A Generalization of Theory on the Evolution of Modifier Genes

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A Generalization of Theory on the Evolution of Modifier Genes

Lee Altenberg

DISCUSSION, excerpt

UNDERLYING HOMOLOGY BETWEEN PHENOTYPE DISTRIBUTION MODELS AND MODIFIER MODELS

An important class of models in the literature are those which are concerned with the evolution of phenotypes. In these models, selection is determined by the phenotype, while the genotype, rather than directly specifying fitnesses, specifies probabilities that individuals have different phenotypes. These models have been posed to investigate situations where the fitnesses of the phenotypes are dependent on their frequencies. They include models of sex ratio, altruism, behavioral contests, habitat selection, and general frequency dependent selection. What is quite significant is that in each of these models there are two kinds of polymorphic equilibria that bear strong resemblance with two of the kinds of polymorphisms that occur with modifiers: viability-analogous, tensor product polymorphisms and balanced mixture modifier polymorphisms. The table below lists these examples and the nomenclature that has been used in the different papers for these two kinds of polymorphism.

TABLE 13. HOMOLOGY BETWEEN MODIFIER AND PHENOTYPE DISTRIBUTION MODELS. AME/HW = "Allelic Marginal Equivalence, Hardy-Weinberg"
Topic	Phenotypes	AME/HW	Balanced Mixture	References
		Type of Polymorphism
		Term used for polymorphism:
Sex Ratio	male, female	"symmetric"	"asymmetric"	Eshel & Feldman (1982)
Sex Ratio	male, female		"even sex ratio"	Karlin and Lessard (1983)
Kin Selection	selfish, altruist	"viability-analogous"	"structural"	Uyenoyama et al (1981)
Behavioral Contests	hawk, dove	"not ESS"	"ESS"	Maynard Smith (1981)
Habitat Preference	habitat choice	"symmetric"	"asymmetric"	Rausher (1984)
Frequency Dependent Selection	phenotype	"genotype"	"phenotypic"	Lessard (1984)
Frequency Dependent Selection	phenotype		"equilibriated fitnesses"	Slatkin (1979)

In correspondence between modifier and phenotype distribution models is as follows:

MODIFIER MODEL PHENOTYPE DISTRIBUTION MODEL
selected type phenotype
transformation type genotype
transformation probabilities phenotype distribution

MODIFIER MODEL	PHENOTYPE DISTRIBUTION MODEL
selected type	phenotype
transformation type	genotype
transformation probabilities	phenotype distribution

In these models, the two kinds of polymorphism can be characterized this way:

Viability-analogous, tensor product equilibria. The gene frequencies are the same in each phenotypic class, and therefore the phenotype distribution is the same for each gene. Thus the tensor product and viability-analogous properties refer to the same thing.
Balanced mixture equilibria. The fitnesses of each phenotype are equal. But different genes may have different phenotype distributions, and different phenotypic classes may have different gene frequencies.

In the different cases in the literature, each model has different means by which the phenotypic fitnesses become equal for the balanced mixture equilibria:

In the case of behavioral contests, the population is at an equilibrium of classic game theory, where the payoffs for each behavior are the same.
In the habitat selection case, soft selection in each habitat results in culling in those habitats that are over their carrying capacity, and filling-out in habitats that are under their carrying capacity. At a balanced mixture equilibrium, each habitat is filled to the same extent with respect to its carrying capacity, and this is achieved by a balance mixture of different genotypes that prefer different habitats.
In the sex ratio case, it may be possible to define a value for the "fitness" of each sex, and at equilibrium the fitnesses for the two sexes are equal. This fitness is defined in diploid models by allowing random mating between all combinations of the two sexes and assigning fertility values of zero to matings of the same sex. Sex ratios of one-to-one give equal fitnesses for each sex. A different method must be employed for the haplodiploid model of Eshel and Feldman (1982). The gene frequencies among each sex can be different.
In the kin selection case, the fitness of a phenotype is not so readily defined. At the analog of the balanced mixture equilibria in these cases, the cost of being altruistic equals the benefit conferred to the recipients discounted by their relatedness, which is the situation where altruism is expected to be neutral.
In the general frequency dependent case, the frequencies of the phenotypes have reached a point where, because of frequency dependence, their fitnesses are all the same, but the gene frequencies within each phenotypic class need not be equal. In Slatkin's model (1979), the generic requirement to obtain a balanced mixture equilibrium is that there be as many alleles as phenotypes, which is analogous to the requirement for the modifier model, in Section 3.(2), that there be as many transformation types as selected types.

The reason that phenotype distribution models and modifier models should share these two kinds of polymorphic equilibria can be understood at least heurisically. In both modifier and phenotype distribution models, selection is not an intrinsic property of the gene, but is induced on it by its association with types that are selected. V.A.T.P. polymorphisms are equilibria because the marginal distribution of selected types associated with each gene are the same. Balanced mixture polymorphisms are equilibria because there are no longer selective differences between the selected types.

But the connection between modifier models and phenotype distribution models can be shown to be even more basic. These two kinds of models actally form a continuum.

In the case of memoriless transformation, when there is no component of perfect transmission in the transformation matrix, a modifier model actually becomes a phenotype distribution model. At this extremum for the transformation matrix, an individual's type becomes irrelevant to the selected types of its offspring, so their selected type need no longer be considered a genotype and can be thought of as a phenotype. The frequency of this phenotype among the offspring will be simply the transformation distribution, which is determined by the modifier locus. So this modifier model is equivalent to a phenotype distribution model where the distribution is controlled by the parents, which have been studied for models where the sex ratio, habitat, or behavior are determined the genotype of the parent.

It has been pointed out to me (Uyenoyama, personal communication) that under non-random mating, the viability-analogous equilibria of phenotype distribution models usually no longer exist. That this occurs would be expected, since non-random mating reveals the action of the segregation-syngamy transformation on the diploid genotypes. This is analogous to modifier models with non-random mating which I have not investigated here.

Although there is both a qualitative homology and a similarity of results between modifier models and the different phenotype distribution models, these models are mathematically distinct, so I am claiming only that in some cases they result in the same behavior.