Frequency-dependent selection remains the most commonly invoked heuristic explanation for the maintenance of genetic variation. For polymorphism to exist, new alleles must be both generated and maintained in the population. Here we use a construction approach to model frequency-dependent selection with mutation under the pairwise interaction model. The pairwise interaction model is a general model of frequency-dependent selection at the genotypic level. We find that frequency-dependent selection is able to generate a large number of alleles at a single locus. The construction process generates multiallelic polymorphisms with a wide range of allele-frequency distributions and genotypic fitness relationships. Levels of polymorphism and mean fitness are uncoupled, so constructed polymorphisms remain permanently invasible to new mutants; thus the model never settles down to an equilibrium state. Analysis of constructed fitness sets reveals signatures of heterozygote advantage and positive frequency dependence.

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SINCE the development of molecular techniques first revealed widespread genetic variability in nature (HUBBY and LEWONTIN 1966; LEWONTIN and HUBBY 1966), population geneticists have been puzzling over just why there are so many alleles. The neutral (KIMURA 1983) and nearly neutral (OHTA 1973; KIMURA 1983) theories hold that the majority of variation is selectively neutral (or nearly so) and thus is a consequence of mutation and random drift. Nevertheless, we have long known that a wide range of natural systems have multiallelic polymorphisms at loci that cannot be assumed to be selectively neutral (KEITH 1983; KEITH et al. 1985; MORIYAMA and POWELL 1996; HAHN 2008). The abundance of selectively maintained variation in nature behooves theoreticians to investigate just what modes of selection could be responsible.

The other side of the selection coin is fitness. If we wish to quantify what "kinds" of selection are best at maintaining variation, we really need to determine what kinds of fitnesses are best at keeping extant alleles from going extinct. Taken together, the fitnesses of all the genotypes in a population compose a "fitness set." The standard approach has been to generate many fitness sets either randomly (LEWONTIN et al. 1978; CLARK and FELDMAN 1986) or according to some preselected pattern (KARLIN 1981) and, using various techniques, to determine which arrangements of fitness are most likely to maintain genetic variation. The potential for variation under a given model is often measured as the volume of total available fitness space that maintains variation. We refer to these types of model as the "parameter-space" approach.

Early work (GILLESPIE 1977; LEWONTIN et al. 1978) found that for constant viability selection coefficients, the potential for a randomly chosen fitness set to maintain more than five alleles is vanishingly small. Even when fitness sets are structured so as to focus on biologically plausible fitness patterns (KARLIN 1981; KARLIN and FELDMAN 1981), the selective maintenance of a fully polymorphic equilibrium remains unlikely. The situation is similarly bleak in models of constant fertility selection (CLARK and FELDMAN 1986). These results do not necessarily suggest that selectively maintained polymorphism is unlikely in nature, but merely that it is unlikely in mathematical models that assume constant viabilities. Few (if any) population geneticists truly believe fitness to be a constant quantity (KOJIMA 1971). Selection is a function of the organism's environment and is likely to change, for example, in space or time.

One straightforward way to model changing selection at the genotype level is to make genotypic fitnesses change with the genetic composition of the population in which they are found. Negative frequency-dependent selection (FDS) (selection in favor of rare alleles) in particular provides a heuristic explanation for the maintenance of variation (CLARKE 1979), and indeed it has been detected in many natural systems (SINERVO and LIVELY 1996; BOND and KAMIL 1998; HUGHES et al. 1999; OLENDORF et al. 2006). Positive FDS (selection in favor of common alleles), which intuition suggests should eliminate variability, is also found in polymorphic natural systems, such as those involving Mullerian mimicry (LANGHAM 2004). The wide variety of FDS regimes found in nature (for a review see SINERVO and CALSBEEK 2006) suggests that an assessment of the ability of FDS to maintain variation should use a general model of frequency dependence.

The most general model of FDS is the pairwise interaction model (PIM), first put forth almost 40 years ago (SCHUTZ and USANIS 1969; COCKERHAM et al. 1972) as a model of selection due to intraspecific competition. In the PIM, genotypic viabilities are determined by a weighted sum of fitnesses in competitive interactions with other genotypes in the population. This formulation allows for a vast number of frequency-dependent viability schemes (including constant viability) to be parameterized by the model. It must be noted that in this study we omit any density-dependent effects of intraspecific competition from our models and focus only on genetic FDS.

