2.1. The Spaces of Hardy-Weinberg

It is not generally realized that the Hardy-Weinberg equilibrium for one biallelic locus was derived both by Hardy and by Weinberg using the frequencies of the diploid genotypes, P, Q, and R, rather than the now familiar $p^{2}\thinspace{:}\thinspace 2pq\thinspace{:}\thinspace q^{2}$ description in terms of the allele frequencies p and q, which was introduced by R. A. Fisher in 1918. But to derive the Hardy-Weinberg equilibrium in terms of $p^{2}$ , $2pq$ , and $q^{2}$ , one begins with arbitrary genotypic frequencies and shows that after a generation of random mating of diploid genotypes followed by Mendelian segregation in the production of gametes, the resultant proportions of offspring genotypes are algebraically equivalent to $p^{2}$ , $2pq$ , and $q^{2}$ . It is sometimes claimed that these proportions can be trivially derived by multiplying the gamete frequencies directly, making no use of the diploid genotypic space, but this depends on the hidden assertion that planktoid mating is identical in outcome with random mating of diploid genotypes followed by Mendelian segregation, a claim that has to be established by first mapping diploid mating frequencies into frequencies of haploid pairs at fertilization. Central to this mapping is the calculation of genic output from genotypic inputs, that is, the mapping of a diploid space into a haploid space. The steps in the derivation of the Hardy-Weinberg equilibrium for a biallelic locus in terms of allele frequencies and the demonstration that this occurs after a single generation of random mating are as follows:

1. 1. In the original mating population the frequencies of the three genotypes are specified as D, H, and R;

2. 2. The six possible types of mating ( $D\times D$ , $D\times H$ , $H\times R$ , etc.) are assumed to occur at random, so the frequencies of these matings are simply the products of the genotypic frequencies ( $D^{2}$ , $2DH$ , $2HR$ , etc.).

3. 3. The offspring ratios from each of these matings are specified. It is still possible to stay in the diploid space by using Mendel's proportions without any commitment to an underlying segregation mechanism. Alternatively, the segregation mapping into the genic space in step (5) below can be carried out here, in which case the expressions for $D^{\prime }$ , $H^{\prime }$ , and $R^{\prime }$ in (4) below will be in terms of allele frequencies p and q. Whether or not the segregation mapping is carried out at this point, however, the, assumption of unbiased segregation in heterozygotes is fundamental, since if there is meiotic drive then Mendel's ratios are incorrect.

4. 4. The sums of the output genotypes $D^{\prime }$ , $H^{\prime }$ , and $R^{\prime }$ are calculated over all matings. These turn out to be

   

5. 5. To turn this genotypic description finally into an allele frequency description $p^{2}\thinspace{:}\thinspace 2pq\thinspace{:}\thinspace q^{2}$ , it must be established that $D+H/ 2$ and $R+H/ 2$ are the allele frequencies p and q. But this requires mapping genotype space into gene space by Mendel's Law of Segregation and this segregation must be unbiased. If it is not, then both parts of the Hardy-Weinberg law—that p remains unchanged from one generation to the next and that D, H, and R can be expressed as $p^{2}$ , $2p$ , and $q^{2}$ with $p=1-q=D+H/ 2$ —are false.

2.2. The Generational Cycle

In addition to the above complications of Hardy-Weinberg, we must take into account the fact that genotypic and genic information alone are not determinative of the various forces operating to change population composition because many of these processes function on phenotypes rather than simply genotypes. Mating patterns, and probabilities of survival and reproduction, although influenced by genes are a consequence of developmental events that are contingent on the environment of the developing organism. In the most general case, environment includes influences of the phenotypes of previous generations by means of cytoplasmic inheritance through the egg. A complete general representation of genetic evolutionary processes then requires not two, but six spaces with sequential transitions within them and mappings from one to the other. (In what follows, mappings refer to translations or re-representations of a population or system in terms of a new set of state variables, i.e., in a new state space, while state transitions refer to dynamical laws that transform the system within a single state space from one step to the next.) These spaces are:

