How to Determine Equilibrium Frequency if Trend Continues

Heterozygote Advantage

In addition to heterozygote advantage under stress, there is a substantial body of data indicating increased mutation, recombination, developmental variability, and phenotypic variability as stresses approach levels where extinctions become a real possibility (Parsons, 1987;

From: Advances in the Study of Behavior , 1998

Balanced Polymorphism☆

R.S. Singh , R.J. Kulathinal , in Reference Module in Life Sciences, 2017

Heterozygote Advantage

Heterotic balance (heterozygous advantage) polymorphisms develop when the fitness of heterozygotes is higher than the fitness of both homozygotes in a given population. A classic case of balanced polymorphism in human populations is sickle cell anemia. A mutation in the hemoglobin gene (β^S) leads to an alteration in the hemoglobin protein such that the homozygote (β^Sβ^S) genotype is effectively lethal because individuals die of anemia due to the characteristic sickling of red blood cells (see Fig. 1a). Normally, this would lead to the elimination of the β^S allele from the population. However, in regions where malaria is prevalent, the incidence of mortality from malaria (caused by Plasmodium falciparum) is relatively higher in normal homozygote (β^Aβ^A) individuals than in heterozygous individuals (β^Aβ^S). Of the three genotypes, β^Aβ^S has the highest fitness by being partly protected from both anemia and malaria. The loss of β^S alleles due to anemia is compensated (at equilibrium) by the loss of β^A alleles from malaria, and thus both alleles are maintained in a state of balanced polymorphism. Such polymorphisms are found in many parts of the world where there is a high incidence of malaria, such as Africa, the Middle East, and India, and many of these regions have fairly high (5–6%) frequencies of the sickle cell allele (see Fig. 1b). Eradication of malaria would lead to the reduction of the β^S allele from human populations, as appears to be the trend in African-American populations in the United States. Thus, the heterozygote advantage is a powerful mechanism in maintaining genetic polymorphisms, even for deleterious alleles; many debilitating human diseases (eg, Tay–Sachs, Gaucher, and Niemann–Pick diseases in Ashkenazi Jews) and some of the highly polymorphic blood group and enzyme genes (eg, the ABO blood groups and glucose-6-phosphate dehydrogenase) are suspected of being cases of present or past selectively maintained balanced polymorphisms.

Fig. 1. Sickle cell anemia in humans and heterozygote advantage. (a) Two red blood cells are shown: the background cell is a normal red blood cell that can transport oxygen at regular levels and the blood cell in the foreground displays the characteristic sickle shape common to the disease and carries a much reduced level of oxygen. (b) The regional presence of intermediate to high malarial loads (red) is superimposed into populations where the sickle cell trait is at appreciable frequencies (yellow) due to the heterozygote advantage at the β-hemoglobin locus. (a) Image copyrighted by the Wellcome Library.

Another example of an overdominant molecular polymorphism is the alcohol dehydrogenase (Adh) gene in natural populations of Drosophila melanogaster. This gene segregates two protein electrophoretic alleles, Adh-F (Fast) and Adh-S (Slow), which show north–south (latitudinal) clinical variation in populations from different continents. DNA sequencing reveals that the Adh-F allele is of recent origin, and a lysine residue in the Adh-S allele has been replaced by a threonine residue in the Adh-F allele. The ADH-F protein shows more enzymatic activity and is produced in larger quantities. DNA sequencing studies of representative samples of these two alleles have shown many silent site polymorphisms and a higher level of nucleotide variation at sites near the amino acid-altering mutation than elsewhere on the protein. The latter observation is expected, because linked polymorphic nucleotide sites cannot segregate freely, and a nucleotide site under balancing selection within the gene, through linkage, will affect levels of the polymorphism at the tightly linked sites. Together, these observations support that Adh polymorphism is maintained by heterozygote advantage.

