A GENERALIZED MODEL OF MUTATION-SELECTIONBALANCE WITH APPLICATIONS TO AGINGDAVID STEINSALTZ+, STEVEN N. EVANS, AND KENNETH W. WACHTERAbstract. A probability model is presented for the dynamics of mutation-selection balance in a haploid infinite-popula tion infinit e-site s setting suffi-ciently general to cover mutation-driven changes in full age-specific demo-graphic schedules. The model accommodates epistatic as well as additiveselective costs. Closed form characterizations are obtained for solutions infinite time, along with proofs of convergence to stationary distributions and aproof of the uniqueness of solutions in a restricted case. Examples are givenof applications to the biodemography of aging.1. IntroductionArguments from the mathematical genetics of mutation-selection balance figurebroadly in evolutionary theories of senescence. Available formal models, however,do not cover cases brought to the fore by recent progress in biodemography [1]. Inthis pap er, we present a rigorous general model encompassing these cases, proveresults concerning existence, uniqueness, and convergence, obtain closed-form rep-resentations for solutions to the model, and give examples of its application toquestions in the demography of aging.The whole mathematical theory of natural selection may be divided into threeparts: positive mutations, neutral mutations, and deleterious mutations. Positivemutations may be thought to add up to an optimal adaptation, at least undersome conditions, and they are generally studied in that context by demographers.Neutral mutations have their primary effects in alleles which drift randomly tofixation. Deleterious mutations, the focal subject for theories of aging and forthis pape r, are expected never to achieve fixation in populations, except, throughfounder effects, in very small populations. Their influence in large populationsderives from their persistent reintroduction and slow meander to extinction.Sir Peter Medawar [2], in 1952, descried an explanation for senescence in theaccumulation of deleterious alleles with age-specific effects, given the declining forceof natural selection with adult age. W. D. Hamilton [3] presented expressionsfor this declining age-specific force, helping others quantify the resulting balance+To whom correspondence should be addressed at Department of Demography, University ofCalifornia, 2232 Piedmont Avenue, Berkeley, CA. 94720-2120; email [email protected] DAVID STEINSALTZ+, STEVEN N. EVANS, AND KENNETH W. WACHTERbetween mutation and selection. B. Charlesworth [4] analyzed the dynamics ofage-specific selection. His work guides the thinking of many experimentalists.At stake are the cumulative effects of numerous mildly deleterious mutationsshowing up at some large collection of loci. In our setting, the genotypes determinefull age-specific schedules of mortality and fertility, and the effects of a mutationhave to be represented as a perturbation of a whole function of age. A rigoroustreatment demands that mutations correspond to points in abstract spaces, suchas function spaces . Relationships between our work and the large literature onmutation and selection reviewed by B¨urger [5] are discussed in Section 8.Up to now, researchers have relied on linear approximations to cost functionsand restricted their representations of the age-sp e cific effects of mutations to styl-ized patterns like step-functions. Intriguing results have been obtained. Some arediscussed in Section 7. The linear analysis, however, can be deceptive, and thestylized patterns are remote from realistic portrayals of gene action. Cases chosenfor analytic tractability give a misleading picture of the full range of possibilities.Our model is an infinite-population, multiple-sites or infinite-sites model in con-tinuous time. The dynamical equation is a fairly standard one, but the space ofmathematical objects to which it applies is novel. Our model allows a highly flex-ible specification of pleiotropic gene action. It is especially suited to demographicapplications with mutant alleles affecting age-specific schedules. The model is ahaploid infinite-population model with no recombination. A parallel model withfree recombination, introduced in Section 9, will be developed in a future paper.Our contribution is to allow large numbers of interacting genes to m ake small con-tributions to a continuum of linked traits. Traditional analyses which recognize in-dividual alleles (thus admitting, in principle, arbitrary configurations of pleiotropy)are amenable only to small numbers of loci; quantitative genetics, which reducesthe contributions of individual genes to a continuum, reduces the complexity ofpleiotropy to covariance matrices.Although multi-locus models without recombination like our own can be formallyimbedded in single-locus mo dels, this imbedding will not generally yield use ful re-sults. When a multilocus model is translated into the single-locus framework, itbrings along an extra structure of transition rates, whose complexity grows ex-ponentially with the number of loci. When the number of loci is large or, as inour model, effectively infinite, this extra structure overwhelms the single-locus in-frastructure. In our function-space setting, the formal embedding itself also posesdifficulties. As a consequence, results for single-locus models are mainly helpful asanalogies.MUTATION-SELECTION MODEL 3Unlike most models of which we are aware, our model comfortably accommodatesepistasis. (A very different approach to epistasis, in the two-allele setting, may befound in [6].) The selective cost of a mutant allele can depend on the configurationof other mutant alleles present in a genome. This property is critical to the studyof senescence, even without special assumptions about interactions among genes,because the fitness costs of cumulative demographic changes are not linear.We are able to obtain closed-form representations of the entire time path ofsolutions to our dynamical equation (Theorem 3.1). Our results are not restricted,like much previous work, to limiting states and equilibrium distributions. We giveproofs of convergence over time (Theorem 4.1), and set machinery into place tocompute rates of convergence and to cope with changing fitness conditions as well.In Section 5, we present some results about the asymptotic behavior of solutions.Theorem 5.1 gives sufficient conditions for the numbers of certain classes of mutantalleles to increase