1. S ₁ a diploid phenotypic space,

2. S ₂ a diploid genotypic space,

3. S ₃ a diploid pair phenotypic space,

4. S ₄ a diploid pair genotypic space,

5. S ₅ a haploid phenotypic space, and

6. S ₆ a haploid genic space.

For the moment we consider only organisms that reproduce in discrete generations, with no overlap between generations. We can then begin to formalize the process of temporal change by choosing some point in the life cycle as the initial condition. For our purposes it is convenient to choose the point of mating pair formation. The sequential transformations and mappings are then described in model stages as follows:

1. 1. The description of the entities in phenotypic space (S ₁) and their mapping into genotypic space (S ₂): ( $S_{1}\rightarrow S_{2}$ ).

2. 2. The formation of mating pairs by mapping pairs of entities in S ₁ and S ₂ into S ₃ and S ₄. The attributes of each individual, in the most general case, must include a list of ancestry relationships with other individuals so that any potential pair can be classified as sibs, cousins or any other descriptor of the ancestry tree. The mapping into mating pairs includes self-fertilizations or obligate outcrossings, and assortative mating by size, behavior, habitat chosen or degree of genetic relationship (inbreeding) or completely random mating in either dioecious or monoecious species, all of which can then be described genotypically ( $S_{1}\rightarrow S_{3}\rightarrow S_{4}$ or $S_{2}\rightarrow S_{4}$ ).

3. 3. Mapping genotypes (diploid genomes) into genes (haploid sequences) by initial gamete formation within each individual, including genetic segregation, recombination, and mutation and meiotic drive, to form gamete potential subpools corresponding to each individual in S ₄ ( $S_{4}\rightarrow S_{6}$ ).

4. 4. Mapping potential gamete subpools into effective gamete subpools taking into account differential fertility of adult gamete producers, gamete motility and survival $S_{6}\rightarrow S_{5}$ .

5. 5. Mapping into diploid genotypes taking into account differential compatibility of sperm with recipient ova $S_{5}\rightarrow S_{6}\rightarrow S_{2}$ .

6. 6. Mapping of diploid genotype into phenotype in development $S_{2}\rightarrow S_{1}$ .

7. 7. Transition of the ensemble of phenotypes by survivorship to mating stage (S ₁).

8. 8. Start of a new generation in S ₁, namely, $S^{\prime }_{1}$ .

While it is true that the system's transition from one generation to the next may be represented in any one of the seven model-stages and six spaces, we need to know the entire loop—and all the parameter values in each of the model stages in that loop—in order to obtain an accurate representation of the chosen state space in the next generation, that is, to get the state transformation equation between generations within any one of the spaces (e.g., if we want to move from (S ₆) to ( $S^{\prime }_{6}$ )). For the transition in allelic space, we must move out of that space, into genotypic space to define the parameters, and back into allelic space in order to characterize the next generation. (We elaborate this point in 5.1 below.) The dynamical problems that arise in the context of several spaces will be elaborated in Section 4, below.

One important claim in the literature has been that any genetical system can be adequately represented in at least two ways: its original, usually genotypic form, and a revised allelic one (Dugatkin and Reeve Reference Dugatkin and Reeve1994; Sterelny Reference Sterelny1996; Kerr and Godfrey-Smith Reference Kerr and Godfrey-Smith2002). In order to assess whether this is correct, we must address the overall issue of how to compare models in population genetics. We propose a new definition of representational adequacy that rests on dynamical and parametric sufficiency, from which equivalences can be derived.

3.1. Parametric Sufficiency

Parameters are properties of objects, and may be properties of more than one object at a time. (Parameters are represented in the models as values that are not variables.) Two models may look as though they should be dynamically equivalent because of the similar appearance and names of their parameters, but real differences in their parameter measures may result in dynamics that are not equivalent. Two models should be considered ‘parametrically equivalent’ if the parameters that apply in one model have a natural representation in the parameters that apply in another. One case in which similar-looking parameters yield very different dynamics concerns allelic and genotypic fitnesses, which has led to much confusion in current controversies. (See Sections 5.1 and 5.3.)

