The level of gene flow considerably influences the outcome of evolutionary processes in structured populations with spatial heterogeneity in selection pressures; low levels of gene flow may allow local adaptation whereas high levels of gene flow may oppose this process thus preventing the stable maintenance of polymorphism. Indeed, proportions of fitness space that successfully maintain polymorphism are substantially larger in spatially heterogenous populations with lower to moderate levels of gene flow when compared to single-deme models. Nevertheless, the effect of spatial heterogeneity on the evolutionary construction of polymorphism is less clear. We have investigated the levels of polymorphism resulting from a simple two-deme construction model, which incorporates recurrent mutation as well as selection. We further compared fitness properties, stability of equilibria, and frequency distribution patterns emerging from the construction approach and compared these to the static fitness-space approach. The construction model either promotes or constrains the level of polymorphisms, depending on the levels of gene flow. Comparison of the fitness properties resulting from both approaches shows that they maintain variation in different parts of fitness space. The part of fitness space resulting from construction is more stable than that implied by the ahistoric fitness-space approach. Finally, the equilibrium allele-frequency distribution patterns vary substantially with different levels of gene flow, underlining the importance of correctly sampling spatial structure if these patterns are to be used to estimate population-genetic processes.

---

THE outcome of selection in habitats with spatial heterogeneity in selection pressures is critically influenced by levels of gene flow (LENORMAND 2002; KAWECKI and EBERT 2004). For example, low levels of gene flow may result in local adaptation within a deme of a metapopulation, whereas high levels of gene flow may cause locally advantageous alleles to be lost. Selection in a metapopulation is more likely to result in the maintenance of genetic variation if the levels of gene flow are low compared to the heterogeneity of the demes (SMITH and HOEKSTRA 1980; KARLIN 1982). Therefore, according to population-genetic models, the potential for heterogeneous habitats to maintain genetic variation is highest in models with low levels of gene flow (FELSENSTEIN 1976; HEDRICK et al. 1976; STAR et al. 2007).

To investigate the potential of spatial heterogeneity to maintain polymorphism, most of these population-genetic models use simple single-locus viability-selection models with a limited number of alleles (mostly two; LEVENE 1953; BULMER 1972; KARLIN 1982) or particular selection schemes (MUIRHEAD 2001; NAGYLAKI and LOU 2001). Yet, for a single-deme model, the proportion of random fitness space that maintains genetic variation becomes vanishingly small for even moderate numbers of alleles (LEWONTIN et al. 1978). While this proportion is higher in a spatial two-deme model, it is still small for a higher number of alleles, a result indicating that the part of fitness space maintaining many alleles remains nonetheless very restricted (STAR et al. 2007). In a different approach to the study of genetic polymorphisms, it has been shown that measures of the size of stable fitness space need not be indicative of how easy it is to construct these polymorphisms in an evolving system where mutations are introduced one by one to a standing stock of alleles (SPENCER and MARKS 1988). Indeed, a simple evolving single-locus model will easily reach the small parts of fitness space that are stable (MARKS and SPENCER 1991). Nevertheless, these construction models may be constrained due to other factors, such as sex-dependent viabilities, lowering the ease with which polymorphisms are constructed (MARKS and PTAK 2000). Such models have not previously been studied in a spatial context; hence it is unknown if different levels of gene flow either constrain or promote the construction of selectively maintained genetic polymorphism in a heterogeneous habitat.

The present study investigates the levels of polymorphism resulting from a construction approach incorporating recurrent mutation and selection in a simple two-deme viability-selection model with different levels of gene flow. The fitness properties of fitness matrices emerging from the construction approach are compared to the fitness properties resulting from a fitness-space approach to see if similar areas of the fitness space are reached. We also analyze the ease with which new mutations invade this model. Since multi-deme viability systems can have polymorphic equilibria that are locally rather than globally stable, the stability of the equilibria resulting from the recurrent mutations is investigated. Finally the equilibrium allele-frequency vectors are investigated to see if particular patterns emerge from the construction approach.

