Match probabilities in a finite, subdivided population

We generalize a recently introduced graphical framework to compute the probability that haplotypes or genotypes of two individuals drawn from a finite, subdivided population match. As in the previous work, we assume an infinite-alleles model. We focus on the case of a population divided into two subpopulations, but the underlying framework can be applied to a general model of population subdivision. We examine the effect of population subdivision on the match probabilities and the accuracy of the product rule which approximates multi-locus match probabilities as a product of one-locus match probabilities. We quantify the deviation from predictions of the product rule by R, the ratio of the multi-locus match probability to the product of the one-locus match probabilities. We carry out the computation for two loci and find that ignoring subdivision can lead to underestimation of the match probabilities if the population under consideration actually has subdivision structure and the individuals originate from the same subpopulation. On the other hand, under a given model of population subdivision, we find that the ratio R for two loci is only slightly greater than 1 for a large range of symmetric and asymmetric migration rates. Keeping in mind that the infinite-alleles model is not the appropriate mutation model for STR loci, we conclude that, for two loci and biologically reasonable parameter values, population subdivision may lead to results that disfavor innocent suspects because of an increase in identity-by-descent in finite populations. On the other hand, for the same range of parameters, population subdivision does not lead to a substantial increase in linkage disequilibrium between loci. Those results are consistent with established practice.     

Remarks on the Evolutionary Effect of Natural Selection

The so-called "Fundamental Theorem of Natural Selection", that the mean fitness of a population increases with time under natural selection, is known not to be true, as a mathematical theorem, when fitnesses depend on more than one locus. Although this observation may not have particular biological relevance, (so that mean fitness may well increase in the great majority of interesting situations), it does suggest that it is of interest to find an evolutionary result which is correct as a mathematical theorem, no matter how many loci are involved. The aim of the present note is to prove an evolutionary theorem relating to the variance in fitness, rather than the mean: this theorem is true for an arbitrary number of loci, as well as for arbitrary (fixed) fitness parameters and arbitrary linkage between loci. Connections are briefly discussed between this theorem and the principle of quasi-linkage equilibrium.     

MODELING THE GENETIC BASIS OF HETEROSIS: TESTS OF ALTERNATIVE HYPOTHESES

Houle (1994) showed that marker-associated heterosis due to general inbreeding depression could not be distinguished from direct overdominance at the marker locus by examining mean genotypic fitnesses, in the one-locus case. Indeed, both hypotheses equally fit the same regression model, referred to as the “adaptive distance model” (Smouse 1986). I here extend the analysis to several loci and to the relationship between marker genotype and variance in fitness. Several predictions differ between the overdominance and inbreeding hypotheses: (1) all locus-specific effects are equal under inbreeding, whereas they are not under overdominance; (2) the adaptive distance model has an increasingly low fit when the number of loci increases, under inbreeding, whereas it always explains the whole variance in fitness under overdominance; (3) a negative relationship is predicted between mean fitness and the variance in fitness, under inbreeding, which is not predicted under overdominance. Some statistical tests are derived from these predictions, that help to identify the genetic basis of heterosis. Simulations show that the power of these tests allows their application to real datasets.     

A two-locus mutation—Selection model and some of its evolutionary implications

The deterministic properties of a two-locus model with mutation and selection have been investigated. The mutation process is unidirectional, and the model is so constructed that the genetic variation at one locus is selectively neutral in the absence of a mutant allele at the other locus. All genotypes with three or four mutant alleles are deleterious, while the double heterozygotes may have the same fitness as the standard genotype. If one of the mutant alleles becomes fixed in the population, then the other locus will show a regular one-locus mutation-selection balance. Such a boundary equilibrium may be unstable or stable in the full two-locus setting. In the symmetric case, which is analyzed in details, the population will either go to one of the two boundary equilibria, or to a fully polymorphic equilibrium at which both the mutant alleles are rare. The origin of reproductive separation between two populations via the fixation of complementary deleterious mutants at different loci, and the fixation of nonfunctional alleles at duplicated loci, are two biological processes which both can be studied with the present model. In the last part of the paper we show how the results from the deterministic analysis can be used to predict how different factors will influence the rates of evolution in these systems.     

