Darwin's theory of evolution through natural selection has been described by Lewontin (1970) as embodying three principles:

1. ``Different individuals in a population have different morphologies, physiologies, and behaviors (phenotypic variation).
2. Different phenotypes have different rates of survival and reproduction in different environments (differential fitness).
3. There is a correlation between parents and offspring in the contribution of each to future generations (fitness is heritable). ...While they hold, a population will undergo evolutionary change.''

One of the major concerns of population genetics is with principle 3. : how do the mechanisms that determine the correlation between parent fitness and offspring fitness impact evolution? The incorporation of Mendelian genetics into models of natural selection is a major contribution of the ``Synthetic Theory'' of evolution. The mechanisms of heredity are important to evolution because they are the causal connection between the characteristics of organisms in one generation and the characteristics of those in the next.

In one sense, the major accomplishment of genetics is the discovery and understanding of how genes are transmitted during reproduction. But what makes genetics non-trivial for evolutionary theory, however, is not that genes are being passed on. Rather, it is the fact that through the mechanisms of transmission, offspring may be different from their parents. Of course, at one extreme, if the characteristics of offspring had nothing to do with those of their parents, then selection could leave no impact on the population, and Darwinian evolution would be impossible. But at the other extreme, were offspring identical to their parents, the mechanisms of heredity would be irrelevant to the evolutionary process. Features of transmission such as segregation, recombination, mutation, and so forth, give certain regularities to the differences between parents and offpring and take a central position in population genetics models of evolution. Differences between parents and offspring are necessary not only to make the mechanisms of heredity important to evolution, but are the sine qua non for the science of genetics itself, from Mendel's experiments with peas to the transformation of Drosophila.

I will refer to any processes contributing to changes between parents and offspring as TRANSFORMATION processes, to emphasize that it is changes during reproduction that make the mechanisms of genetic transmission important to evolution.

The way I would like to characterize population genetics models of evolution will not even refer to genes specifically. In its general form, a micro-evolutionary model concerns populations composed of individuals of different types in different numbers; changes in these numbers over time constitute evolution. An individual's type is

determined by, but not necessarily the same as, the types of its parents. Transformation processes determine the nature of differences between parents and offspring. Selection determines the relative proportion of the next generation that the offspring of a given individual comprise, and this proportion will be determined by the individual's type.

In the context of Darwin's theory of evolution, the importance of transformation is commonly thought to lie with Lewontin's principle 1., above: transformation is the supplier of variation, the ``raw material'' upon which the ``force'' of selection can act. But processes in transmission that cause transformations do not simply introduce variation and then disappear to let selection act alone on the population. They interact continually with selection to affect the composition of the population. In this respect they are also a ``force'' in evolution.

For example, consider a classical case of transformation, the production of overdominant or underdominant heterozygotes from different homozygous parents. The offspring can all be quite different, with respect to selection, from either of their two parents. In the case of a mating of two identical heterozygotes bearing balanced lethal genes, half of the offspring can have very different phenotypes from their parents. Sexual transmission in these cases will not simply introduce variation into the population, but will confound the result of selection on the previous generation by transforming the phenotype between the parents and their offspring. A classic consequence of this is the occurrence of protected polymorphisms when the fittest type is the heterozygote. If the heterozygote could reproduce clonally it would go

to fixation, but because heterozygotes keep producing homozygotes under sexual reproduction, selection is confounded and a polymorphism results.

In deterministic models of evolution, selection and transformation comprise a basic dichotomy of forces that can change the composition of the population. Selection affects the quantity of reproductive output, while transformation affects the content of reproductive output. In infinite populations, if neither selection nor transformation are occurring, the composition of the population will be static. When population sizes are finite, both the quantity and content of reproduction become stochastic, and drift comes in to play.

TRANSFORMATION - SELECTION BALANCES

One can imagine two extrema for the evolutionary processes acting on a population. We can imagine a complete absence of transformations, with all offspring identical to their parents. Here, constant selection will eventually fix the population on the types with the maximal fitness value, or in the case of frequency dependent selection, possible bring the population to a polymorphism where all the types have the same fitness. On the other hand, we can imagine a complete absence of selection, where only transformations can change the composition of the population. In general, we would not expect the composition of types that the population might attain at any equilibria to be the same in the two situations. In real populations, there will be a combination of selection, transformation and drift impacting on the composition of types. This thesis concerns the interaction of selection and transformation.

