A pleiotropic model of phenotypic evolution

A pleiotropic model is presented for deriving the equilibrium genetic variance by mutation and stabilizing selection and the long-term genetic responses to directional selection in the case where mutations have pleiotropic effects on fitnes itself (direct deleterious effect) and on a quantitative trait (phenotypic effect). The equilibrium genetic variance is derived as a general form of the rare-alleles models, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\tilde V_G = \frac{{2n\mu \alpha ^2 }}{{s_u + \alpha ^2 /(2V_s )}}$$ \end{document}, where n is the number of loci, μ is the per-locus mutation rate, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\alpha ^2 $$ \end{document} is the variance of new mutations, Vs is the measure of stabilizing selection, and su is the selection coefficient on mutations by direct deleterious effect. The genetic responses to directional selection is calculated based on the assumption that the genetic variance is kept at an equilibrium by mutation and stabilizing selection but without directional selection, and the directional selection starts to operate on the target trait. The evolutionary rate at the t-th generation after the introduction of the directional selection is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\Delta \bar z(t) = i\tilde V_G e^{ - s_T t} $$ \end{document} , where i is the directional selection intensity, and sT is the total selection coefficient on mutations, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$s_u + \alpha ^2 /(2V_s )$$ \end{document} . The selection limit is\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$R = iV_m /s_T^2 $$ \end{document} , where Vm is the mutational variance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$(2n\mu \alpha ^2 )$$ \end{document}. The pleiotropic effects of genes reduce both the evolutionary rate and the selection limit.