Isolation by Distance in a Quantitative Trait

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.     

Spike trains in a stochastic Hodgkin-Huxley system

We consider a standard Hodgkin-Huxley model neuron with a Gaussian white noise input current with drift parameter mu and variance parameter sigma(2). Partial differential equations of second order are obtained for the first two moments of the time taken to spike from (any) initial state, as functions of the initial values. The analytical theory for a 2-component (V,m) approximation is also considered. Let mu(c) (approximately 4.15) be the critical value of mu for firing when noise is absent. Large sample simulation results are obtained for mumu(c), for many values of sigma between 0 and 25. For the time to spike, the 2-component approximation is accurate for all sigma when mu=10, for sigma>7 when mu=5 and only when sigma>15 when mu=2. When mumu(c), most paths show similar behavior and the moments exhibit smoothly changing behavior as sigma increases. Thus there are a different number of regimes depending on the magnitude of mu relative to mu(c): one when mu is small and when mu is large; but three when mu is close to and above mu(c). Both for the Hodgkin-Huxley (HH) system and the 2-component approximation, and regardless of the value of mu, the CV tends to about 1.3 at the largest value (25) of sigma considered. We also discuss in detail the problem of determining the interspike interval and give an accurate method for estimating this random variable by decomposing the interval into stochastic and almost deterministic components.     

The influence of spatial inhomogeneities on neutral models of geographical variation : I. Formulation

The influence of spatial variation in the carrying capacity and migration rate of a geographical barrier on the one-dimensional stepping-stone model is studied. The monoecious, diploid population is subdivided into an infinite linear array of panmictic colonies that exchange gametes. In each deme, the rate of self-fertilization is equal to the reciprocal of the number of individuals in that deme. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus in the absence of selection; every allele mutates to new alleles at the same rate. In the diffusion approximation, a partial differential equation that incorporates spatial (and temporal) variation in the carrying capacity and migration rate is derived for the probability of identity. Transition conditions that simultaneously take into account discontinuities in the carrying capacity and migration rate are established: the probability of identity is continuous, but its partial derivatives are not, their ratio being a simple function of the carrying capacities and migrational variance on the two sides of the inhomogeneity. The partial derivatives of the probability of identity are continuous across a geographical barrier, whereas the probability of identity itself has a discontinuity proportional to the partial derivative at the barrier, the constant of proportionality being a measure of the difficulty of crossing the barrier.     

A simulation-based evaluation of methods for inferring linear barriers to gene flow

Different analytical techniques used on the same data set may lead to different conclusions about the existence and strength of genetic structure. Therefore, reliable interpretation of the results from different methods depends on the efficacy and reliability of different statistical methods. In this paper, we evaluated the performance of multiple analytical methods to detect the presence of a linear barrier dividing populations. We were specifically interested in determining if simulation conditions, such as dispersal ability and genetic equilibrium, affect the power of different analytical methods for detecting barriers. We evaluated two boundary detection methods (Monmonier's algorithm and WOMBLING), two spatial Bayesian clustering methods (TESS and GENELAND), an aspatial clustering approach (STRUCTURE), and two recently developed, non-Bayesian clustering methods [PSMIX and discriminant analysis of principal components (DAPC)]. We found that clustering methods had higher success rates than boundary detection methods and also detected the barrier more quickly. All methods detected the barrier more quickly when dispersal was long distance in comparison to short-distance dispersal scenarios. Bayesian clustering methods performed best overall, both in terms of highest success rates and lowest time to barrier detection, with GENELAND showing the highest power. None of the methods suggested a continuous linear barrier when the data were generated under an isolation-by-distance (IBD) model. However, the clustering methods had higher potential for leading to incorrect barrier inferences under IBD unless strict criteria for successful barrier detection were implemented. Based on our findings and those of previous simulation studies, we discuss the utility of different methods for detecting linear barriers to gene flow.     

