host-dependent symbiont, the density of transposons first rises, then subsequently declines over evolutionary time. Here we show that these observations can be accounted for by a class of generic stochastic models for the evolution of genomes in the presence of continuous selection and gene duplication. The models use a fitness function that allows for partial contributions from multiple gene copies, is an increasing but bounded function of copy number, and is optimal for one fully adapted gene copy. We use Monte Carlo simulation to show that the dynamics result in an initial rise in gene copy number followed by a subsequent falloff due to adaptation to the new environmental parameters. These results are robust for reasonable gene duplication and mutation parameters when adapting to a novel target sequence. Our model provides a generic explanation for the dynamics of microbial transposon density following a large environmental change such as host restriction.

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

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.