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Genetics, Vol. 178, 1473-1489, March 2008, Copyright © 2008
doi:10.1534/genetics.107.082131
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* Allan Wilson Centre for Molecular Ecology and Evolution, Department of Zoology, University of Otago, Dunedin 9054, New Zealand and The Murray-Darling Freshwater Research Centre, CSIRO Land and Water, Wodonga, Victoria 3689, Australia
1 Corresponding author: The Murray-Darling Freshwater Research Centre, CSIRO Land and Water, P.O. Box 991, Wodonga, VIC 3689, Australia.
E-mail: rick.stoffels{at}csiro.au
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ABSTRACT |
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 TOP ABSTRACT THE MODELRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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, and incorporate it, as well as the degree of recognition degeneracy, m, into a model of heterozygote advantage that utilizes a set-theoretic definition of fitness. Counterintuitively, we show that levels of polymorphism are positively related to m and that a high level of recognition degeneracy is necessary for polymorphism at MHC loci under heterozygote advantage. Increasing reduces levels of polymorphism considerably. Hence, if intersection advantage is significant for MHC genotypes, then heterozygote advantage may not explain the very high levels of polymorphism observed at MHC genes.
Some theoretical studies have shown that heterozygote advantage may lead to the levels of polymorphism that we see in real populations of MHC genes (MARUYAMA and NEI 1981; TAKAHATA and NEI 1990; TAKAHATA et al. 1992), but these models assume very high levels of symmetry in selection, which implies minimal or even no variance in the fitness of homozygotes and no variance in the fitness of heterozygotes. The biological validity of highly symmetric selection has been questioned, because it ignores the contribution made by individual MHC molecules (DE BOER et al. 2004). Given that MHC alleles are codominantly expressed (ABBAS and LICHTMAN 2006) and that individual alleles affect the host's fitness when exposed to a particular pathogen (JEFFERY and BANGHAM 2000; NIKOLICH-
UGICH et al. 2004), significant variation—and hence asymmetry—in genotype fitness may exist within host populations. This variation is important with respect to the heterozygote advantage hypothesis of MHC polymorphism because models of asymmetric heterozygote advantage do not easily lead to a high level of polymorphism (LEWONTIN et al. 1978; SPENCER and MARKS 1988; MARKS and SPENCER 1991; HEDRICK 2002). Consequently, evolutionary immunologists have called for fitness functions that more accurately capture the biology of the gene products, so that the validity of the hypotheses of MHC polymorphism may be more rigorously assessed.
Using allele-based fitness functions DE BOER et al. (2004) and BORGHANS et al. (2004) concluded that heterozygote advantage is not a valid explanation of MHC polymorphism. They showed that a high level of polymorphism is possible only if the fitnesses of all MHC alleles are very similar, which, they claimed, contradicts what we see in reality, and so heterozygote advantage fails to explain the high degree of polymorphism of the MHC. By contrast, we present a large body of evidence implying that the fitness of MHC alleles may actually be quite similar. Before we present this evidence, however, we must introduce some terminology that we use to describe the relationship between a pathogen strain and an MHC molecule.
Throughout this article we will define an MHC molecule by its pathogen "recognition set." By saying that an MHC molecule "recognizes" a pathogen strain, we mean that the pathogen has at least one epitope that binds to the peptide-binding groove of that MHC molecule with an appropriate affinity and/or conformation to activate clonal expansion of a T-cell lineage. We assume an MHC molecule has some finite "recognition set," which is the set of pathogen strains recognized by that MHC molecule. Because two MHC alleles can have disjoint peptide-binding sets but both recognize the same pathogen strain and hence have the same fitness under single-strain infection (see Figure 1 and below), the fitness of an MHC molecule is (partially) defined by its recognition set and not by the set of peptides that it binds. If the recognition sets of MHC molecules are broad, then the specificity of the MHC–pathogen interaction is low and variation in MHC allele fitness is low. By contrast, if recognition sets are narrow, then the specificity of the MHC–pathogen interaction is high and variation in MHC allele fitness is high.
