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CROP BREEDING, GENETICS & CYTOLOGY

Selection Dynamics and Limits under Additive x Additive Epistatic Gene Action

Department of Agronomy, Iowa State University, 1208 Agronomy Hall, Ames, IA 50011-1010

^* Corresponding author (jjannink{at}iastate.edu)

	ABSTRACT

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Theory predicts that response to selection under additive gene action with finite population size will decline over cycles of selection. In practice, this decline is not always observed. Selection under additive x additive epistatic gene action was examined to determine if it could condition greater and longer-term response to selection than additive gene action relative to the initial response in the base population. By use of transition matrix simulation, the ratio of the long-term to the initial selection response under epistatic gene action was found to be more than double that under additive gene action for effective population sizes of N = 2, 4, and 8. Furthermore, the selection half-lives under epistatic and additive models approximated 2N and 1.4N, respectively, indicating that selection response lasts longer under epistatic than additive gene action. Analysis showed that under the epistatic model, selection affected the mean in two distinct ways. First, selection generated a positive covariance in the allele content among selected individuals. This effect was strengthened by linkage between interacting loci. Second, selection shifted allele frequencies to increase covariance in allele frequencies among selection lines. This effect was weakened by linkage between interacting loci. At intermediate allele frequencies, initial selection response was due entirely to the first effect and was stronger for small than for large N. Consequently, by the third cycle of selection, response was 76% greater for N = 2 than for N = 8 under an infinitesimal epistatic model with unlinked loci and selection intensity of 50%. This differentiation in response between population sizes may lead to an experimental test of the importance of epistatic gene action affecting a trait.

	INTRODUCTION

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
PREDICTIONS OF SHORT-TERM RESPONSE to selection use statistical approaches that summarize the joint effect of the many loci that underlie a trait using parameters of phenotypic and additive genetic variances (Falconer and Mackay, 1996; Hill and Rasbash, 1986; Turelli and Barton, 1990). Predictions of long-term response, however, require consideration of the underlying genetic details (e.g., number of loci, allelic frequency, linkage, mode of action, and effect distribution, Hill and Rasbash, 1986; Lewontin, 1977). In addition, through the action of drift and chance allele fixation, these factors interact with population size in determining long-term selection response and the selection limit.

In Robertson's seminal paper on limits to selection (1960), formulas were derived to predict the selection limit assuming the additive infinitesimal model. Characteristics of this model are that a trait is influenced by many independent loci, each with a small and additive effect. The infinitesimal effect of each locus results in the property that selection does not affect allele frequencies so that the decline in genetic variance in the population is due solely to drift. The limiting response is then modeled in terms of the probability of fixation of the favorable allele (Robertson, 1960). These assumptions lead to the prediction that the selection limit will be 2N times the initial response to selection in the base population, where N is the effective size of the selected population (Robertson, 1960). Further work has addressed the question of linkage among loci in the infinitesimal model, the general conclusion being that linkage has small effects on the selection limit, reducing it relative to the infinitesimal model (Hill and Robertson, 1966; Robertson, 1970). Finally, Hill and Rasbash (1986) have examined models assuming independent and additive-effect loci but in which the number of loci, the distributions of allele effects and frequencies are flexible. As fewer loci affect the trait (increasing model departure from the infinitesimal model), initial allele effect and frequency distributions become irrelevant as all favorable alleles become fixed at the limit. Given a symmetrical initial allele frequency distribution, additive genetic variance and selection response decline monotonically with cycles of selection at a rate that increases with increasing departure from the infinitesimal model (Hill and Rasbash, 1986).

Experimental results show that infinitesimal model predictions are not always adequate. Analyses of selection experiments using allozymes and DNA markers clearly show shifts in marker allele frequencies that are greater than predicted on the basis of drift alone (e.g., Brown, 1971; DeKoeyer et al., 2001; Stuber et al., 1980). Observations of allele frequency shifts caused by selection are inconsistent with the infinitesimal model which predicts that drift alone acts to shift frequencies. More obviously, QTL analyses of experimental populations regularly find loci with large rather than infinitesimal effects (e.g., Edwards et al., 1987; Kianian et al., 1999; Paterson et al., 1988). These types of result clearly point away from the infinitesimal model as a viable predictive model and toward models parameterized with finite locus numbers (Hill and Rasbash, 1986). Analyses of selection experiments, however, often contradict the finite-locus model prediction of monotonic declines in additive variance and selection response over cycles of selection (see Weber and Diggins, 1990). Mutation and the release of variation through recombination have been invoked to explain the failure of genetic variance to decline as predicted under additive models (Falconer and Mackay, 1996; Lande, 1975; Robertson and Hill, 1983). An alternative possibility is that epistatic gene action may be converted to additive genetic variance as selection progresses. The phenomenon of conversion of epistatic to additive variance as a consequence of population bottlenecks or genetic drift has been addressed theoretically (Cheverud and Routman, 1996; Goodnight, 1988) and experimentally (Bryant and Meffert, 1993; Bryant and Meffert, 1996; Cheverud et al., 1999). Under epistasis, the shift in allele frequency at one locus caused by a bottleneck can change the marginal effect of alleles at an interacting locus. The interacting locus may subsequently generate additive genetic variance. Selection results in allele frequency shifts caused both by differential fitness and by drift in sampling the selected population. Selection could therefore bring about similar effects as drift in dynamically changing additive genetic variance derived from epistatic gene action.

