Rules for assortative mating in relation to selection for linear merit functions

Summary This study examined how assortative mating (without selection) based on linear combinations of two traits could be used to change genetic parameters so as to increase efficiency of selection. The efficiency of the Smith-Hazel index for improvement of multiple traits is a function of phenotypic and genetic variances and covariances, and of the relative economic values of the traits involved. Assortative mating is known to change genetic variances and covariances. Recursive formulae were derived to obtain these variances and covariances after t generations of assortative mating on linear combinations (mating rules) of phenotypic values for two traits, with a given correlation between mates. Selection efficiency after t generations of assortative mating without selection was expressed as a function of random mating genetic parameters, economic values, the mating rule, and the correlation between mates. Selection efficiency was maximized with respect to the coefficients in the mating rule. Because the objective function was nonlinear, a computer routine was used for maximizing it. Two cases were considered. When random mating heritabilities for the two traits were h_X²=0.25 and h_Y²=0.50, the genetic correlation r_XY=-0.60, and the economic values were a_X=3 and a_Y=1, continued assortative mating based on the optimal mating rule for 31 generations (with a correlation of 0.80 between mates) increased selection efficiency by 29%. Heritabilities changed to 0.38 and 0.66, respectively, and the genetic correlation became – 0.79. When h_X²=0.60, h_Y²=0.60, r_XY=– 0.20, a₁=1 and a₂=1, 36 generations of continued assortative mating with the optimal mating rule increased the efficiency of selection by 17%, heritabilities became h_X²= h_Y²=0.71, and the genetic correlation changed to 0.25. Only three generations of assortative mating were required to change the sign of the genetic correlation.     

Two-trait selection response with marker-based assortative mating

 Marker-based assortative mating (MAM) – the mating of individuals that have similar genotypes at random marker loci – can increase selection response for a single trait by 3–8% over random mating (RM). Genetic gain is usually desired for multiple traits rather than for a single trait. My objectives in this study were to (1) compare MAM, phenotypic assortative mating (PAM), and RM of selected individuals for improving two traits and (2) determine when MAM will be most useful for improving two traits. I simulated 20 generations of selecting 32 out of 200 individuals in an F₂ population. The individuals were selected based on an index (SI) of two traits and were intermated by MAM, PAM, or RM. I studied eight genetic models that differed in three contrasts: (1) weight, number of quantitative trait loci (QTL), and heritability (h ²) for each trait; (2) linkage of QTL for each trait; and (3) trait means of the inbred parents of the F₂. For SI and the two component traits, MAM increased short-term selection response by 5–8% in six out of the eight genetic models. The MAM procedure was least effective in two genetic models, wherein the QTL for one trait were unlinked to the QTL for the other trait and the parents of the F₂ had divergent means for each trait. The loss of QTL heterozygosity was much greater with MAM than with PAM or RM. Consequently, the advantage of MAM over RM dissipated after 5–7 generations. Differences were small between selection responses with PAM and RM. The MAM procedure can enhance short-term selection response for two traits when selection is not stringent, h ² is low, and the means of the parents of the F₂ are equal for each trait. Received: 10 June 1998 / Accepted: 5 August 1998     

Effects of genetic architecture on the evolution of assortative mating under frequency-dependent disruptive selection

We consider a model of sympatric speciation due to frequency-dependent competition, in which it was previously assumed that the evolving traits have a very simple genetic architecture. In the present study, we numerically analyze the consequences of relaxing this assumption. First, previous models assumed that assortative mating evolves in infinitesimal steps. Here, we show that the range of parameters for which speciation is possible increases when mutational steps are large. Second, it was assumed that the trait under frequency-dependent selection is determined by a single locus with two alleles and additive effects. As a consequence, the resultant intermediate phenotype is always heterozygous and can never breed true. To relax this assumption, here we add a second locus influencing the trait. We find three new possible evolutionary outcomes: evolution of three reproductively isolated species, a monomorphic equilibrium with only the intermediate phenotype, and a randomly mating population with a steep unimodal distribution of phenotypes. Both extensions of the original model thus increase the likelihood of competitive speciation.     