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Thus, a paradox emerges. Population geneticists have shown that selection shapes polymorphism in MHC genes, but at the same time immunologists have shown that they possess a great degree of functional redundancy. Here we provide a reappraisal of the heterozygote advantage hypothesis of MHC polymorphism using a simple, single-locus model of asymmetric selection. We build on the work of DE BOER et al. (2004) and BORGHANS et al. (2004) by utilizing a fitness function that makes allowances for the dual requirements of allele-specific fitness and degeneracy in pathogen recognition sets. To this end, we employ a set-theoretic approach to defining the fitness of MHC alleles. This approach allows us to address two particular aspects of MHC polymorphism under heterozygote advantage. The first is the effect of the degree of degeneracy in pathogen recognition sets among MHC alleles. If each MHC molecule recognizes and presents a large proportion of the total set of pathogens to T-cells, then the host population may not need a large number of MHC molecules to maintain immunity to the pathogen community. Here we test this hypothesis.
Second, we parameterize our model to control for the form of the fitness profile of genotypes under single-strain infection. Consider the following interaction between two pathogen strains, X and Y, and two MHC alleles, R and r. Suppose allele R contains only X in its recognition set while allele r contains only Y. Under single-strain infection with pathogen X, what are the relative fitnesses—as measured by, for example, pathogen density and blood cell counts—of the three host genotypes RR, Rr, and rr? Figure 2, A and B, presents two alternative fitness profiles under single-strain infection. For the fitness profile in Figure 2A we assumed that genotype RR obtains an advantage from expressing two alleles that both recognize X, while for the profile in Figure 2B we assumed that RR does not obtain any benefit from expressing two alleles that contain X in their recognition sets. Empirically derived fitness profiles under single-strain infection often vary between the two extremes of Figure 2, A and B (PENN et al. 2002; MCCLELLAND et al. 2003; WEDEKIND et al. 2005, 2006) so we introduce a parameter, , that enables us to control for the relative benefit that a genotype obtains by expressing two alleles that recognize a pathogen strain when infected with that strain. We call this benefit "intersection advantage" since it is the proportional benefit obtained from pathogen strains in the intersection—i.e., the overlapping portion—of the alleles' recognition sets in a diploid genotype (see THE MODEL discussed below). Heterozygote advantage emerges under coinfection (Figure 2, E and F) when the corresponding alleles have opposite fitness profiles under single-strain infection, as has been experimentally demonstrated (MCCLELLAND et al. 2003). Also the degree of heterozygote advantage under coinfection is dependent on the shape of the fitness profile under single-strain infection, and hence on the degree of intersection advantage, so the inclusion of this parameter is pivotal to the rigorous assessment of the heterozygote advantage hypothesis of MHC polymorphism maintenance. Finally, under single-strain infection, = 1 means that alleles have an additive effect, which corresponds to a dominance coefficient of 
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THE MODEL |
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TOPABSTRACTTHE MODELRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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l 100, vl is drawn from a uniform distribution, U[0,1].
We denote the set of n MHC alleles in the host population as 
Suppose that allele ai codes for an MHC molecule that recognizes some subset of V. Denote this subset as Vi. This subset has size m for all i; m is a parameter. For ease of explanation, we refer to Vi as an MHC allele's recognition set.
We assign fitnesses to individual alleles. Let vi,k be the kth element from the set Vi and 
be the bth element from 
; then
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Therefore, when = 0, the fitness of each homozygote is equal to the sum of the weights in its allele's recognition set, and the fitness of each heterozygote is equal to the sum of the weights in the union of its alleles' recognition sets. Here, is the degree of intersection advantage and represents the proportional benefit that a genotype obtains by having two alleles that recognize a pathogen strain when infected with that strain (0 1). If = 0, the homozygote obtains no benefit from having two copies of an allele and a heterozygote obtains no improvements in fitness from the elements in . By contrast, if = 1, the fitness benefit that a host genotype obtains in the presence of a pathogen strain, vi, is directly proportional to the number of alleles that it carries that recognize that strain.
We assume a monoecious, randomly mating population with discrete, nonoverlapping generations. We also assume that the pathogen community, V, is constant for each individual simulation. By making this assumption, we effectively assume that all hosts are infected by all pathogens before finding a mate, that there is no variance in pathogen abundance, and that pathogens do not evolve. Of course, this assumption is artificial, albeit necessary, since we wanted to isolate the effects of heterozygote advantage on polymorphism construction. That is, if we allowed the pathogen community to vary, then we would no longer be studying polymorphism maintenance due to heterozygote advantage alone, but instead studying the combined effects of heterozygote advantage and variation in selective pressures, which are separate hypotheses of polymorphism maintenance in the MHC (e.g., HEDRICK 2002). Furthermore, if we allowed the pathogen community to evolve, then we would naturally have a coevolutionary model that would necessarily incorporate frequency-dependent fitness. Since frequency-dependent selection may also maintain polymorphism in the MHC (e.g., BORGHANS et al. 2004), it is a hypothesis that competes with the heterozygote advantage hypothesis of MHC polymorphism and we would then be confounding our treatment of heterozygote advantage.