Parameter-space approaches to modeling FDS (ASMUSSEN and BASNAYAKE 1990; TROTTER and SPENCER 2007) find that general FDS results in a greater potential for maintaining variation for any given number of alleles than the constant viability model. Nevertheless, the proportion of random sets that keep more than five alleles remains very low.

For polymorphism to exist new alleles must be both generated and maintained in the population. Most work has focused on the latter issue. Recall that for a system with n alleles, if we ignore mutation and drift and assume constant selection, the eventual fate of the system (e.g., fixation of one allele, extinction of one or more alleles, full polymorphism) depends entirely on the fitnesses of the genotypes in the population. Under FDS, the end state of the system depends on both the fitness set and the starting allele-frequency vector (ASMUSSEN and BASNAYAKE 1990). TROTTER and SPENCER (2007) showed that very few fitness sets keep all n original alleles for all initial allele frequencies.

In the natural world, allele frequencies are the product of a historical process of mutation and selection. The static equilibrium focus of the parameter-space approach ignores this dynamic of mutation–selection balance. It follows, then, that to extend our understanding of the maintenance of variation by FDS we must include a historical mutational process in our models.

One way to model mutation is to start with a monomorphism and bombard it with mutant alleles over many generations (SPENCER and MARKS 1988). Depending on their relative fitness, the mutant alleles either invade the growing polymorphism or are repelled to extinction. Similar models have been applied to the problem of ecological community assembly, where historical invasion processes can have significant impacts on the extant community structure (GINZBURG 1979; TAYLOR 1989; NEE 1990; AKCAKAYA and GINZBURG 1991). We refer to this class of model as the construction approach.

Each generation of a model using the construction approach has three steps: mutation, reproduction (including selection), and extinction check. Starting from an initial monomorphism, each generation mutant alleles are added to the system at some low frequency according to a mutation rate, µ. Second, all allele frequencies are iterated according to their fitnesses. Finally, alleles that have fallen below some threshold frequency are removed from the system and considered "extinct."

Early construction models found that constant selection can generate large numbers of alleles when combined with a mutational process (SPENCER and MARKS 1988). This type of approach has been used to investigate the dynamics of the invasion/extinction process, the effect of different mutational fitness distributions (MARKS and SPENCER 1991; SPENCER and MARKS 1992), and selection in spatially structured populations (STAR et al. 2007).

In this study, we continue our investigations of the potential for variation under FDS (see TROTTER and SPENCER 2007). We continue to use the PIM of FDS, this time incorporating a mutational process into the model. We model mutation by starting with single initial allele, which is then bombarded by novel mutant alleles with PIM fitnesses. By adding mutation to the model we can measure the potential for polymorphism in two new ways: First, we can observe the model for some arbitrary number of generations and record the dynamics of allele numbers and frequencies over that time period. Second, we can stop the mutation process at some point and continue iterating allele frequencies to assess the maximum number of alleles at the stable equilibrium for that fitness set (at the time mutation was stopped).

This dynamic style of modeling allows us to ask a variety of questions about the generation and maintenance of polymorphism. In equilibrium-based models, FDS fitnesses tend to maintain more alleles than constant fitnesses (TROTTER and SPENCER 2007) but it is not at all clear that this increased potential for polymorphism under FDS extends to construction models. Does assuming PIM fitness, using a construction approach, lead to polymorphisms with more alleles than those produced using constant fitness? What kinds of PIM fitnesses best facilitate the invasion of an existing polymorphism by new alleles? Which of these also inhibit the extinction of extant alleles? We are also interested in the number and frequency distribution of alleles generated by this construction approach to the PIM. Throughout we use constant-viability construction model results as a basis for comparisons.

We assume a large, isolated, diploid, monoecious population with random mating. We follow the dynamics of allele frequencies at a single locus. Each replicate run of the model begins with a monomorphism. Every generation, a new mutant allele, A_j, is added to the existing n-allele system at low frequency (p_j = 10^–6, j = n + 1). The addition of the jth allele always results in j new mutant genotypes. When generating mutant fitnesses, we assume all genotypic fitnesses are frequency dependent. We implement frequency-dependent selection by using the general PIM.