Here, transition matrix simulation is used to explore selection response under additive versus additive x additive epistatic gene action and examine the differences between these models in magnitude and longevity of selection response. Selection responses under the epistatic model are evaluated to assess whether epistasis can reconcile emerging results from QTL studies on the distribution of allele effects with classical results from selection experiments congruent with the infinitesimal model. Finally, the different selection effects on the population mean under the epistatic model are analyzed to see whether selection can provide a test of the importance of epistatic gene action in determining genotypic values of a trait.

	MATERIALS AND METHODS

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Models of Gene Action
Loci A and B are biallelic with alleles designated A/a and B/b, respectively, having frequencies P(A) = p_A and P(B) = p_B. Two modes of gene action are considered here to translate genotype into genotypic value (Table 1) : standard additive action or the pure additive x additive epistatic action given by Cheverud and Routman (1996)(Table 1). Given the two-locus, biallelic model, there are 10 possible genotypes. Coupling and repulsion heterozygotes, however, are assumed to have the same genotypic value and are collapsed in Table 1. Table 1 does not specify allele frequencies within a population. Table 1 can be used in a straightforward way to calculate the marginal effects of alleles at one locus conditional on allele frequencies at the other locus. This exercise shows that the marginal effect at an additive locus is always a. At an epistatic locus, however, marginal effects at one locus vary from - to depending on the interacting locus allele frequency.

Finite Population Limits And Dynamics
Selection for high phenotypic value was simulated starting from a population in Hardy-Weinberg equilibrium with initial allele frequencies p_A = p_B = 0.5. Transition matrix modeling of drift or selection has typically only accounted for allele frequencies, and therefore has always assumed linkage and Hardy-Weinberg equilibria (e.g., Cheverud and Routman, 1996; Hill and Rasbash, 1986). These assumptions were relaxed by a transition matrix containing the probabilities of going from one set of N two-locus diploid parents to another over the course of a selection cycle. Given that each parent could be one of 10 genotypes, the function relating N to the number of possible parental sets, P_N is P_N = . Thus, for N = 2, 4, and 8 there are 55, 715, and 24 310 distinct sets, respectively, and the transition matrix must contain P²_N cells.

Transition probabilities were calculated as follows. The parents of a set were randomly mated, including the possibility of selfing, to produce a line population [following the nomenclature of Falconer and Mackay (1996), a subpopulation started from a sample of parents is a line. From a population biology perspective, an alternative nomenclature would be to refer to the set of lines as a metapopulation and to each line as a deme (Goodnight, 2000)]. The creation of gametes for random mating allowed for the possibility of recombination, with recombination frequency r. With the chosen gene mode of action, the genetic variance present in the line population was calculated. The sum of genetic and error variance determined the phenotypic variance. A truncation point was calculated so as to include a percentage SP of the line population in the selected tail of the phenotypic distribution. A genotype probability vector g was then defined by calculating the percentage of each genotype in the selected tail, accounting for the genotype's mean, the error variance, and the truncation point. The probability of sampling each possible set of parents in the next cycle of selection was determined using the multinomial distribution and the vector g. The selection limit was established by iterating cycles of selection until the remaining genetic variance was 100 000 times smaller than the initial genetic variance.

The simulation incorporates two approximations. First, the mixture of normal distributions that results from the sum of different genetic means with a normal error was assumed identical to a single normal distribution with variance equal to the sum of genetic and error variances. Second, even though finite populations were simulated, the selection intensity (that is, the standardized deviation between the population mean and the mean of selected individuals) assumed was that of a large population. The selection intensity for a given selected percentage of the population declines with population size (Falconer and Mackay, 1996). For SP = 50%, the selection intensity at N = 2 will be approximately 15% lower than at N = 8. Since the simulations assumed the intensities to be equal, they therefore overstate slightly the gain from selection to be expected at N = 2 relative to N = 8.