The model analyzed here is based on the well-known system of recurrence equations for constant-viability selection at a single locus, but then adapted for two demes. Thus, the frequency of the ith allele (i = 1, 2,..., k), A_i, in the dth deme (d {1, 2}), after selection is given by

(1)

in which is the current frequency of A_i in the dth deme and ) is the fitness of the A_iA_j genotypes in the dth deme. Migration follows selection, and a proportion (m) of the frequency vector p_i,d is divided over both demes, giving the new frequency of A_i in deme d,

(2)

where if d = 1 and vice versa. The model is initiated with a single allele with a frequency of 1.0 and a homozygote fitness of 0.5 in both demes. Each generation, a new allele (A_k+1) is added to a random deme with an initial allele frequency of and all new fitnesses ( for i = 1, ..., k + 1 and d = 1, 2) are independently drawn from the uniform distribution on [0, 1]. Because each genotype has a unique fitness, we are essentially studying an "infinite alleles" model. Mutations occur not just at a single site, but anywhere within a gene and these changes lead to different gene products from which the different fitnesses arise. Any allele is considered extinct if . The model was run for seven different levels of gene flow (m {0, 0.01, 0.05, 0.1, 0.2, 0.5, 1.0}) up to 10,000 generations with 1000 replicates for each level of gene flow. To compare the results with a homogeneous habitat, we also ran 1000 replicates using a single-deme model with the same initial settings and extinction definition described above.

After 10,000 generations, the fitness sets, the number of alleles present (n), and allele-frequency vectors were recorded for further analysis. The fitness data were analyzed for levels of heterozygote advantage and for correlation of fitness values between demes to investigate the extent of balancing selection due to heterozygote advantage and local adaptation. To investigate the ease with which mutations invade the model, we recorded the number of successfully invading mutations and their persistence time. The distributions of allele-frequency vectors were analyzed and the Ewens–Watterson test (EWENS 1972; WATTERSON 1978) was used to discover if the allele frequencies were detectably different from those expected under the neutral hypothesis.

A substantial part of the fitness space maintaining fully polymorphic equilibria in a two-deme model leads to locally stable rather than globally stable equilibria (STAR et al. 2007). To investigate the stability of the equilibria resulting from the construction approach, each recorded fitness set was evaluated with 250 random initial allele-frequency vectors that were generated using the "broken-stick" method (MARKS and SPENCER 1991). Each of these 250 runs was iterated until equilibrium, or until any allele became extinct at . Fitness sets that led to the maintenance of all recorded alleles for all initial allele-frequency vectors were defined as type I fitness sets. Those that led to the maintenance of all alleles for <250 of the runs were defined as type II fitness sets. The proportion of frequency vectors leading to a fully polymorphic equilibrium, as a measure of the size of domain of attraction, was also recorded for type II fitness sets. Furthermore, since the model is stopped after an arbitrary time, a number of fitness sets may have one or more transient alleles (MARKS and SPENCER 1991). These fitness sets never maintained the recorded total number of alleles and were defined as type III fitness sets.

Several properties emerging from the construction approach were compared with those from a fitness-space approach (STAR et al. 2007). This comparison is not straightforward since the construction model results in higher levels of polymorphism. Comparisons of heterozygote advantage, fitness correlations between demes and on distributions of allele-frequency vectors were made for those simulations for which n = 5, the only level of n for which enough replicates were obtained in both approaches for each value of m. Comparisons of the size of domain of attraction for type II fitness sets were made using data from the fitness-space approach averaged over the number of alleles (n).

Gene flow has a strong effect on the average number of alleles maintained after 10,000 generations (Figure 1 and Table 1). In the two-deme model, more than twice as many alleles are maintained on average in absence of gene flow compared to the number maintained at high levels (m 0.5). All post hoc comparisons [Tukey's honestly significant difference (HSD) test] were significant (P < 0.0001) except for the comparison between the highest two levels of migration (0.5 and 1.0). Thus, in comparison with the single-deme model, more alleles are maintained for lower and intermediate levels (m < 0.2) of gene flow, while for higher levels (m 0.2) of gene flow, fewer alleles are maintained.

View larger version (10K): In this window In a new window Download PPT slide   FIGURE 1.— The average number of alleles (n) maintained after 10,000 generations as a function of gene flow in the two-deme model. The shaded symbol indicates the result for the single-deme model. The error bars are 95% confidence intervals. Gene flow has a strong significant (single-factor ANOVA, F7,7992 = 1569, P < 0.0001) effect on the average number of alleles maintained. Data from the single-deme model were added as an additional level to the factor.

 

View this table: In this window In a new window   TABLE 1 The number of simulations leading to n alleles for different levels of gene flow

 

Heterozygote advantage:

Viability selection is capable of maintaining high to moderate levels of polymorphism in a single deme if the fitness sets have a specific structure (LEWONTIN et al. 1978; KARLIN 1981). An example of such a specific structure is a form of heterozygote advantage whereby the mean fitness of heterozygotes is greater than the mean fitness of homozygotes. We defined the amount of heterozygote advantage in the fitness sets after 10,000 generations, as , where the averaging is over all fitnesses and both demes. We use as a heuristic for the relative importance of heterozygote advantage as a form of balancing selection in the two-deme model. Individual runs of the construction model result in different numbers of alleles present (n) after 10,000 generations and we analyzed data only for combinations of n and m that had at least 10 replicates (Table 1).