Preferential mating in symmetric multilocus systems: limits for multiallelism and for many Loci

Models in which general forms of preferential mating have been superimposed on the framework of the symmetric heterozygosity selection regime have been examined previously with respect to the existence and local stability of a central polymorphic equilibrium. The results are now extended to produce the limiting form of the stability conditions in two cases: First, where the number of alleles per locus is assumed to be very large; second, where the number of loci affecting the character is very large. It is argued that some type of frequency dependence in the mating pattern must be included, and a particular case is examined in detail. It is shown that multiallelism is ambiguous in its effect on stability, while an increasing number of loci, at least under zero linkage, leads to a simple stability condition which is analogous to the one-locus heterosis principle. Assortative mating appears to be more likely to produce a stable central polymorphism under high levels of allelism than is sexual selection, but is relatively very much weaker than sexual or viability selection if the number of loci involved is large.     

One- and multi-locus multi-allele selection models in a random environment

Summary We deduce conditions for stochastic local stability of general perturbed linear stochastic difference equations widely applicable in population genetics. The findings are adapted to evaluate the stability properties of equilibria in classical one- and multi-locus multi-allele selection models influenced by random temporal variation in selection intensities. As an example of some conclusions and biological interpretations we analyse a special one-locus multi-allele model in more detail.This work was supported in part by Stiftung Volkswagenwerk.     

Hardy-Weinberg equilibria in random mating populations

The structure of multiloci random mating populations is examined. Sufficient conditions for the existence of stable local Hardy-Weinberg equilibria for n loci and an arbitrary number of alleles per locus, are then derived for specified situations under the assumption of multiplicative gene action between loci. It is shown that a stable Hardy-Weinberg equilibrium can not be a local maximum of the mean fitness function with multiplicative gene action between loci. The stability of Hardy-Weinberg type border points and the condition for the increase of newly introduced genes are topics on which some n-loci results are also obtained for an arbitrary number of alleles per locus in systems that allow Hardy-Weinberg equilibria.     

Variational principles in evolution

For a one-locus selection model, Svirezhev introduced an integral variational principle by defining a Lagrangian which remained stationary on the trajectory followed by the population undergoing selection. It is shown here (i) that this principle can be extended to multiple loci in some simple cases and (ii) that the Lagrangian is defined by a straightforward generalization of the one-locus case, but (iii) that in two-locus or more general models there is no straightforward extension of this principle if linkage and epistasis are present. The population trajectories can be constructed as trajectories of steepest ascent in a Riemannian metric space. A general method is formulated to find the metric tensor and the surface in the metric space on which the trajectories, which characterize the variations in the gene structure of the population, lie. The local optimality principle holds good in such a space. In the special case when all possible linkage disequilibria are zero, the phase point of then-locus genetic system moves on the surface of the product space ofn higher dimensional unit spheres in a certain Riemannian metric space of gene frequencies so that the rate of change of mean fitness is maximum along the trajectory. In the two-locus case the corresponding surface is a hyper-torus.     

Parental sex discrimination and intralocus sexual conflict

Intralocus sexual conflict occurs when populations segregate for alleles with opposing fitness consequences in the two sexes. This form of selection is known to be capable of maintaining genetic and fitness variation in nature, the extent of which is sensitive to the underlying genetics. We present a one-locus model of a haploid maternal effect that has sexually antagonistic consequences for offspring. The evolutionary dynamics of these maternal effects are distinct from those of haploid direct effects under sexual antagonism because the relevant genes are expressed only in females. Despite this, we find the same opportunity for sexually antagonistic polymorphism at the maternal effect locus as at a direct effect locus. Thus, sexually antagonistic maternal effects may underlie some natural genetic variation. The model we present permits alternative interpretations of how the genes are expressed and how the fitness variation is assigned, which invites a theoretical comparison to models of both imprinted genes and sex allocation.     