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Susceptibility and Response to Infection

Michael F. Murray , in Emery and Rimoin's Principles and Practice of Medical Genetics (Sixth Edition), 2013

39.2 Heterozygous Advantage and Homozygous Disadvantage

The notion of heterozygous advantage with respect to infectious disease is typified by the example of sickle-cell disease and malaria (7). It is important to note that under the selective pressure of malaria there is the complementary phenomenon of homozygous disadvantage, such that both the person homozygous for the normal β-globin allele and the person homozygous for the mutant β-globin allele are at a disadvantage despite the fact that the β-globin^S/β-globin^S genotype offers additional antimalarial advantage compared to the heterozygote. With regard to human leukocyte antigen (HLA) heterozygosity there is evidence for advantages (see HLA discussion later). However, while this phenomenon has been suggested for other common recessive alleles, such as CFTR ΔF508, a clear proof has been somewhat elusive (8).

In 2010, an important new example of heterozygous advantage with respect to selective pressure of infection in Africa was recognized. A link was made between an unlikely infectious-disease-associated gene, apoprotein L1 (APOL1), and an increased risk of renal disease in African-Americans (9). Researchers went on to prove that two common variant alleles of this gene (i.e. APOL1 G1 or APOL1 G2), which are present in West African populations, are associated with protection against African trypanosomiasis, but when found in the homozygous state (G1/G1 or G2/G2) appear to be a risk for renal failure. Like the sickle hemoglobin–malaria situation, the scenario of APOL1-trypanosomiasis appears to be one where away from the selective pressure associated with infection (e.g. North America), there is a genetic disadvantage but no clear advantage to carrying the APOL1 alleles. Unlike the sickle cell–malaria scenario, the homozygous disadvantage of the variants do not likely exert pressure on reproductive fitness in Africa since renal disease typically occurs later in life, beyond the life expectancy of most native Africans.

The work of Lyons and colleagues (10) suggests that more examples may exist, including examples of homozygous disadvantage for invasive bacterial infection. In their study of microsatellite markers in African children who died of invasive bacterial infection and controls they found that homozygosity at a relatively small sample of markers was associated with significant increases in the odds ratio for mortality due to invasive bacteria, these markers segregated differently in some cases based on Gram-positive vs. Gram negative bacteria. The most significant odds ratio (40.7; 95% confidence interval (CI), 4.28–387) was observed for Gram-negative infection in those children who were homozygous for both a site at 7q31 (D7S486) as well as a second site at 16p13.3 (D16S423). The mapping of the pathogenic genetic variants linked to this recessive disease risk has not yet been reported.

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Homology Effects

Sarah P. Otto , Paul Yong , in Advances in Genetics, 2002

B. Heterozygote advantage

The first case that we shall consider is heterozygote advantage. We assume that alleles A and a perform different functions such that the fitness of an individual is higher when both functions are performed. To simplify matters, we assume that all individuals carrying both A and a alleles (regardless of the number of them) have a fitness of 1(= W56 = W25 = W35 = W45 = W16 = W26 = W36), that individuals with only A alleles have a reduced fitness of 1 – s (= W55 = W15), and that individuals with only a alleles have a reduced fitness of 1 – t (= W66 = W46). This model differs from that examined by Spofford (1969), who focused on the case of a dimeric enzyme taking into account the probability that each type of dimer would be produced. In addition, the following derivation estimates the effective selection coefficient acting on a new duplicate, whereas Spofford focused on a numerical analysis of the dynamics of the duplicate gene.

Before the appearance of the duplication, the frequency of allele A approaches $\hat{x}$ ₅ = t/(s + t), at which point the mean fitness of the population is $\hat{W}$ = 1 – st/(s + t) (Crow and Kimura, 1970). Without loss of generality, we label the alleles such that s ≤ t, so that the frequency of allele A is greater than or equal to $\frac{1}{2}$ . Under these assumptions, it can be shown that the leading eigenvalue describing the initial rate of spread of a gene duplicate is equal to:

(A3) ${\lambda }_L=\frac{1}{\hat{W}}-\frac{1-\hat{W}+r}{2\hat{W}}\left[1-\sqrt{1-\frac{4r{\hat{x}}_6\left(1-\hat{W}\right)}{{\left(1-\hat{W}+r\right)}^2}}\right]$