Since the simulation model contained no stochastic component, only one simulation was run for each set of conditions. Unless otherwise specified, all results presented below were parameterized with the recombination frequency between the loci r = 0.5, the percent of the population selected SP = 50%, the error variance V_e = 500, and the allele substitution effects from Table 1 a = 1 and = 2. These substitution effects were chosen because they cause a population at equilibrium with p_A = p_B = 0.5 to show a mean of zero, a genetic variance of one and a range of four under both additive and epistatic models (all genotypic values and variances are in arbitrary units). Simulations were performed with parental sets of size N = 2, 4, and 8.

For each selection cycle, the mean of the selected parents across the P_N possible sets was calculated, each set being weighted by its probability of occurrence in the cycle. Similarly, the mean and variance among the P_N line populations was calculated for the population mean, linkage disequilibrium between loci, and additive genetic variance. Cheverud and Routman (1996) provide formulas to calculate the additive genetic variance arising from loci with epistatic modes of gene action. These formulas, however, assume linkage equilibrium and Hardy-Weinberg equilibrium. In the simulations presented, loci were rarely in linkage equilibrium in the line populations. Additive genetic variance was therefore calculated as four times the variance attributable to multiple regression of the mean of an individual's progeny on its content of A and B alleles, as follows. For locus A, the content of A alleles, c_Ai, takes values 1, 0.5, or 0 for individual i of genotype AA, Aa, or aa, respectively. The content of B alleles, c_Bi, is similarly defined. The breeding value of individual i, bv_i, is the mean genotypic value of its progeny. If u is the vector , and S is the variance-covariance matrix of c_A and c_B across individuals in the population, then V_A = 4u ^tS^-1u. The multiplier 4 arises because an individual contributes to only half of the genotype of its progeny. For populations at linkage and Hardy-Weinberg equilibrium, this calculation was numerically verified to give the same result as Cheverud and Routman's (1996). At the same time, it preserves Fisher's original notion of additive variance as the regression of genotypic value on gene content (Lynch and Walsh, 1998). To separate the effect of selection on linkage disequilibrium from its effect on allele frequencies, the mean of the line population, had it been at linkage equilibrium, was also calculated.

	RESULTS AND DISCUSSION

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Selection Limits under Additive and Epistatic Models
A first analysis compared the effects of additive and pure additive x additive epistatic modes of gene action on the Castle-Wright ratio, R²/V_A, where R is the difference between lines divergently selected for a trait and V_A is its additive genetic variance in the base population. Under the assumptions of large population size (and thus no drift during selection), additive gene action, equal allelic effects among loci, and initial allele frequencies of 0.5, this ratio is eight times the number of loci affecting the selected trait, and forms the basis for the Castle-Wright estimator of that number (Falconer and Mackay, 1996; Lynch and Walsh, 1998). Under additive gene action, as initial allele frequencies diverge from 0.5, the additive genetic variance present in the base population is reduced while the limiting range from divergent selection remains the same. In consequence, the Castle-Wright ratio is high at extreme allele frequencies. Under two-locus additive x additive epistatic gene action, additive genetic variance is reduced either when both allele frequencies are extreme, or when they are both intermediate (Cheverud and Routman, 1996). Consequently, the Castle-Wright ratio is high both at extreme and at intermediate allele frequencies. Because of these differing effects of allele frequencies under the two modes of gene action, in a range of intermediate frequencies, the Castle-Wright ratio is greater under epistatic than under additive action, dramatically so as allele frequencies approach 0.5 (Fig. 1) . Thus, assuming intermediate initial allele frequencies and large population size, epistatic gene action could condition a much greater and longer-term response to selection than additive action. We further investigated this possibility under finite population sizes using simulation.

View larger version (37K): [in this window] [in a new window]   Fig. 1. Comparison of Castle-Wright ratios under pure additive x additive epistasis (CWAxA) versus under additive (CWAdd) gene action. Allele frequencies only go to 0.5 in the graph because it is symmetrical. Positive values on the Z axis indicate greater limiting response to selection relative to initial additive variance under epistatic than additive gene action.

View this table: [in this window] [in a new window]   Table 2. Initial response to selection, R0, and limiting response, R, and ratio between the two, in two-locus additive versus epistatic genetic models as a function of the recombination frequency between loci, r, and the populations size N. The variance in response among lines is given in parentheses.