Under the PIM, each genotype A_iA_j has distinct fitnesses () in its interactions with the other genotypes A_kA_l in the population. These values are referred to individually as interaction fitnesses and collectively as fitness sets. A_iA_j is assumed to be equivalent to A_jA_i, and thus . All interaction fitnesses are drawn randomly from the uniform distribution [0, 1]. The consequences of drawing fitnesses from different distributions will be discussed elsewhere. We assume random mixing, such that intergenotypic interactions occur in proportion to genotype frequencies. Overall genotypic fitnesses, , are linear functions of the interaction fitnesses with all other genotypes in the model, weighted by the frequencies of the interacting genotypes:

The marginal fitness of A_i is a sum of fitnesses for all genotypes involving A_i, weighted by their frequencies: .

The single initial allele A₁ is assigned intermediate fitness, w₁ = 0.5. The frequency of allele A_i at generation t + 1 is calculated using the standard equation

(1)

where is the frequency of allele at generation t, and is the mean fitness of the population at generation t. The change in allele frequency after selection is . After allele frequencies are iterated, alleles whose frequencies fall below 10^–6 are considered to be extinct and are removed from the system. Each generation we recorded the numbers, ages, and frequencies of all alleles and also the mean fitness. These per-generation measures were used to calculate summary statistics to characterize the overall range of behaviors of the model. After 10,000 such generations had passed, we recorded snapshot fitness sets, numbers, and frequencies of alleles.

To successfully invade a polymorphism, a new mutant allele must be fitter on average than the existing alleles in the system (Equation 1). In models using constant viability, the mean fitness of the system increases after each successful invasion. As repeated invasions drive the mean fitness toward its maximum of 1, the upper bound of the uniform distribution of mutational fitnesses, new mutants are less and less likely to be more fit than average, and so invasions become rarer. SPENCER and MARKS (1988) found that the mutant-invasion process in constant-selection construction models "settles down" after 10⁴ generations, after which point invasions are rare but not impossible.

No such settling down occurs in the PIM construction model. The data discussed this chapter were taken from simulations which ran for 10⁴ generations. Smaller numbers of replicates were also run for 10⁵, 10⁶, and even 10⁷ generations. While long periods of relative stability can occur (see Figure 1), regardless of timescale, the system invariably returns to its dynamic fluctuating behavior. Measurements taken after 10,000 generations thus represent a snapshot of a population in a perpetual state of flux. Some of the alleles captured in that snapshot will be firmly established, others are new invaders, some are on their way to becoming common, and others on their way to extinction. To separate the transitory from the permanent, after 10,000 generations had passed, the mutation process was halted. The final fitness sets and allele frequencies were then allowed to continue iterating, in the absence of mutation, until equilibrium was reached (either a fixation equilibrium or all |p_i| < 10^–8).

View larger version (50K): In this window In a new window Download PPT slide   FIGURE 1.— Levels of polymorphism (bottom solid lines) and mean fitness (top, shaded lines) over time from randomly chosen sample runs of the PIM construction model.

 
The snapshot statistics taken after 10⁴ generations provide us with an instantaneous sample of the model's behavior, analogous to a single sample one might take from an evolving population in nature. The equilibrium statistics indicate how much of the snapshot variation is transient and how much is likely to be permanent.

Allele numbers:

PIM polymorphism construction is a dynamic process of mutant invasion and extinction of extant alleles. The snapshot and equilibrium numbers of alleles in polymorphisms produced by both PIM and constant-viability construction approaches are illustrated in Figure 2. Snapshot numbers indicate the level of polymorphism present in the model run after 10,000 generations. Equilibrium numbers suggest how many of the snapshot alleles would be permanent in the absence of mutation.

View larger version (12K): In this window In a new window Download PPT slide   FIGURE 2.— Snapshot and equilibrium numbers of alleles produced by 1000 replicate runs each of the PIM and constant-viability models of polymorphism construction.

 
Across 1000 replicate runs of each model, the PIM produced snapshot polymorphisms with 7.4 alleles on average (range, 1–59), while constant selection produced snapshot polymorphisms with 5.45 alleles (range, 1–10). These results agree with the intuitive consensus that FDS is more likely to maintain large amounts of variation: FDS polymorphisms tend to have more alleles than constant-selection polymorphisms in the snapshot taken at the end of runs. Nevertheless, while the PIM was capable of generating very large numbers of alleles, moderate numbers of alleles was a more common end result: 488 PIM runs finished with >6 final alleles present, but only 189 runs finished with >10 alleles.