Variance Heterogeneity in Saccharomyces cerevisiae Expression Data: Trans-Regulation and Epistasis

Here, we describe the results from the first variance heterogeneity Genome Wide Association Study (VGWAS) on yeast expression data. Using this forward genetics approach, we show that the genetic regulation of gene-expression in the budding yeast, Saccharomyces cerevisiae, includes mechanisms that can lead to variance heterogeneity in the expression between genotypes. Additionally, we performed a mean effect association study (GWAS). Comparing the mean and variance heterogeneity analyses, we find that the mean expression level is under genetic regulation from a larger absolute number of loci but that a higher proportion of the variance controlling loci were trans-regulated. Both mean and variance regulating loci cluster in regulatory hotspots that affect a large number of phenotypes; a single variance-controlling locus, mapping close to DIA2, was found to be involved in more than 10% of the significant associations. It has been suggested in the literature that variance-heterogeneity between the genotypes might be due to genetic interactions. We therefore screened the multi-locus genotype-phenotype maps for several traits where multiple associations were found, for indications of epistasis. Several examples of two and three locus genetic interactions were found to involve variance-controlling loci, with reports from the literature corroborating the functional connections between the loci. By using a new analytical approach to re-analyze a powerful existing dataset, we are thus able to both provide novel insights to the genetic mechanisms involved in the regulation of gene-expression in budding yeast and experimentally validate epistasis as an important mechanism underlying genetic variance-heterogeneity between genotypes.     

The Maintenance of Genetic Variability in Two-Locus Models of Stabilizing Selection

The maintenance of genetic variability at two diallelic loci under stabilizing selection is investigated. Generations are discrete and nonoverlapping; mating is random; mutation and random genetic drift are absent; selection operates only through viability differences. The determination of the genotypic values is purely additive. The fitness function has its optimum at the value of the double heterozygote and decreases monotonically and symmetrically from its optimum, but is otherwise arbitrary. The resulting fitness scheme is identical to the symmetric viability model. Linkage disequilibrium is neglected, but the results are otherwise exact. Explicit formulas are found for all the equilibria, and explicit conditions are derived fro their existence and stability. A complete classification of the six possible global convergence patterns is presented. In addition to the symmetric equilibrium (with gene frequency 1/2 at both loci), a pair of unsymmetric equilibria may exist; the latter are usually, but not always, unstable. If the ratio of the effect of the major locus to that of the minor one exceeds a critical value, both loci will be stably polymorphic. If selection is weak at the minor locus, the more rapidly the fitness function decreases near the optimum, the lower is this critical value; for rapidly decreasing fitness functions, the critical value is close to one. If the fitness function is smooth at the optimum, then a stable polymorphism exists at both loci only if selection is strong at the major locus.     

Mendel's First Law: partisan interests and the parliament of genes

Mendel’s First Law requires explanation because of the possibility of ‘meiotic drivers’, genes that distort fair segregation for selfish gain. The suppression of drive, and the restoration of fair segregation, is often attributed to genes at loci unlinked to the drive locus—such genes cannot benefit from drive but do suffer its associated fitness costs. However, selection can also favour suppressors at loci linked to the drive locus, raising the question of whether suppression of drive usually comes from linked or unlinked loci. Here, I study linked and unlinked suppression in a two-locus model with initial stable polymorphism at the drive locus. I find that the invasion rate of suppressors is a decreasing function of the recombination fraction between the drive and suppressor loci. Surprisingly, the relative likelihood of unlinked vs. linked suppression increases with the strength of drive and is insensitive to the fitness costs of the driver allele. I find that the chromosomal position of the driver influences how rapidly it is suppressed, with a driver in the middle of a chromosome suppressed more rapidly than a driver near the tip. When drive is strong, only a small number of chromosomes are required for suppression usually to derive from unlinked loci. In contrast, when drive is weak, and especially when suppressor alleles are associated with fitness costs, suppression will usually come from linked loci unless the genome comprises many chromosomes.Subject terms: Evolutionary genetics, Population genetics     