Under our assumptions that s, t, $\hat{x}$ ₅, $\hat{x}$ ₆ > 0, this leading eigenvalue is strictly greater than one. Thus, there will always be selection favoring the spread of a duplication. When the duplicate genes are tightly linked (r ≈ 0), the leading eigenvalue becomes 1/ $\hat{W}$ , which is greater than one under our assumptions. In this case, there is a slight caveat: if the first haplotype to appear is either AA or aa, the duplication will not be positively selected until either the Aa or aA haplotype is produced by mutation, conversion, or recombination. Increasing the recombination rate above zero always reduces the leading eigenvalue and hence slows the initial spread of the duplicate gene. When selection is weak (s, t ≪ 1) and when the genes are not very tightly linked (r> 0), the eigenvalue is approximately:

(A4) ${\lambda }_L\approx 1+{\hat{x}}_5\left(1-\hat{W}\right)=1+\frac{s{t}^2}{{\left(s+t\right)}^2}$

Equation (A4) indicates that the indirect or effective selection acting on the duplicate has the same order of magnitude as selection acting directly on the A and a alleles. Furthermore, selection for the duplicate is strongest when the two functions are approximately of equal benefit(s ≈ t). Comparing the strength of selection on the duplicate (λ _L – 1), the duplicate experiences (s +1)/t times the amount of selection when linkage is tight than when linkage is loose, assuming weak selection. This reaches a maximum of a twofold difference when s ≈ t, indicating that a tandem duplication carrying both A and a alleles is twice as likely to fix within a population (replacing λ _L – 1 for the selection coefficient in equation 1) and will spread twice as quickly compared to an unlinked gene duplication. The advantage of tandem duplications is that the most fit chromosomes, i.e., ones bearing both A and a alleles, are unlikely to be broken apart by recombination. This may partially explain why tandem gene duplications are common in many gene families, although another obvious explanation is that the frequency with which duplications appear in tandem is higher as a result of unequal crossing over.

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Population Genetics

Brian Charlesworth , in Encyclopedia of Biodiversity (Second Edition), 2013

Segregational Load

Similar calculations can be performed for models of balancing selection, yielding estimates of the segregational load. In the case of heterozygote advantage, the load due to a single locus is st/(s+t) (Charlesworth and Charlesworth, 2010). Equation [4] can be used to determine the segregational load contributed by a large number of polymorphic loci with independent effects. This can be considerable, even if selection is weak. For example, 10,000 loci each with s=t=0.001 would yield a mean fitness of only 0.0067. This is so low that only a very high fecundity species would be able to produce the two surviving offspring per adult needed to maintain itself. This implies that either most molecular variation has very slight or no effects on fitness, or that the assumption of multiplicative fitnesses is unrealistic.

An extreme alternative to multiplicative fitnesses is truncation selection. Genotypes at a set of loci are assumed to be ordered with respect to their fitnesses as determined by the multiplicative fitness model; a fixed proportion of the population, containing the set of genotypes with the highest fitnesses, is allowed to survive. This is equivalent to assuming that individuals compete for a limiting resource, and that only the fittest succeed. Under these conditions, a much larger number of loci can be exposed to selection for a given total L than with multiplicative fitnesses, for the same selection intensity per locus. Less extreme forms of departure from multiplicativity can have similar but smaller effects on the total load.

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Genetic Variation, Maintenance of

H.G. Spencer , in Encyclopedia of Evolutionary Biology, 2016

Problems with the Selectionist Hypothesis

Both hypotheses have theoretical and empirical weaknesses. For example, the intuition underlying the balance school suggests that selection in the form of heterozygote advantage underlies the observed levels of genetic variation. Interestingly, this model is unique among the classical, one-locus, two-allele models of constant viability selection in maintaining both alleles at a stable equilibrium: heterozygote advantage is both necessary and sufficient to ensure that any population with allele frequencies close to those at this equilibrium will evolve toward that equilibrium. All other classical models with constant viabilities (e.g., deleterious recessive) predict that genetic variation should eventually be depleted. (Compare Figures 1 and 2.)