Let pi and 
be the frequencies of allele ai at times t and t + 1, respectively; the allele dynamics are then described by the usual recursion equations:
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and 
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We conducted simulations with allele introduction and selection. This nonequilibrium, "constructionist" approach (following SPENCER and MARKS 1993) has proved very useful in the analysis of polymorphism maintenance in the past (SPENCER and MARKS 1988, 1992, 1993; MARKS and SPENCER 1991). Researchers utilizing this constructionist approach have shown that polymorphism is far more easily generated and maintained via a simple process of allele introduction and selection than equilibrium-based approaches would suggest (SPENCER and MARKS 1993). Therefore, simulations were initiated with a single allele and new alleles were introduced at one of two per-locus rates (µL = 2µ, where µ is the per-gene mutation rate): 10–5 and 10–6. Here we consider these allele-introduction rates to represent the combined effects of point mutation and recombination, both of which are important to the generation of MHC diversity (MARTINSOHN et al. 1999; OHTA 1999; RICHMAN et al. 2003; CONSUEGRA et al. 2005; REUSCH and LANGEFORS 2005; SCHASCHL et al. 2006). We ran simulations with three different effective population sizes (Ne) of 103, 104, and 105, so that the rate at which new alleles were added to the population, µP, was µP = µLNe; new alleles were introduced when, for generation t, t mod 
(the combinations of µL and Ne used here ensured that µP was an integer). We also ran simulations in which mutations were introduced at random time intervals at the same mean rate, but there was no notable difference in results. New alleles were introduced with a frequency of (2Ne)–1, and any pi that fell below (2Ne)–1 was eliminated from the population. Here, we assume that all alleles in the host population recognize the same number, m, of pathogens and simulate allele introduction and selection with nine levels of m: 10, 20, ..., 90, which correspond to fractions 0.1, 0.2, ..., 0.9. respectively. The parameter m represents the degree of degeneracy in pathogen recognition by MHC molecules. Although this model contains a finite number of alleles, there is an extremely large number of distinct combinations of the vi's for any given value of m: 100!/[m!(100 – m)!]. Four values of the parameter are simulated for each m-value: 0, 0.2, 0.4, and 0.8. All simulations were run with and without drift. Genetic drift in a population of n alleles was simulated by taking a sequence of n – 1 conditional binomial samples each generation (see GENTLE 2003, p. 198). Drift took place after selection. Twenty replicate simulations were run for each m––Ne–µL combination, both with and without drift. A new pathogen community, V, was drawn for each replicate simulation.
After each simulation was run for 105 generations, we measured five quantities of particular interest. The first quantity is the number of alleles, n(A). For the second quantity, we measured the mean pairwise strength of selection across all genotypes. We defined the relative fitness of a genotype as 
. Selection strength, sij, is equal to 
and has domain [0,1]. We then take the average of the n(n + 1)/2 sij values as our measure of the strength of selection. For the third quantity, as a measure of the average proportionate heterozygote advantage (
) relative to the fittest homozygote, we defined 
and then 
, and then calculated the mean across the n(n – 1)/2 heterozygotes. For the fourth quantity, we calculated the expected heterozygosity: 
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We compare levels of heterozygosity and polymorphism with those expected under neutrality. Levels of heterozygosity under neutrality can be obtained from KIMURA and CROW (1964). In addition, we constructed a simple neutral computational model, which was similar to THE MODEL outlined above, in that alleles were introduced at a per-locus rate of µL to an originally monomorphic locus, which was then subject to genetic drift without selection. Because levels of H and nA are so variable in small populations under neutrality, we ran more replicates for the smaller population sizes: 104, 103, and 200 replicates for Ne = 103, 104, and 105, respectively. Our computational estimates of heterozygosity agree very well with analytic estimates from KIMURA and CROW (1964), so we can have some confidence that our genetic drift algorithm is correct (APPENDIX A).