Heterozygosity in pin-thrum plants or with partial sex linkage

A two locus model is constructed for selection of a gene closely linked to the S locus in pin-thrum plants or to the sex determining part of the Y chromosome. Using this model, conditions for stability at the equilibrium point which is predicted by one-locus theory when there is heterozygotic superiority are derived. If the recombination value is small, it is found that this equilibrium point is unstable and that the gene frequencies go to a new stable equilibrium point at which the population has a higher average fitness. A few simple cases of selection and the implication of these to the theory of the evolution of the Y chromosome are discussed.     

Estimation of effective population size and migration rate from one- and two-locus identity measures

Standard methods for inferring demographic parameters from genetic data are based mainly on one-locus theory. However, the association of genes at different loci (e.g., two-locus identity disequilibrium) may also contain some information about demographic parameters of populations. In this article, we define one- and two-locus parameters of population structure as functions of one- and two-locus probabilities for the identity in state of genes. Since these parameters are known functions of demographic parameters in an infinite island model, we develop moment-based estimators of effective population size and immigration rate from one- and two-locus parameters. We evaluate this method through simulation. Although variance and bias may be quite large, increasing the number of loci on which the estimates are derived improves the method. We simulate an infinite allele model and a K allele model of mutation. Bias and variance are smaller with increasing numbers of alleles per locus. This is, to our knowledge, the first attempt of a joint estimation of local effective population size and immigration rate.     

THE FITNESS CONSEQUENCES OF MULTIPLE-LOCUS HETEROZYGOSITY UNDER THE MULTIPLICATIVE OVERDOMINANCE AND INBREEDING DEPRESSION MODELS

There is a growing body of literature suggesting that the fitness of an individual increases with the observed number of heterozygous loci. Broad theoretical considerations indicate that under various sorts of balancing selection, this is what one should generally expect in a population of multiple-locus genotypes. To date, however, it has not been possible to distinguish between two potential explanations of the phenomenon. The first explanation is that the loci examined are themselves responsible for the fitness differences observed (or, equivalents, are very closely linked to those that do). The genetic variation in question is thought to be maintained in polymorphic equilibrium by some form of balancing selection. The second explanation assumes that the observed loci are themselves selectively irrelevant but that their heterozygosity reflects that of the total genome. Genomic heterozygosity is thought to be predictive of fitness, being an obverse measure of generalized inbreeding depression. We provide a formal derivation of an explicit relationship between fitness and multiple-locus genotype for a simple form of the first explanation, the multiplicative overdominance model. The inbreeding depression model is a degenerate special case of this more general formulation. A formal estimation and testing framework is constructed that should facilitate evaluation of the two models with empiric data on heterozygosity and fitness.     

Transgene integration,organization and interaction in plants

It has been appreciated for many years that the structure of a transgene locus can have a major influence on the level and stability of transgene expression. Until recently, however, it has been common practice to discard plant lines with poor or unstable expression levels in favor of those with practical uses. In the last few years, an increasing number of experiments have been carried out with the primary aim of characterizing transgene loci and studying the fundamental links between locus structure and expression. Cereals have been at the forefront of this research because molecular, genetic and cytogenetic analysis can be carried out in parallel to examine transgene loci in detail. This review discusses what is known about the structure and organization of transgene loci in cereals, both at the molecular and cytogenetic levels. In the latter case, important links are beginning to be revealed between higher order locus organization, nuclear architecture, chromatin structure and transgene expression.     