Effects on phenotypic variability of directional selection arising through genetic differences in residual variability

In standard models of quantitative traits, genotypes are assumed to differ in mean but not variance of the trait. Here we consider directional selection for a quantitative trait for which genotypes also confer differences in variability, viewed either as differences in residual phenotypic variance when individual loci are concerned or as differences in environmental variability when the whole genome is considered. At an individual locus with additive effects, the selective value of the increasing allele is given by ia/sigma + 1/2 ixb/sigma2, where i is the selection intensity, x is the standardized truncation point, sigma2 is the phenotypic variance, and a/sigma and b/sigma2 are the standardized differences in mean and variance respectively between genotypes at the locus. Assuming additive effects on mean and variance across loci, the response to selection on phenotype in mean is isigma2(Am)/sigma + 1/2 ixcov(Amv)/sigma2 and in variance is icov(Amv)/sigma + 1/2 ixsigma2(Av)/sigma2, where sigma2(Am) is the (usual) additive genetic variance of effects of genes on the mean, sigma2(Av) is the corresponding additive genetic variance of their effects on the variance, and cov(Amv) is the additive genetic covariance of their effects. Changes in variance also have to be corrected for any changes due to gene frequency change and for the Bulmer effect, and relevant formulae are given. It is shown that effects on variance are likely to be greatest when selection is intense and when selection is on individual phenotype or within family deviation rather than on family mean performance. The evidence for and implications of such variability in variance are discussed.     

Barriers to the spread of neutral alleles in the cytonuclear system

Neutral alleles can eventually pass a hybrid zone and their initial clines generated by a pure diffusion process dissipate with time, irrespective of the presence or absence of physical barriers. However, the transient neutral clines at the nuclear or organelle sites can be reinforced by the cytonuclear disequilibrium generated by diploid seed and haploid pollen dispersal. In this study, the spread of a neutral allele in an ecological zone of hermaphrodite plants is examined under three cytonuclear systems for genomes with contrasting modes of inheritance (paternal, maternal, and biparental inheritance). The results show that the transient neutral clines can exhibit the spatial pattern similar to the selective clines from separate genomes although discordance between them exists. The spread of a neutral allele is not only related to the vectors of seed and pollen dispersal but also to the mode of its inheritance. Pollen dispersal facilitates the direct effects of the selective organelle sites with paternal inheritance on the spread of a neutral nuclear allele. It also enhances its indirect effects on the spread of a neutral organelle allele with maternal inheritance via modifying the cytonuclear disequilibrium. A positive relationship exists between the barriers to the spread of selective nuclear (or organelle) and neutral organelle (or nuclear) alleles. An asymmetric barrier to the spread of the neutral alleles exists on the two sides of the physical barrier, given the presence of symmetric barrier to the spread of the selective alleles. These theoretical predictions highlight the effects of cytonuclear disequilibrium on the spread of a neutral allele and draw attention to our empirical cline analysis with neutral markers.     

Separation of time scales, fixation probabilities and convergence to evolutionary stable states under isolation by distance

To a first order of approximation, selection is frequency independent in a wide range of family structured models and in populations following an island model of dispersal, provided the number of families or demes is large and the population is haploid or diploid but allelic effects on phenotype are semidominant. This result underlies the way the evolutionary stability of traits is computed in games with continuous strategy sets. In this paper similar results are derived under isolation by distance. The first-order effect on expected change in allele frequency is given in terms of a measure of local genetic diversity, and of measures of genetic structure which are almost independent of allele frequency in the total population when the number of demes is large. Hence, when the number of demes increases the response to selection becomes of constant sign. This result holds because the relevant neutral measures of population structure converge to equilibrium at a rate faster than the rate of allele frequency changes in the total population. In the same conditions and in the absence of demographic fluctuations, the results also provide a simple way to compute the fixation probability of mutants affecting various ecological traits, such as sex ratio, dispersal, life-history, or cooperation, under isolation by distance. This result is illustrated and tested against simulations for mutants affecting the dispersal probability under a stepping-stone model.     

Influence of mutational and sampling factors on the estimation of demographic parameters in a "continuous" population under isolation by distance

In numerous species, individual dispersal is restricted in space so that "continuous" populations evolve under isolation by distance. A method based on individual genotypes assuming a lattice population model was recently developed to estimate the product Dsigma2, where D is the population density and sigma2 is the average squared parent-offspring distance. We evaluated the influence on this method of both mutation rate and mutation model, with a particular reference to microsatellite markers, as well as that of the spatial scale of sampling. Moreover, we developed and tested a nonparametric bootstrap procedure allowing the construction of confidence intervals for the estimation of Dsigma2. These two objectives prompted us to develop a computer simulation algorithm based on the coalescent theory giving individual genotypes for a continuous population under isolation by distance. Our results show that the characteristics of mutational processes at microsatellite loci, namely the allele size homoplasy generated by stepwise mutations, constraints on allele size, and change of slippage rate with repeat number, have little influence on the estimation of Dsigma2. In contrast, a high genetic diversity (approximately 0.7-0.8), as is commonly observed for microsatellite markers, substantially increases the precision of the estimation. However, very high levels of genetic diversity (>0.85) were found to bias the estimation. We also show that statistics taking into account allele size differences give unreliable estimations (i.e., high variance of Dsigma2 estimation) even under a strict stepwise mutation model. Finally, although we show that this method is reasonably robust with respect to the sampling scale, sampling individuals at a local geographical scale gives more precise estimations of Dsigma2.     