Levels of gene flow and the number of alleles have an interaction effect on the levels of (Figure 2). For the two-deme model, this interaction effect is mainly due to a varying response to lower numbers (n 4) of alleles and higher levels (m > 0.1) of gene flow (Figure 2a). The interaction effect is also due to the comparison to the single-deme model, as its levels of heterozygote advantage increase differently for increasing numbers of alleles. For higher numbers (n 5) of alleles, levels of heterozygote advantage in the two-deme model are more affected by gene flow than by n and the interaction effect diminishes (Figure 2, b–d). Gene flow has a larger effect on the amount of in the construction model when compared to the fitness-space approach for the same number of alleles. These results show that, for a similar level of polymorphism, heterozygote advantage is more important in the construction approach when compared to the fitness-space approach for higher levels of gene flow while the opposite is true for lower levels of gene flow.

View larger version (16K): In this window In a new window Download PPT slide   FIGURE 2.— Levels of heterozygote advantage () as a function of gene flow (m) for fitness sets containing n alleles after 10,000 generations. The shaded symbols indicate the results for the single-deme model whereas the open symbols indicate those for the fitness-space approach (see text for explanation). Standard errors are very small and are omitted for clarity. The dashed line indicates zero heterozygote advantage, which is the average for randomly generated fitnesses. Gene flow (m) and the number of alleles (n) have a significant interaction effect (ANCOVA, F7,7914 = 134, P 0.0001, m as factor, n as covariate) on .

 

Disruptive selection:

As a measure of disruptive selection, we calculated Pearson's correlation coefficients () between the fitnesses of the genotypes in the two demes for each fitness set with n 2. Negative correlation coefficients imply local adaptation and disruptive selection. Only combinations of n and m were analyzed for which at least 10 replicates were found.

Gene flow and number of alleles have an interaction effect on Pearson's correlation coefficients (Figure 3). As in the patterns found for levels of heterozygote advantage, the interaction effect of gene flow and number of alleles diminishes with an increasing number of alleles. The general pattern that emerges is that at low levels of gene flow, disruptive selection is prominent across all levels of polymorphism. As gene flow increases, fitnesses become more correlated, and disruptive selection no longer emerges as an important form of balancing selection. Interestingly, as with the patterns in levels of heterozygote advantage, the construction approach and the fitness-space approach result in completely different levels of disruptive selection relative to the amount of gene flow.

View larger version (11K): In this window In a new window Download PPT slide   FIGURE 3.— Pearson's correlation coefficients () between two demes for fitness values as a function of gene flow (m) for fitness sets with n 2. The error bars are standard error intervals. The dashed line indicates zero correlation, which is the average for randomly generated sets of fitnesses. The open symbols indicate the results for the fitness space approach (see text for explanation). Gene flow (m) and the number of alleles (n) have a significant interaction effect (ANCOVA, F6,6924 = 34.1, P 0.0001, m as factor, n as covariate) on .

 
The positive correlations at moderate to high levels of gene flow are indicative of the requirement for heterozygote advantage to maintain polymorphism as the spatial structure begins to dissipate. Note also that we generated higher levels of polymorphism (n > 8) only when disruptive selection was evident, possibly indicating that disruptive selection is a particularly important form of balancing selection for higher levels of polymorphism. Furthermore, these results are not due to high numbers of rare alleles: A similar pattern was found when performing this analysis using only the fitness values of genotypes with common alleles (i.e., with frequencies >5%; data not shown).

Invasions:

The effect of gene flow on the number of mutations that successfully invaded the system and their persistence was investigated using two different approaches. We counted the total number of invasions and calculated the ratio of long-term mutants (i.e., persistent for >1000 generations) over short-term mutants (i.e., persistent for <1000 generations).

Only mutants that became extinct before 10,000 generations were used in the persistence analysis.