The number of cycles of selection required to reach half of R, known as the half-life of the selection response also differed between additive and epistatic models (Table 3) . Robertson (1960) predicted a half-life of t = 1.4N under an additive model. The simulation results agreed with this prediction (Table 3). For the population sizes simulated, response half-life under the epistatic model was greater than 1.4N. Under the additive model, additive genetic variance declined by a factor close to the drift expectation of 1 - per cycle so that response declined exponentially over cycles of selection (Fig. 2) . Under the epistatic model, the response per cycle was initially small but increased over cycles of selection (Fig. 2). Population means under the epistatic and additive models were equal after about t = N cycles of selection, but response per cycle at that time was greater under the epistatic model than under the additive model (Fig. 2). It appears that the lag required to build up additive variance under the epistatic model was sufficient to increase response half-life under that model. The comparative dynamics of selection under epistatic versus additive models was very little affected by population size (Fig. 2). The extension of the response half-life conditioned by epistasis represents a possible mechanism explaining results obtained in long-term selection experiments in which additive genetic variance and selection response fail to decline as would be predicted under additive gene action (see Weber and Diggins, 1990).

View larger version (18K): [in this window] [in a new window]   Fig. 2. Response to selection in the population mean under additive and epistatic models.

[1]

Under linkage and Hardy-Weinberg equilibrium,

[2]

Combining Eq. [1] and [2] gives:

Expanding the terms (1 - p_B)² and grouping by (1 - p_A)² and p²_A gives

[3]

If we relax the assumption of linkage equilibrium then

>

Squaring these gamete frequencies to obtain the genotype frequencies results in terms containing only allele frequencies that are identical to those found in µ_eq, terms in D² that cancel out, and crossproducts between allele frequencies and D such that the mean at linkage disequilibrium is:

[4]

Finally, to find the expectation of the mean in a cycle, we take the expectation of Eq. [4] which is:

[5]

In Eq. [5], cov(p_A, p_B) is the covariance in allele frequencies across lines and [5] follows because E(p_Ap_B) = cov(p_A, p_B) + E(p_A)E(p_B).

Eq. [5] shows that selection can affect the cycle mean in three different ways: by generating linkage disequilibrium, generating a covariance across lines in the frequencies of the loci involved, or shifting the expectations of the allele frequencies themselves. These different possibilities play out differently depending on the initial allele frequencies and on the approach to allelic fixation as selection cycles progress.

For a population in linkage and Hardy-Weinberg equilibrium with p_A = p_B = 0.5, neither drift nor selection affect E(p_A) or E(p_B). At intermediate allele frequencies, the marginal affect of the A and B alleles are zero (Cheverud and Routman, 1995). Once drift has shifted allele frequencies from their intermediate position, selection favors a joint increase in the frequencies of A and B in lines where they are above 0.5 and a joint decrease in the frequencies of A and B in lines where they are below 0.5. Thus, no overall shift in the expectations of p_A and p_B occurs. In contrast, the initial effect of selection on linkage disequilibrium is strong. Individuals homozygous for specific combinations of alleles have the highest genotypic values (Table 1) and consequently selection for high phenotypic value will result in positive disequilibrium between these alleles. This effect is transient, however, given that allele fixation progresses over cycles of selection and linkage disequilibrium only occurs when both loci are polymorphic. Linkage disequilibrium most strongly affects the parents sampled from selected populations giving the mean of these samples the greatest boost (Fig. 3) . These parents are then random-mated, leading to a decrease in linkage disequilibrium in the process of generating line populations. The line populations therefore have a lower mean than their parents (Fig. 3). Finally, we can envision allowing random mating within lines to bring them to linkage equilibrium, which would further decrease the mean.

View larger version (16K): [in this window] [in a new window]   Fig. 3. Response to selection of the expected population mean in the parental sample, random-mated line population, and in line populations at linkage equilibrium under additive by additive epistasis with effective population size N = 8.

[6]

Equation [6] follows because p_A = c_Ai and p_B = c_Bi and each parent represents an independent sample of c_A and c_B from the line population. Equation [6] implies that large selected populations are less effective at transmitting a given covariance in allelic content to the next generation than are small selected populations. It also explains why initial selection response decreases with increasing population size as discussed earlier (Table 2). Interestingly, the transformation of cov(c_A, c_B)_t to cov(p_A, p_B)_t+1, and therefore a permanent shift in the expected mean, can occur independently of the existence of additive genetic variance in the trait. That is, this shift in the expected mean can occur in a population where no additive variance is present.