Adaptation is the phenomenon most usually considered as the main effect of selection. Transformation has no single effect. A number of familiar biological phenomena are evolutionary effects of transformation:

THE NATURE OF TYPES THAT CAN EVOLVE

The discussion of micro-evolutionary models thus far has referred only to genotypes as the types involved in evolution. But from the way

I have characterized the general micro-evolutionary model, any domain of information about an organism satisfying two criteria will be involved in evolution and should be considered a part of an individual's type. This information

1. must be determined by the organism's parents, and
2. must either affect the organism's fitness or help determine the types of its offspring.

Now, there are untold numbers of details in an individual organism's existence that determine what and how much it reproduces -- its genetic constitution, embryonic environment, mutations, cultural heritage, chance encounters with mates, predators, prey, pathogens, accidents, weather, and so forth. But only some of this information is both determined by its parents and will be transmitted to the offspring or affect its fitness.

Some important examples of non-genetic information satisfying these criteria include geographical, micro-habitat, and cultural information. An organism's location or micro-habitat can both affect its reproductive output and affect the location or micro-habitat of its offspring. An organism's cultural heritage is also a domain of information that can affect its reproductive output, and that can be transmitted to its offpring.

Transformation in these cases takes the form of migration or dispersal in the case of organismal location, or shifts in habitats, hosts or diet from parents to offspring in the case of micro-habitat. Changes in cultural traits during the process of learning are transformations. In each of these processes, the offspring types bear a relation to, but need not be identical with, the parental types.

THE EVOLUTION OF TRANSFORMATIONS

The main task of this thesis is to develop a theoretical framework for understanding how transformation processes have themselves evolved. The general modifier approach to this question is to extend the domain of types to include the transformation process itself as an aspect of an organism's type. Thus, within the actual type of the parents there will be information on how these types are to be changed in producing offspring. The evolutionary question becomes: what happens to variation for transformation types when it is introduced into the population? Variation in transformations need not directly alter an organism's fitness, but what is of interest is how selection can come to be induced on transformation types.

Within this framework, it is possible that transformations also act on the transformation types, or even that the transformation type controls its own transformation. This would be the case, for example, if a mutator gene caused itself to mutate into other mutator alleles. Such dynamics will profoundly alter the evolution of the transformation types. In this thesis, transformation will be excluded for the most part from acting on the transformation types themselves, so that the means by which selection can come to be induced on transformation types can be seen clearly.

A fundamental assumption of the treatment here is that the processes of selection and transformation do not happen simultaneously but occur during disjoint intervals of time during the life cycle. This allows the life cycle to be formalized as an alternating sequence of selection events and transformation events. This is the appropriate representation for processes such as reproduction, which occurs as

discrete events, but it is not as good for processes such as migration, which may be a continuous process concurrent with selection during an organsism's life.

FORMALIZING TRANSFORMATION PROCESSES

The basic model I will use for transformation processes will be that of a mapping from the parental types to the probabilities of producing each possible offspring type. For now, the ``type'' , with its entire genetic and other information, will be represented by a single index: i, j, k, etc.. Instead of putting all the complexity of genetic, cultural or other information that satisfies the above two criteria into the specification of the individual's type, this information is embedded within the structure of the transformation probabilities. This will clarify the point in analysis when more of the structure of the mapping must be specified, and allows the exploration of generalities that would be foregone had the models been restricted too early.

It is difficult to obtain models that include the complete variety of life cycles of different organisms. A division must be made between asexual and sexual organisms, since the ``topology of descent'' is purely branching in asexual, but involves anastomosing in sexuals. In cultural transmission, more than two individuals may be involved in determining a given cultural trait of an individual. In this thesis I examine only pair mating, where two individuals determine another's type.

Actually, sexual organisms that are purely selfing or apomictic also have pure branching descent. Population genetics models are often classified as sexual vs. asexual, or haploid vs. diploid, but these classifications overlap in terms of topology of descent. This is shown below:

It will be seen later that sexual haploid models are often subcases of diploid models, where special constraints have been placed on selection and transformation.