Separation of time scales, fixation probabilities and convergence to evolutionarily stable states under isolation by distance

To a first order of approximation, selection is frequency independent in a wide range of family structured models and in populations following an island model of dispersal, provided the number of families or demes is large and the population is haploid or diploid but allelic effects on phenotype are semidominant. This result underlies the way the evolutionary stability of traits is computed in games with continuous strategy sets. In this paper similar results are derived under isolation by distance. The first-order effect on expected change in allele frequency is given in terms of a measure of local genetic diversity, and of measures of genetic structure which are almost independent of allele frequency in the total population when the number of demes is large. Hence, when the number of demes increases the response to selection becomes of constant sign. This result holds because the relevant neutral measures of population structure converge to equilibrium at a rate faster than the rate of allele frequency changes in the total population. In the same conditions and in the absence of demographic fluctuations, the results also provide a simple way to compute the fixation probability of mutants affecting various ecological traits, such as sex ratio, dispersal, life-history, or cooperation, under isolation by distance. This result is illustrated and tested against simulations for mutants affecting the dispersal probability under a stepping-stone model.     

The effect of epistasis on the structure of hybrid zones

Within hybrid zones that are maintained by a balance between selection and dispersal, linkage disequilibrium is generated by the mixing of divergent populations. This linkage disequilibrium causes selection on each locus to act on all other loci, thereby steepening clines, and generating a barrier to gene flow. Diffusion models predict simple relations between the strength of linkage disequilibrium and the dispersal rate, sigma, and between the barrier to gene flow, B, and the reduction in mean fitness, W. The aim of this paper is to test the accuracy of these predictions by comparison with an exact deterministic model of unlinked loci (r = 0.5). Disruptive selection acts on the proportion of alleles from the parental populations (p,q): W = exp[-S(4pq)beta], such that the least fit genotype has fitness e-s. Where beta < 1, fitness is reduced for a wide range of intermediate genotypes; where beta > 1, fitness is only reduced for those genotypes close to p = 0.5. Even with strong epistasis, linkage disequilibria are close to sigma 2p'ip'j/rij, where p'i, p'j are the gradients in allele frequency at loci i, j. The barrier to gene flow, which is reflected in the steepening of neutral clines, is given by [formula: see text] where r, the harmonic mean recombination rate between the neural and selected loci, is here 0.5. This is a close approximation for weak selection, but underestimates B for strong selection. The barrier is stronger for small beta, because hybrid fitness is then reduced over a wider range of p. The widths of the selected clines are harder to predict: though simple approximations are accurate for beta = 1, they become inaccurate for extreme beta because, then, fitness changes sharply with p. Estimates of gene number, made from neutral clines on the assumption that selection acts against heterozygotes, are accurate for weak selection when beta = 1; however, for strong selection, gene number is overestimated. For beta > 1, gene number is systematically overestimated and, conversely, when beta < 1, it is underestimated.     

Spatial variation and density-dependent dispersal in competitive coexistence

It is well known that dispersal from localities favourable to a species' growth and reproduction (sources) can prevent competitive exclusion in unfavourable localities (sinks). What is perhaps less well known is that too much emigration can undermine the viability of sources and cause regional competitive exclusion. Here, I investigate two biological mechanisms that reduce the cost of dispersal to source communities. The first involves increasing the spatial variation in the strength of competition such that sources can withstand high rates of emigration; the second involves reducing emigration from sources via density-dependent dispersal. I compare how different forms of spatial variation and modes of dispersal influence source viability, and hence source-sink coexistence, under dominance and pre-emptive competition. A key finding is that, while spatial variation substantially reduces dispersal costs under both types of competition, density-dependent dispersal does so only under dominance competition. For instance, when spatial variation in the strength of competition is high, coexistence is possible (regardless of the type of competition) even when sources experience high emigration rates; when spatial variation is low, coexistence is restricted even under low emigration rates. Under dominance competition, density-dependent dispersal has a strong effect on coexistence. For instance, when the emigration rate increases with density at an accelerating rate (Type III density-dependent dispersal), coexistence is possible even when spatial variation is quite low; when the emigration rate increases with density at a decelerating rate (Type II density-dependent dispersal), coexistence is restricted even when spatial variation is quite high. Under pre-emptive competition, density-dependent dispersal has only a marginal effect on coexistence. Thus, the diversity-reducing effects of high dispersal rates persist under pre-emptive competition even when dispersal is density dependent, but can be significantly mitigated under dominance competition if density-dependent dispersal is Type III rather than Type II. These results lead to testable predictions about source-sink coexistence under different regimes of competition, spatial variation and dispersal. They identify situations in which density-independent dispersal provides a reasonable approximation to species' dispersal patterns, and those under which consideration of density-dependent dispersal is crucial to predicting long-term coexistence.     