A Numerical Simulation of the One-Locus, Multiple-Allele Fertility Model

Numerical simulations were performed to determine the equilibrium behavior of the one-locus fertility model in which fitness is considered as a property of a pair of mating diploids. A series of patterns of "fertility matrices" were considered for a single locus with two to six alleles. From these simulations, 19 different statistics were collected that characterize, at equilibrium, the heterozygosity, the mean fitness and the fate of populations begun at the allele-frequency centroid. For more than one-half of the trajectories produced by random fertility matrices, there was a decrease in the mean fitness at some time on the way to equilibrium. The mean number of alleles maintained at equilibrium increased only slightly with matrix dimension. Despite the potential for fertility models to display multiple stable equilibria, random fertility models maintain fewer distinct stable points than do random one-locus viability models. Pleiotropic models were also considered with fertility and viability selection operating sequentially within each generation. Most of the equilibrium statistics (with the exception of mean fertility) for the pleiotropic model were intermediate between the corresponding random viability and fertility models.     

On the evolution of genetic incompatibility systems. V. Origin of sporophytic self-incompatibility in response to overdominance in viability

Conditions for the origin of partial sporophytic self-incompatibility (SSI) are obtained from two quantitative models, which differ with respect to the determination of offspring viability. Offspring viability depends solely on the source (self or nonself) of the fertilizing pollen in the first model, which describes changes only at a primitive S-locus itself. Two loci evolve in the second model: overdominant viability selection maintains an arbitrary number of alleles at one locus, with SSI under the control of a separate locus. In both cases, the origin of SSI requires that the relative change in the numbers of offspring derived by the two reproductive modes compensate for the twofold cost of outcrossing. In the first model studied, the viability of inbred offspring fully determines the relative change in the numbers of inbred and outbred offspring produced. In the second model, the relative change in offspring numbers depends in addition on associations between the S-locus and the viability locus. Because these two-locus associations are comparable in magnitude to the differences between the viabilities of inbred and outbred offspring, SSI can arise under less restrictive conditions than expected from the one-locus model. Greater allelic multiplicity at the viability locus facilitates the origin of SSI by reducing the relative viability of inbred offspring. Tight linkage between the S-locus and the viability locus and high rates of receipt of self-pollen promote the generation and maintenance of associations between the S-locus and the viability locus. In populations in which more than two viability alleles are maintained, the active S-allele can invade even in the absence of linkage with the viability locus. The present study establishes that incompatibility systems can arise in response to identity disequilibrium between a modifier of incompatibility and a locus subject to overdominant viability selection; in particular, compensation for the twofold cost of outcrossing does not require preexisting gametic level disequilibria.     

Strategies for conditional two-locus nonparametric linkage analysis

In this article we deal with two-locus nonparametric linkage (NPL) analysis, mainly in the context of conditional analysis. This means that one incorporates single-locus analysis information through conditioning when performing a two-locus analysis. Here we describe different strategies for using this approach. Cox et al. [Nat Genet 1999;21:213-215] implemented this as follows: (i) Calculate the one-locus NPL process over the included genome region(s). (ii) Weight the individual pedigree NPL scores using a weighting function depending on the NPL scores for the corresponding pedigrees at speci fi c conditioning loci. We generalize this by conditioning with respect to the inheritance vector rather than the NPL score and by separating between the case of known (prede fi ned) and unknown (estimated) conditioning loci. In the latter case we choose conditioning locus, or loci, according to prede fi ned criteria. The most general approach results in a random number of selected loci, depending on the results from the previous one-locus analysis. Major topics in this article include discussions on optimal score functions with respect to the noncentrality parameter (NCP), and how to calculate adequate p values and perform power calculations. We also discuss issues related to multiple tests which arise from the two-step procedure with several conditioning loci as well as from the genome-wide tests.     

The effect on neutral gene flow of selection at a linked locus

The effect on gene flow at a neutral locus of a selective cline at a linked locus is investigated. A diffusion approximation for a two-locus island model is derived in which only one locus is subject to selection. The moments of the stationary distribution are obtained and compared to the corresponding moments from a one-locus, neutral island model. This comparison yields an effective migration rate. The effective migration rate is always less than the actual migration rate, but this effect is seen to be small for weak selection and loose linkage in the case of adult migration. The importance of selection at linked loci to the question of genetic differentiation in a subdivided population is discussed.