Gene flow has a strong effect on both the number of invasions and the ratio of long-/short-term invasions (Figure 4). For the number of invasions, most post hoc comparisons (Tukey's HSD test) were significant, except for the comparison m = 0 vs. 0.01 and for the ratio of long-/short-term mutants, most of these comparisons were significant, except for the comparisons m = 0.1 vs. 1.0 and m = 0.2 vs. 0.5. These results show that the two-deme model has more invading mutants compared to the single-deme model for any level of gene flow and this number of invasions peaks at an intermediate level of gene flow (m = 0.02). Nevertheless, in general, higher invasion rates need not result in higher levels of polymorphism after 10,000 generations. Indeed, the lower ratio of long-/short-term mutants in the two-deme model for most levels of gene flow (m > 0.05) shows that the increased number of invasions is mainly the result of an increase in short-term mutants rather than long-term mutants. Consequently, after 10,000 generations, most of these short-term mutants have become extinct and do not influence the final level of polymorphism. In contrast, even though a strong reduction is observed in the number of invasions for lower levels of gene flow (m < 0.2), proportionally more of these invasions are long-term mutants, resulting in higher levels of polymorphism after 10,000 generations compared to the single-deme model.

View larger version (13K): In this window In a new window Download PPT slide   FIGURE 4.— The square root of the number of invasions (circles, left y-axis) and the square root of the ratio of long-/short-term mutants (triangles, right y-axis) as a function of gene flow. The shaded symbols indicate the results for the single-deme model. The error bars are 95% confidence intervals. Gene flow has a strong significant effect on the number of invasions (single-factor ANOVA, F7,7992 = 1350, P 0.0001) and the ratio of long-/short-term mutants (single-factor ANOVA, F7,7992 = 589, P 0.0001). Both responses were square-root transformed to comply with the ANOVA assumption of homoscedasticity of residuals.

 

Stability of equilibria:

The stability of the fitness sets was investigated using two different approaches: First, the proportion of simulations leading to type I, II, or III fitness sets was recorded for each level of gene flow. Type I fitness sets maintain the recorded polymorphic equilibrium for all of the 250 initial random allele-frequency vectors, type II maintain this equilibrium for some of these vectors, and type III fitness sets never maintain the recorded polymorphic equilibrium. As such, type I fitness sets can be considered the most stable, type II fitness sets moderately stable, and type III fitness sets the least stable. Second, the proportion of initial allele-frequency vectors leading to equilibrium for type II fitness sets was recorded as a measure of the size of domain of attraction. A type II fitness set with a larger domain is considered more stable. The effect of gene flow on domain size was investigated for all levels of gene flow for which at least 10 type II fitness sets were found.

Gene flow has a particularly strong effect on the types of fitness sets that maintain polymorphism; most fitness sets are type I for high levels of gene flow whereas for low levels of gene flow most were type II (Table 2). Because any fully polymorphic equilibrium is globally stable for a single-deme model, no type II fitness sets exist for this model. The proportion of type III fitness sets decreases more moderately with higher levels of gene flow, but is higher in the single-deme model when compared to the two-deme model for the highest levels of gene flow. In analyzing the size of the domain of attraction for type II fitness sets, only a sufficient number was found for m 0.2. For these fitness sets most initial frequency vectors lead to a fully polymorphic equilibrium (Figure 5). Furthermore, the proportion of initial frequency vectors leading to equilibrium increases for higher levels of gene flow.

View this table: In this window In a new window   TABLE 2 The proportion of simulations leading to type I, II, or III fitness sets for different levels of gene flow

 

View larger version (9K): In this window In a new window Download PPT slide   FIGURE 5.— Size of domain of attraction measured by the proportion of initial allele-frequency vectors leading to fully polymorphic equilibrium as a function of gene flow for type II fitness sets. The error bars are standard error intervals. For the construction model, gene flow has a significant (Kruskal–Wallis, = 182, P < 0.0001) effect on the size of domain of attraction.

 
Overall, both the proportion of type I fitness sets increases and the size of domain of attraction enlarges with higher levels of gene flow, thus the fitness sets are more stable for these levels of gene flow. The size of the domain of attraction in the construction model is larger in comparison to the fitness-space approach where only the minority of initial frequency vectors leads to fully polymorphic equilibrium for m 0.2 (Figure 5) (STAR et al. 2007).

Frequency distributions:

The distribution of allele frequencies after 10,000 generations was summarized by calculating the index I, the sum of squared differences of the frequencies from the mean (LEWONTIN et al. 1978). Two indices of I were calculated for each final allele-frequency vector; one index was calculated using the allele-frequency vector (p_i,d) taken from one of the demes, the other index was calculated using the pooled frequency vectors over both demes. The latter index was used to investigate the frequency distributions that emerge if the underlying spatial structure is ignored. The index I is given by for the frequencies in a single deme and by for the pooled allele frequencies over both demes and only frequency vectors for which n > 1 were used for both indexes. If all alleles have equal frequency, , and if one allele is close to fixation . We are mainly interested in the effect of gene flow on the index I. Nevertheless, the number of alleles also affects the values of this index, not the least by increasing the maximum possible I values for increasing number of alleles. Only combinations of n and m were analyzed for which at least 10 replicates were found.

