Hardyweinberg For Two Loci

The original Hardy-Weinberg model assumed a genetic architecture of one autosomal locus with two alleles. We will now consider a slightly more complicated genetic architecture of two autosomal loci, each with two alleles (say A and a at locus 1 and B and b at locus 2). Otherwise, we will retain all other assumptions of the original Hardy-Weinberg model. However, there is one new assumption. Recall from Chapter 1 that our second premise is that DNA can mutate and recombine. We will retain the Hardy-Weinberg assumption of no mutation, but we will allow recombination (either independent assortment if the two loci are on different autosomes or crossing over if they are on the same autosome).

Because our main interest is on whether or not evolutionary change occurs, we will start with the gene pool and go to the next generation's gene pool (Figure 2.4), rather than going from adult population to adult population as in Figures 2.1 and 2.3. Given two loci with two alleles each and the possibility of recombination between them, a total of four gamete types are possible (AB,Ab, aB, and ab). The gene pool is characterized by four gamete frequencies (Figure 2.4), symbolized by gxy, where x indicates the allele at locus 1 and y indicates the allele at locus 2. Just as p and q sum to 1, these four gamete frequencies also sum to 1 because they define a probability distribution over the gene pool. The transition from this gene pool to the zygotes is governed by the same population structure (rules of uniting gametes) as given in the single-locus Hardy-Weinberg. In particular, the assumption of random mating means that gametes are drawn independently from the gene pool, with the probability of any given gamete type being equal to its frequency. The probability of any particular genotype is simply the product of its gamete frequencies, just as in the single-locus Hardy-Weinberg model. In Figure 2.4 we are not keeping track of the paternal or maternal origins of any gamete, so both types of heterozygotes are always pooled and therefore the product of the gamete frequencies for heterozygous genotypes is multiplied by 2. For example, the frequency of the genotype AB/Ab is 2gABgAb. Note that there are two types of double heterozygotes, AB/ab (the cis double heterozygote with a random-mating frequency of 2gABgab) and Ab/aB (the trans double-heterozygote with a random-mating frequency of 2gAbgaB). Although the cis and trans double heterozygotes share the double-heterozygous genotype, completely different gamete types produce the cis and trans double-heterozygosity. As we will soon see, the cis and trans double-heterozygous genotypes contribute to the gene pool in different ways. Hence, we will keep the cis and trans double-heterozygote classes separate.

The rules for uniting gametes in the two-locus model are the same as for the single-locus model, the only difference being that there are now 10 genotypic combinations. As with the single-locus model, if we know the gamete frequencies and know that the mating is at random (along with the other population structure assumptions of Hardy-Weinberg),

Derivation Hardy Weinberg

Figure 2.4. Derivation of Hardy-Weinberg law for two autosomal loci with two alleles each: A and a at locus 1 and B and b at locus 2.In going from gametes to zygotes, solid lines represent gametes bearing the AB alleles and are assigned the weight gAB, dashed lines represent gametes bearing the Ab allele and are assigned the weight gAb, grey lines represent gametes bearing the aB gametes and are assigned the weight gaB, and dotted lines represent gametes bearing the ab alleles and are assigned the weight gat,. In going from adults to gametes, solid lines represent Mendelian transition probabilities of 1 for homozygotes, dashed lines represent Mendelian transition probabilities of j for single heterozygotes, dotted lines represent nonrecombinant Mendelian transition probabilities of ¿(1 - r) (where r is the recombination frequency between loci 1 and 2) for double heterozygotes, and grey lines represent recombinant Mendelian transition probabilities of ^r for double heterozygotes.

Figure 2.4. Derivation of Hardy-Weinberg law for two autosomal loci with two alleles each: A and a at locus 1 and B and b at locus 2.In going from gametes to zygotes, solid lines represent gametes bearing the AB alleles and are assigned the weight gAB, dashed lines represent gametes bearing the Ab allele and are assigned the weight gAb, grey lines represent gametes bearing the aB gametes and are assigned the weight gaB, and dotted lines represent gametes bearing the ab alleles and are assigned the weight gat,. In going from adults to gametes, solid lines represent Mendelian transition probabilities of 1 for homozygotes, dashed lines represent Mendelian transition probabilities of j for single heterozygotes, dotted lines represent nonrecombinant Mendelian transition probabilities of ¿(1 - r) (where r is the recombination frequency between loci 1 and 2) for double heterozygotes, and grey lines represent recombinant Mendelian transition probabilities of ^r for double heterozygotes.

we can predict the zygotic genotype frequencies. If we further assume that there are no phenotypic differences that affect viability, mating success, or fertility, we can also predict the next generation's adult genotype frequencies from the gamete frequencies.

