Altenberg, L. 2011. An Evolutionary Reduction Principle for Mutation Rates at Multiple Loci. *Bulletin of Mathematical Biology* 73: 1227–1270, DOI: 10.1007/s11538-010-9557-9. Submitted September 13, 2009, Accepted: June 4, 2010.

The modifier gene model.

A model of mutation rate evolution for multiple loci under arbitrary selection is analyzed. Results are obtained using techniques from Karlin (1982) that overcome the weak selection constraints needed for tractability in prior studies of multilocus event models.

A multivariate form of the reduction principle is found: reduction results at individual loci combine topologically to produce a surface of mutation rate alterations that are neutral for a new modifier allele. New mutation rates survive if and only if they fall below this surface — a generalization of the hyperplane found by Zhivotovsky et al. (1994) for a multilocus recombination modifier. Increases in mutation rates at some loci may evolve if compensated for by decreases at other loci. The strength of selection on the modifier scales in proportion to the number of germline cell divisions, and increases with the number of loci affected. Loci that do not make a difference to marginal fitnesses at equilibrium are not subject to the reduction principle, and under fine tuning of mutation rates would be expected to have higher mutation rates than loci in mutation-selection balance.

Other results include the nonexistence of ‘viability analogous, Hardy- Weinberg’ modifier polymorphisms under multiplicative mutation, and the sufficiency of average transmission rates to encapsulate the effect of modifier polymorphisms on the transmission of loci under selection. A conjecture is offered regarding situations, like recombination in the pres- ence of mutation, that exhibit departures from the reduction principle. Constraints for tractability are: tight linkage of all loci, initial fixation at the modifier locus, and mutation distributions comprising transition probabilities of reversible Markov chains.

The reduction principle is an important issue for evolution and it is well documented in the paper. ... I do not know if Samuel Karlin, whose work has inspired the author, had noticed that a transition matrix for an ergodic irreducible Markov chain is symmetrizable if and only if the chain is reversible, but this is certainly of interest.— Anonymous reviewer

First off I like the paper. It is very clean and thought through and technically correct.— Anonymous reviewer

Consideration of generalizations of the reduction principle of evolutionary modifications of transmission parameters is a worth while endeavour. Formally, the present model and the ensuing results provide a non-trivial generalization.— Anonymous reviewer