Figure 1. Mean fitness of the population (above) and change in frequency of allele A ₁ (below) under an example of heterozygote advantage with 2 alleles. The fitness of A ₁ A ₁ homozygotes is 0.9, that of A ₂ A ₂ homozygotes 0.8, compared to A ₁ A ₂ heterozygotes with a fitness of 1.0. Note that for frequencies of A ₁ below ²/₃, the change in frequency of A ₁ is positive (i.e., it increases); above this value the change is negative (i.e., A ₁ becomes rarer). Hence, A ₁ moves closer to the stable equilibrium frequency of ²/₃ every generation. At the same time A ₂ moves toward a stable equilibrium frequency of ¹/₃. Since both alleles are present at this equilibrium, it is polymorphic. Its stability means that this form of selection maintains variation in the population. This equilibrium frequency also maximizes the population's mean fitness.

Figure 2. Mean fitness of the population (above) and change in frequency of allele A ₁ (below) under an example of deleterious recessive with 2 alleles. The fitness of A ₁ A ₁ homozygotes is 0.8, compared to A ₁ A ₂ heterozygotes and A ₂ A ₂ homozyotes with fitnesses of 1.0. Note that for all non-trivial frequencies of A ₁, the change in frequency of A ₁ is negative (i.e., A ₁ becomes rarer). Hence, A ₁ moves closer to the stable equilibrium frequency of 0 every generation. At the same time A ₂ moves toward a stable equilibrium frequency of 1. Since only A ₂ is present, this equilibrium is monomorphic, not polymorphic. Its stability means that this form of selection does not maintain variation in the population. Nevertheless, this equilibrium frequency also maximizes the population's mean fitness.

Unfortunately, when this basic model is extended to more than two alleles, the situation is less clear. For a start, the very meaning of 'heterozygote advantage' becomes ambiguous. We could mean simply that each heterozygote is fitter than its two corresponding homozygotes, or we could require the stronger condition that all heterozygotes are fitter than the fittest homozygote. Ironically, whatever our definition, it turns out that heterozygote advantage is neither necessary nor sufficient to maintain all the alleles at a stable equilibrium. In other words, it is possible to find fitness parameters that do not fit our definition that afford a stable fully polymorphic equilibrium, as well as other fitnesses that do meet our definition that nevertheless lead to the elimination of one or more alleles. Lewontin et al. (1978) exhibit counterexamples to both the necessity and sufficiency of 'heterozygote advantage' in maintaining even 3 alleles at equilibrium.

A related issue is that the sorts of fitnesses that are able to maintain larger number of alleles at an equilibrium are highly unusual. In a simulation study, Lewontin et al. (1978) generated random sets of n(n+1)/2 viabilities for a population with n alleles (and hence n(n+1)/2 different genotypes) for n=2, 3, 4, 5, and 6 in different simulations, and asked what proportion of these sets allowed a stable equilibrium with all n alleles present. For n=2, heterozygote advantage is both necessary and sufficient and thus we know the answer is 1/3, since there are 3 viabilities needed (for the two homozygotes and one heterozygote) and the chance that the heterozygote fitness in the largest of the 3 viabilities is 1/3. For n>2, however, there is no intuitive explanation. Simulations showed that the proportion rapidly becomes minute: for n=3, the proportion was ~0.042, for n=5, it was 6×10^–5, and for n=6, not one in 100   000 sets yielded a stable 6-allele polymorphism. Note that the viability sets that did maintain all n alleles did not necessarily display heterozygote advantage, however defined (except for n=2). Clearly, being able to maintain n alleles for larger values of n is a very unusual property of fitness sets.