In contrast to the snapshot results, the PIM equilibrium polymorphisms have fewer alleles on average (3.4) than do the constant-viability equilibria (4.6). The PIM shows a very drastic decrease in the number of alleles after iteration from snapshot to equilibrium. The iteration from snapshot to equilibrium effectively cuts out rare transient alleles, of which there are many more in the PIM snapshot polymorphisms than in their constant-viability counterparts.

Also notable, six PIM runs finished with monomorphism, while monomorphism is all but impossible to achieve in the constant-viability model.

Potential for polymorphism:

Historically (LEWONTIN et al. 1978; ASMUSSEN and BASNAYAKE 1990; STAR et al. 2007; TROTTER and SPENCER 2007), the "potential" for polymorphism under any given model has been measured as the proportion of random initial allele frequencies and randomly generated fitness sets that maintain all alleles. In the context of the construction approach, then, one could measure a sort of potential as the proportion of runs that had the same numbers of final and equilibrium alleles. For example, in the PIM construction model we found that 134 runs reached generation 10⁴ with five alleles. Of these 134 fitness sets, 44 maintained all five alleles at equilibrium, while the remaining 90 fitness sets lost at least one allele before reaching equilibrium. Measures of this potential for variation in constructed, and randomly generated, fitness sets with n = 2, ... , 7 are found in Figure 3. It is important to note that the two measures of potential illustrated in Figure 3 are not exactly equivalent. The constructed fitness sets were iterated to equilibrium from a single starting allele-frequency vector, namely the allele frequencies after 10,000 generations of mutation and selection. Randomly generated fitness sets were iterated to equilibrium from a large number of random allele frequencies.

View larger version (7K): In this window In a new window Download PPT slide   FIGURE 3.— Proportion of PIM fitness sets that maintained all snapshot alleles at equilibrium, starting from the snapshot allele-frequency vector. Open circles are data from Figure 1A in TROTTER and SPENCER (2007).

 
In the parameter-space approach, the proportion of sets that maintain all alleles decreases drastically as n increases. The region of parameter space that maintains polymorphism is minuscule for n > 4 (TROTTER and SPENCER 2007). In our constructed sets, the proportion that kept all alleles decreases approximately linearly with increasing n, a much less precipitous drop-off than that observed for random sets. The increased potential for polymorphism in constructed sets thus agrees with earlier studies (SPENCER and MARKS 1988) that found that selection tends to preserve alleles with fitnesses in those regions of fitness space that are most likely to maintain variation.

As an additional measure of potential for variation under the construction approach, each snapshot fitness set was iterated to equilibrium from 1000 randomly generated starting allele-frequency vectors (as in STAR et al. 2007). The proportion of random vectors that maintain all snapshot alleles for a particular fitness set gives an estimate of the domain of attraction for the fully polymorphic equilibrium of that set. STAR et al. (2007) use this method to partition their equilibrium fitness sets into three classes as follows: Type I fitness sets maintain all alleles from all start vectors and can be considered to have globally stable equilibria. Type II fitness sets maintain all alleles from only a subset of all start vectors and can be considered to have locally stable equilibria. Type III fitness sets lose at least one allele from all start vectors, implying that some of their snapshot alleles were transients. Numbers of types I, II, and III fitness sets from the constant and PIM construction models can be found in Table 1. Nearly half of all PIM snapshot equilibria contained at least one transient allele. Less than 1% of PIM sets were globally stable, but 24.4% of sets kept all alleles from the snapshot allele-frequency vector.

View this table: In this window In a new window   TABLE 1 Numbers of types I, II, and III fitness sets from 1000 runs each of the constant-viability and PIM construction models

 

Allele-frequency distributions:

We recorded the number and frequencies of alleles in the polymorphism after 10,000 generations and at the eventual equilibrium. We wished to see if the construction approach to the PIM left any particular signature in the allele-frequency distributions of the polymorphisms it produced. Here we use as a measure of centrality of allele frequencies. If all alleles present are equally frequent, and . If one or two alleles are very common and the others rare, I will approach its maximum value of n – 1/n.

