The evolution of genetic systems has been studied through the use of modifier gene models, in which a selectively neutral gene controls genetic transmission of other genes under selection. Analytical studies to obtain a general understanding of the dynamics have faced tradeoffs between different features for tractability. No study has been able to obtain results for arbitrary selection on multiple loci undergoing multiple genetic transformation events. Here techniques from Karlin (1982) are applied to a multilocus model of mutation with multivariate control over mutation rates, and the reduction principle is found to operate. Constraining assumptions are that mutation distributions follow the transition probabilities of reversible Markov chains, and that all loci are tightly linked. The extension of the reduction principle is shown topologically to require a manifold of mutation rate alterations that are neutral for a new modifier allele , below which will cause a new modifierallele to increase when rare, and above which cause it to go extinct. This manifold is the same structure as found in a multivariate multilocus model of recombination modification by Zhivotovsky, Feldman, and Christiansen ( 1994 ). A discussion of the near-equilibrium models that depart from the reduction principle concludes with a conjecture about the structural causes .
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 presence 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 .
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The evolution of genetic systems has been studied through the use of modifier gene models, in which a selectively neutral gene controls genetic transmission of other genes under selection. Analytical studies to obtain a general understanding of the dynamics have faced tradeoffs between different features for tractability. No study has been able to obtain results for arbitrary selection on multiple loci undergoing multiple genetic transformation events. Here
A model of mutation rate evolution for multiple loci under arbitrary selection is analyzed. Results are obtained using
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are applied to a multilocus model of mutation with multivariate control over mutation rates, and the reduction principle is found to operate. Constraining assumptions are that mutation distributions follow the transition probabilities of reversible Markov chains, and that all loci are tightly linked. The extension
that overcome the weak selection constraints needed for tractability in prior studies of multilocus event models. A multivariate form
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shown topologically to require a manifold
found: reduction results at individual loci combine topologically to produce a surface
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, below which will cause a new modifierallele to increase when rare, and above which cause it to go extinct. This manifold is the same structure as found in a multivariate multilocus model of recombination modification by Zhivotovsky, Feldman, and Christiansen (
. New mutation rates survive if and only if they fall below this surface - a generalization of the hyperplane found by Zhivotovsky et al. (
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). A discussion of the near-equilibrium models that depart
) 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 presence of mutation, that exhibit departures
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concludes with a conjecture about the structural causes
. 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