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RESULTS |
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TOPABSTRACTTHE MODELRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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= 0. Recall that the intersection advantage, , determines the proportionate benefit that a host obtains by having two alleles that recognize a pathogen when infected with that pathogen. Thus, at = 0, a host obtains no further benefit from having a second allele that also recognizes that pathogen. We therefore draw the reader's attention to points connected by the solid line in Figure 3, which shows how mean levels of polymorphism n(A) and heterozygote advantage () vary as a function of both recognition degeneracy (m) and degree of intersection advantage ().
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). This relationship was particularly strong in the absence of genetic drift (Figure 3, a–c) and was consistent across rates of allele introduction (µL; APPENDICES A–D). The mechanisms underlying this relationship are as follows: At low levels of m there is greater variance in the composition of the alleles' recognition sets than that expected at high levels of m. Thus there is the potential for much more variance in allele fitnesses and a more asymmetric form of selection at low levels of m. Selection strength across genotypes is then strongest at low levels of m (Figure 4; APPENDIX A), and small sets of alleles dominate the population of MHC molecules. As recognition degeneracy increases, the compositions of recognition sets become increasingly similar, which in turn lowers selection strength (Figure 4; APPENDIX A), makes selection more symmetric, and enables the coexistence of more alleles. However, while weak selection, or near neutrality, is apparently a requirement for high levels of MHC polymorphism under heterozygote advantage, complete selective neutrality generally results in very low levels of polymorphism (see "Neutral" expectations in APPENDICES A–D).
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Levels of polymorphism were severely affected by genetic drift, even for very large population sizes (Ne = 105; Figures 3 and 4; APPENDIX A). The highest mean level of MHC polymorphism recorded was 
233 alleles; this occurred in the absence of genetic drift with µL = 10–5, Ne = 105, = 0, and m = 90, while the highest mean level recorded in the presence of drift was 43 alleles, which occurred at the same parameter values (compare APPENDICES C and D). As a consequence of the negative relationship between selection and recognition degeneracy, m, the loss of polymorphism due to genetic drift is greatest at high levels of recognition degeneracy. This relationship is clearly demonstrated in Figure 4. Interestingly, the greatest net loss of polymorphism due to genetic drift occurred at the largest population size (Ne = 105; Figure 4). This result may, at first, seem counterintuitive. However, at high levels of recognition degeneracy selection becomes very weak, which means that new alleles either do not easily invade (CROW and KIMURA 1970, p. 422) or invade but are easily lost from the population. Because large, finite populations are subject to more frequent introductions of alleles, under such weak selection the proportion of successful invasions may be negatively correlated with population size in the presence of drift. Alternatively, allele invasion rates may not vary with population size, but the pull of the attractor about the polymorphic equilibrium may be negatively correlated with the number of alleles in the population and hence negatively correlated with population size also (e.g., KIMURA and CROW 1964). Thus, the average lifetime of alleles may be negatively correlated with population size, which may result in a relatively greater loss of polymorphism to genetic drift in larger populations.
Intersection advantage:
Intersection advantage, , had surprisingly complex effects on both the statistical properties of fitness sets and polymorphism. The most obvious effects of increasing are to reduce polymorphism and mean levels of heterozygote advantage (Figure 3; APPENDIX A). It is obvious that m and have an interactive effect on both nA and ; the relative increase in polymorphism due to increasing m is greatest for low levels of (Figure 3, a–f; APPENDIX A), and the relative reduction in with decreasing is greatest at low levels of m (Figure 3, g–i; APPENDIX A). Since increasing lowers mean heterozygote advantage and levels of polymorphism, one might expect that the lower levels of polymorphism are a consequence of diminished heterozygote advantage and that high levels of heterozygote advantage are necessary for high levels of polymorphism in the MHC. However, the full explanation is more complicated. By our definition of , when < 0, 
, on average, so that when < 0, a homozygote is most fit, directional selection will result, and polymorphism will vanish. LEWONTIN et al. (1978) obtained a similar necessary condition for a stable polymorphic equilibrium under overdominant selection: mean heterozygote fitness must be greater than mean homozygote fitness for a polymorphic equilibrium to occur (
). Thus the condition > 0 may be necessary for high levels of polymorphism to evolve, but one needs only to observe that the highest levels of polymorphism evolve under the lowest levels of heterozygote advantage (at high m-levels) to see that high levels of heterozygote advantage alone are not sufficient for high levels of polymorphism (Figure 3). Therefore, in addition to the effect that has on mean levels of heterozygote advantage, must affect some other statistical property of the fitness structure of the population, which in turn affects levels of polymorphism.
