Expected genetic contributions and their impact on gene flow and genetic gain

Long-term genetic contributions (r(i)) measure lasting gene flow from an individual i. By accounting for linkage disequilibrium generated by selection both within and between breeding groups (categories), assuming the infinitesimal model, a general formula was derived for the expected contribution of ancestor i in category q (mu(i)(q)), given its selective advantages (s(i)(q)). Results were applied to overlapping generations and to a variety of modes of inheritance and selection indices. Genetic gain was related to the covariance between r(i) and the Mendelian sampling deviation (a(i)), thereby linking gain to pedigree development. When s(i)(q) includes a(i), gain was related to E[mu(i)(q))a(i)], decomposing it into components attributable to within and between families, within each category, for each element of s(i)(q). The formula for mu(i)(q) was consistent with previous index theory for predicting gain in discrete generations. For overlapping generations, accurate predictions of gene flow were obtained among and within categories in contrast to previous theory that gave qualitative errors among categories and no predictions within. The generation interval was defined as the period for which mu(i)(q), summed over all ancestors born in that period, equaled 1. Predictive accuracy was supported by simulation results for gain and contributions with sib-indices, BLUP selection, and selection with imprinted variation.