Gene flow and the number of alleles have an interaction effect on I on the basis of the allele-frequency vectors taken from one of the demes (Figure 6). For the pooled frequency vectors, no significant interaction effect was detected, but both gene flow and the number of alleles separately influenced this index (Figure 6). Post hoc comparisons (Bonferroni) between levels of gene flow for the pooled index show that the distributions of frequencies for these vectors in the two-deme model are less skewed for any level of gene flow when compared to the frequency vectors of the single-deme model (Table 3). The distribution of pooled allele frequencies is most even for the lowest (m < 0.05) and the highest (m > 0.2) levels of gene flow. Gene flow appears to have a larger effect on the distribution of frequency vectors taken from a single deme than on the pooled frequency vectors, but the significant interaction effect with the number of alleles prevents further analysis using post hoc tests. Nevertheless, large differences exist between distribution of frequency vectors taken from a single deme and the pooled frequencies, especially for low (m < 0.1) levels of gene flow. In comparison to the fitness-space approach (STAR et al. 2007), the construction approach results in more skewed frequency distributions for both types of indices, mainly for lower levels of gene flow.

View larger version (17K): In this window In a new window Download PPT slide   FIGURE 6.— Distribution of allele-frequency vectors measured by the square root of index I for n alleles as a function of gene flow (m). The index I was calculated using allele frequencies from one of the demes (black symbols), pooled frequencies from both demes (white symbols) and frequencies from a single deme (gray symbols). The red symbols indicate the results for the fitness-space approach (see text for explanation). The error bars are standard error intervals. For the construction model, m and n have a significant interaction effect (ANCOVA, F7, 7914 = 4.69, P 0.0001, m as factor, n as covariate) on the single index I. For the pooled index I, no significant interaction effect was detected. Both m (F7,7921 = 108, P 0.0001) and n (F1,7921 = 109, P 0.0001) have a significant effect on the pooled index I. Both indexes were square-root transformed to comply with the ANCOVA assumption of normality of residuals.

 

View this table: In this window In a new window   TABLE 3 Significance values for multiple comparisons for the index I between levels of gene flow for pooled frequency vectors using Bonferroni

 

Neutrality:

The Ewens–Watterson test compares the observed level of homozygosity to that which is expected under neutrality. This test was used to examine if the allele-frequency vectors after 10,000 generations were detectably different from those expected under the neutral hypothesis. Again for each run of the simulation, neutrality was examined using two measurements that are similar as described for the indices I above; one measurement used the allele-frequency vector (p_i,d) taken from one of the demes, the other used the pooled frequency vectors from both demes. Following the procedure described by MARKS and SPENCER (1991) and SPENCER and MARKS (1992), from each of these two population frequency vectors per run, 200 sample frequency vectors were taken, each with a sample size of 200 genes. For each of these sample frequency vectors, the sample homozygosity () was calculated and compared to the lower and upper critical points of Ewens sampling distributions (EWENS 1972; WATTERSON 1977, 1978). Each single sample vector was considered a binomial experiment with a constant probability of rejection. Then, if significantly >5% of the sample vectors for a particular population vector are rejected as nonneutral, that population vector is detectably different from a vector expected under the neutral hypothesis. This detection occurs when >9% of the sample vectors are rejected (binomial distribution with N = 200 and P = 0.05).

Gene flow has a considerable effect on the proportion of vectors that deviate from neutrality (Figure 7). For allele-frequency vectors taken from one of the demes, the proportion of vectors that deviate from neutrality due to in the lower critical region increases with higher levels of gene flow. This proportion is lowest for an intermediate gene flow (m = 0.1) for the pooled allele-frequency vectors. Only for lower (m < 0.2) levels of gene flow and allele-frequency vectors taken from one of the demes, substantial numbers of vectors were detected that deviate from neutrality due to in the upper critical region. The probability of calculating either lower or higher -values depends on the level of skew of allele-frequency distributions. A skewed distribution will more likely result in higher -values and an even distribution will more likely result in lower -values. Thus both the level of gene flow and the manner in which the final allele-frequency vectors are sampled have a large effect on the outcome of the Ewens–Watterson test as both have a large influence on the skew of frequency distributions. Overall, the proportions of allele-frequency vectors that are detectably different from those expected under neutrality are low.

