Modelling Effects at Multiple Loci

Modelling Effects at Multiple Loci I. The Complex Phenotype - traits are often influenced by more than one locus, and their effects are not necessarily independent. In other words, there are often epistatic interactions between loci...the value of an allele may depend on the genotype at other loci

Modelling Effects at Multiple Loci I. The Complex Phenotype - traits are often influenced by more than one locus, and their effects are not necessarily independent. In other words, there are often epistatic interactions between loci...the value of an allele may depend on the genotype at other loci - For example, suppose 'A' and 'B' each contribute a unit of growth, and suppose there is selection for intermediate size (2 units of growth, total).

Modelling Effects at Multiple Loci I. The Complex Phenotype - traits are often influenced by more than one locus, and their effects are not necessarily independent. In other words, there are often epistatic interactions between loci ...the value of an allele may depend on the genotype at other loci - For example, suppose 'A' and 'B' each contribute a unit of growth, and suppose there is selection for intermediate size. There will be three adaptive genotypes: AABB = 4 units AABb = 3 units AAbb = 2 units AaBB = 3 units AaBb = 2 units Aabb = 1 unit aaBB = 2 units aaBb = 1 unit aabb = 0 units

Modelling Effects at Multiple Loci I. The Complex Phenotype However, a population full of heterozygotes is impossible, so the population will either move to "fixation" at AAbb OR aaBB. AAbb = 2 units AaBb = 2 units aaBB = 2 units

Modelling Effects at Multiple Loci I. The Complex Phenotype However, a population full of heterozygotes is impossible, sothe population will either move to "fixation" at AAbb OR aaBB. AAbb = 2 units AaBb = 2 units aaBB = 2 units So, whether ‘A’ or ‘a’ will be selected for (an increase in frequency) depends on the frequency of ‘B’. If ‘B’ is at low frequency, ‘A’ will be selected for. If ‘B’ is at high frequency, ‘a’ will be selected for.

Modelling Effects at Multiple Loci I. The Complex Phenotype II. "Linkage" Equilibrium

Modelling Effects at Multiple Loci I. The Complex Phenotype II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model - Suppose: f(A) = p1 and f(a) = q1 f(B) = p2 and f(b) = q2

Modelling Effects at Multiple Loci I. The Complex Phenotype II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model - Suppose: f(A) = p1 and f(a) = q1 f(B) = p2 and f(b) = q2 - Expected frequency of AB haplotype (gamete) = a = p1* p2

Modelling Effects at Multiple Loci I. The Complex Phenotype II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model - Suppose: f(A) = p1 and f(a) = q1 f(B) = p2 and f(b) = q2 - Expected frequency of AB haplotype (gamete) = a = p1*p2 f(Ab) = b = p1q2 f(aB) = c = q1p2 f(ab) = d = q2q2

Modelling Effects at Multiple Loci I. The Complex Phenotype II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model - Suppose: f(A) = p1 and f(a) = q1 f(B) = p2 and f(b) = q2 - Expected frequency of AB haplotype (gamete) = a = p1*p2 f(Ab) = b = p1q2 f(aB) = c = q1p2 f(ab) = d = q2q2 - In essence, this becomes the equilibrium model for two loci; when a population reaches linkage equilibrium, the frequency of haplotypes will be equal to the product of the independent gene frequencies….

Modelling Effects at Multiple Loci I. The Complex Phenotype II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model Why wouldn’t this occur? Suppose you did a test cross: AaBb x aabb Haplotypes produced by the AaBb parent should occur at frequencies of: AB = ¼ Ab = ¼ aB = ¼ ab = ¼

Modelling Effects at Multiple Loci I. The Complex Phenotype II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model Why wouldn’t this occur? Suppose you did a test cross: AaBb x aabb Haplotypes should occur at frequencies of: AB = ¼ Ab = ¼ aB = ¼ ab = ¼ Why could you get: AB = 50% Ab = 0 aB = 0 ab = 50%