The similarities to the single-locus model end when we advance to the transition from the next generation's adult population to the gene pool of the next generation (Figure 2.4). At this point, some new rules are encountered in producing gametes that did not exist at all in the single-locus model (Figure 2.1). As before, homozygous genotypes can only produce gametes bearing the alleles for which they are homozygous (this comes from the assumptions of normal meiosis and no mutations). As before, genotypes heterozygous for just one locus produce two gamete types, with equal frequency as stipulated by Mendel's first law. However, genotypes that are heterozygous for both loci can produce all four gamete types, and the probabilities are determined by a combination of Mendel's first law and recombination (Mendel's second law of independent assortment if the loci are on different chromosomes or the recombination frequency if on the same chromosome). Hence, the transition from genotypes to gametes requires a new parameter, the recombination frequency r, which is 1 if the loci are on different chromosomes and 0 < r < 2 if the loci are on the same chromosome.

The addition of recombination produces some qualitative differences with the single-locus model. First, in the single-locus model, an individual could only pass on gametes of the same types that the individual inherited from its parents. But note from Figure 2.4 that the cis double heterozygote AB/ab, which inherited the cis AB and ab gamete types from its parents, can produce not only the cis gamete types, each with probability 1(1 - r), but also the trans gamete types Ab and aB, each with probability 1 r. Similarly, the trans double heterozygote can produce both cis and trans gamete types (Figure 2.4). Thus, recombination allows the double heterozygotes to produce gamete types that they themselves did not inherit from their parents. This effect of recombination is found only in the double-heterozygote class, but this does not mean that recombination only occurs in double heterozygotes. Consider, for example, the single heterozygote AB/Ab. If no recombination occurs in meiosis, this genotype will produce the gamete types AB and Ab with equal frequency. Hence, the total probability of gamete type AB with no recombination is 1 (1 - r), and similarly it is 2(1 - r) for Ab. Now consider a meiotic event in which recombination did occur. Such a recombinant meiosis also produces the gamete types AB and Ab with equal frequency, that is, with probability 2 r for each. However, in the recombinant AB gamete the A allele that is combined with the B allele originally came from the Ab gamete that the AB/Ab individual inherited from one of its parents. Hence, recombination has occurred, but because we do not distinguish among copies of the A alleles, we see no observable genetic impact. Hence, the total probability of an AB gamete, regardless of the source of the A allele, is 2(1 - r) + 2r = 2, and the total probability of an Ab gamete, regardless of the source of the A allele, is 2 (1 - r) + 1 r = 2. Thus, recombination is occurring in all genotypes but is observable only in double heterozygotes.

The qualitative difference from the single-locus model that causes some genotypes to produce gamete types that they themselves did not inherit leads to yet another qualitative difference: The two definitions of gene pool given in Chapter 1 are no longer equivalent. If we define the gene pool as the shared genes of all the adult individuals, we obtain the gamete frequencies from the pool of gametes produced by their parents (the gxy's in Figure 2.4). On the other hand, if we define the gene pool as the population of potential gametes produced by all the adult individuals, the effects of recombination enter and we obtain the g'xy's in Figure 2.4. To avoid any further confusion on this point, the term "gene pool" in this book will always refer to the population of potential gametes unless otherwise stated. The general population genetic literature often does not make this distinction because in the standard single-locus Hardy-Weinberg model it is not important. Quite frequently there is a time difference of one generation among the models of various authors depending upon which definition of gene pool they use (usually implicitly). Therefore, readers have to be careful in interpreting what various authors mean by gene pool when dealing with multilocus models or other models in which these two definitions may diverge.

The most important qualitative difference from the single-locus model involves the potential for evolution. As seen before, the single-locus Hardy-Weinberg model goes to equilibrium in a single generation of random mating and then stays at equilibrium, resulting in no evolution. To see if this is the case for the two-locus model, we now use equation 2.1 to calculate the gamete frequency of the AB gamete using the weights implied by the arrows in Figure 2.4 going from adults to gametes:

SAb = 1 ■ Sab + [email protected]) + -¿VgABgaB) + 2(1 - r)(2gABgab) + 2r(2gAbgaB)

= gAB [gAB + gAb + gaB + (1 - r)gab] + rgAbgaB (2 5)

= gAB [gAB + gAb + gaB + gab] + rgAbgaB - rgABgab = gAB + r(gAbgaB - gABgab) = gAB - rD

where D = (gABgab - gAbgaB)- The parameter D is commonly known as linkage disequilibrium. However, because it can exist for pairs of loci on different chromosomes that are not linked at all, a more accurate but more cumbersome term is gametic-phase imbalance. Because the term linkage disequilibrium dominates the literature, we will use it throughout the book, but with the caveat that it can be applied to unlinked loci.