Before exploring the additional effects that has on polymorphism construction, recall that in our model the set of all genotypes—and hence fitnesses—available to the host population is finite and defined a priori by the set of weights from which alleles are drawn, V, and the parameters (m, ) that determine how the recognition sets of individual alleles are transformed into the fitnesses of diploid genotypes. It is therefore possible, for any combination of V, m, and , to determine the distribution of the set of all genotype fitnesses available to the host population before any mutation, selection, and genetic drift takes place. We call these distributions of available genotype fitnesses "preselection fitness distributions" (PFDs) and each combination of V, m, and results in a unique PFD. As explained below, we argue that has its greatest impact on polymorphism construction through the direct effect that it has on the PFDs.
To determine the additional effects that intersection advantage has on polymorphism construction, we conducted an analysis of invasion dynamics, which showed that increasing enabled certain alleles to dominate the population and precluded all other new alleles from invading. Such a dynamic might result if there were a positive correlation between and the variance of the PFD. That is, higher variance in the PFD may make it easier for selection to favor small subsets of particularly fit alleles, which may dominate the population and preclude most alleles from invading, thus lowering levels of polymorphism. We therefore needed to know (1) if the standard deviation of the homozygote and heterozygote PFDs increases as increases; (2) if increasing results in selection pushing mean homozygote and heterozygote fitnesses closer to their maximum or farther into the right tail of the PFDs; and (3) if increasing lowers the invasion rate. To this end, we constructed PFDs for homozygotes and heterozygotes using the fitness determination algorithm described above: We constructed a single set of 100 pathogen weights, V, from which we constructed alleles and, subsequently, a very large number of both heterozygotes and homozygotes for m = 20, 80, 90 and = 0, 0.2, 0.8. We then determined the standard deviation of both the homozygote and heterozygote PFDs.
After determining the PFDs, we then wanted to determine how selection transforms mean heterozygote and homozygote fitness and contrast these transformed fitnesses with the PFDs. Following the same algorithm outlined in THE MODEL, we then constructed a polymorphism by bombarding an originally monomorphic locus with new alleles drawn from exactly the same pathogen community, V, used to construct the PFDs above and, in turn, by subjecting that locus to selection (no genetic drift). The parameters used for this polymorphism construction were Ne = 104; µL = 10–6; m = 20, 80, 90; and = 0, 0.2, 0.8. After 105 generations, we recorded mean heterozygote and homozygote fitness and the proportion of the PFD greater than or equal to these means after selection (
) for each of the nine pairings of m and . We also determined invasion rate of new alleles (
) for each of the above simulations: An introduced allele was deemed a successful invasion if its frequency increased from 1/(2Ne) after one generation of selection.
The results of these analyses are given in Figure 5 (homozygotes) and Figure 6 (heterozygotes). All PFDs of homozygotes and most fitness distributions for heterozygotes were normal (Figures 5 and 6). Heterozygote PFDs at high values of m and low values of were skewed to the right and more multimodal (Figure 6, d, e, g, and h). Within such distributions, fitness values in the long left tail have many elements in the intersection of the allele recognition sets, while the reverse is true for those values at the right of the distribution. The fitness class with the highest frequency in Figure 6g represents the upper limit of heterozygote fitness at the values of and m (the fitness for this class is simply the sum of the elements in V, since the size of the set intersection is at its minimum, 
, and the intersection advantage is zero).
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. Increasing consistently increases the standard deviation of homozygote PFDs (Figure 5), but increases the standard deviation of heterozygote PFDs only at high values of recognition degeneracy (m; Figure 6, d–i). However, the increase in standard deviation of homozygote PFDs with may be great enough to increase variation in the marginal fitnesses of alleles, which is what ultimately determines variance in the rate of change of allele frequencies. Therefore, these patterns generally imply that the standard deviations of PFDs are positively related to . Also, it is worth noting that decreasing m increases the standard deviation of the PFDs, particularly for heterozygotes, across all levels of intersection advantage (Figures 5 and 6), which is concordant with our implied mechanisms underlying effects of m on polymorphism (see above). Now it remains to be shown that is negatively related to the proportion of the homozygote/heterozygote PFD greater than or equal to the mean homozygote/heterozygote fitness after selection and that is negatively related to the invasion rate of new mutants.
