Genetic variance components and heritability of multiallelic heterozygosity under inbreeding

Abstract

The maintenance of genetic diversity in fitness-related traits remains a central topic in evolutionary biology, for example, in the context of sexual selection for genetic benefits. Among the solutions that have been proposed is directional sexual selection for heterozygosity. The importance of such selection is highly debated. However, a critical evaluation requires knowledge of the heritability of heterozygosity, a quantity that is rarely estimated in this context, and often assumed to be zero. This is at least partly the result of the lack of a general framework that allows for its quantitative prediction in small and inbred populations, which are the focus of most empirical studies. Moreover, while current predictors are applicable only to biallelic loci, fitness-relevant loci are often multiallelic, as are the neutral markers typically used to estimate genome-wide heterozygosity. To this end, we first review previous, but little-known, work showing that under most circumstances, heterozygosity at biallelic loci and in the absence of inbreeding is heritable. We then derive the heritability of heterozygosity and the underlying variances for multiple alleles and any inbreeding level. We also show that heterozygosity at multiallelic loci can be highly heritable when allele frequencies are unequal, and that this heritability is reduced by inbreeding. Our quantitative genetic framework can provide new insights into the evolutionary dynamics of heterozygosity in inbred and outbred populations.

Figure 1

Qualitative explanation of heritability of heterozygosity. (a) A situation of random mating where offspring genotypes are produced according to the allele frequencies in the population. At most 50% of the offspring are expected to be heterozygous (dark gray areas). This maximum of 50% heterozygous offspring occurs when allele frequencies are equal (i.e. p=q). (b) A situation where a heterozygous individual mates with a random individual from the population. In this case, half of the offspring are expected to be heterozygous (dark gray areas), regardless of the allele frequencies in the population. Thus, for unequal allele frequencies (i.e. p≠q), matings involving a heterozygous individual produce more heterozygous offspring than random matings. Hence, heterozygosity is heritable when allele frequencies are unequal, or put another way, heterozygosity is heritable because of the presence of rare alleles.

Figure 2

Derivation of additive and dominance genetic variance of heterozygosity for one locus with two alleles with frequencies p and q (see text for details). Genotypic values G_ij of 1 are assigned to heterozygotes and 0 to homozygotes, and thus the mean genotypic value μ of the population equals 2pq. Gray circles represent genotypic values and their surface area represents genotype frequencies. Predicted values of a least-squares regression (black line) weighted by genotype frequencies of G_ij on the number of A₁ alleles represent breeding values (black dots) and the slope of this regression represents the average effect of allelic substitution α. Variance in breeding values represents the additive genetic variance. Dominance deviations (gray vertical lines) are differences between breeding values and genotypic values G_ij, and their variance represents dominance genetic variance. Panel a shows an example of unequal allele frequencies, in which case there exists additive genetic variance in heterozygosity, causing heritability of heterozygosity. Panel b shows the case of equal allele frequencies, leading to no additive genetic variance, but considerable dominance genetic variance.

Figure 3

Expected genetic variance components and heritability of heterozygosity for a locus with two alleles. Note that additive genetic variance is maximal at allele frequencies of (which is at p≈0.146 and ≈0.854), whereas heritability approaches maximal values at highly unequal allele frequencies.

Figure 4

Expected genetic variance components and heritability of heterozygosity for a locus with three alleles in a randomly mating, infinitely large population are shown in simplex plots. Along each side of the triangle, one allele has a frequency of 0%, which increases with distance away from the triangle side until reaching a frequency of 100% at the opposite corner. Colors indicate the values of heritability, total genetic, additive and dominance genetic variance, respectively. Note that the range of values is different for heritability than for the three genetic variance components. Note that the color key is different for genetic variance to allow for better visibility of patterns.

Figure 5

Relationship between variance in allele frequencies and heritability or genetic variance components of heterozygosity for different numbers of alleles. Note that the relationships are similar in shape for different numbers of alleles, but high variance in allele frequencies can only be reached when few alleles are present, leading to compressed curves for higher numbers of alleles. Although theoretically possible, many variances in allele frequencies shown here are unlikely or impossible in reasonably sized populations (e.g., <1000 individuals) because they may require some extremely low allele frequencies.

Figure 6

Relationship between mean heterozygosity and heritability or genetic variance components of heterozygosity for different numbers of alleles. Although theoretically possible, many data points shown here are unlikely or impossible in reasonably sized populations (e.g. <1000 individuals) because they may require some extremely low allele frequencies.

Figure 7

Plots of all genetic (co)variance components and heritability of heterozygosity under inbreeding. Values for a set of inbreeding coefficients are colored according to (l). (a and d) A randomly mating population with inbreeding because of genetic drift. (b and e) Experimental populations where one or several lines are inbred to a known and identical degree and descending from an infinitely large randomly mating population. These distinctions are not necessary for the other panels. Heritability (h²) of heterozygosity approaches 0, but is not defined for complete inbreeding (F=1) because then there is no genetic variance for heterozygosity as all individuals are completely homozygous (a and b). Genetic variance in heterozygosity (d and e) decreases with inbreeding. When genetic variance is partitioned into two components only (Equations (9) and (10)), both additive (c) and dominance (f) genetic variance decrease with inbreeding. When genetic variance is partitioned into five or six components (Equation (8)), the contribution from additive genetic variance because of random mating (g) increases with inbreeding, whereas the corresponding dominance variance (h) decreases. The covariance between the additive effect of alleles and their homozygous dominance deviations (i) becomes increasingly negative with inbreeding, whereas the dominance variance because of inbreeding (j) becomes increasingly positive. The inbreeding depression effect (k) is maximal for F=0.5 and 0 for F=0 and F=1, which means that some curves fall on top of each other (thus they are dashed).