The infinitesimal model with dominance

Abstract

The classical infinitesimal model is a simple and robust model for the inheritance of quantitative traits. In this model, a quantitative trait is expressed as the sum of a genetic and an environmental component, and the genetic component of offspring traits within a family follows a normal distribution around the average of the parents' trait values, and has a variance that is independent of the parental traits. In previous work, we showed that when trait values are determined by the sum of a large number of additive Mendelian factors, each of small effect, one can justify the infinitesimal model as a limit of Mendelian inheritance. In this paper, we show that this result extends to include dominance. We define the model in terms of classical quantities of quantitative genetics, before justifying it as a limit of Mendelian inheritance as the number, M, of underlying loci tends to infinity. As in the additive case, the multivariate normal distribution of trait values across the pedigree can be expressed in terms of variance components in an ancestral population and probabilities of identity by descent determined by the pedigree. Now, with just first-order dominance effects, we require two-, three-, and four-way identities. We also show that, even if we condition on parental trait values, the "shared" and "residual" components of trait values within each family will be asymptotically normally distributed as the number of loci tends to infinity, with an error of order 1/M. We illustrate our results with some numerical examples.

Keywords: dominance; infinitesimal model.

Fig. 1.

Three- and four-way identities. Lines indicate identity by descent between genes. See the main text for further explanation.

Fig. 2.

Changes of the mean and variance of the additive part of the trait, the dominance part, and their sum over 50 generations of neutral evolution. The top row shows a single replicate, while the bottom row shows the average over 300 replicates using the same sequence of individuals spanning the 50 generations. The left column shows the means ($\overline{G}=\overline{A}+\overline{D}$, $\overline{A}$, $\overline{D}$; black, or middle curve; blue, or bottom curve; red, or top curve), while the right column shows the variance components ($V_G=\mathtt{V}\mathtt{a}\mathtt{r}(G)$, $V_A=\mathtt{V}\mathtt{a}\mathtt{r}(A)$, $V_D=\mathtt{V}\mathtt{a}\mathtt{r}(D)$, $V_{A,D}=\mathtt{C}\mathtt{o}\mathtt{v}(A,D)$; black, or top curve; blue, or middle top curve; red, or middle bottom curve; purple, or bottom curve). On the right, solid lines show the total variances and covariance, while the dashed lines show the genic component. These differ through the contribution of linkage disequilibrium, which generates substantial variation. The genic component changes smoothly, as expected with a large number ($M=$1,000) of loci. With $M=$1,000 loci, we expect the infinitesimal model to be accurate for about $\sqrt{M}\sim 30$ generations. Simulations are made on a single pedigree with 30 individuals; variance components are measured relative to the ancestral population. The predicted values for these means and variances under the infinitesimal model are given in Equations (12)–(14) (note that the identity coefficients $F_{ii}$ increase through time due to genetic drift).

Fig. 3.

The relation between the dominance deviation and the probability of identity of the two genes within an individual. There is one point for the average over $1,000$ replicates for each of the 30 individuals in generations 5, 10, 20, 40 (black, or left-most group of points; blue, or second left-most group; purple, or second right-most group; red, or right-most group). (Recall that the pedigree is fixed, so identities are the same for each replicate.) The mean of D decreases as $\iota F_{ii}=-0.53F_{ii}$ (solid line), in accordance with Equation (12).

Fig. 4.

The variance and covariance of A and D versus identity $F_{ii}$ for individuals in the pedigree. As in Fig. 3, there are 30 points in each generation, one corresponding to each of the 30 individuals in the population. Generations 5, 10, 20, 40 (black, or left-most group of points; blue, or second left-most group; purple, or second right-most group; red, or right-most group). Here, again we use the shorter notation $V_A=\mathtt{V}\mathtt{a}\mathtt{r}(A)$, $V_D=\mathtt{V}\mathtt{a}\mathtt{r}(D)$, $V_{A,D}=\mathtt{C}\mathtt{o}\mathtt{v}(A,D)$ and the theoretical predictions were derived in Equations (13) and (14).

Fig. 5.

The variance and covariance within families between the residual additive and dominance deviations $R_A$ and $R_D$ ($V_{R_A}=\mathtt{V}\mathtt{a}\mathtt{r}(R_A)$, $V_{R_D}=\mathtt{V}\mathtt{a}\mathtt{r}(R_D)$, $V_{R_A,R_D}=\mathtt{C}\mathtt{o}\mathtt{v}(R_A,R_D)$). One hundred pairs of parents were chosen at random from the ancestral population and from each one thousand offspring were generated. The within-family variances obtained in this way were averaged over 10 replicates (with the same pedigree and parents). Each of the 100 points in each plot corresponds to one pair of parents. The five outliers are families produced by selfing. The blue lines (or top lines) show a least-squares regression; the red lines (or bottom lines) are the theoretical predictions [see Equation (8)]. The two lines exactly coincide in the plot on the right.

Fig. 6.

Comparison between a neutral population (dashed lines) and one subject to truncation selection (solid lines). Top row: change in means relative to the initial value ($\overline{G}=\overline{A+D}$, $\overline{A}$,$\overline{D}$; black, or top curves; blue, or middle curves; red, or bottom curves); middle: variances, including linkage disequilibria (top to bottom: $V_G=\mathtt{V}\mathtt{a}\mathtt{r}(A+D)$, $V_A=\mathtt{V}\mathtt{a}\mathtt{r}(A)$, $V_D=\mathtt{V}\mathtt{a}\mathtt{r}(D)$, $V_{A,D}=\mathtt{C}\mathtt{o}\mathtt{v}(A,D)$; black, blue, red, purple). The bottom row is the changes to genic variances with time against predictions of the infinitesimal model. The values are averages over 300 replicates for the neutral case, 1,000 for the selected case, made with the same pedigree. There are $M=1,000$ loci, and thus we expect the infinitesimal model to be accurate for about $\sqrt{M}\sim 30$ generations. Selection is made within families; for each offspring, two individuals are generated from the corresponding parents, and the one with the larger trait value retained.

Fig. 7.

Convergence of the variance components at 50 generations, as the number of loci increases from $M=100$ to $M={10}^4$ (same notation as in Fig. 6). Simulations with 50% truncation selection are compared with neutral simulations (solid, dashed lines). The replicate simulations were generated as in Fig. 6 (see main text). Regressions of the log absolute difference between selected and neutral variance components against $\ln (M)$ have slopes $-$0.62, $-$0.72, $-$0.70, $-$0.66 for $V_G$, $V_A$, $V_D$, $V_{A,D}$, respectively (see supplementary material for details). Thus, convergence is somewhat faster than $\sqrt{M}$.

Fig. 8.

The distributions of the residual (top row: $R_A$ , $R_D$) and shared (bottom row: $\mathcal{A}$, $\mathcal{D}$) components of phenotype ($M=1,000$ loci); for each, the cumulative distribution function is plotted as standard deviations of a Gaussian, z, so that a normal distribution appears as a straight line. These are calculated from families of 1,000 offspring, from multiple pairs of parents, each replicated 10 times, drawn after 20 generations without selection. The residuals are calculated by subtracting values from the family mean, and pooling across the 10 replicates. Thus, for each family there are 10,000 values; the cumulative distribution function is shown for 10 pairs of parents, in 10 colors. The shared component is calculated by taking the mean of each family, and pooling across 100 pairs of parents and across the 10 replicates. Thus, for each plot there are 1,000 points. There is now some deviation from a Gaussian.

Fig. B1.

All possible four-way identities. The dots represent the four genes across the two parents (each parent corresponding to a row) and lines indicate identity (c.f. Abney et al. 2000).