The level of intersection advantage, , is negatively related to both the proportion of the homozygote PFDs greater than or equal to the mean homozygote fitness after selection (; Figure 5) and the proportion of the heterozygote PFDs greater than or equal to mean heterozygote fitness after selection (; Figure 6). That is, increasing causes selection to push both heterozygote and homozygote fitnesses toward their maximum values. For example, at m = 90, >9% and <1% of the PFDs are greater than or equal to mean homozygote fitness after selection for = 0 and = 0.8, respectively (Figure 5, g and i). The increase in mean heterozygote fitness due to increasing is more severe: At m = 90, >46% and <1% of the PFDs are greater than or equal to mean heterozygote fitness after selection for = 0 and = 0.8, respectively (Figure 6, g and i). Finally, it is clear from data presented in Figure 5 that invasion rate () declines with increases in intersection advantage across all levels of recognition degeneracy.
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DISCUSSION |
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TOPABSTRACTTHE MODELRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
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The utilization of a set-theoretic definition of fitness enabled us to control the amount of degeneracy among the sets of pathogens recognized by MHC molecules. Here we have defined recognition degeneracy (m) such that the proportion of pathogens bound by each MHC molecule is equal to m/100. Therefore, recognition degeneracy is positively correlated with the degree of functional similarity among MHC molecules. Consequently, one might expect a negative relationship between recognition degeneracy and polymorphism. That is, if recognition degeneracy is high and each MHC molecule recognizes a large proportion of the pathogens to which the host population is susceptible, then that host population may require only low levels of polymorphism to provide complete protection from the pathogen community. By contrast, we have shown that the correlation between recognition degeneracy and polymorphism is actually positive.
One could suggest that the mechanisms underlying this relationship are as follows: Low levels of m result in only small subsets of the pathogen community being recognized by each MHC molecule. If we assume that the MHC population is exposed to a pathogen community of variable virulence, then low levels of m result in higher variation in the fitnesses of both MHC alleles and genotypes. As a result of this increased fitness variation at low levels of recognition degeneracy, selection intensity and selective asymmetry increases, and fewer alleles can coexist. This result is concordant with the analytic results of LEWONTIN et al. (1978), who showed that similarity among heterozygote fitnesses—and hence a high level of symmetry—is a necessary condition for a stable polymorphism under overdominant selection (see triangle inequality on p. 160, LEWONTIN et al. 1978). Thus, as m increases, so does symmetry, and selection weakens, polymorphism increases, and the MHC population approaches neutrality. However, while near neutrality appears to be necessary for high levels of MHC polymorphism under heterozygote advantage, complete neutrality generally results in very low levels of polymorphism (see "Neutral" levels of polymorphism in APPENDIX A).
So it appears that if heterozygote advantage can explain the levels of polymorphism observed at certain MHC loci, then recognition degeneracy needs to be high. For recognition degeneracy to be high, pathogen strains must contain many epitopes and/or the peptide-binding groove of MHC molecules must bind overlapping sets of epitopes (Figure 1). There is some evidence that both of these conditions are satisfied in reality. For example, DOOLAN and colleagues have shown that Plasmodium falciparum may contain hundreds of epitopes capable of binding MHC molecules and activating T-cells in the human population (DOOLAN et al. 2003; see also DOOLAN et al. 1997, 2000). This phenomenon is not limited to multicellular pathogens. Multiple epitopes have been identified in hepatitis B virus (NAYERSINA et al. 1993; REHERMANN et al. 1995; BERTONI et al. 1997), hepatitis C virus (KOZIEL et al. 1995; SCHULZE ZUR WIESCH et al. 2005), HIV (ROWLAND-JONES et al. 1998; CROTZER et al. 2000), Epstein–Barr virus (KHANNA et al. 1998), and influenza A virus (JAMESON et al. 1998; GIANFRANI et al. 2000; BOON et al. 2002). These contemporary findings challenge the traditional view of strict immunodominance, whereby only a very small fraction of the potential epitopes contained in complex antigenic proteins elicit T-cell activation (e.g., YEWDELL and BENNINK 1999).