3.2. Dynamical Sufficiency

The concept of dynamical sufficiency concerns what state space and variables are sufficient to describe the evolution of a system given the parameters being used in the specific system. What happens to the frequency of the variable over time? In a simple allelic model, this question becomes: Can we describe the changes in the frequency of allele A over time, with the information that we have, which includes the state space (variables) and the parameters (fitnesses, population size, etc.)?

Consider a set of objects $\Omega _{1}=\left\{O_{1},\, O_{2},\,\ldots ,\, O_{m}\right\}$ (e.g., genotypes $O_{1}=AA$ , $O_{2}=Aa$ , $O_{3}=aa$ ) whose frequencies are given by $\kappa =\left\{k_{1},\, k_{2},\,\ldots ,\, k_{m}\right\}$ , and a set of variables $G=\left\{g_{1},\, g_{2},\,\ldots ,\, g\,_{n}\right\}$ that represents the frequencies of another set of objects $\Omega _{2}=\left\{O_{1},\, O_{2},\,\ldots ,\, O_{n}\right\}$ . For example, $\Omega _{2}=\left\{A,\, a\right\}$ . Then $\Omega _{2}$ is dynamically sufficient for $\Omega _{1}$ if the functions {f _i} and {h _i} are such that we can write the recursions

with

In the one-locus diploid case with selection on AA, Aa, aa in terms of fitnesses w ₁₁, w ₁₂, w ₂₂ (stage 7 above) we have

Then after selection

In this case, the allele frequencies G in the allelic state space are dynamically sufficient to study the evolution of genotype frequencies $\kappa $ . These equivalences depend, however, on the parameters, w ₁₁, w ₁₂, w ₂₂, that could only be determined in the genotypic space.

Increasing attention has recently been paid to the phenomenon of epigenetics, which includes a variety of biological processes that act on genes and may be transmitted between generations, but not according to any of the rules of genetic inheritance. Formal models for the evolution of epigenetic objects or properties (Feldman and Cavalli-Sforza Reference Feldman and Cavalli-Sforza1976, Reference Cavalli-Sforza and Feldman1981; see also Jablonka and Lamb Reference Jablonka and Lamb2005) utilize an additional state space S ₇, the phenogenotype, in which changes may occur during organismal development.

4.1. Example—Dynamical Insufficiency: Allelic Frequencies and Self-Fertilization

Again $\Omega _{1}=\{ AA,\, Aa,\, aa\} $ , $\kappa =\{ x,\, y,\, z\} $ , $\Omega _{2}=\{ A,\, a\} $ , $G=\{ p_{A},\, 1-p_{A}\} $ . With random mating and no selection, after the first generation,

The allele frequencies are constant, $p^{t+1}=p^{t}$ , $t\geq 0$ , as are the genotype frequencies after the initial generation. With self-fertilization we have

Again $p^{t+1}=x^{t+1}+y^{t+1}/ 2=p^{t}$ , and allele frequencies are again constant over time. But the heterozygote frequency y ^t tends to zero at the rate 1/2 per generation. Thus, in the case of selfing, allele frequencies do not give the dynamical properties of the full system, which converges to a mixed population of AA and aa homozygotes with frequencies p, $1-p$ , respectively. Partial self-fertilization would have a similar qualitative difference between allelic and genotypic evolution.

The lesson from such cases is not, however, that genotypic space is itself somehow magically sufficient or transparently representative of the evolutionary system, for genotypic parameters, particularly the fitness parameter, are most often averages, estimated in a number of ways.