Distributions of I-values for the polymorphisms with five alleles present in the PIM construction model after 10,000 generations, at equilibrium, as well as five-allele polymorphisms from the PIM parameter-space approach and random allele frequencies can be found in Figure 4. Random allele frequencies were generated by breaking the interval (0, 1) into n pieces, using n – 1 uniform random numbers (broken-stick method; see HOLST 1980).

View larger version (14K): In this window In a new window Download PPT slide   FIGURE 4.— The proportion of allele-frequency vectors falling within successive shells around the centroid of the allele-frequency state space. Only vectors for which n = 5 are included. Data for the parameter-space model are taken from TROTTER and SPENCER (2007).

 
Polymorphisms produced by the PIM parameter-space approach are in general slightly more centered than expected by chance (Figure 4, dotted line). Polymorphisms produced by the PIM construction approach, both at snapshot and at equilibrium, however, are generally less centered than expected by chance. The I-values of the equilibrium polymorphisms tend to be more centered than those of the snapshot polymorphisms.

Mean fitness:

The relationship between allele number and mean fitness in the constant and PIM construction models, as seen in two randomly chosen model runs, is illustrated in Figure 5. In the constant-selection model, mean fitness rapidly asymptotes to a value near 1 during an initial burst of successive mutant invasion. This positive relationship between allele number and mean fitness eventually breaks down after 100 generations. After this time, highly fit mutants begin to drive other extant alleles to extinction, resulting in a drop in allele number but continuing small increases in mean fitness. The extremely high mean fitness finally makes the polymorphism resistant to invasion.

View larger version (9K): In this window In a new window Download PPT slide   FIGURE 5.— Levels of polymorphism (solid line) and mean fitness (dotted line) as functions of time for 1000 generations each from single-example runs of the constant-fitness construction model and the PIM construction model.

 
In the PIM results mean fitness and allele number appear tightly coupled when viewed over long timescales (Figure 1). In these trajectories, it appears that allele number and mean fitness are negatively correlated. However, a closer look at shorter timescales (Figure 5) shows that mean fitness and allele number are actually much less tightly coupled in the PIM than in the constant-viability model. The initial spike in allele number is accompanied by an increase in mean fitness. Another sharp spike in allele number between 500 and 600 generations is accompanied by a decrease in mean fitness. Nevertheless, the steady increase in allele number between 600 and 800 generations coincides with a plateau in the mean fitness.

In the snapshot results of both models, there is a slight negative relationship between allele number and mean fitness (constant selection model, r = –0.327, P < 0.001; PIM, r = –0.301, P < 0.001). This relationship is much stronger in the equilibrium results for constant viabilities (r = –0.675, P < 0.001) and much weaker in the equilibrium results for the PIM (r = –0.138, P < 0.001), where it appears that there is no strong relationship between mean fitness and allele number. Equilibrium polymorphisms with many alleles are only marginally less fit than those with few alleles.

Analysis of fitness sets:

We now turn to the question of what fitnesses are best at maintaining variation under the PIM. To facilitate comparisons across sets with different n, we have summarized the information contained in the fitness sets by partitioning the interaction-fitness values from each set into classes. The fitness-class values themselves are the means of all interaction fitnesses that fall into that class. The classes are partitioned according to the homozygosity and the degree of allele sharing between the interacting genotypes. The class of homozygote-by-unlike-homozygote interactions is characterized by C_ii,jj, that of homozygote-by-like-heterozygote interactions by C_ii,ij, etc. The final class is composed of all genotypes' fitnesses in interactions with their own genotypes, C_ij,ij.

Interaction-class means for constructed PIM polymorphisms of all n > 3 follow the same general pattern. Fitness-class data for PIM constructed and parameter-space polymorphisms of size n = 5 can be found in Figure 6. We chose to use the n = 5 case as it is indicative of the general pattern found in fitness sets for both approaches and provides the largest number of snapshot replicates from both the PIM and the constant-viability model.

View larger version (15K): In this window In a new window Download PPT slide   FIGURE 6.— Fitness-class data for both constructed and parameter-space PIM polymorphisms with n = 5 alleles.

