The Evolutionary Reduction Principle for Linear Variation in Genetic Transmission

The evolution of genetic systems has been analyzed through the use of modifier gene models, in which a neutral gene is posited to control the transmission of other genes under selection. Analysis of modifier gene models has found the manifestations of an evolutionary reduction principle: in a population near equilibrium, a new modifier allele that scales equally all transition probabilities between different genotypes under selection can invade if and only if it reduces the transition probabilities. Analytical results on the reduction principle have always required some set of constraints for tractability: limitations to one or two selected loci, two alleles per locus, specific selection regimes or weak selection, specific genetic processes being modified, extreme or infinitesimal effects of the modifier allele, or tight linkage between modifier and selected loci. Here, I prove the reduction principle in the absence of any of these constraints, confirming a twenty-year-old conjecture. The proof is obtained by a wider application of Karlin's Theorem 5.2 (Karlin in Evolutionary biology, vol. 14, pp. 61-204, Plenum, New York, 1982) and its extension to ML-matrices, substochastic matrices, and reducible matrices.