The effects of habitat fragmentation via forestry plantation establishment on spatial genotypic structure in the small marsupial carnivore, Antechinus agilis

Dispersal is an important influence on species' distributions, patch colonization and population persistence in fragmented habitat. We studied the impacts of habitat fragmentation resulting from establishment of an exotic pine plantation on dispersal of the marsupial carnivore, Antechinus agilis. We applied spatial analyses of individual multilocus microsatellite genotypes and mitochondrial haplotypes to study patterns of gene flow in fragmented habitat and natural habitat 'control' areas, and how this is affected by the spatial dispersion of habitat patches, the presence of corridors and a 'mainland' source of migrants. Spatial analysis of molecular variance and partial Mantel tests confirmed the absence of cryptic barriers to gene flow in continuous habitat, which if present would confound the comparison of genetic structures in fragmented vs. unfragmented habitats. Spatial genotypic structure suggested that although dispersal was male-biased in both habitat types, fragmentation restricted dispersal of males more than that of females and the degree of restriction of male dispersal was dependent on the geographical isolation of the patch. The scale of positive genotypic structure in fragmented habitat was restricted to the two closest patches for females and the three closest patches for males. Our results provide evidence for significantly increased gene flow through habitat corridors relative to that across the matrix and for significantly lower gene flow between 'mainland' unfragmented habitat and habitat patches relative to that within either habitat type, suggesting a behavioural barrier to crossing habitat interfaces.     

Pollen dispersal in spatially aggregated populations

We perform a theoretical study of effective pollen dispersal within plant populations exhibiting intraspecific spatial aggregation. We simulate nonuniform distributions of individuals by means of a Poisson cluster process and use an individual-based spatially explicit model of pollen dispersal to assess the effects of different aggregation patterns on the effective pollen pool size (N(ep)) and the axial variance of pollen dispersal (sigma (p)). Results show clear interactions between clumping and both N(ep) and sigma (p), whose precise form and intensity depend on the relative spatial scale of aggregation to pollen dispersal range. If clump size is small relative to dispersal range, clumping results in lower N(ep) and sigma (p) than in randomly distributed populations. Interestingly, by contrast, aggregation may actually enlarge N(ep) and has minimum impact on sigma (p) if clump size is near or above the scale of dispersal. High intraclump to global density ratios enhance the sensitivity of both N(ep) and sigma (p) to clumping, while leptokurtic pollen dispersal generates sharper reductions of both N(ep) and sigma (p) for small clump sizes and stronger increments of N(ep) for larger clump sizes. Overall, our results indicate that isolation-by-distance models in plants should not ignore the effects of intraspecific spatial aggregation on effective dispersal.     

Mechanisms of speciation and faunal enrichment in Atlantic parrotfishes

Relationships based on mtDNA and nDNA sequences were used to assess effects of two major geographic barriers (the >30 myo Atlantic ocean and the approximately 11 myo Amazon-Orinoco outflow) on speciation among Atlantic parrotfishes (Sparisoma and Nicholsina). Allopatric distributions of sister taxa implicate isolating actions of both barriers in all recent speciation in these fishes, with no clear indications that any speciation resulted from other mechanisms. Molecular clock estimates of the timing of lineage splits indicate that both barriers acted by limiting dispersal well after they formed, although the Amazon barrier also may have been a vicariance agent. Fluctuations in sealevel, climate, and ocean-current dynamics over the past approximately 10 my likely produced marked variation in the effectiveness of both barriers, but particularly the Amazon barrier, allowing intermittent dispersal leading to establishment and allopatric speciation. A dynamic Amazon barrier represents a major engine of West Atlantic faunal enrichment that has repeatedly facilitated bidirectional dispersal, allopatric speciation, and remixing of the Caribbean and Brazilian faunas.     