 
In the parameter-space approach, homozygote fitnesses tend to be lower than heterozygote fitnesses (implying heterozygote advantage), and interaction fitnesses decrease as allele sharing increases (giving a strong suggestion of negative FDS; see TROTTER and SPENCER 2007). In both PIM construction results, the signal for heterozygote advantage remains strong but negative FDS is not evident. Interaction fitnesses with like homozygotes (C_ii,ii, C_ij,jj) are maximal, not minimal, in the constructed fitness sets. High-fitness interactions with similar genotypes are indicative of positive FDS, which is a counterintuitive mechanism for maintaining alleles.

In the multiallelic case, positive self-FDS can be overwhelmed by the fitness impacts of other alleles in the system. In other words, the fitness of a given allele may be more strongly dependent on the frequency of a different allele than on its own frequency. Further analysis of constructed fitness sets that had five alleles after 10,000 generations reveals that for all alleles, fitness is often strongly correlated with the frequency of the most common allele (A₁ in most cases) than with any other allele frequency.

Flavors of frequency dependence:

Another method of finding patterns in unwieldy PIM fitness sets is to search the constructed sets for common "flavors" of selection such as negative FDS, positive FDS, and heterozygote advantage and disadvantage. We define our flavors on the basis of schemes defined by relationships between fitness classes. A genotype will have higher fitness at low frequency if it has low fitness in interactions with like genotypes. We consider a fitness set to be under negative FDS if self–self interactions (C_ii,ii and C_ij,ij) are the lowest fitness-class values. To define strict negative FDS, we extend the above definition to include low fitness in interactions between genotypes that share similar alleles (C_ii,ii, C_ii,ij, C_ij,jj, C_ij,ik). A fitness set was considered to contain heterozygote advantage if all heterozygote fitness-class values were greater than all homozygote fitness-class values. Positive FDS schemes are the mirror images of the negative FDS schemes, and homozygote advantage is the opposite of heterozygote advantage.

Frequencies of the different flavors of FDS in constructed, parameter space, and random fitness sets can be found in Table 2. Again, Table 2 lists only results for fitness sets where n = 5 although the general pattern of flavor for each model holds true for any n > 5. The fitness schemes we used accounted for between 22 and 32% of fitness sets, while the majority of sets had no obvious intuitive pattern to their fitnesses.

View this table: In this window In a new window   TABLE 2 Frequencies of selection schemes in snapshot, equilibrium, parameter-space equilibrium, and random fitness sets where n = 5 alleles

 
Negative FDS was more common in the parameter-space results than in the construction approach results, while the reverse was true of positive FDS. The prevalence of positive FDS may seem counterintuitive at first, as we usually think of positive FDS as favoring monomorphism. In the multidimensional fitness space of the PIM, the total fitness of an allele under positive self-FDS is likely to be more strongly dependent on the frequency of a nonself allele. The constructed fitness sets also revealed a stronger signal for heterozygote advantage than did the parameter-space fitness sets.

Incorporating history into the analysis of polymorphism maintained by FDS reveals several exciting features. First, the PIM construction approach can produce high levels of polymorphism. Within 10⁴ generations, PIM construction produced as many as 59 alleles, and one model run that went for 10⁷ generations had polymorphisms of up to 70 alleles. To our knowledge, this is the first model to predict such large numbers of alleles without specifically invoking negative FDS.

The parameter-space approach tells us that the volume of the total PIM parameter space that has positive potential for variation is infinitesimally small for n > 5 (TROTTER and SPENCER 2007). The equilibrium model results provide a means of comparing the construction and parameter-space approaches to modeling the maintenance of variation. The number of constructed PIM fitness sets that maintained all their snapshot alleles at equilibrium was orders of magnitude larger than one would expect had we iterated randomly generated fitness sets to equilibrium from the same initial n. Thus, while the total volume of parameter space maintaining variation is small, the polymorphism construction process evolves fitnesses and allele frequencies into those regions of the parameter space. This finding begs the question, What is it about PIM fitnesses that enables such high numbers of alleles? Since the PIM is based on the premise that selection is a consequence of competition, we can gain insight from comparisons with other competition-based models. Our models of the maintenance of genetic variation under the PIM have much in common both with species-coexistence models in community ecology and with phenotypic-evolution models of evolutionary game theory.