View larger version (8K): In this window In a new window Download PPT slide   FIGURE 7.— Proportion of simulations with allele-frequency vectors that were significantly different from neutrality as a function of gene flow. Samples were taken from one of the demes (triangles) and from pooled frequency vectors from both demes (circles). Frequency vectors were rejected if 18 of the generated sample homozygosity () values were lower (solid symbols) or higher (open symbols) than the critical points of Ewens' sampling distribution (see text for explanation).

 

Spatial heterogeneity has long been considered a potential solution to the problem of the maintenance of genetic variation (LEWONTIN 1974; KASSEN 2002). Here we show that this solution need not always work: A spatial construction approach demonstrates that the simulated environmental heterogeneity can either constrain or promote the levels of polymorphism, depending on the level of gene flow between the demes. High levels of gene flow decrease the levels of polymorphism when compared to the single-deme construction approach. For these high levels of gene flow, heterozygote advantage emerges as the dominant form of balancing selection that maintains polymorphism; substantial disruptive selection is not evident since fitness sets become positively correlated between the demes. Thus, for high levels of gene flow, the evolutionary process selects mutants that have fitness sets with heterozygote advantage and similarly structured fitness values in both demes. The probability of randomly generating fitness sets satisfying these particular requirements in a two-deme model is lower when compared to the single-deme model, since two fitness sets (one set for each deme) of similar structure need be generated instead of only one. The lower probability of attaining heterozygote advantage in both demes results in lower levels of polymorphism after 10,000 generations compared to a single-deme model. A similar result has been reported for a sex-dependent viability-selection model (MARKS and PTAK 2000). Indeed, that particular sex-dependent viability-selection model is a special case of our two-deme model with the highest level (m = 1) of gene flow: Males and females can be considered separate demes with complete mixing each generation.

Moving from a single-deme to a two-deme model with high levels of gene flow gives contrasting results using the construction and fitness-space approaches. In the latter the potential to maintain polymorphism increases due to a combination of heterozygote advantage and disruptive selection (LEWONTIN et al. 1978; STAR et al. 2007); in the former the level of polymorphism decreases and disruptive selection is not apparent. These contrasting results suggest that, for a given level of polymorphism, the regions of fitness space reached by a population under a more natural process of mutation and selection are quite different from those regions implied using the more static fitness-space approach for high levels of gene flow.

For low levels of gene flow, results are more similar when comparing levels of polymorphism in the construction approach and the potential to maintain variation in the fitness-space approach (STAR et al. 2007). The construction approach maintains substantially higher levels of polymorphism and the fitness-space approach results in more potential to maintain variation when each of these different approaches are compared to their equivalent single-deme models. Nevertheless, for comparable levels of polymorphism, both approaches have fitness sets with strikingly different fitness properties. Heterozygote advantage is the dominant form of balancing selection in the fitness-space approach, while disruptive selection emerges as the important form of balancing selection in the construction approach for these lower levels of gene flow. Thus, for both lower and higher levels of gene flow, the process of mutation and selection in the construction approach results in the maintenance of polymorphism in a region of fitness space that is, on average, different from that implied by the fitness-space approach.

Interestingly, for the construction approach, disruptive selection emerges as a result of the evolutionary process, rather than due to any fitness restriction or forced trade-off between the demes, as fitnesses are randomly assigned between the two demes. This emergence of disruptive selection is similar to the emergence of heterozygote advantage as a result of evolutionary processes in single-deme models (GINZBURG 1979; TURELLI and GINZBURG 1983; SPENCER and MARKS 1988; MARKS and SPENCER 1991). We suggest that generating negatively correlated fitnesses between the demes will make the construction of polymorphism easier for low levels of gene flow, and harder for higher levels of gene flow. The opposite effect might be observed by generating more positively correlated fitnesses between the demes. The latter model would become more similar to a spatially subdivided model without spatial heterogeneity in selection pressures (e.g., MUIRHEAD 2001). We are currently investigating a model in which correlated fitness sets are generated.

Invasibility and stability:

The evolutionary process allows for more invasions in the two-deme model compared to the single-deme model, especially for intermediate levels of gene flow. Nevertheless, most of this increase in invasions is due to an increase in short-term mutants, which may have little effect on the levels of polymorphism after 10,000 generations. After this number of generations, stability analysis shows that the alleles present in the majority of the replicate runs are part of true polymorphic equilibria, and the final levels of polymorphism are not heavily influenced by the presence of large numbers of transient mutants.