Effects of assortative mating for pupa weight on genetic parameters of unselected Tribolium castaneum

Summary Effects of random (R) or positive assortative (A) mating for pupal weight (PW) on genetic parameters of pupation time (PT), pupal and larval weights (LW) were studied in unselected populations of Tribolium castaneum. Two groups, each with 50 males mated to 100 females in each of 5 replicates, were either R-mated or A-mated for 3 generations. Genetic parameters were estimated from covariances between sibs (R group) or by an iterative method (A group). Estimates of heritability in R and A groups were 0.30±0.12 and 0.39±0.02 (PW); 0.26±0.13 and 0.49±0.04 (LW); and 0.39±0.10 and 0.25±0.03 (PT). Estimates of genetic correlations in the R group were –0.21±0.23 (PW and LW); 0.45±0.10 (PW and PT); and –0.77±0.14 (LW and PT). Those in the A group were 0.27±0.10 (PW and LW); 0.15±0.14 (PW and PT); the genetic correlation between LW and PT was not estimable in this group. Within-family variances (grams squared) of PW by generation (1, 2, and 3) were, respectively: 0.048 (R) and 0.047 (A); 0.054 (R) and 0.041 (A); and 0.050 (R) and 0.046 (A). In agreement with theory, estimates of heritability of PW and LW were larger in the A group. Estimates of genetic correlations in the A group were inconsistent with expectations from theory. Assortative mating tended to decrease within-family variance of PW.     

A general linear model for the genotypic covariance between relatives under assortative mating

A linear model for the genotypic covariance between relatives under assortative mating comprising the classical linear model and the model of selective assortative mating is proposed. The general conditions on the genetical and developmental mechanisms of quantitative characters, as well as on selection and the mating system, on which the model is based, are explicitly stated and discussed. A classification of different relationships is presented and it is shown that these conditions are sufficient to obtain the genotypic covariance between relatives only if the relationship is a combination of descendant-ancestor, full sib, Type 1 and Nth uncle-niece relationships. All the traditional relationships, i.e., those for which the covariances of the relatives have been obtained in the literature, fall into this category. These conditions also ensure that the regression of the individual's genotypic value on the genotypic value or phenotype of any of its ancestors is always linear.Paper No. 6619 of the Journal Series of the North Carolina Agricultural Research Service, Raleigh, North Carolina. This investigation was supported in part by NIH Research Grant No. GM 11546 from the National Institute of General Medical Sciences     

Sexual size dimorphism and assortative mating for morphological traits in Passer domesticus

In this study, assortative mating for different morphological traits was studied in a captive population of house sparrows (Passer domesticus). Males were larger than females. Assortative mating was found for tail length, wing length and general body size. Males with larger badge size mated with females with longer tails. The strongest assortative mating occurred for tail length (r=0.77), and this assortative mating remained significant after controlling for wing length, mass and tarsus length, suggesting that it was not an artefact of assortative mating for body size. The possibility of sexual selection for tail length in the house sparrow is discussed.     

Convergence of genotypic frequencies for differential selfing and positive assortative mating at a biallelic locus

For a biallelic model of differential self-fertilization and differential positive assortative mating based on genotype, it is shown that the genotypic frequencies converge for all sets of mating system parameters. Overdominance and underdominance with respect to the parameters are necessary but not sufficient conditions for global convergence to a polymorphic equilibrium and local attractiveness of both the fixation states, respectively. There are cases of overdominance and underdominance for which one fixation state is globally attractive. The relationship of the result to those known from the classical viability selection model are briefly discussed. For the multiallelic version, it is shown that after the first generation all of the homozygote frequencies are always in excess of the corresponding Hardy-Weinberg proportions if at least one homozygote rate of self-fertilization or assortment probability is positive.     

Assortative mating through a mechanism of sexual selection

We propose that assortative mating can arise through a mechanism of sexual selection by active female choice of partners based on a 'self-seeking like' decision rule. Using a mathematical model, we show that a gene can be selected that make females to choose mates that are similar to themselves with respect to an arbitrary tag, even if two independent and unlinked genes determine the preference and the tag. The necessary requisite for this process to apply is an asymmetry between partners, such that the female can choose the male but this one must always accept to mate. The fitness advantage is due to linkage disequilibrium built up between both genes. Simulations have been run to check the algebraic results and to analyse the influence of several factors on the evolution of the system. Any factor that favors linkage disequilibrium also favors the evolution of the preference allele. Moreover, in a large population subdivided in small subpopulations connected by migration, the assortative mating homogenizes the population genotypic structure for the tags in contrast to random mating that maintains most of the variation.     