These findings were interpreted as showing that the likelihood of such sets occurring in nature was minute. But such a conclusion is a logical fallacy: the size of parameter space is not a measure of likelihood, especially in a system that evolves over time. In another simulation study, Spencer and Marks (1988) showed that recurrent mutation at a locus could lead to the occasional successful invasion of a population by a new allele and the building up of polymorphism over time. Indeed, after 10   000 generations, with just one mutation per generation, the simulated populations possessed an average of more than 5 alleles (Marks and Spencer, 1991). These ideas have been applied to various other forms of selection (e.g., frequency-dependent selection, spatially variable selection, differential selection on males and females) and the overall conclusion is similar: variation is constructed over time under simple combinations of mutation and selection. Nevertheless, it is not clear if these models are sufficiently realistic. For example, one possible disconnect is that many of the simulations generated some form of heterozygote advantage and yet convincing cases of this form of selection acting in natural populations are extremely rare.

A different theoretical problem arises when variation at many loci is considered. Under simple models of selection such as heterozygote advantage at a single locus, the mean fitness of the population is lower than that of the ideal (heterozygous) genotype. This reduction in fitness is called the 'segregational load,' a form of 'genetic load.' When multiple loci are considered, it seems reasonable to assume that effects of each locus are independent and so fitnesses combine multiplicatively. Doing so, however, leads to an absurdity if large numbers of loci have variation maintained by these forms of selection: the mean fitness becomes ridiculously small compared to that of the ideal, multiply-heterozygous types (see Lewontin (1974) and Kimura (1983) for the details of this criticism). Different forms of selection, such as frequency-dependent selection, are less subject to this problem, but it has not been satisfactorily answered.

Finally, as mentioned above, convincing cases of genetic variation in natural populations that are actively maintained by selection are rare. In part, this absence of evidence reflects the difficulty of measuring selection in the wild, but if it is due to a real dearth of examples, then the selectionist hypothesis needs significant modification.

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Natural Selection*

K.E. Holsinger , in Encyclopedia of Genetics, 2001

Stabilizing Selection

Stabilizing selection occurs when heterozygous individuals are the most likely to survive. For that reason this fitness pattern is also referred to as heterozygote advantage. As with disruptive selection, if a population happened to start with an allele frequency exactly equal to:

$p*=(w_{12}-w_{22})/(2w_{12}-w_{11}-w_{22})$

the allele frequency would not change. When heterozygotes are more likely to survive than either homozygote, however, p* is a stable equilibrium. Selection causes small departures from p* to become even smaller with time. Moreover, the allele frequency in the population will evolve toward p* regardless of the initial allele frequency, as long as both alleles are initially present. In Figure 2, for example, w ₁₁ = 0.72, w ₁₂ = 0.9, and w ₂₂ = 0.81, and the population evolves toward p* = 0.33 regardless of whether the initial allele frequency is 0.01 or 0.99.

Figure 2. Dynamics of heterozygote advantage.

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Favism

T. Isbir , ... A.B. Dalan , in Brenner's Encyclopedia of Genetics (Second Edition), 2013

G6PD deficiency is one of the most common human enzyme defects in the world carried on the X chromosome. The main mutation is sex linked – that is, it is carried on the X chromosome, and probably maintained by heterozygote advantage in females, but males have the disadvantage as they carry only one X chromosome. Adult female heterozygotes usually have a double population of RBCs, some with and some without the enzyme deficiency, with an average of about 50% of each. There are over 300 allelic variants for the disease and there are mainly five classes of the disease according to enzyme activity:

Class I – associated with chronic nonspherocytic hemolytic anemia

Class II – severely deficient enzyme (<10% residual activity)

Class III – moderately deficient enzyme (10–60% residual activity)

Class IV – normal enzyme activity (60–150%)

Class V – increased activity

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Heterozygote and Heterozygosis

D.E. Wilcox , in Encyclopedia of Genetics, 2001

Heterozygote Advantage

In some circumstances, the effects of a recessive mutation can affect the phenotype and thus reproductive fitness of heterozygotes. This is not always a negative effect as can be seen in the condition human sickle-cell anemia. Sickle-cell carriers have a heterozygote advantage over the reproductive fitness of normal homozygotes in some environments. In most populations, sickle-cell anemia is a rare mutation, but in malarial regions of Africa as many as one in three of the population are carriers of the mutation in the hemoglobin gene. The presence of the mutant hemoglobin in heterozygotes interferes with the malarial parasite's life cycle. Heterozygotes are therefore more resistant to the debilitating effects of malaria than the normal homozygotes. This heterozygote advantage in many sickle-cell carriers outweighs the severe reproductive disadvantage of the rarer sickle-cell homozygotes. This maintains the mutation in this population at a high frequency as a polymorphism.