LINKAGE Modelling Effects at Multiple Loci I. The Complex Phenotype II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model Why wouldn’t this occur? Suppose you did a test cross: AaBb x aabb Haplotypes should occur at frequencies of: AB = ¼ Ab = ¼ aB = ¼ ab = ¼ Why could you get: AB = 50% Ab = 0 aB = 0 ab = 50% LINKAGE

Modelling Effects at Multiple Loci I. The Complex Phenotype II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model - In essence, this becomes our equilibrium model for two loci. BUT: TWO LOCI, EVEN IF THEY ARE CLOSELY LINKED, WILL CROSS-OVER. OVER TIME, EVEN WITH CLOSELY LINKED GENES, THE FREQUENCIES OF HAPLOTYPES WILL REACH THE ‘EQUILIBRIUM’ VALUES (ACTING LIKE THEY ASSORT INDEPENDENTLY) IF PANMIXIA OCCURS.

Modelling Effects at Multiple Loci I. The Complex Phenotype II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model - In essence, this becomes our equilibrium model for two loci. BUT: TWO LOCI, EVEN IF THEY ARE CLOSELY LINKED, WILL CROSS-OVER. OVER TIME, EVEN WITH CLOSELY LINKED GENES, THE FREQUENCIES OF HAPLOTYPES WILL REACH THE ‘EQUILIBRIUM’ VALUES (ACTING LIKE THEY ASSORT INDEPENDENTLY) IF PANMIXIA OCCURS. SO: Recombination drives a population towards Linkage Equilibrium, at a rate dependent upon the distance between genes. Closely linked genes will take longer to equilibrate, all things being equal, than distantly linked or independently assorting genes.

Modelling Effects at Multiple Loci I. The Complex Phenotype II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium

Modelling Effects at Multiple Loci I. The Complex Phenotype II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium - Linkage DISequilibrium is measured as "D" D = ad - bc. If D = 0, the pop is in equilibrium. If D is not zero, then D becomes a measure of deviation from the equilibrium. Suppose F(A) = F(B) = 0.5 Expected Observed F(AB) = a = 0.25 F(AB) = a = 0.5 F(Ab) = b = 0.25 F(Ab) = b = 0.0 F(aB) = c = 0.25 F(aB) = c = 0.0 F(ab) = d = 0.25 F(ab) = d = 0.5 D = (0.25)*(0.25) – (0.25)*(0.25) = 0 D = (0.5)*(0.5) – (0.0)*(0.0) = 0.25

II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium

II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium - Linkage!! (without sufficient time to equilibrate haplotype frequencies)

II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium - Linkage!!! without sufficient time to equilibrate haplotype frequencies. - Selection!!! Some haplotypes occur more frequently than random because these combos confer a selective advantage.

II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium - Linkage!!! without sufficient time to equilibrate haplotype frequencies. - Selection!!! Some haplotypes occur more frequently than random because these combos confer a selective advantage. Curiously, selection will only create linkage disequilibrium if the alleles act "epistatically".

II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium Papilio memnon – females mimic different Toxic model species. TOXIC MODELS females male

II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium Papilio memnon – females mimic different Toxic model species. In the simplest terms for the hind wing: T (swallowTAIL) > t (no tail) and C (open pattern) > c (solid) pattern TOXIC MODELS females MODELS are either Tailed-Open or Solid-Tailless male

II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium Papilio memnon – females mimic different Toxic model species. In the simplest terms for the hind wing: T (swallowTAIL) > t (no tail) and C (open pattern) > c (solid) pattern TOXIC MODELS females MODELS are either Tailed-Open or Solid-Tailless So, fitness at T locus depends on the fitness at the C locus. If C is advantageous (tailed model), then T is, too (‘open’). If c is advantageous (untailed model), then T is NOT. So, fitness at T depends on fitness at C. male

II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium Papilio memnon – females mimic different Toxic model species. In the simplest terms for the hind wing: T (swallowTAIL) > t (no tail) and C (open pattern) > c (solid) pattern TOXIC MODELS females MODELS are either Tailed-Open or Solid-Tailless So, TC haplotype or tc haplotype will dominate in a constant disequilibrium. male

