One of the ubiquitous features of models of evolution is that they do not model everything evolving at once. Some traits of the organisms being modeled are proposed to have heritable variation, and a model is constructed for the effects of these these traits on the reproductive cycle of the organism in order to examine what happens to the heritable variation over time. Other traits, however, are taken as a fixed background in which this evolution is to be occurring.
The traits whose evolution is being modeled are typically traits affecting physiology, resource utilization, interactions with other organisms and interactions with conspecifics-- traits which affect fitness. The heritable variation for these traits is posited to occur at genetic loci. Background features of the population include features such as sexual reproduction, the genetic and mating systems, mutation processes, and the population structure. Genetic and mating systems include features of reproduction that determine inheritance, such as meiosis, chromosome number, recombination, alternation of generations, gender, level of inbreeding, assortative mating, and so forth. These are usually taken as fixed and appear in the structure of the models. A common task of the theoretician is to see how the evolutionary fate of the heritable variation is a function of the state of the background features of the population. Some classical examples of this are the question of how linkage affects the evolution of two loci, and how haplodiploidy affects the evolution of eusociality.
Although these background features are usually separated logically from those features in the model that are evolving under selection, in fact there is no necessary biological dividing line between them in real organisms. These background features comprise some of the most elaborate and interesting phenomena to be found in biology. Meiosis and sex require a long line of complex structures and processes, from chromosomal structures, to cell physiology, to morphology and behavior. On this basis alone one would believe that these features have been shaped by evolution, that they are not in fact fixed.
The purpose of modifier gene models is to study how these background features themselves might have evolved, and their approach is to ask what happens when there is heritable variation for these features. Many background features have the same biological origins as selected features. Genetic systems and mutation processes are mediated by enzymes, regulatory genes, and other cellular components for which genetic variation has been found. Similarly, morphological and behavioral features under genetic control may affect the mating systems and the spatial structure of a population through the same means that they can affect resource utilization, predator avoidance and other aspects of fitness.
There is ample documentation both of specific loci affecting features such as recombination (Catcheside, 1977) and mutation (e.g. Cox and Gibson, 1974) and of phylogenetic variation and change of features such as sexual versus asexual reproduction (Templeton, 1982), mating systems (Bateson, 1983) and so forth. The requisite genetic variation therefore appears to exist for these features to evolve.
The first modifier model to gain attention was a model for the evolution of dominance proposed by Fisher (1928). However, forty years passed before this approach for traits affecting fitness was first applied, by Nei (1967), to one of the major background features not involved directly in fitness, recombination. In his model, there are two loci undergoing viability selection, and a neutral third locus, the modifier, which controls the rate of recombination between them.
Before the modifier gene approach was developed, the explanations for the evolution of non-fitness features paralleled the kind of ``group selection'' or optimization ideas in use to explain altruism and population regulation. The essence of these approaches was to examine some statistic of the population, adopted as a measure of its ``success'' , such as mean fitness, genetic load, or response rates to selection, and see how differences in the non-fitness trait would affect it. These non-genetic approaches have been applied to the evolution of mutation rates (Kimura, 1960), recombination (Muller, 1964; Turner, 1967; Eshel and Feldman, 1970), and sexual reproduction (Fisher, 1930; Crow and Kimura, 1965; Eshel, 1971). Although these statistics were not incorporated into actual dynamic models of evolutionary change, they were a first attempt to understand how natural selection can gain a hold on features which have no intrinsic effect on an organism's fitness.
This is what may have delayed the formulation of modifier gene models-- control over these background features does not give modifier alleles any intrinsic selective differences. In the absence of any pleiotropic effects, these are neutral genes which can be selected only by becoming associated with genes that are under selection, that is, through hitchhiking.
Since Nei's model was formulated, modifier gene models of a number of non-fitness features have been analyzed, including
Some regularities emerge, but there are many complexities which have remained unexplained in the literature.
Models of modifier genes can be divided into two major classes, those dealing with populations that have reached equilibrium in their genetic composition and those dealing with populations that are in transient phases, due to fluctuating selection, drift, or through the recent introduction of advantageous genes. I will describe mainly the results from the equilibrium models, which are the topic of this thesis. Some of the main results are summarized in Table 1 .
To understand the long term evolution of a given feature, one wants to know whether evolution will produce any inexorable trends toward certain phenotypes. Feldman (1972) first developed the theoretical methodology to investigate this question for the evolution of recombination. His approach was to ask whether, in Nei's (1967) modifier gene model, populations with a given rate of recombination were immune at equilibrium to the invasion of new modifier genes that changed the recombination rate, and what requirements were there on the recombination rate produced by the new modifier to enable it to invade the population. His result was that populations with linkage disequilibrium between the two selected loci could be invaded by new modifier alleles which reduced the recombination rate, and were immune to invasion by modifier alleles that increased the recombination rate.
