The infinitesimal model with dominance.

The article focuses on extending the classical infinitesimal model of quantitative genetics to include dominance effects in diploid populations. The infinitesimal model traditionally assumes that a quantitative trait is the sum of many additive Mendelian factors of small effect, resulting in normally distributed offspring traits around the parental mean with variance independent of parental traits. This work rigorously justifies that when dominance is incorporated, the genetic component of offspring traits within families remains asymptotically multivariate normal as the number of loci tends to infinity, but now requires consideration of two-, three-, and four-way identity-by-descent probabilities determined by the pedigree. The model partitions the genetic trait into a shared family component and a residual Mendelian segregation component, both normally distributed even after conditioning on parental trait values, with explicit formulas for means and variances expressed in terms of classical quantitative genetics parameters and identity coefficients. The authors provide error bounds on the normal approximation, show that the infinitesimal model with dominance holds over timescales proportional to the square root of the number of loci, and illustrate their results with numerical simulations. This extension offers a mathematically grounded framework for modeling polygenic traits with dominance, preserving the robustness and predictive power of the infinitesimal model in evolutionary and breeding contexts.