Similarly, the other three gamete types can be obtained from equation 2.1 as

gAb = 1 ' gAb + 2(2gABgAb) + 2(2gAbgab) + 2(1 - r)(2gAbgaB) + 2r(2gABgab)

gaB = 1 ' gaB + 2^gABgaB) + ^gaBgab) + 2(1 - r )(2gAbgaB) + 2 r (2gABgab)

gab = 1 ' gab + 2 (2gAbgab) + 2^gaBgab) + 2(1 - r )(2gABgab) + 2r (2gAbgaB)

At this point, we can now address our primary question: Is evolution occurring? Recall our definition from Chapter 1 of evolution as a change in the frequencies of various types of genes or gene combinations in the gene pool. As is evident from equations 2.5 and 2.6, as long as r > 0 (that is, some recombination is occurring) and D = 0 (there is some linkage disequilibrium), gxy = g'xy: Evolution is occurring! Thus, a seemingly minor change from one to two loci results in a major qualitative change of population-level attributes.

No evolution occurs in this model if r = 0. In that case, the two-locus model is equivalent to a single-locus model with four possible alleles. Thus, some multilocus systems can be treated as if they were a single locus as long as there is no recombination. On the other hand, recombination can sometimes occur within a single gene. As mentioned in

Chapter 1, the genetic variation within a 9.7-kb segment of the lipoprotein lipase (LPL) gene in humans was shaped in part by about 30 recombination events (Templeton et al. 2000a). Thus, in some cases the evolutionary potential created by recombination must be considered even at the single-locus level. In the case of LPL, we are looking at two or more different polymorphic nucleotide sites within the same gene and not, technically speaking, at different loci. However, the qualitative evolutionary potential is still the same as long as the polymorphic sites under examination can recombine, regardless of whether those sites are single nucleotides within a gene or traditional loci.

No evolution also occurs in this model if D = 0. Here, D will equal zero when the two-locus gamete frequencies are the product of their respective single-locus allele frequencies. To see this, let pA be the frequency of the A allele at locus 1 and pB the frequency of the B allele at locus 2. These single-locus allele frequencies are related to the two-locus gamete frequencies by pA = gAB + gAb pB = gAB + gaB (2.7)

Now consider the product of the A and B allele frequencies:

= gAB + gABgaB + gABgAb + gAbgaB = gAB (gAB + gaB + gAb) + gAbgaB = gAB (1 — gab) + gAbgaB = gAB — gABgab + gAbgaB = gAB — D

Solving equation 2.8 for D yields

and similar equations can be derived in terms of the other three gamete frequencies. Equation 2.9 suggests another biological interpretation of D; it is the deviation of the two-locus gamete frequencies from the product of the respective single-locus allele frequencies. Equation 2.9 also makes it clear that D will be zero when the two-locus gamete frequency is given by the product of the respective single-locus allele frequencies. This can also be seen by evaluating the original formula for linkage disequilibrium under the assumption that the two-locus gamete frequencies are given the product of their respective allele frequencies: D = gABgab — gAbgaB = (pApB )(papb) — (pApb)(papB) = pApBpapb — pApbpapB = 0.

The two-locus gamete frequencies will be products of the single-locus allele frequencies when knowing what allele is present at one locus in a gamete does not alter the probabilities of the alleles at the second locus; that is, the probabilities of the alleles at the second locus are simply their respective allele frequencies regardless of what allele occurs at the first locus. When D = 0, knowing which allele a gamete bears at one locus does influence the probabilities of the alleles at the second locus. In statistical terms, D = 0 means that there is no association in the population between variation at locus 1 with variation at locus 2. When D = 0, equations 2.5 and 2.6 show that the gamete frequencies (and hence the genotype frequencies) are constant, just as they were in the single-locus Hardy-Weinberg model. Thus, when D = 0 the population is at a nonevolving equilibrium, given the other standard Hardy-Weinberg assumptions. We can now understand why D is called disequilibrium. When D is not zero and there is recombination, the population is evolving and is not at a two-locus Hardy-Weinberg equilibrium. The larger D is in magnitude, the greater this deviation from two-locus equilibrium.

Evolution occurs when r > 0 and D = 0, and we now examine the evolutionary process induced by linkage disequilibrium in more detail. From Figure 2.4 or equations 2.5 and 2.6, we see that linkage disequilibrium in the original gene pool (gABgab - gAbgaB) influences the next generation's gene pool. Similarly, the linkage disequilibrium in the next generation's gene pool will influence the subsequent generation's gene pool. The linkage disequilibrium in the next generation's gene pool in Figure 2.4 is

= [(gAB - rD)(gab - rD) - (gaB + rD)(gAb + rD)] (2.10)

Using equation 2.10 recursively, we can see that D2 (the linkage disequilibrium in the gene pool two generations removed from the original gene pool) is D(1 - r )2. In general, if we start with some initial linkage disequilibrium, say D0, then Dt, the linkage disequilibrium after t generations of random mating, is

Equation 2.11 reveals that the evolution induced by linkage disequilibrium is both gradual and directional, as illustrated in Figure 2.5. Because r < 1, the quantity (1 - r)t goes to

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