In addition to the production of covariance in allelic content by selection within a single cycle, the effects of selection have effects on that covariance that persist over several cycles. Persistent effects occur because selection generates linkage disequilibrium, which can be transmitted through meioses. For a random-mated population, gametes are in Hardy-Weinberg equilibrium and cov(c_A, c_B) = 1/2D. This follows because D is equivalent to the covariance between alleles in gametes, and, in a population in Hardy-Weinberg equilibrium, individual genotypes result from the random sampling of two gametes. The efficacy of both gametic and zygotic phase disequilibria in causing a permanent covariance in allele frequencies depends on the retention of polymorphism at both loci. This efficacy is therefore transient and it will last longer for large than for small N.

The effects of covariance in individual allele contents require polymorphism at both loci. On the contrary, under the epistatic model, the ability of selection to increase the covariance of allele frequencies by shifting allele frequencies within a cycle is enhanced as those frequencies move to extremes and even reach fixation at one of the two interacting loci. In particular, use of Table 1 to calculate the marginal effect of allele B conditional on the frequency of allele A shows that higher frequencies of allele A cause allele B to have a more positive effect. With selection for high phenotypic values, a high frequency of allele A within a line will lead selection to shift the frequency of allele B upward. The converse is also true for low frequencies of the alleles within a line. Allele frequency shifts caused by selection therefore tend to increase the covariance in allele frequencies across lines. The divergence across lines in the marginal effects of specific alleles at each locus exemplifies Goodnight's (2000) concept of differentiated "local average effects" of alleles among demes of a metapopulation. The transmission of this effect to the next cycle is straightforward: in the expectation, parents will have the same allele frequencies as the selected line population from which they were sampled. These allele frequency shifts depend on and can be predicted using knowledge of additive genetic variance present in the population (Cheverud and Routman, 1996). Note, however, that the feedback between an allele's frequency and its marginal effect alter the time course of additive genetic variance relative to the case of pure drift previously described (Cheverud and Routman, 1996). In the quasi-infinitesimal model studied here, selection-driven feedback caused additive genetic variance to peak earlier and at a lower level than the prediction under drift alone, though the difference was small (data not shown).

Because of the divergent requirements for polymorphism between the allelic frequency shift and allelic content covariance effects of selection, these two effects have different dynamics over cycles of selection (Fig. 4) . Response due to allelic covariance is strongest initially and tapers off while the response due to allelic shift increases over a longer period (Fig. 4). Furthermore, the two modes are affected differently by linkage between the interacting loci. Linkage between loci increases the response due to allelic covariance because linkage disequilibrium is dissipated more slowly when loci are linked and it therefore builds to higher levels. Higher linkage disequilibrium in turn contributes to higher allelic covariance. With large population sizes, the response due to allelic covariance can build over cycles as selection progressively increases linkage disequilibrium. With small population sizes, the build-up of linkage disequilibrium is cut short by allele fixation. Consequently, response to selection due to allelic covariance is greater under linkage in large rather than small populations (Fig. 4). Linkage between loci causes a reduced response attributable to allelic frequency shifts (Fig. 4). Presumably, this effect occurs because linkage increases the probability of joint fixation of the alleles, which stops the effects of selection. This effect is less severe under large than small population sizes (Fig. 4). A consequence of these dynamics is that linkage increases the overall response more in large than in small populations. In fact, for each population size, an optimal recombination frequency between loci exists that maximizes R. Lower recombination excessively reduces the allelic shift response while higher recombination excessively reduces the allelic covariance response. Optimal recombination frequencies were 0.42, 0.30, and 0.18 for N = 2, 4, and 8 respectively. While recombination rate is not under the control of the breeder, this observation confirms that the effect of the average recombination frequency between interacting loci is complex. Furthermore, assuming that recombination frequency between loci is under genetic control, selection on that frequency will depend not only on the nature of the epistatic interaction (Phillips et al., 2000) but on the effective size of the population in which selection takes place. These optimal recombination frequencies were remarkably unaffected by the selection intensity (data not shown).

View larger version (18K): [in this window] [in a new window]   Fig. 4. Response to selection per cycle in the expected population mean under additive x additive epistasis as a function of recombination frequency, r. Within each graph, the upper and lower pair of curves show, respectively, the response due to shifts in allele frequencies and due to covariance in allele content (see text).

View larger version (18K): [in this window] [in a new window]   Fig. 5. Response to selection in the population mean under epistatic (A) and additive (B) models.

	CONCLUSIONS

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

	NOTES

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Journal Paper No. J-19814 of the Iowa Agriculture and Home Economics Experiment Station, Ames, IA, Project No. 3571, and supported by Hatch Act and State of Iowa.

Received for publication April 15, 2002.

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