CENTRAL EQUILIBRIA IN MULTILOCUS SYSTEMS

This paper is a continuation of the paper “Central Equilibria in Multilocus Systems I,’’ concentrating on existence and stability properties accruing to central H-W type equilibria in multilocus bisexual systems acted on by generalized nonepistatic selection forces coupled to recombination events. The stability conditions are discussed and interpreted in three perspectives, and the influence of sexual differences in linkage relationships together with sex-dependent selection is appraised. In this case we deduce that the stability conditions of the H-W polymorphism in the bisexual model coincide exactly with the conditions for the corresponding monoecious model, provided that the recombination distribution imposed is that of the arithmetic mean of the male and female recombination distributions. A second concern has the same recombination distribution for both sexes, but contrasting selection regimes between sexes. It is then established that, with respect to discerning the relevance of the H-W equilibrium, there is an equivalent monoecious selection regime which is an appropriate “weighted combination” of the male and female selection forms. Finally, in the case where the selection and recombination structures are both sex dependent, a hierarchy of comparisons is elaborated, seeking to unravel the nature of selection-recombination interaction for monoecious uerszu diocecious systems. HIS paper delineates existence criteria and stability properties accruing to central equilibria in multilocus bisexual systems acted on by generalized nonepistatic selection forces coupled to recombination events. An elaborate discussion of the corresponding monoecious model occurs in KARLIN and LIBERMAN (1979). Consider a trait determined by several loci. The bisexual selection regime comprises a pair of fitness value arrays in the form of generalized nonepistatic viability effects based on a prescribed set of intrinsic loci fitness matrices. Moreover, the linkage relationships (i.e., set of recombination-segregation frequencies) may differ between the sexes. Selection effects are formidable to establish, let alone when these are sex dependent. Identification of many cases of sex-dependent selection differentials in regard to quantitative and polygenic traits is well recognized. These cover Supported in part by Public Health Service Grant GM 10452-15 and National Science Foundation Grant MCS 7680624-AOl . * Department of Mathematics, Stanford University, Stanford, California 94305. Department of Statistics, Tel-Aviv University, Tel-Aviv, Israel and Department of Mathematics, Stanford University, Stanford, California 94305. Genetics 91: 79’3816 April, 1979. 800 S . KARLIN AND U. LIBERMAN traits such as stature, weight, a host of disease susceptibilities, metabolic rates, etc. Selection differences can be mediated by hormonal factors, environmental correlates and sex limited gene expression. Having in mind that the variation in polygenic traits can often be associated with a few to several major loci. entailing some heterogeneous background, the analyses of the appropriate bisexual multilocus systems may lend insights into mechanisms of polygenic inheritance. There are multilocus traits connected with color pattern, disease incidence, multiple enzyme functions entailing sex differences in selection expression. etc. For example, genomic complexes that show sex-dependent viabilities include cases related to mimicry and/or crytography based on sex-limited gene expression correlated with predation activities, the t-allele system where males are lethal and females normal, and male recombining genes that cause sterility. Another significant contrast between the sexes relates to the recombination distributions, where “more” recombination (diminished linkage) tends to occur for the heterogametic sex as compared to the rate of crossover events during homogametic meiosis. There are documented cases where recombination in females is unrestricted, while male recombination entails absolute linkage or limited numbers of crossover events. The phenomenon of reduced recombination rates for males (or, more accurately, for the heterogametic sex) as compared to iemales is colloquially called Haldane’s Rule (HALDANE 1922). A classic extreme example is manifested in Drosophila populations where usually no recombination for male segregation appears, while varying levels of positive recombination outcomes are realized during female meiosis. Genetical and cytological studies over these past several decades have documented significant numbers of animals with achiasmatic meiosis in one sex, usually the heterogametic sex. These include cases in protozoa, mites, copepods, (having heterogametic females) , mantids, a couple of grasshoppers, an assortment of Lepidoptera including Bombyx (again the female sex is heterogametic) and various other insects, including Drosophila (see WHITE 1973). The same is true in many plant species, including