When can fitness epistasis be ignored in a polygenic trait at equilibrium?

Abstract

Although many phenotypic traits are determined by a large number of genetic variants, the behavior of allele frequencies in a polygenic trait is not completely understood. The problem is especially challenging when the quantitative trait of interest is under epistatic selection as the allele frequency at a locus is affected by those at other loci. Here, we consider a panmictic, diploid finite population evolving under stabilizing selection and symmetric mutations when the population is in linkage equilibrium. In the stationary state, using a diffusion theory, we calculate the marginal distribution of allele frequency, and find parameter regimes where fitness epistasis can not be ignored for an accurate description of the frequency distribution. For such parameters, the mean deviation in the phenotypic optimum and genic variance are, however, found to be well captured even when epistatic interactions are neglected. Thus, while the presence of epistasis may not be evident in phenotypic quantities, it can strongly affect the allele frequency distribution.We also find that the allele frequency distribution at a locus is unimodal if its effect size is below a threshold effect and bimodal otherwise; these results are the stochastic analog of the deterministic ones where the stable allele frequency becomes bistable when the effect size exceeds a threshold. Our analytical results are verified against Monte Carlo simulations and numerical integration of a Langevin equation.

Figures (7)

• Figure 1: Diagram depicting how each evolutionary force shapes the genotypic configuration of individuals in a Wright-Fisher process implemented in Monte Carlo simulations. Here, we consider $N=4$ diploid individuals with $L=3$ loci to show the simulation steps from generation $t$ to $t+1$.
• Figure 2: Marginal distribution for weak mutation ($4 N \mu < 1$) at a locus with effect size (a) $\gamma_i=0.01$, (b) $\gamma_i=0.05$ and (c) $\gamma_i=0.7$. The parameters are $N=1000$, $s=0.05$, $\mu=0.00002, L=200$, $\bar{\gamma}=0.1$, and $z_o=1$. The deterministic threshold size ${\hat{\gamma}}=0.056$ for these parameters whereas stochastically, there is no threshold effect. The points are obtained from MC simulations and the red solid line shows the analytical expression (\ref{['psimain']}) where $\kappa_2 \approx 4.8$ for the effect sizes used in this plot. Here, $2 N s {\bar{\gamma}}^2=1$ and as both $\kappa_2$ and $2 N s \kappa_2$ are large, (\ref{['epiig']}) is satisfied and the marginal distribution (\ref{['indmarg']}) shown by black dashed line matches the distribution $\psi$.
• Figure 3: Marginal distribution for strong mutation ($4 N \mu > 1$) and strong selection ($N s {\bar{\gamma}}^2 > 2$) at a locus with effect size (a) smaller ($\gamma_i=0.1$), (b) just below ($\gamma_i=0.23$), (c) just above ($\gamma_i=0.3$) and (d) larger ($\gamma_i=0.7$) than the threshold size ${\hat{\gamma}}_N(L)=0.24$. The parameters are $N=1000$, $s=0.1$, $\mu=0.001, L=200$, $\bar{\gamma}=0.7$, and $z_o=1$. For the set of effects used here, there were $137$ large-effect loci. The points are obtained by solving (\ref{['Langevin']}) numerically, and the red solid line shows the analytical expression (\ref{['psimain']}) where $\kappa_2 \approx 235$ for the effect sizes used in this plot. Here, as both $\kappa_2$ and $2 N s \kappa_2$ are large, (\ref{['epiig']}) is satisfied and the marginal distribution (\ref{['indmarg']}) shown by black dashed line matches the distribution $\psi$.
• Figure 4: Marginal distribution for strong mutation ($4 N \mu > 1$) and weak selection ($N s {\bar{\gamma}}^2 < 2$) at a locus with effect size (a) smaller ($\gamma_i=0.05$), (b) close to ($\gamma_i=0.4$) and (c) larger ($\gamma_i=0.5$) than the threshold size ${\hat{\gamma}}_N(L) \approx 0.42$. The parameters are $N=1000$, $s=0.1$, $\mu=0.002, L=1000$, $\bar{\gamma}=0.08$, and $z_o=2$. For the set of effects used here, there were $7$ large-effect loci. The points are obtained by solving (\ref{['Langevin']}) numerically, and the red solid line shows the analytical expression (\ref{['psimain']}) where $\kappa_2 \approx 1.24$ for the effect sizes used in this plot. Here, although $2 N s \kappa_2 \approx 248$ is large, as $\kappa_2$ is not very large, (\ref{['epiig']}) is not satisfied and the marginal distribution (\ref{['indmarg']}) shown by black dashed line does not match the distribution $\psi$ at loci except when the effect size is small.
• Figure 5: Mean genic variance (\ref{['gvdef']}) for $L=2$ for (a) weak and (b) strong mutation. In all the figures, $N=1000, z_o=0, \gamma_i=0.5$ (equal effects). In the legend, the line denoted by 'Exact' is obtained by numerically integrating the double integral of the joint distribution in the result (\ref{['Ec2def']}) of diffusion theory, whereas 'Bulmer' shows the result (\ref{['varB']}) obtained by Bulmer, while 'MC' and 'EM', respectively, represent the data obtained using Monte-Carlo simulations and numerical integration of Langevin equation via Euler-Maruyama method.