4.2. Fertility Selection

The issue of types mating with types arises similarly with fertility selection. Here the types of matings produce different numbers of offspring, which means that the fertility-level fitness parameters are properties of the mating pairs. Bodmer (Reference Bodmer1965) shows that it is not possible to represent the results of differential fertility among mating types in general in terms of allelic state space. Having given the equations that relate the proportions of genotypes in one generation (right-hand side of the equations) to those same genotypes in the next generation (left-hand side of equations), he then remarks: “These equations are intractable as they stand, and allow no simple expression of the right-hand sides in terms of gene frequencies” (Bodmer Reference Bodmer1965, 412). The population dynamic is then necessarily presented in genotypic space, not in terms of genic types. Most often, planktoid mating is simply assumed—that is, the random union of gametes—obviating the need for any additional parameters beyond fertilities of the mating pairs of genotypes.

4.3. Continuously Breeding Population

A much greater complication of prediction occurs when diploid organisms with overlapping generations and continuous breeding are considered. In addition to the age specific mortality and fecundity schedules of the different genotypes, it is essential to specify the growth rate of the population as a whole, particularly whether the population is expanding, stable in size or shrinking. For example, in a growing population, genotypes with reproduction at earlier ages increase in frequency since, relative to those that postpone reproduction, they are contributing their genotypes while the population is at a smaller size. The same genotypes will decrease in frequency in a shrinking population, since it is advantageous to wait until the population has decreased in total size before adding copies of the genotype (Charlesworth and Giesel Reference Charlesworth and Giesel1972). Thus the complete age-specific mortality and fecundity schedules are insufficient to predict gene frequency change and there is no ‘genotype fitness parameter’ at all in this model; genotypic fitnesses change continuously over time. The only way to arrive at answers about the evolutionary state of the system is to grind through the mathematical dynamics of the model. Note that, in this model, the biological character of the genetic types does not change, however we choose to model them. The changes in frequencies occur because the types are part of a population that is either growing larger or smaller. There is no reason to think that this sort of context-dependence is unusual; rather, it seems to be the rule. In that case, parameters that describe the context must be integrated into genotypic fitnesses, with the result that separating the context and genotypic dynamics from a forteriori allelic evolution becomes impossible.

Moreover, dependence on population growth or decline is not the only kind of context-dependence that matters, in looking at genotypic fitnesses. Lewontin demonstrated that fitnesses in genetics, including genotypic fitnesses, depend on what other genotypes are present in the population at that time (Reference Lewontin1974). Thus, we must conclude that the fixed genotypic fitnesses that we are accustomed to seeing in population genetic models are at best approximations, as close to the truth as we can afford at a given level of mathematical precision.

5.1. Dugatkin and Reeve's Formulation of Allelic Equivalence

Dugatkin and Reeve claim:

They do this with a pair of inequalities (Reference Dugatkin and Reeve1994, 109). Dugatkin and Reeve claim that their inequality (2) “encompasses both broad-sense individual selection and any form of trait-group selection that one may care to envision.” Moreover, they conclude, “If broad-sense individual selection, genic selection, and trait-group selection all can be represented by a single condition based only on allele frequencies, then they cannot fundamentally differ from one another” (Reference Dugatkin and Reeve1994, 109; their emphasis). But there are serious problems here. The first is that their inequalities (1) and (2) hide completely the causes of why the numbers of alleles change.

Their inequalities (1) and (2) state

where p _i and $p^{\prime }_{i}$ are the frequencies of the allele in group i before and after selection, respectively, and N _i and $N^{\prime }_{i}$ are the corresponding group sizes. Of course, inequality (DR) is misleading and illustrates the importance of both dynamic and parametric sufficiency. Hidden in (DR) is a ‘fitness’ of $p_{i}$ , namely, $N_{i}/ \Sigma N_{i}$ , but N _i changes to $N^{\prime }_{i}$ over a generation, which requires an additional dynamic relationship and assumptions on the rule of demographic change.