Conformation of the denatured DNA molecule in solutions of different ionic strengths]

The conformation of the denatured DNA molecule of different molecular weights in the solutions of various ionic composition was studied by the methods of viscometry, light scattering and flow birefringence. Formaldehyde purified from metallic ions with the help of ionites was used for fixation of the denatured state of the DNA molecule. It has been shown that theories developed for flexible macromolecules are in a sufficient accordance with hydrodynamical and optical data. The unperturbed dimensions, equilibrium rigidity of the macromolecule in solutions of different ionic strengths, mu, were determined. In the range of mu greater than or equal to 0.005 the length of Kuhn's segment (A) is equal to approximately 40 A and its value increases with an increase of mu. At mu 0.001 A approximately 60 A and mu 0.0005 A approximately 85 divided by 100 A. A relation between intrinsic viscosity and molecular weight of the denatured DNA molecule was established. Data on the flow birefringence in the solutions of the denatured DNA have shown that the sigh of optical anisotrophy of the macromolecule depends on the ionic strength. The observed dependency may be explained only by assuming that ionic strength influences the equilibrium orientation of nitrogen base planes with respect to the main chain of the macromolecule.     

Plant dispersal, neighbourhood size and isolation by distance

A theoretical relationship between isolation by distance or spatial genetic structure (SGS) and seed and pollen dispersal is tested using extensive spatial-temporal simulations. Although for animals Wright's neighbourhood size N(e) = 4pisigma(2)(t) has been ascertained also, where sigma(2)(t) is the axial variance of distances between parents and offspring, and it was recently confirmed that N(e) = 4pi(sigma(2)(f) + sigma(2)(m))/2 when dispersal of females and males differ, the situation for plants had not been established. This article shows that for a very wide range of conditions, neighbourhood size defined by Crawford's formula N(e) = 4pi(sigma(2)(s) + sigma(2)(p)/2) fully determines SGS, even when dispersal variances of seed (sigma(2)(s)) and pollen sigma(2)(p)) differ strongly. Further, self-fertilization with rate s acts as zero-distance pollen dispersal, and N(e) = 4pi[sigma(2)(s) + sigma(2)(p)(1 - s)/2] fully determines SGS, for most cases where there are both likely parameter values and substantial SGS. Moreover, for most cases, there is a loglinear relationship, I(1) = 0.587 - 0.117 ln(N(e)), between SGS, as measured by I(1), Moran's coefficient for adjacent individuals, and N(e). However, there are several biologically significant exceptions, namely for very low or large N(e), SGS exceeds the loglinear values. There are also important exceptions to Crawford's formula. First, plants with low seed dispersal, high outcross pollen dispersal and high selfing rate show larger SGS than predicted. Second, in plants with very low (near zero) seed dispersal, selfing decreases SGS, opposite expectations. Finally, in some cases seed dispersal is more critical than pollen dispersal, in a manner inconsistent with Crawford's formula.     

Effective Sizes and Dynamics of Uniparentally and Diparentally Inherited Genes

Models to determine the temporal dynamics and spatial heterogeneity for maternally and paternally inherited genes were derived for populations that may or may not exhibit spatial subdivision. Results were compared to those for diparentally inherited genes. The models permit definition of parameters for mean and variance of litter sizes, breeding group (subpopulation) sizes, and numbers of female mates per male, dispersal rates, and multiple paternity. Exact solutions for asymptotic effective size and spatial divergence (F(LS)) for maternal and paternal genes are derived. It is shown that solutions for effective size and F(LS) are transformations of the same quadratic equation. When compared to values for diparentally inherited genes, it is shown that effective sizes for maternal genes may be considerably higher when female dispersal is low as in many mammalian taxa. Likewise, effective sizes for paternal genes may be higher than for diparentally inherited traits when male dispersal is relatively low, as in many species of birds. The traditional assumption that the effective size for maternal genes is approximately equal to the number of females is seldom realized. Spatial heterogeneity and temporal dynamics of genes are inextricably linked as is shown by the interdependency of effective size and spatial heterogeneity.     