Incipient allochronic speciation due to non-selective assortative mating by flowering time, mutation and genetic drift

We model the evolution of flowering time using a multilocus quantitative genetic model with non-selective assortative mating and mutation to investigate incipient allochronic speciation in a finite population. For quantitative characters with evolutionary parameters satisfying empirical observations and two approximate inequalities that we derived, disjunct clusters in the population flowering phenology originated within a few thousand generations in the absence of disruptive natural or sexual selection. Our simulations and the conditions we derived showed that cluster formation was promoted by limited population size, high mutational variance of flowering time, short individual flowering phenology and a long flowering season. By contrast, cluster formation was hindered by inbreeding depression, stabilizing selection and pollinator limitation. Our results suggest that incipient allochronic speciation in populations of limited size (satisfying two inequalities) could be a common phenomenon.     

Differential contribution of stature phenotypes to assortative mating in parents of Philadelphia Black and White school children

Assortative mating for stature, weight, and the ponderal and Quetelet's indices is considered in a large sample of Philadelphia Blacks and Whites. The husband-wife correlation for stature is positive and significant in Whites (r = 0.34, d.f. = 382), but negligible in Blacks (r = 0.06, d.f. = 420). Correlations for weight and the body indices are positive and significant in both samples, those for body indices showing some dependency on the husband-wife correlations for stature and weight. When couples are grouped into statural mating combinations on the basis of short (S), medium (M) and tall (T), White spouses' statures show an approximately linear relationship to one another. The distribution of Black spouses' statures, however, is not completely independent, even though the husband-wife correlation is close to zero. There are elements of both positive and negative assortative mating among Blacks, resulting in an excess of certain mating types over that expected on the basis of chance. These mating types are usually those in which the husband is shorter than the wife, except at the heterogeneous extremes of the bivariate array.     

The effect of positive assortative mating on genetic parameters in a simulated beef cattle population

Summary Two simulated data sets, representing random mating and positive assortative mating in a beef cattle population over 10 rounds of mating, were each composed of 100 replicates. Three correlated traits were considered; calving ease (CE), 200 day weight (WW) and postweaning gain (PG). All selection practiced in the simulation was random. Positive assortative mating, which was based on parental WW phenotypic records, increased the progeny additive genetic variance of WW. The absolute values of genetic covariances and correlations between WW with CE and PG were also increased by positive assortative mating. Variances or covariances did not reach their expected equilibrium values due to overlapping generations, low replacement rates and only 10 rounds of mating.The financial assistance of Agriculture Canada and the Natural Sciences and Engineering Research Council of Canada are gratefully acknowledged     

Positive assortative mating with selection restrictions on group coancestry enhances gain while conserving genetic diversity in long-term forest tree breeding

Selection and mating principles in a closed breeding population (BP) were studied by computer simulation. The BP was advanced, either by random assortment of mates (RAM), or by positive assortative mating (PAM). Selection was done with high precision using clonal testing. Selection considered both genetic gain and gene diversity by "group-merit selection", i.e. selection for breeding value weighted by group coancestry of the selected individuals. A range of weights on group coancestry was applied during selection to vary parent contributions and thereby adjust the balance between gain and diversity. This resulted in a series of scenarios with low to high effective population sizes measured by status effective number. Production populations (PP) were selected only for gain, as a subset of the BP. PAM improved gain in the PP substantially, by increasing the additive variance (i.e. the gain potential) of the BP. This effect was more pronounced under restricted selection when parent contributions to the next generation were more balanced with within-family selection as the extreme, i.e. when a higher status effective number was maintained in the BP. In that case, the additional gain over the BP mean for the clone PP and seed PPs was 32 and 84% higher, respectively, for PAM than for RAM in generation 5. PAM did not reduce gene diversity of the BP but increased inbreeding, and in that way caused a departure from Hardy-Weinberg equilibrium. The effect of inbreeding was eliminated by recombination during the production of seed orchard progeny. Also, for a given level of inbreeding in the seed orchard progeny or in a mixture of genotypes selected for clonal deployment, gain was higher for PAM than for RAM. After including inbreeding depression in the simulation, inbreeding was counteracted by selection, and the enhancement of PAM on production population gain was slightly reduced. In the presence of inbreeding depression the greatest PP gain was achieved at still higher levels of status effective number, i.e. when more gene diversity was conserved in the BP. Thus, the combination of precise selection and PAM resulted in close to maximal short-term PP gain, while conserving maximal gene diversity in the BP.Communicated by O. Savolainen     

Assortative mating for anthropometric traits in parents of Punjabi twins

The marital correlations between 97 pairs of parents of Punjabi twins reveal positive phenotypic assortative mating for body traits while almost random mating with respect to cranio-facial traits. There is no evidence of any significant negative assortative mating for any of the 50 traits. The results have been compared with those from other world populations. The data contradict the earlier reported hypothesis that assortative mating is associated with lowered fertility.     