For a true haploid dynamic, we would usually write, for one locus with alleles A and a and p _i the frequency of A in population i,

where $\overline{w}^{i}=p_{i}w^{i}_{1}+w^{i}_{2}\left(1-p_{i}\right)$ , and $w^{i}_{1}$ and $w^{i}_{2}$ represent the fitnesses of alleles A and a in population i, and the prime indicates the next generation. For the diploid one-locus case, we again write

where

is the mean fitness in group i, and $w^{i}_{11},\, w^{i}_{12},\, w^{i}_{22}$ are the fitnesses (viabilities) of AA, Aa, aa in group i. As in equation (1) it may seem that we can write

where $\tilde{\overline{w}}^{i}=p_{i}w^{i}_{1}+\left(1-p_{i}\right)w^{i}_{2}$ . But now we are dealing with haploid allelic parameters, $\tilde{w}^{i}_{1},\,\tilde{w}^{i}_{2}$ , for the fitnesses of A, a in population i. Of course, although these appear to be haploid parameters they are actually expressible as

which involve genotypic parameters (as well as allelic frequencies). In other words, the apparently purely-allelic parameters depend crucially on genotypic fitnesses—but these values are completely hidden. The assumption of Mendelian transmission is also made in deriving (3) (see also Godfrey-Smith and Lewontin Reference Godfrey-Smith and Lewontin1993). For this two-allele diploid case, if meiotic drive were introduced, one more parameter, k, giving the probability that A is produced by Aa heterozygotes, would be necessary.

Thus, we note that in describing the algebra, many discussions of allelic models and their algebra obscure the origins of that algebra and all the information it contains and represents. A common move is to infer that the genic state space, and its basic entity, the allele, have a metaphysically fundamental and autonomous character (Sterelny Reference Sterelny1996). This is both biologically and mathematically problematic. Biologically, the changes represented are dependent on all changes in the entire generational cycle reviewed above, represented in the variety of spaces and parameters. Mathematically, because genic space is dynamically insufficient for representing many system changes, and because in diploids there are no allelic transition laws within allelic space with allelic parameters, there is nothing autonomous about it; thus, it cannot support the metaphysical inferences based on its supposed autonomy (see Section 4 above and Lloyd Reference Lloyd2005). But there is another account available which might be thought to avoid some of the above problems.

5.2. Kerr and Godfrey-Smith's Formulation of Allelic Equivalence

Echoing Dugatkin and Reeve's claims of equivalence, which they endorse, Kerr and Godfrey-Smith (Reference Kerr and Godfrey-Smith2002) similarly appear to be claiming parametric sufficiency for allelic fitnesses in the diploid one-locus population genetic model, but in a different fashion. They introduce interchangeable sets of parameters {α_i, β_i} and {π_i, φ_i} to describe the expected numbers of alleles in groups of size n. In the standard diploid population genetic model with Mendelian segregation, groups are of size 2 and the relative fitnesses of AA, Aa, and aa are in Kerr and Godfrey-Smith's notation $\pi _{0}=2\beta _{0}=w_{11}$ , $\pi _{1}=\alpha _{1}+\beta _{1}=w_{12}$ , $\pi _{2}=2\alpha _{2}=w_{22}$ . An additional parameter, φ₁, analogous to k in the case of meiotic drive, completes the four-parameter specification. Note that in the Mendelian case, three parameters are necessary to describe the fitnesses of the genotypes. The statement by Kerr and Godfrey-Smith that “the fitnesses of, for example, allele A in a given genotypic context is given by the α terms” and therefore “the dynamics of the population genetic model can be redescribed using the fitnesses of alleles” (Reference Kerr and Godfrey-Smith2002, 499), gives the impression that only two parameters, properties of alleles A and a are sufficient, and is misleading.

In their footnote 27 they admit that their use of the term allelic fitness departs from the standard marginal fitness of alleles (given by the $\tilde{w}_{i}$ above). But their “context-dependent fitness” requires that a parameter be assigned to an allele for each “genotypic environment” in which it appears. Thus, in terms of parametric sufficiency, there are no allelic properties that are independent of genotypic properties. In other words, in their formulation, allelic fitnesses are not parametrically sufficient whereas genotypic fitnesses are.