Evolution of dispersal polymorphism and local adaptation of dispersal distance in spatially structured landscapes

Many organisms show polymorphism in dispersal distance strategies. This variation is particularly ecological relevant if it encompasses a functional separation of short‐ (SDD) and long‐distance dispersal (LDD). It remains, however, an open question whether both parts of the dispersal kernel are similarly affected by landscape related selection pressures. We implemented an individual‐based model to analyze the evolution of dispersal traits in fractal landscapes that vary in the proportion of habitat and its spatial configuration. Individuals are parthenogenetic with dispersal distance determined by two alleles on each individual's genome: one allele coding for the probability of global dispersal and one allele coding for the variance σ of a Gaussian local dispersal with mean value zero. Simulations show that mean distances of local dispersal and the probability of global dispersal, increase with increasing habitat availability, but that changes in the habitat's spatial autocorrelation impose opposing selective pressure: local dispersal distances decrease and global dispersal probabilities increase with decreasing spatial autocorrelation of the available habitat. Local adaptation of local dispersal distance emerges in landscapes with less than 70% of clumped habitat. These results demonstrate that long and short distance dispersal evolve separately according to different properties of the landscape. The landscape structure may consequently largely affect the evolution of dispersal distance strategies and the level of dispersal polymorphism.     

Dispersal, Environmental Correlation, and Spatial Synchrony in Population Dynamics

Many species exhibit widespread spatial synchrony in population fluctuations. This pattern is of great ecological interest and can be a source of concern when the species is rare or endangered. Both dispersal and spatial correlations in the environment have been implicated as possible causes of this pattern, but these two factors have rarely been studied in combination. We develop a spatially structured population model, simple enough to obtain analytic solutions for the population correlation, that incorporates both dispersal and environmental correlation. We ask whether these two synchronizing factors contribute additively to the total spatial population covariance. We find that there is always an interaction between these two factors and that this interaction is small only when one or both of the environmental correlation and the dispersal rate are small. The interaction is opposite in sign to the environmental correlation; so, in the normal case of positive environmental correlation across sites, the population synchrony will be lower than predicted by simply adding the effects of dispersal and environmental correlation. We also find that population synchrony declines as the strength of population regulation increases. These results indicate that dispersal and environmental correlation need to be considered in combination as explanations for observed patterns of population synchrony.     

Effects of genetic drift on variance components under a general model of epistasis

We analyze the changes in the mean and variance components of a quantitative trait caused by changes in allele frequencies, concentrating on the effects of genetic drift. We use a general representation of epistasis and dominance that allows an arbitrary relation between genotype and phenotype for any number of diallelic loci. We assume initial and final Hardy-Weinberg and linkage equilibrium in our analyses of drift-induced changes. Random drift generates transient linkage disequilibria that cause correlations between allele frequency fluctuations at different loci. However, we show that these have negligible effects, at least for interactions among small numbers of loci. Our analyses are based on diffusion approximations that summarize the effects of drift in terms of F, the inbreeding coefficient, interpreted as the expected proportional decrease in heterozygosity at each locus. For haploids, the variance of the trait mean after a population bottleneck is var(delta(z)) = sigma(n)k=1 FkV(A(k)), where n is the number of loci contributing to the trait variance, V(A(1)) = V(A) is the additive genetic variance, and V(A(k)) is the kth-order additive epistatic variance. The expected additive genetic variance after the bottleneck, denoted (V*(A)), is closely related to var(delta(z)); (V*(A)) = (1 - F) sigma(n)k=1 kFk-1V(A(k)). Thus, epistasis inflates the expected additive variance above V(A)(1 - F), the expectation under additivity. For haploids (and diploids without dominance), the expected value of every variance component is inflated by the existence of higher order interactions (e.g., third-order epistasis inflates (V*(AA. This is not true in general with diploidy, because dominance alone can reduce (V*(A)) below V(A)(1 - F) (e.g., when dominant alleles are rare). Without dominance, diploidy produces simple expressions: var(delta(z)) = sigma(n)k=1 (2F)kV(A(k)) and (V(A)) = (1 - F) sigma(n)k=1 k(2F)k-1V(A(k)). With dominance (and even without epistasis), var(delta(z)) and (V*(A)) no longer depend solely on the variance components in the base population. For small F, the expected additive variance simplifies to (V*(A)) approximately equal to (1 - F)V(A) + 4FV(AA) + 2FV(D) + 2FC(AD), where C(AD) is a sum of two terms describing covariances between additive effects and dominance and additive X dominance interactions. Whether population bottlenecks lead to expected increases in additive variance depends primarily on the ratio of nonadditive to additive genetic variance in the base population, but dominance precludes simple predictions based solely on variance components. We illustrate these results using a model in which genotypic values are drawn at random, allowing extreme and erratic epistatic interactions. Although our analyses clarify the conditions under which drift is expected to increase V(A), we question the evolutionary importance of such increases.