Two-locus models. Two-locus models. Two-locus models. Two-locus models. Consider two loci, A and B, each with two alleles:

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1 The human genome has ~30,000 genes. Drosophila contains ~10,000 genes. Bacteria contain thousands of genes. Even viruses contain dozens of genes. Clearly, one-locus models are oversimplifications. Unfortunately, the math gets quite complicated even with only two loci. Nevertheless, two-locus model helps us determine what properties of the one-locus model are unique and might not apply to the real world. Consider two loci, A and B, each with two alleles: Locus Allele Frequency A B A 1 p A1 A 2 p A2 p B1 p B2 If A and B are on the same chromosome, there are four possible chromosome types: Chromosome Frequency (1) A 1 x 1 There are two important new concepts in the twolocus model: recombination and linkage disequilibrium (2) A 1 x 2 (3) A 2 (4) A 2

2 Recombination Example: Recombination occurs during meiosis in sexual organisms to generate gametes carrying new combinations of alleles. We specify the rate of recombination between two loci by r. (Recombination may occur in any individual but it only changes the type of offspring produced if the parent is a double heterozygote) Linkage disequilibrium Linkage disequilibrium, on the other hand, measures whether an allele at one locus is associated (or correlated) with an allele at a second locus. Linkage disequilibrium, D, is measured by x 1 - x 2. Positive D (x 1 > x 2 ) implies that chromosomes A 1 (x 1 ) and A 2 ( ) are more common than expected. Negative D (x 1 < x 2 ) implies that chromosomes A 1 (x 2 ) and A 2 ( ) are more common than expected. Linkage disequilibrium In a randomly mating population, linkage disequilibrium measures the difference between observed and expected chromosome frequencies D = x 1 p A1 p B1 D = x 2 p A1 p B2 D = p A2 p B1 D = p A2 p B2

3 Linkage disequilibrium: Example Among 15 diploid individuals (30 chromosomes): 10 solid square - solid circle x l = 10/30 7 solid square - hollow circle x 2 = 7/30 5 hollow square - solid circle = 5/30 8 hollow square - hollow circle = 8/30 What is D? x l - x 2 = 0.05 Which alleles are found together more frequently than expected? Relationship between r and D Recombination (r) is a measure of distance and equals the probability that a gamete contains a chromosomal combination not present in the parents. Linkage disequilibrium (D) is a measure of association and describes which chromosomal combinations are disproportionately common. Model with no selection If chromosomes are equally fit, chromosome frequencies change from one generation to the next according to: x 1 = x 1 rd x 2 = x 2 + rd x 3 = + rd x 4 = rd p A1 = x 1 + x 2 = x 1 + x 2 = p A1 Point one: In the absence of selection, allele frequencies remain constant, but not chromosome frequencies. } Model with no selection If all chromosomes are equally fit, genotype frequencies change from one generation to the next according to: x 1 = x 1 rd x 2 = x 2 + rd x 3 = + rd x 4 = rd D = (1 r)d Point two: Linkage disequilibrium decays at rate r every generation.

4 Model with no selection After one generation, linkage disequilibrium becomes: D = (1 r)d After t generations, linkage disequilibrium will be: D[t] = (1 r) t D[0] After enough time has passed, there should be very little linkage disequilibrium. (Unlike Hardy-Weinberg, linkage equilibrium (D=0) is not achieved in a single generation) Model with no selection: example Linkage disequilibrium was measured between several pairs of loci in Drosophila melanogaster. The statistical evidence for a non-zero D value is here plotted as a function of distance between the pair of loci Model with no selection: example Model with selection: Fitness Alleles at most pairs of loci are not significantly associated. The pairs of loci that do have significant associations tend to be close to each other on the chromosome (low r). Assume fitness of a genotype doesn t depend on which chromosome comes from which parent (e.g., w 14 = w 41 )

5 Recursion equations Two neutral loci (no selection) Disequilibrium decays at a rate r every generation. Allele frequencies do not change over time. Chromosome frequencies do change as the disequilibrium decays. Selected locus (A) evolves as in the one-locus selection model. Allele frequencies at neutral locus (B) do not stay constant if there is linkage disequilibrium! Genetic hitchhiking Green/yellow polymorphism at a neutral locus (squares)

6 Is there any disequilibrium at this point? A beneficial red mutant appears at a nearby gene (circles) Why didn t the yellow allele fix? Is there any disequilibrium at this point? As the red allele rises in frequency over time, the yellow allele hitchhikes along because of the initial disequilibrium.

7 Genetic hitchhiking: Allele frequencies change at a neutral locus because of an association (D) with a selected locus. If A1 is favorable and D is positive, B1 increases in frequency. If A1 is favorable and D is negative, B1 decreases in frequency. The extent of hitchhiking is large only when D is large and r is small (association decays slowly). Two selected loci (Not fully analysed) It is conjectured that with purely directional selection the favorable genotype rises to fixation (NOT PROVEN). With heterozygous advantage, polymorphic equilibria can occur -- linkage disequilibrium may or may not be present at equilibrium. Without directional selection, chromosome frequencies can cycle under some fitness schemes. Two selected loci (Not fully analysed) Heterozygote advantage at both loci: If fitnesses at the two loci ADD together to determine fitness (additive model), the one-locus equilibrium predicts the two-locus equilibrium. If fitnesses at the two loci MULTIPLY together to determine fitness (multiplicative model), the one-locus equilibrium predicts the two-locus equilibrium ONLY when r is large. When r is small, the population approaches a polymorphic equilibrium where linkage disequilibrium is maintained indefinitely. Mean fitness can decline over time! (Karlin & Carmelli 1975)

8 Important concepts to remember Important concepts to remember Recombination (r) measures the distance between loci in terms of the probability of producing non-parental chromosome combinations. Linkage disequilibrium (D = x 1 - x 2 ) measures associations among alleles at different loci. In the absence of selection, linkage disequilibrium decays over time at rate r. With selection, linkage disequilibrium may be generated and maintained by selection even at equilibrium. Linkage disequilibrium between a selected and a neutral locus can cause alleles at the neutral locus to change in frequency (hitchhiking). Mean fitness need not increase.