Gene flow has a very strong influence on level of stability of the fitness sets found after 10,000 generations and on the proportion of frequency vectors that lead to fully polymorphic equilibria. Both the proportion of more stable fitness sets and the average size of domain of attraction increase for higher levels of gene flow. A form of heterozygote advantage appears to be the sole form of balancing selection as the spatial structure begins to vanish and this form of balancing selection is more likely to have a globally stable equilibrium in the absence of strong environmental heterogeneity (LEWONTIN et al. 1978; KARLIN 1982). The higher stability of the spatial model at higher levels of gene flow is most likely due to this increase in average levels of heterozygote advantage in combination with constraints on the fitness sets of mutations that can successfully invade.

Regardless of the level of gene flow, the fitness sets resulting from the construction approach have a larger size of domain of attraction when compared to the fitness-space approach (STAR et al. 2007). Successfully invading alleles necessarily have to increase their frequency from low frequencies and therefore will be more likely to have fitness properties that will protect them from becoming extinct at these low frequencies. These results suggest that the nonequilibrium mutation-selection process selects for fitness sets that are part of a more stable portion of fitness space than is expected by the ahistoric fitness-space approach.

Frequency distributions and neutrality:

Allele frequency distributions resulting from balancing selection often result in distributions that are fairly even (LEWONTIN et al. 1978). In a single deme the frequency distributions emerging from the construction approach are more skewed when compared to the distributions resulting from the fitness-space approach (MARKS and SPENCER 1991). In contrast, the two-deme models result in a whole range of frequency distributions that can be more even or more skewed, depending on the level of gene flow. For low levels of gene flow, ignoring the spatial structure of the two-deme model (by pooling the allele frequencies from both demes) results in more even frequencies, underestimating the skew of frequency distributions in each deme at these levels of gene flow. Thus, if allele-frequency distributions are to be used as signatures of certain population-genetic processes, then it may be necessary to understand the underlying spatial structure of the population.

Frequency distributions resulting from selection often cannot be distinguished from the neutral theory using the Ewens–Watterson test (GILLESPIE 1991; MARKS and SPENCER 1991; SPENCER and MARKS 1992). In our two-deme model, gene flow has a strong effect on these frequency distributions and as such on the outcome of the Ewens–Watterson test. Higher levels of gene flow cause continuous mixing of the frequencies between the demes. This mixing prevents the skewed frequency distributions leading to high sample homozygosity () and increases even frequency distributions leading to low sample . Similarly, for low levels of gene flow, frequency distributions become more even by pooling the frequencies from both demes effectively ignoring the spatial structure of the two-deme model. Such pooling obviously leads to an underestimation of the number of vectors that can be distinguished from the neutral theory due to high sample . Traditionally, frequency distributions that are considered nonneutral due to a low sample homozygosity () are regarded as evidence for selection for heterozygotes, whereas frequency distributions rejected due to high sample are regarded as evidence for selection against heterozygotes (MANLY 1985). These explanations should be regarded with caution, especially if the underlying spatial structure is unknown. Furthermore, both the large influence of gene flow on the frequency distributions and ignoring the spatial structure further reduce the already low power of the Ewens–Watterson test. These results show that allele frequency distributions alone are not likely to be sufficient to determine the presence of selection in natural populations, an argument made previously by GILLESPIE (1997) and KREITMAN (2000). Moreover, it would seem that screens for selection based on a skewed frequency distribution may well be underestimating the amount of selection.

In summary, whether the spatial construction approach leads to higher or lower levels of polymorphism when compared to its single-deme equivalent is critically determined by the level of gene flow. Regardless of gene flow, the construction and fitness-space approaches result in fitness sets with both qualitatively and quantitatively different properties. Furthermore, the resulting polymorphic equilibria from the construction approach are more stable than a static fitness-space approach might suggest. Most patterns of allele frequencies resulting from selection in the two-deme model cannot be properly distinguished from those expected under the neutral theory, especially when spatial structure is ignored when sampling these frequencies.

This work was supported by the Allan Wilson Centre for Molecular Ecology and Evolution, the University of Otago postgraduate scholarships, and the Marsden Fund of the Royal Society of New Zealand.

BULMER, M. G., 1972 Multiple niche polymorphism. Am. Nat. 106: 254–257.[CrossRef]

EWENS, W. J., 1972 Sampling theory of selectively neutral alleles. Theor. Popul. Biol. 3: 87–112.[CrossRef][Medline]

FELSENSTEIN, J., 1976 The theoretical population genetics of variable selection and migration. Annu. Rev. Genet. 10: 253–280.[CrossRef][Medline]

GILLESPIE, J. H., 1991 The Causes of Molecular Evolution. Oxford University Press, New York.