Higher-level properties must be specified even in the case of meiotic drive where, in addition to the three genotypic fitnesses of AA, Aa, and aa, a fourth parameter is needed to specify the chance, k, that Aa produces gamete A. Marginal fitnesses of alleles A and a can be legitimately computed, but they are not parametrically sufficient because they are ‘stated’ or ‘defined’ in terms of w _AA, w _Aa, w _aa, and k. Empirically, of course, it is these four biological parameters that can be measured, whereas Kerr and Godfrey-Smith's α_i and β_i confound genotypic and gametic contributions to fitness. It is certainly not parsimonious to introduce allelic properties of genotypes that are made up of combinations of genotypic properties (of genotypes) and allelic properties (of gametes), which together make up a parametrically sufficient set of property values for use in the dynamics of the evolutionary process.

5.3. Group Selection/Altruism's Claimed Reduction to Allelic Formulation

One of Dugatkin and Reeve's most important and widely-repeated claims has been that “new-group selection” approaches are mathematically equivalent to kin- and individual-selection models (Sterelny Reference Sterelny1996, 577; Sober and Wilson Reference Sober and Wilson1998, 57, 98–99; Sterelny and Griffiths Reference Sterelny and Griffiths1999, 168–169, 172; Kerr and Godfrey-Smith Reference Kerr and Godfrey-Smith2002, 479, 508; Waters Reference Waters2005, 312). But Cavalli-Sforza and Feldman showed the contrary in Reference Cavalli-Sforza and Feldman1978, and the paper cited by Dugatkin and Reeve in their own support by Uyenoyama and Feldman (Reference Uyenoyama and Feldman1980) reflected on this very result.

Consider the following model in which $\gamma $ is the fitness loss by ‘doing altruism’ and $\beta $ measures the gain in fitness. In the case of parent-to-offspring altruism where AA is an altruist, Aa is an altruist with probability h, and aa is selfish, and the fitness loss and gain are additive, we have, writing x, y, z for AA, Aa, aa frequencies and p, q for allele frequencies of A, a, with primes indicating the next generation,

where $\overline{\phi }$ is the sum of the right sides. This is a case where if we add $x^{\prime }+y^{\prime }/ 2$ to get $p^{\prime }$ , the frequency of A in the next generation, it cannot be expressed solely in terms of p (Cavalli-Sforza and Feldman Reference Cavalli-Sforza and Feldman1978). The allele frequencies are not dynamically sufficient; genotype frequencies must be used.

Later, Uyenoyama and Feldman (Reference Uyenoyama and Feldman1980) showed what the consequences of this dynamical insufficiency of allelic frequencies could be in generating different outcomes in the genotypic analysis from those obtained under the ‘allelic sufficiency’ assumption that $x=p^{2}$ , $y=2pq$ , $z=q^{2}$ in 5a, b, c). Thus, in true kin selection modeling there is neither dynamical sufficiency of alleles nor parametric sufficiency, because we need both genotypic fitnesses and donor and recipient fitness (or group-context) information. In other words, these models involve an interaction between genotypes, and not just a simple downward effect on genic frequency or even on genotypic frequency. Thus, these early models show that evolutionary dynamics of allele frequencies that are derived by approximating genotype frequencies are not equivalent to the dynamics of the genotypes when they are correctly counted.

Thus, empirical differences between systems in nature are involved in the models, and underlie the reasons that higher-level fitness and other parameters are introduced into the models in the first place, for the sake of dynamical sufficiency unattainable through other means (see Sections 5.1 –5.3). To emphasize this important result: the claim that no higher-level parameters are needed for the empirical sufficiency of multi-level or “new group selection” models is untrue. The biology involved in these models demands the measurement of higher-than-individual-genotype values; specifically, the properties of genotypes must be measured within their genotypic context.

Thus, we conclude that there are fundamental differences between many of the models that have explicitly been claimed to be ‘mathematically equivalent’ or ‘empirically equivalent’. These models are neither dynamically nor empirically equivalent, and thus are not equivalent in any meaningful way.