hermaphroditic species, where particularly meiosis occurs at differing times with respect to male and female gamete production. In human populations, linkage-mapping . via segregation and pedigree analysis also tends to confirm diminished intrachromosoma1 recombination frequencies (e.g., in chromosome I ) for males as against females (MCKUSICK and RUDDLE 1977). Within the plant and animal domains, many successful species have dispensed with intrachromosomal recombination in one sex, incorporating a sufficiency of recombination in the other sex. On the other hand, the great majority of plants and animals do not display any clear-cut sex-related linkage differentials. A recent review of contrasts in rates of recombination between the sexes is given in CALLAN and PERRY (1977). Apparently, if there is a difference in crossing over between the sexes, it is lower in the heterogametic sex in those species where morphologically differentiated X and Y chromosomes (or an XO mechanism) exist. The evidence is based on genetic crosses in some cases and on cytological analysis (from absence of chiasmata or lower number of chiasmata) in other cases. NONEPISTATIC SELECTION REGIMES I1 801 Where selection in the two sexes takes the form of a generalized nonepistatic structure based on the same intrinsic fitnesses and stipulating that these component fitness matrices exhibit “overdominant selection” (signifying that the marginal fitness matrix at each individual locus in isolation would maintain a stable polymorphism), then a common “central” Hardy-Weinberg (H-W) equilibrium, independent of the specific recombination distributions, is pervasive in the bisexual system. However, the stability endowments of this polymorphic H-W equilibrium are markedly influenced by the recombination mechanisms interacting with the genotypic fitness differentials invoked in the male and female populations. Our present objective is to discern, in qualitative and quantitative terms, the stability nature of the H-W “central” polymorphism in its dependence on the bisexual selection forces and recombination distributions. In the monoecious onesex case, we recall that in the presence of pure additive selection any non-zero recombination already ensures local stability (and likely global stability) of the H-W equilibrium. [For the two-locus, two-allele case, see KARLIN and FELDMAN (1970), and (for the multiallele situation), KARLIN and LIBERMAN (1978). ] On the other hand, under pure multiplicative viability effects, the H-W equilibrium is stable only provided that “enough” recombination contingencies are feasible. The precise statement and development of these facts are set forth in KARLIN and LIBERMAN (1979). With this background, it is of interest to unravel the balances attendant to alternative nonepistatic selection forms associated with the two sexes and to show how they account for the degree and strength of stability concomitant to the H-W equilibrium. In more specific terms we treat the following problems: (1) With the same selection regime but disparate patterns of recombination rates, how does the stability of the H-W polymorphism in the dioecious case compare with the “corresponding” monoecious case? Is there an “equivalent-effective” monoecious recombination distribution? (2) Considering the model involving sex-dependent selection intensities but with similar linkage relationships for both sexes, is there an “equivalent” monoecious selection form effective with respect to the stability analysis of the central H-W polymorphism? (3) How does the dioecious model carrying selection regimes rm and rf in males and females, respectively, compare with the monoecious model acted on by the selection regime , with reference to the “amount” of recombination needed to maintain the H-W polymorphism where no sex differences in linkage relationships are assumed. (4) What is the outcome due to coupling contrasting selection regimes for the sexes? Generallly. what kind of selection-recombination interaction accrues due to bisexuality ? Analysis of bisexual models in the past has focused primarily on one-locus, two-allele models [e.g., OWEN (1953) ; BODMER (1965) ; MBRAT (1969) ; MANDEL (1971) ; KIDWELL et al. (1977)l . In this context the realization of multiple stable polymorphisms was uncovered by OWEN (1953). Some contributions to the study of selection balance for dioecious populations (one-locus, multiallele cases) are 2 802 S. KARLIN AND U. LIBERMAN contained in CANNINGS (1967) and KARLIN (1978). The analysis of bisexual linkage selection interaction in multilocus systems is sparse and mostly of recent vintage. GEIRINGER (1 948) and BENNETT (1 954) previously confined their efforts to the case of no selection differentials operating on both sexes. Each established the property that one can treat the population as a monoecious population by replacing the recombination rates with the arithmetic mean of the separate recombination frequencies. STROBECK (1 975) proved the valid