Karlin and McGregor (1974) applied this approach to modifiers of a number of classical ``background'' features. In an attempt to offer a principle that would explain, in general, the evolution of modifier genes, they proposed that the initial increase in the population of a new modifier allele would be governed by a ``mean fitness principle'' , which I paraphrase:
If the effect of a new modifier allele, were it fixed in the population, would cause the population's equilibrium mean fitness to increase, then this new allele will increase in frequency when introduced into the population. If the effect would mean a decrease in the mean fitness, the allele would be excluded from the population.
What they were proposing, in effect, was that the intuitively appealing mean fitness statistic actually predicted the dynamic behavior of modifier genes in the population. This principle works in some cases, but unfortunately is violated in several cases that have been discovered subsequently, including the basic recombination modifier model (Feldman et al., 1980, for the examples in Karlin and Carmelli, 1975), and modifiers of segregation distortion (Thomson and Feldman, 1974). No one has offered any refinement to the principle that would explain these counterexamples.
Nevertheless, one regularity emerges from a number of modifier models. Karlin and McGregor (1974) noted that for modifiers of recombination, mutation, and migration, the modifier always evolves to reduce these processes in populations at equilibrium. Subsequent treatments of modifier models have found this property to generalize to arbitrary viability selection regimes in random mating populations at equilibrium, for modifiers of recombination (Feldman, et al., 1980), mutation (Holsinger and Feldman, 1983 b), and migration (Teague, 1977; Asmussen, 1983). The possibility that these cases manifest some sort of reduction principle has been conjectured by Feldman (Feldman et al., 1980). These cases are shown in Table 1.
From Table 1, we see that in quite a number of cases, the reduction principle cannot be operating. With recombination modifiers, an increase in recombination can evolve when mutation, migration, segregation distortion or partial selfing are added to the model. These are all results for equilibrium populations. For transient phase populations, increases or decreases in the modified process can evolve.
In this thesis I develop a framework for population genetics models in which the models of modifier genes in the literature become special cases of a more general model. The essential idea is that processes incorporated in population genetics models can be dichotomized into selection processes and transformation processes, that is, changes in number and changes in kind.
Chapter 1 develops this idea. The way that recombination, mutation, migration and other non-fitness features can be seen to be transformation processes will be formalized. Within the dichotomy of selection and transformation, modifiers can be seen to be a natural complement to selected loci: they are loci for variation in transformation processes rather than selection processes.
In Chapter 2, I examine some very basic properties of population genetics models from the perspective of this dichotomy. The point of this chapter that is most important for the theory of modifier genes is that transformation processes allow a standing genetic variance in fitness in a population at a stable genetic equilibrium. Another way to put this is that the population can be made to bear a stable genetic load.
In Chapter 3, I explore what happens to variation in the population for transformation processes. The mathematical reason for the reduction principle- i.e., why the reduction of recombination, mutation, and migration is a common result in several models-- is shown for these and a more general class of transformation processes, and is extended to some models for modifiers of sexual reproduction and cultural transmission. The reason that recombination and mutation rate increases evolve in some models in the literature is found to reside in the nature of the variation in transformations. A concept of ``affine'' variation in transformations is developed, which accounts for these situations and predicts others where increases in transformation may evolve.
The ``dark horse'' among principles offered to explain the evolution of modifier genes must certainly be Kimura's (1960) ``Principle of Minimum Genetic Load'' , which proposes that modifiers would evolve to minimize the total genetic load in the population, including mutational and substitutional loads. Kimura offerred this principle not as a result, but as a premise upon which he derived ``optimum'' degrees of dominance and optimal mutation rates, and he gave no justification for it. No one has ever pursued this principle further. Although genetic load arguments have often been invoked without justification, we will find that the results in this thesis lead one back to Kimura'a ``Principle of Minimum Genetic Load'' in a restricted form: how a modifier gene evolves depends on the nature of the variation it controls, but as long as it is not itself being transformed, it may be that inevitable effect is to reduce the equilibrium genetic load of the population.
Furthermore, we can address what is something of a myth about neutral modifier genes: that selection on them will be very weak, and that any intrinsic pleiotropic fitness effects will overwhelm the selection due to modifier effects (Wright, 1964). In Chapter 3, we will see that this is not a necessary property of modifier genes, but rather, that the strength of the induced selection acting on a neutral modifier can be on the order of the genetic variance in fitness in the population.
Chapter 4 is a discussion of these results in a general context. One point made is that models of genetic control of phenotype distributions, such as sex ratio, habitat preference, propensity to perform altruism, and so forth, bear a fundamental homology with modifier gene models, and I discuss some of the common features of both.
Although the results I obtain here generalize much about the theory of modifier genes, they comprise the bare surface of what is easily tractable within the broad scope of of ``selection processes'' and ``transformation processes'' which includes phenomena as diverse as sexual reproduction, cultural transmission, migration, and host shifts of organisms. The theory analyzed here will be restricted to infinite populations, near genetic equilibrium, in constant environments. Nevertheless, I hope that by describing the general scope of this framework, it may facilitate further extensions of the theory.