Heterozygote advantage fails to explain the high degree of polymorphism of the MHC

Abstract

Major histocompatibility (MHC) molecules are encoded by extremely polymorphic genes and play a crucial role in vertebrate immunity. Natural selection favors MHC heterozygous hosts because individuals heterozygous at the MHC can present a larger diversity of peptides from infectious pathogens than homozygous individuals. Whether or not heterozygote advantage is sufficient to account for a high degree of polymorphism is controversial, however. Using mathematical models we studied the degree of MHC polymorphism arising when heterozygote advantage is the only selection pressure. We argue that existing models are misleading in that the fitness of heterozygotes is not related to the MHC alleles they harbor. To correct for this, we have developed novel models in which the genotypic fitness of a host directly reflects the fitness contributions of its MHC alleles. The mathematical analysis suggests that a high degree of polymorphism can only be accounted for if the different MHC alleles confer unrealistically similar fitnesses. This conclusion was confirmed by stochastic simulations, including mutation, genetic drift, and a finite population size. Heterozygote advantage on its own is insufficient to explain the high population diversity of the MHC.

Appendix

Standard population genetical theory (Nagylaki 1992) implies that all alleles i present at equilibrium need to have the same marginal fitness, $w_{i} = {\sum\nolimits_{j = 1}^n {p_{j} f_{{ij}} } }$, where p _i is the frequency of allele i, and f _ij is the genotypic fitness. From this condition one obtains $f_{i} {\left( {1 - {\left( {1 - \lambda f_{i} } \right)}p_{i} - \lambda \bar{f}} \right)} = f_{j} {\left( {1 - {\left( {1 - \lambda f_{j} } \right)}p_{j} - \lambda \bar{f}} \right)}$ for all alleles i and j. Here, $\bar{f} = {\sum\nolimits_{k = 1}^n {p_{k} f_{k} } }$ is the weighted mean allelic fitness contribution. By means of the last identity, one can express all equilibrium frequencies p _j by the same p _i . Summing up the resulting n equations, one obtains

$$ p_{i} = \frac{{1 - \lambda \bar{f}}} {{1 - \lambda f_{i} }}{\left( {1 - \frac{{n - 1}} {n}\frac{{\hat{f}}} {{f_{i} }}} \right)}, $$

(3)

where $\hat{f} \equiv n/{\sum\nolimits_{j = 1}^n {f^{{ - 1}}_{j} } }$ is the harmonic mean of the n allelic fitness contributions. Since the leading term is strictly positive, the frequency p _i is positive if and only if $f_{i} > \frac{{n - 1}} {n}\hat{f}$. This shows that MHC alleles with a too low fitness contribution f _i cannot persist at equilibrium. Note that this condition for persistence is independent of the choice of λ and the basis fitness value β (Weissing and Van Boven 2001; Van Boven and Weissing 2001).

For a novel allele with fitness contribution f _n+1 to invade into an established polymorphism of n alleles, its marginal fitness has to exceed the marginal fitness of the other alleles (Weissing and Van Boven 2001). Writing w _n+1>w _i , this yields

$$ {\sum\limits_{j = 1}^n {p_{j} {\left( {f_{{n + 1}} + f_{j} - \lambda f_{{n + 1}} f_{j} } \right)} > {\sum\limits_{j = 1}^n {p_{j} } }{\left( {f_{i} + f_{j} - \lambda f_{i} f_{j} } \right)} - p_{i} f_{i} {\left( {1 - \lambda f_{i} } \right)}} }, $$

(4)

which can be simplified into

$$ f_{{n + 1}} > f_{i} {\left( {1 - p_{i} \frac{{1 - \lambda f_{i} }} {{1 - \lambda \bar{f}}}} \right)}. $$

(5)

Substituting p _i from Eq. 5 gives Eq. 2 in the Results.

To obtain Fig. 1, we substitute f _i =(1−s)ⁱ⁻¹ into Eq. 2 (Results), where φ≡(1−s); one can test the invasion of the n+1^th allele, and simplify to obtain

$$ {\sum\limits_{i = 0}^n {{\left( {1 - s} \right)}^{i} = \frac{{1 - {\left( {1 - s} \right)}^{{n + 1}} }} {s}} } > n, $$

(6)

which can be used to solve the critical s for invasion into any polymorphism of n alleles.

In the asymmetric scenario of Takahata and Nei (1990) heterozygote genotypes all had a fitness f _ij =1, whereas the homozygote genotypic fitnesses were set to random values f _ii =f _i . Requiring the same marginal fitness $w_{i} = {\sum\nolimits_{j = 1}^n {p_{j} f_{{ij}} } }$ for all alleles yields p _i (1−f _i )=p _j (1−f _j ) for all j. By the same procedure one now obtains

$$ p_{i} = \frac{1} {n}\frac{{1 - \bar{f}}} {{1 - f_{i} }}, $$

(7)

which is strictly positive for$0 < \bar{f},\;f_{i} < 1$. Since the marginal fitness of a novel invading allele is one, whereas that of the established alleles is always smaller than one, new alleles can always invade, and poor alleles will never go extinct.