Papilio memnon (palatable) females ttcc TTCC Toxic Models On Left Papilio memnon male

D = ad - bc = .25 - 0 = 0.25.... not in equilibrium In the simplest terms: T (swallowTAIL) > t (no tail) and C >c, where CC, Cc is coloration for a tailed model and cc is coloration for a tailless model. So, fitness at t locus depends on the fitness at the C locus. If CC is advantageous, then TT is, too. If cc is advantageous, then TT is NOT. So, fitness at T depends on fitness at C. So, TC haplotype and tc haplotypes will dominate in a constant disequilibrium. f(T) = p1 = .5 f(t) = q1 = .5 f(C) = p2 = .5 f(c) = q2 = .5 Expected Observed a = p1*p2 .25 .5 b = p1*q2 c = q1*p2 d = q1*q2 D = ad - bc = .25 - 0 = 0.25.... not in equilibrium

II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium - Linkage!!! without sufficient time to equilibrate haplotype frequencies. - Selection: Some haplotypes occur more frequently than random because these combos confer a selective advantage. Curiously, selection will only create linkage disequilibrium if the alleles act "epistatically". If allelic effects are multiplicative or additive (like the size example), then the system will proceed to equilibrium.

II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium - Linkage!!! without sufficient time to equilibrate haplotype frequencies. - Selection: Some haplotypes occur more frequently than random because these combos confer a selective advantage. Curiously, selection will only create linkage disequilibrium if the alleles act "epistatically". If allelic effects are multiplicative or additive (like the size example), then the system will proceed to equilibrium. In the case of size, it goes to fixation for AAbb or aaBB, each of which is ad - bc = 0 = linkage equilibrium. So selection, in this case, does NOT create a disequilibrium.

D = ad - bc = 0 - 0 = 0.... in equilibrium In the case of size, ii goes to fixation for AAbb or aaBB, and ad - bc = 0 = linkage equilibrium. So selection, in this case, does NOT create a disequilibrium. For example, if the population fixes at AAbb: f(A) = p1 = 1.0 f(a) = q1 = 0.0 f(B) = p2 = 0.0 f(b) = q2 = 1.0 Expected Observed a = p1*p2 0.0 b = p1*q2 1.0 c = q1*p2 d = q1*q2 D = ad - bc = 0 - 0 = 0.... in equilibrium

II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium - Linkage!!! without sufficient time to equilibrate haplotype frequencies. - Selection: Some haplotypes occur more frequently than random because these combos confer a selective advantage. - Drift: Random fluctuation in the relative frequencies of haplotypes can cause disequilibrium.

II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium - Linkage!!! without sufficient time to equilibrate haplotype frequencies. - Selection: Some haplotypes occur more frequently than random because these combos confer a selective advantage. - Drift: Random fluctuation in the relative frequencies of haplotypes can cause disequilibrium. - Non-random mating: If A individuals preferentially mate with B individuals, then AB will be more frequent than expected by chance.

II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium 4. Effects

II. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium 4. Effects Hitchhiking - selection at one locus can drive changes in linked genes before recombination eliminates disequilibrium. This can work to prevent the acquisition of advantageous alleles at a locus if it is closely linked to a deleterious allele at a second locus. However, this effect declines as the distance between genes (and thus the rate of recombination) increases.

B. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium 4. Effects 5. The Utility of Linkage Disequilibrium

B. "Linkage" Equilibrium 1. A two-Locus Equilibrium Model 2. Deviations from "Linkage" Equilibrium 3. Causes of Linkage Disequilibrium 4. Effects 5. The Utility of Linkage Disequilibrium - By examining whether an allele is in disequ. with a neutral marker, we can draw inferences about its selective value. For example, if an allele is in disequ, we can infer it is relatively new (or recombination would have reduced the disequ), or it is probably of selective value.

OK!!!