A Stochastic Model of Gene-culture Coevolution Suggested by the "culture Historical Hypothesis" for the Evolution of Adult Lactose Absorption in Humans

A stochastic model of gene-culture coevolution, suggested by the "culture historical hypothesis" of Simoons and McCracken, is presented. According to this hypothesis, adult lactose absorption, believed to be an autosomal dominant trait, attained a high frequency in some human populations due to the positive selection pressure induced by culturally determined milk use in those populations. Two-dimensional Kolmogorov backward equations with appropriate boundary conditions are derived for the ultimate fixation probability of milk users, of the gene for adult lactose absorption, and of both jointly, and for the average time until fixation of the gene. These boundary value problems are solved numerically by the Gauss-Seidel method. I define a theoretical measure of the correlation between gene and culture in terms of the three ultimate fixation probabilities. Monte Carlo simulations are conducted to check and extend the numerical results and also to obtain the first arrival time at gene frequency 0.70, which is approximately the highest observed frequency in any population. Two results that pertain to the culture historical hypothesis are obtained. First, the incomplete correlation observed between adult lactose absorption and milk use does not necessarily constitute evidence against the hypothesis. Second, for the postulated genetic change to have occurred within the 6000-year period since the advent of dairying, either the effective population size was of the order of 100, or, if it was of larger order, the selection coefficient probably had to exceed 5%.