Moreover, it appears that the binding groove of MHC molecules has relatively flexible requirements for peptide binding, and thus any given epitope may be bound by many different MHC molecules (Figure 1; e.g., SINIGAGLIA et al. 1988; PANINA-BORDIGNON et al. 1989; BARBER et al. 1995; KOZIEL et al. 1995; SIDNEY et al. 1995; BERTONI et al. 1997; DOOLAN et al. 1997; KHANNA et al. 1998; SOUTHWOOD et al. 1998; CROTZER et al. 2000; DOOLAN et al. 2000; GIANFRANI et al. 2000; SIDNEY et al. 2001; DIAZ et al. 2005; SCHULZE ZUR WIESCH et al. 2005). The mechanisms behind the flexibility of the peptide-binding groove are not entirely clear. The commonly held view is that there are only one or two key anchor pockets in the binding groove of each MHC molecule and that a peptide only needs to be of the right length and have one or two matching residues in the corresponding position(s) to successfully bind to the groove. The composition of these key anchor pockets may allow the many thousands of HLA (human MHC) alleles to be classified into 1 of 10 supertypes, each of which shares the same peptide-binding motif (SIDNEY et al. 1996, 2005; SETTE and SIDNEY 1998; CASTELLI et al. 2002; BURROWS et al. 2003). However, the rules governing peptide binding may be much more complex than the supertype classification scheme suggests, as the anchor pockets do not completely determine the requirements for peptide binding. Indeed, it has been shown that non-anchor residues may strongly influence binding affinity of some antigen–MHC complexes (e.g., NAYERSINA et al. 1993; CHEN et al. 1994; KAST et al. 1994), and high-affinity binding may occur without any involvement of anchor residues (e.g., SCOTT et al. 1998). Moreover, some epitopes bind with MHC molecules from separate supertypes (see epitope A4 in Figure 1; THIMME et al. 2001; SIDNEY et al. 2005).
Intersection advantage:
Intersection advantage, , represents the proportional fitness benefit that a genotype obtains by having two alleles that both recognize a pathogen when infected by that pathogen. Intersection advantage is an important parameter in our models of heterozygote advantage as it provides a simple way for the theory of MHC polymorphism to relate empirically derived fitness profiles of genotypes under single-strain infection (e.g., PENN et al. 2002; MCCLELLAND et al. 2003; WEDEKIND et al. 2006) to those more complex profiles that emerge under multi-strain coinfection over an organism's lifetime. Since the fitness profiles of MHC genotypes under single-strain infection imply significant intersection advantage, and since intersection advantage has strong influences on polymorphism construction under heterozygote advantage, it is clearly a parameter worthy of theoretical investigation.
One effect of intersection advantage is to alter the form of selection. In particular, as increases, the form of selection operating at MHC genes shifts from balancing to directional. A simple thought experiment illustrates why: Suppose we order the 100 weights in our pathogen set, V, from least to most virulent and that there exists one allele—call this allele af—in the population that recognizes the m most virulent pathogens. Then, for this level of m, this allele has maximum fitness. Now, if 
, then, when forming heterozygotes, it is possible to pair af with other alleles that recognize some subset of the remaining 100-m pathogens and heterozygotes will therefore be more fit than the fittest homozygote (afaf; see Equations 1 and 2). Balancing selection will then emerge in the form of heterozygote advantage, resulting in polymorphism. Now suppose that 
; then the fitness of homozygote afaf will be 1.8 times the sum of the m most virulent antigens (see Equation 2). A heterozygote that, in addition to those antigens bound by af, also recognizes some subset of the 100-m pathogens least virulent antigens will not easily obtain a higher fitness than afaf. Consequently, heterozygote advantage will vanish and the population should become less polymorphic. Obviously, if 
, then it would be impossible for any genotype to be more fit than afaf and selection will be directional (and purifying), favoring only the afaf genotype.
Our results confirm that, in this case, our intuition is generally correct. Increasing decreases both mean heterozygote advantage and levels of polymorphism. It would therefore be easy to conclude that the decline in polymorphism with increasing intersection advantage is caused by declining heterozygote advantage alone. However, this conclusion would be false, as we know from our examination of the effects of recognition degeneracy that high levels of heterozygote advantage are not necessary for high levels of polymorphism. Indeed, the highest levels of polymorphism may occur at the lowest levels of mean heterozygote advantage, as discussed above. So, in addition to 's effect on mean levels of heterozygote advantage, exactly how does increasing result in decreased levels of polymorphism?