Ecological models of multispecies coexistence, for example, are in some ways analogous to parameter-space models of multiple-allele polymorphism. Each of these aims to investigate the relationships between species (or alleles) that best facilitate coexistence (polymorphism). In models of multispecies community assembly, intransitive competitive interactions increase the likelihood of multispecies coexistence (LAIRD and SCHAMP 2006). In a perfectly intransitive competitive network, all competitors win and lose to an equal number of competing strategies (A beats B beats C beats A) as in rock–paper–scissors (RPS), whereas in a transitive/hierarchical network, some strategies are universally superior (A beats B and C, B beats C.) Several naturally polymorphic systems have been identified where variation is maintained by genetic RPS games (SHUSTER and WADE 1991; SINERVO and LIVELY 1996; SINERVO et al. 2007). In our construction approach to the PIM, each new mutant's interaction fitnesses are drawn from the uniform distribution on [0, 1], so that on average half its interactions will have values >0.5 (a competitive "win") and the other half <0.5 (a competitive "loss"). Since this distribution of fitnesses will tend to produce equal numbers of wins and losses across rows in the fitness matrix, the way in which we have generated mutant fitnesses could thus be interpreted as promoting a kind of weak intransitivity at the level of genotypic fitness. At the level of marginal allelic fitness, however, the intuitive argument for implicit intransitivity disappears.

At a glance, the PIM fitness matrix is also reminiscent of the payoff matrix of evolutionary game theory (MAYNARD SMITH 1982). Game theoretic approaches to FDS assume that, in the long term, the genetic information underlying phenotypic evolution can be ignored. Population genetic models concern themselves more with the short-term genetic changes that go on between long-term (stable phenotypic equilibrium) "stops." It is interesting, then, that the FDS construction model rarely settles down to any long-term stable state. Our PIM population remains in a constant intermediate state between fixations, with monomorphisms being exceptionally rare. The initially monomorphic founder allele remained in the population after 10,000 generations in only 1 replicate. Indeed, the turnover of alleles is so consistent and rapid that one fails to see how a long-term phenotypic equilibrium would come about at all.

In constructed polymorphisms with constant selection, the mean fitness of the population necessarily increases with each successful mutant invasion. Eventually, successive mutant invasions drive the mean fitness so high as to make subsequent mutant invasions all but impossible. In the FDS model, the level of polymorphism and the population mean fitness are less closely coupled. When plotted across generations, the mean fitness and the level of polymorphism exhibit booms and crashes that are not necessarily coincident in time (Figure 5). There is actually a weak negative relationship between mean fitness and allele number, but this is notably skewed by the small number of observations of high allele numbers. The uncoupling of the mean fitness and the allele number is part of what allows the FDS polymorphism to remain permanently invasible. Low mean fitness increases the possibility of a new mutant being sufficiently fit to invade the polymorphism.

If a new mutant invades that has negative impacts on other allelic fitnesses, as the mutant's frequency increases so too will the magnitude of its negative impact on other alleles, depressing the mean fitness and potentially resulting in extinctions. The extinction of unfit alleles then allows the mean fitness to rebound. Alternatively, if a low-fitness allele goes extinct, but had largely positive interactions with the other alleles in the system, its loss may cause the mean fitness of the remaining alleles to drop. The lack of relationship between allele number and mean fitness, then, makes more sense once one realizes that the addition of a new mutant can either increase or decrease mean fitness and that an extinction can similarly either increase or decrease mean fitness.

Another interesting result of the PIM construction approach is the wide range of snapshot and equilibrium allele-frequency distributions produced (Figure 4). Recall that if all alleles are present at equal frequencies, I = 0, and we say the distribution is central. Larger I-values indicate skewed allele-frequency distributions. The skewness of snapshot allele frequencies could be explained by the presence of rare transient alleles. One would expect that the subsequent iteration from snapshot to equilibrium would remove any transient alleles and produce more centered () allele frequencies. Somewhat counter to expectation, equilibrium distributions of alleles remain far less centered than expected by chance. Our results here mirror the findings of the constant-viability construction approach studies done in the 1990s (SPENCER and MARKS 1988, 1992, 1993; MARKS and SPENCER 1991). They found that constant-viability construction approaches produce allele-frequency distributions that are less central than expected by chance (MARKS and SPENCER 1991; SPENCER and MARKS 1992, Figure 1). Constant-viability parameter-space approaches predict that highly polymorphic allele-frequency distributions will be more central than expected by chance (LEWONTIN et al. 1978, Figure 3) and so are also much more central than constructed polymorphisms. In the case of the PIM, the constructed polymorphisms are much less even than predicted by the parameter-space models. The distributions of the equilibrium constructed polymorphisms suggest that unevenness cannot be explained away by the influence of rare transient alleles. The more complex fitness structure of the PIM allows for richer equilibrium behavior and a wide variety of equilibrium allele-frequency distributions.