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Population and Evolutionary Genetics

Leon E. Rosenberg , Diane Drobnis Rosenberg , in Human Genes and Genomes, 2012

Heterozygote Advantage

As just stated, in many autosomal recessive traits homozygotes for the mutant allele have reduced fitness (f   <   1 to f   =   0) compared to heterozygotes or those homozygous for the normal allele. There are a few disorders where environmental conditions result in the fitness of heterozygotes being greater than that for either homozygous genotype. This is termed heterozygote advantage . It is important because even a slight heterozygote advantage may act to increase the frequency of the mutant allele in the population—even if the mutant allele causes major reduction in fitness in homozygotes in that population. Those situations in which natural selection acts, at the same time, to maintain a deleterious allele in the gene pool and to remove it from the pool are termed balanced polymorphisms.

Sickle cell anemia is the best-known example of heterozygote advantage and balanced polymorphism in humans. As discussed in Chapter 12 and again in Figure 18.1, the frequency of heterozygotes for hemoglobin S (genotype AS) and the frequency of homozygotes for the mutation (SS) is higher in areas where malaria occurs than in non-malarious areas. This is explained as follows. In regions where malaria is endemic (as in West Africa), homozygotes (AA) for the normal allele are susceptible to malaria; many become infected and some die, leading to reduced fitness. SS homozygotes are even more reproductively disadvantaged because of their severe hematologic disease, their fitness approaching zero. Heterozygotes (AS), however, have red blood cells that resist infection by the malaria parasite (Plasmodium falciparum) and burst more quickly when infected, thereby leading to the death of the parasites before they infect other cells. In this situation, the fitness of heterozygotes is greater than that for either homozygote. Accordingly, heterozygotes reproduce at a higher rate. Over time the S allele has reached a frequency of 20% (or more) in some areas of West Africa (meaning that as many as one in three people are carriers for the mutant allele). This gene frequency is much higher than can be accounted for by mutation alone. Under these circumstances, Hardy-Weinberg equilibrium is significantly perturbed. When black Africans move to countries like the United States where malaria no longer exists, one would predict that the frequency of the S allele would decline over time. It may already be beginning to do so.

Heterozygote advantage has been shown to exist for other genotypes affecting the red blood cell. Deleterious alleles for the thalassemias, certain other hemoglobinopathies, and glucose-6-phosphate dehydrogenase (G6PD)—an enzyme erythrocytes use to extract energy from glucose—are all thought to be maintained at higher than expected frequencies because they provide protection against malaria. Heterozygote advantage may also explain the high frequency of CF in whites and of a number of lipid storage diseases, including Tay-Sachs, in Ashkenazi Jews. The environmental forces responsible in these situations remain unclear.

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Hardy–Weinberg Equilibrium and Random Mating

J. Lachance , in Encyclopedia of Evolutionary Biology, 2016

Natural Selection

Natural selection modifies allele and genotype frequencies and these effects depend on both the magnitude and type of selection present. Selection results in departures from Hardy–Weinberg proportions whenever genotypic fitnesses are non-multiplicative (i.e., w _AB ² ≠ w _AA × w _BB ) (Lachance, 2008 ). Not surprisingly, overdominance (heterozygote advantage) results in an excess of heterozygotes compared to Hardy–Weinberg expectations, and underdominance (heterozygote disadvantage) results in an excess of homozygotes. Strong directional selection, such as when one allele is a recessive lethal, leads to marked departures from Hardy–Weinberg expectations. However, weak directional selection has only modest effects on genotype frequencies, and detecting these effects can require sample sizes that are larger than 10  000 individuals (Lachance, 2009). Note that the effects of natural selection tend to be locus-specific rather than genome-wide.

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