Assortative mating for a quantitative character

Phenotypic assortative mating is investigated for a character determined by additive loci without dominance and a stochastically independent environment. Conditional-expectation arguments are used to calculate the equilibrium values of the phenotypic variance and the correlation between sundry relatives. For the latter, it suffices to suppose that the regressions of an individual's genotype on his phenotype and of his phenotype on that of his mate are linear. For the former, linearity of the regression of the allelic effects on the phenotype is also posited. The biological implications of these assumptions are discussed.Supported by National Science Foundation Grant DEB81-03530     

The dynamics of sexually antagonistic coevolution and the complex influences of mating system and genetic correlation

Sexual conflict has been proposed to be a mediator of speciation but recent theoretical work, as well as empirical studies, suggests that sexual conflict may also be able to prevent speciation and to preserve genetic polymorphism within a species. Here, we develop a population genetic model and study the effects of sexual conflict in a polymorphic population. The morphs mate assortatively based on different sexually antagonistic traits and females are assumed to suffer a cost when the proportion of matching males is high. We consider the model in two different mating systems; promiscuity and polygyny. Our results show that genetic polymorphism may be maintained through negative frequency dependent selection established by assortative mating and female conflict costs. However, the outcome significantly differs between mating systems. Furthermore, we show that indirect selection may have profound effects on the evolutionary dynamics of a sexual conflict.     

A variable selection index for the compensation of correlated genetic change

Summary A selection index for two traits has been constructed which allows partial restriction for one of the traits. The index is used in a situation where correlated response to selection in one sex is compenstated for by selection for other traits in the opposite sex. A numerical example is given.     

Emergence and loss of assortative mating in sympatric speciation

We have studied an agent model which presents the emergence of sexual barriers through the onset of assortative mating, a condition that might lead to sympatric speciation. In the model, individuals are characterized by two traits, each determined by a single locus A or B. Heterozygotes on A are penalized by introducing an adaptive difference from homozygotes. Two niches are available. Each A homozygote is adapted to one of the niches. The second trait, called the marker trait has no bearing on the fitness. The model includes mating preferences, which are inherited from the mother and subject to random variations. A parameter controlling recombination probabilities of the two loci is also introduced. We study the phase diagram by means of simulations, in the space of parameters (adaptive difference, carrying capacity, recombination probability). Three phases are found, characterized by (i) assortative mating, (ii) extinction of one of the A alleles and (iii) Hardy-Weinberg like equilibrium. We also make perturbations of these phases to see how robust they are. Assortative mating can be gained or lost with changes that present hysteresis loops, showing the resulting equilibrium to have partial memory of the initial state and that the process of going from a polymorphic panmictic phase to a phase where assortative mating acts as sexual barrier can be described as a first-order transition.     

Sexual selection and assortative mating by size and their roles in the maintenance of a polymorphism in Swedish Littorina saxatilis populations

Two independent components of mating behaviour, sexual selection and assortative mating, were studied in two allopatric morphs, one sheltered boulder shore form (S-morph) and one exposed cliff shore form (E-morph), of Littorina saxatilis from the west coast of Sweden. Sexual selection was studied by comparing the sizes of copulating and non-copulating snails in the field. Size assortative mating was studied by collecting copulating pairs in the field, while assortative mating between morphs was investigated by bringing the pure morphs together in intermediary habitats and then noting the matings. The S-morph mated randomly in relation to size in two of the studied populations and exhibited a trend towards size assortative mating in a third, while the E-morph showed size assortative mating in both studied populations. The microdistribution of sizes of snails on the shores could not explain all the size assortative mating found, and instead it is argued that a size-based mate rejection behaviour also contributes to the assortative mating in at least some of these populations. There was sexual selection on size in both males and females in the S-morph, with large individuals being favoured as mates. In contrast, copulating snails of the E-morph were smaller than non-copulating ones. The significantly different sexual selection intensities between the two morphs may help to explain the size differences between them. There was random mating between the E- and the S-morphs of L. saxatilis, which suggests no incipient reproductive isolation between morphs on Swedish rocky shores. This is in agreement with earlier studies of Swedish populations, but is in contrast to the situation found in other geographical areas.