GILLESPIE, J. H., 1997 Junk ain't what junk does: neutral alleles in a selected context. Gene 205: 291–299.[CrossRef][Medline]

GINZBURG, L. R., 1979 Why are heterozygotes often superior in fitness. Theor. Popul. Biol. 15: 264–267.[CrossRef]

HEDRICK, P. W., M. E. GINEVAN and E. P. EWING, 1976 Genetic-polymorphism in heterogeneous environments. Annu. Rev. Ecol. Syst. 7: 1–32.[Medline]

KARLIN, S., 1981 Some natural viability systems for a multiallelic locus—a theoretical study. Genetics 97: 457–473.[Abstract/Free Full Text]

KARLIN, S., 1982 Classifications of selection migration structures and conditions for a protected polymorphism. Evol. Biol. 14: 61–204.

KASSEN, R., 2002 The experimental evolution of specialists, generalists, and the maintenance of diversity. J. Evol. Biol. 15: 173–190.[CrossRef]

KAWECKI, T. J., and D. EBERT, 2004 Conceptual issues in local adaptation. Ecol. Lett. 7: 1225–1241.[CrossRef]

KREITMAN, M., 2000 Methods to detect selection in populations with applications to the human. Annu. Rev. Genomics Hum. Genet. 1: 539–559.[CrossRef][Medline]

LENORMAND, T., 2002 Gene flow and the limits to natural selection. Trends Ecol. Evol. 17: 183–189.[CrossRef]

LEVENE, H., 1953 Genetic equilibrium when more than one ecological niche is available. Am. Nat. 87: 331–333.[CrossRef]

LEWONTIN, R. C., 1974 The Genetic Basis of Evolutionary Change. Columbia University Press, New York.

LEWONTIN, R. C., L. R. GINZBURG and S. D. TULJAPURKAR, 1978 Heterosis as an explanation for large amounts of genic polymorphism. Genetics 88: 149–169.[Abstract/Free Full Text]

MANLY, B. F. J., 1985 The Statistics of Natural Selection. Chapman & Hall, London.

MARKS, R. W., and S. E. PTAK, 2000 The maintenance of single-locus polymorphism. V. Sex-dependent viabilities. Selection 1: 217–228.[CrossRef]

MARKS, R. W., and H. G. SPENCER, 1991 The maintenance of single-locus polymorphism. 2. The evolution of fitnesses and allele frequencies. Am. Nat. 138: 1354–1371.[CrossRef]

MUIRHEAD, C. A., 2001 Consequences of population structure on genes under balancing selection. Evolution 55: 1532–1541.[CrossRef][Medline]

NAGYLAKI, T., and Y. A. LOU, 2001 Patterns of multiallelic polymorphism maintained by migration and selection. Theor. Popul. Biol. 59: 297–313.[CrossRef][Medline]

SMITH, J. M., and R. HOEKSTRA, 1980 Polymorphism in a varied environment: How robust are the models? Genet. Res. 35: 45–57.[Medline]

SPENCER, H. G., and R. W. MARKS, 1988 The maintenance of single-locus polymorphism. 1. Numerical-studies of a viability selection model. Genetics 120: 605–613.[Abstract/Free Full Text]

SPENCER, H. G., and R. W. MARKS, 1992 The maintenance of single-locus polymorphism. 4. Models with mutation from existing alleles. Genetics 130: 211–221.[Abstract]

STAR, B., R. J. STOFFELS and H. G. SPENCER, 2007 Single-locus polymorphism in a heterogeneous two-deme model. Genetics 176: 1625–1633.[Abstract/Free Full Text]

TURELLI, M., and L. R. GINZBURG, 1983 Should individual fitness increase with heterozygosity. Genetics 104: 191–209.[Abstract/Free Full Text]

WATTERSON, G. A., 1977 Heterosis or neutrality. Genetics 85: 789–814.[Abstract/Free Full Text]

WATTERSON, G. A., 1978 Homozygosity test of neutrality. Genetics 88: 405–417.[Abstract/Free Full Text]

Communicating editor: M. W. FELDMAN

This article has been cited by other articles:

				B. Star, M. V. Trotter, and H. G. Spencer Evolution of Fitnesses in Structured Populations With Correlated Environments Genetics, July 1, 2008; 179(3): 1469 - 1478. [Abstract] [Full Text] [PDF]

International Access Link

Online ISS 1091-6490
Copyright 2008 by the Genetics Society of America
phone: 412-268-1812   fax: 412-268-1813   email: genetics-gsa{at}andrew.cmu.edu