Before answering the above question, we need to be reminded that our set-theoretic model utilizes a very large, yet finite, number of alleles; each allele is an m combination from a finite set of pathogen weights, V (see THE MODEL). The finite-alleles approach is important as it allows determination of the preselection distribution of available fitnesses, which, in turn, affects polymorphism construction. It turns out that limits levels of polymorphism not only through the effect that it has on mean levels of heterozygote advantage, but also through the effect that it has on the variance of the distribution of available genotype fitnesses and hence on the degree of asymmetry in selection: Increasing the variance of available genotype fitnesses enables small subsets of maximally fit alleles to dominate the polymorphism and preclude other mutants from invading. This result is consistent with the general theory of heterozygote advantage and the maintenance of polymorphism, which has shown that variation in heterozygote fitnesses severely limits the ability of overdominant selection to maintain polymorphism (LEWONTIN et al. 1978; MARKS and SPENCER 1991).
The empirical foundation for intersection advantage comes from experimentally derived fitness profiles under single-strain infection (PENN et al. 2002; MCCLELLAND et al. 2003; WEDEKIND et al. 2005, 2006), which have shown that immune responses of heterozygotes may be intermediate between the responses of both corresponding homozygotes. Coupled with the general knowledge that MHC alleles are codominantly expressed (ABBAS and LICHTMAN 2006), such a response pattern among genotypes may imply that genotypes containing two MHC molecules that recognize a pathogen strain may obtain some fitness advantage over genotypes containing only one MHC molecule that recognizes that strain when infected by that strain. The molecular mechanisms underlying this pattern are currently unknown, but may have something to do with the quantity of MHC molecules expressed on the surface of antigen-presenting cells; perhaps the magnitude of the T-cell response is directly proportional to the quantities of matching MHC molecules on the surface of the cell (CHEN et al. 1994; TYNAN et al. 2005), which in turn may be proportional to levels of expression of each MHC allele. Irrespective of the mechanisms underlying what we have called intersection advantage, the results that we present here clearly show that the ability of heterozygote advantage to maintain polymorphism at MHC genes depends crucially on the levels of intersection advantage. More specifically, as intersection advantage increases, polymorphism declines.
Conclusion:
The results that we present here show that, for heterozygote advantage to lead to high levels of polymorphism at the MHC, two requirements may need to be satisfied. First, the pathogen recognition sets of MHC molecules should be degenerate. As discussed above, there is some evidence that this is indeed the case. However, there has been no direct examination of the degree of degeneracy in pathogen recognition sets of MHC molecules. Determining the degree of recognition degeneracy would require identification of the major pathogen strains with which a certain host MHC locus interacts and then estimating the proportion of those pathogen strains recognized by each MHC allele. Again, we say an MHC molecule "recognizes" a pathogen strain when that strain produces at least one epitope that binds to that molecule with an appropriate affinity and/or conformation to activate T-cells. Thus, determining the degree of recognition degeneracy is theoretically possible, but would be extremely laborious.
Second, intersection advantage may need to be low for heterozygote advantage to explain very high levels of polymorphism at MHC loci. Currently, experimentally derived fitness profiles imply that intersection advantage may be high (Figure 2, A, C, and E; PENN et al. 2002; MCCLELLAND et al. 2003; WEDEKIND et al. 2005, 2006), which in turn leads to low levels of polymorphism. Therefore, the current experimental data, coupled with the model results presented here, imply that heterozygote advantage may not be as important to the maintenance of polymorphism at MHC loci as is currently perceived.
However, both the degree of pathogen recognition degeneracy (m) and intersection advantage () influenced polymorphism construction through the effect that they had on selective symmetry or on the variance in homozygote and heterozygote fitnesses; selective symmetry is positively correlated with m, but negatively correlated with . Increasing the variance in fitness leads to heterozygote advantage being less symmetric and lowers the level of polymorphism maintained. Our simple model may exaggerate the influence of m and on asymmetry, because we have not included a parameter to control for background fitness in our fitness function (1), and increasing levels of background fitness should lower variance in fitness among genotypes and make heterozygote advantage more symmetric. Nevertheless, the model presented here serves as an important heuristic, clearly showing that the incorporation of salient features of the MHC–pathogen interaction affect the degree of symmetry in heterozygote advantage in unobvious ways, which in turn has a strong influence on the properties of polymorphism construction.
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. Here, µL = 10–6, 
, where 
is the mean level of selection obtained in the absence of genetic drift (deterministic) and 
is the mean level of selection in the presence of genetic drift (stochastic). Loss of polymorphism is simply the difference between the mean level of polymorphism in the absence of drift and the mean level of polymorphism in the presence of genetic drift.
, standard deviation of PFD; 