One of the challenges of multiallelic models is producing measures for comparing between systems with different n alleles. We are interested in the kinds of fitnesses that best maintain variation, but the sorts of fitness relationships that are possible in systems of 10 alleles differ from those in systems of n = 3 and also from those of n = 30. The partitioning of fitness sets into classes (C_ij,kl) and flavors allows us to compare fitness sets across all n, and indeed the general patterns shown in Figure 6 and Table 2 apply to all constructed fitness sets with n > 5.

In the context of fitness classes, negative FDS could be produced by very low values in the C_ii,ii, C_ij,ij, and C_ij,jj classes. Logically, if a genotype fares poorly in interaction with similar genotypes, it will be less fit when its component alleles are common. By extension, negative FDS will occur whenever the values of interaction fitnesses are negatively correlated with the numbers of alleles shared between interacting genotypes. Biologically, these kinds of interactions could be caused by mate choice, disease transmission, resource limitation, etc. Parameter-space approaches find just such a pattern of negative FDS in the overall means of fitness-class values (Figure 6). Surprisingly, the self-interaction classes were maximal in both the snapshot and the equilibrium constructed fitness sets (Figure 6). This kind of fitness pattern suggests positive FDS, which was found in 5% (snapshot) and 9%(equilibrium) of constructed fitness sets (Table 2). In the PIM construction model, fitnesses are favored that facilitate generating variation more than those that maintain it. A high positive self-interaction fitness serves to nudge fitness higher at the initial stages of invasion.

We are accustomed to thinking of FDS in terms of accepted patterns like negative FDS (allele fitness negatively correlated with allele frequency) and its opposite, positive FDS. Our analysis of standard flavors of FDS, however, classified at most 30% of the constructed PIM polymorphisms. We postulate three explanations for polymorphism that our analyses would not have uncovered. One possible explanation for the maintenance of larger numbers of alleles could be the presence of smaller, core networks (a three-allele RPS interaction, for example) with other alleles maintained by weaker negative FDS with the main vertex alleles. Similarly, one could also postulate a large polymorphism where all fitnesses depend most strongly on the frequency of one or a few "keystone" alleles. Finally, the snapshot polymorphisms could owe their existence to "ghosts of interactions past" from earlier in the construction process of invasion and selection.

The only noteworthy difference between the snapshot and the equilibrium fitness sets was the increased signal for heterozygote advantage in equilibrium fitness sets (Table 2). In single-locus constant-viability models, average superiority of heterozygotes () is a necessary (but not sufficient) condition for polymorphism (GINZBURG 1979). Homozygote fitnesses were higher in snapshot than in equilibrium sets, while heterozygote fitnesses were higher in equilibrium sets. This result reconfirms the finding of parameter-space approaches that heterozygote advantage is beneficial for the maintenance of alleles—presumably the alleles lost in between the snapshot and the equilibrium states of the construction model were those that did not exhibit heterozygote advantage.

This implementation was the first construction approach under FDS. It is encouraging that this construction approach to the PIM, which borrows from both ecological (GINZBURG et al. 1988; AKCAKAYA and GINZBURG 1991) and evolutionary construction (SPENCER and MARKS 1988, 1992; MARKS and SPENCER 1991) traditions, is able to explain larger amounts of variation than its progenitors.

An obvious feature of this model is the assumption of independently distributed mutant fitnesses. A more natural way to model mutation would be to assume each mutant allele to be descended from an existing parent allele (SPENCER and MARKS 1993), and such investigations are already underway. Further modeling projects will address the effects of variable mutation rate, as well as finite population size, drift, and spatial structure on the potential for polymorphism under FDS.

The authors thank Bastiaan Star for helpful discussion and Lev Ginzburg and an anonymous reviewer for comments on the manuscript. This work was supported by the Marsden Fund of the Royal Society of New Zealand (contract U00315) and by the Allan Wilson Centre for Molecular Evolution and Ecology. M.V.T. was the recipient of a scholarship from the Division of Sciences of the University of Otago.

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