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Two Locus genetics is concerned with Haplotype frequencies

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    Haplotype means a combination of alleles at more
    than one locus. For two loci with 2 alleles, A1, A2, B1, and B2. There are 4 haplotypes, A1B1, A1B2, A2B1, A2B2.

    A diploid individual’s genotype will be something like A1B1/A1B2. It has two haplotypes, one inherited from each parent, just as a one-locus genotype contains two genes from the two parents. If the A and B loci are on the same chromosome, each haplotype is a gene combination on a chromosome but haplotypes can also be specified for loci on different chromosomes.

    Frequencies of haplotypes may or may not be in
    linkage equilibrium

    in the absence of selection and in random mating the frequencies of haplotypes will be find by hardy Weinberg equilibrium this also lead to the discovery of linkage equilibrium.

    Linkage Equilibrium : Linkage equilibrium describes the situation in which the haplotype frequencies in a population have the same value that they would have if the genes at each locus were combined at random.

    The haplotype frequencies observed differ from expected frequencies by factor D which is a crucial part of the observation it is called the correction factor. we add and subtract it from frequencies. The observed haplotype frequencies are given as

    Haplotype Frequency in population
    A1B1 = a = p1q1 + D
    A1B2 = b = p1q2 − D
    A2B1 = c = p2q1 − D
    A2B2 = d = p2q2 + D

    total frequencies add up to 1 i.e.

    a + b + c + d = 1. (Also, p1q1 + p1q2 + p2q1 +
    p2q2 = 1,

    and the sum of the two +D and two –D factors is zero. The important term
    to understand is D; it is a measure of “linkage disequilibrium.” Linkage equilibrium is when D = 0 and means that the alleles at the two loci are combined independently.

    D measure deviation from linkage equilibrium. If D > 0, A1 is more often found with B1 (and less often with B2) than would be expected if alleles at the two loci were combined at random and the population contains an excess of A1 B1 (and of A2B2) haplotypes.

    Papilio memnon example:

    Papilio memnon is an example of high linkage disequilibrium. If Clarke and
    Sheppard are correct, the allele T+ is almost always combined with the other alleles W1,F1, E1, and B1 rather than with W2, W3, or W4 (and equivalent alleles at the other loci). There is a large excess of the haplotypes T+W1F1E1B1, T–W2F2E2B2, T–W3F3E3B3, etc.,
    The linkage disequilibrium in P. memnon, as we have seen, is caused by selection.

    Frequency of recombinant haplotypes:

    The frequency of haplotypes is given as a in one generation and a’ in the next generation. In the absence of natural selection frequency of haplotypes changes but the frequency of each gene remains constant.

    In homozygous ( single or double) haplotype combination the frequency remain same and there will be no recombination but in double heterozygous recombination changes the haplotype frequency.

    When recombination takes place in an A1B1/A2B2 individual, the number of A1B1 haplotypes is decreased. When it takes place in an A1B2/A2B1 individual, the number of A1B1 is increased. To be exact, half the genes of an A1B1/A2B2 double heterozygote are A1B1; when recombination
    hits between the loci the frequency of A1B1 decreases by an amount −1/2. Similarly, recombination in an A1B2/A2B1 individual increases the frequency of A1B1 by an amount +1/2. The frequency of A1B1/A2B2 heterozygotes in the population is 2ad and of A1B2/A2B1
    is 2bc.

    The frequency of A1B1/A2B2 heterozygotes in the population is 2ad and of A1B2/A2B1 is 2bc.

    The frequency at which the alleles at two loci are recombined per generation is defined as r. (r can theoretically have any value up to a maximum of 0.5 if the loci are on different chromosomes; r is between 0 and 0.5 for loci on the same chromosome depending on how tightly linked they are.

    a’= a-1/2r2ad + 1/2r2bc a′ = a − r(ad − bc)

    Now, the expression (ad − bc) is simply equal to the linkage disequilibrium D.

    a′ = p1q1 + D − rD

    a′ − p1q1 = (1 − r)D

    The right side of the equation is also called as the linkage disequilibrium so we consider it as D’

    now D’=(1-r)D

    In the absence of selection and in an infinite random mating population, the amount of linkage disequilibrium undergoes exponential decay at a rate equal to the recombination rate between the two loci (Figure 8.2). In other words, the difference between the actual frequency of a haplotype such as A1B1 (a) and the random proportion (p1q1) decreases each generation by a factor equal to the recombination rate between the loci. Over time, any non-random genetic associations will disappear; recombination will
    destroy the association. The higher the rate of recombination, the more rapid the destruction. The highest possible value of r is 1/2, which is true when the two loci are on different chromosomes. Genic associations persist longer for tightly linked loci on the same chromosome, as we would intuitively expect.

    Reference , Mark Ridley Evolution 4th edition

    linkage equilibrium

    linkage disequilibrium

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