Two-Locus Models

Two-Locus Models

The human genome is thought to have about 40,000 genes, Drosophila about 13,000 genes, and even bacteria contain thousands of genes per cell!
Clearly, one-locus models of selection are highly simplified depictions of how populations evolve over time.
Exact analyses with multiple loci are, however, extremely difficult if not impossible to obtain.
Even with only two loci, the dynamics are complicated and not completely understood. Results are limited in scope, focusing on particular fitness schemes.
Nevertheless, results from two-locus models are very important in determining what properties of the one-locus model are unique and might not apply to the real-world situation in which a number of loci collectively interact to guide the formation of the individual.
Consider two loci, A and B, with two alleles each: A₁, A₂ and B₁, B₂, at frequencies p_A₁, p_A₂ and p_B₁, p_B₂.
There are four possible combinations of these alleles on a chromosome:
Chromosome type: A₁ B₁ A₁ B₂ A₂ B₁ A₂ B₂
Frequency: x₁ x₂ x₃ x₄
[Note: x₁ + x₂ + x₃ + x₄ = 1]
There are two important new concepts in the two-locus model: Recombination and linkage disequilibrium.
http://www.zoology.ubc.ca/~otto/Pictures/hr.jpgRecombination

http://www.zoology.ubc.ca/~otto/Pictures/hr.jpg
Recombination occurs during meiosis in sexual organisms to generate gametes carrying new combinations of alleles. For example:
http://www.zoology.ubc.ca/~otto/PopG...e3/Meiosis.jpgWe specify the rate of recombination between two loci by r.
[Note: Recombination may occur in any individual but it only changes the type of offspring produced if the parent was a double heterozygote.]
http://www.zoology.ubc.ca/~otto/Pictures/hr.jpgLinkage Disequilibrium

http://www.zoology.ubc.ca/~otto/Pictures/hr.jpg
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 x₁ x₄ - x₂ x₃.
Positive D implies that the chromosomes A₁ B₁ (x₁) and A₂ B₂ (x₄) are more common than expected.
Negative D implies that the chromosomes A₁ B₂ (x₂) and A₂ B₁ (x₃) are more common than expected.
Linkage disequilibrium also measures the difference between observed and expected chromosome frequencies:
|D| = |OBS - EXP| D = x₁ - p_A₁ p_B₁
- D = x₂ - p_A₁ p_B₂
- D = x₃ - p_A₂ p_B₁
D = x₄ - p_A₂ p_B₂
http://www.zoology.ubc.ca/~otto/Pictures/hr.jpgComparing r and D

http://www.zoology.ubc.ca/~otto/Pictures/hr.jpg
Recombination rate (r) is a measure of the distance between two loci and equals the probability that a gamete contains a chromosomal combination not found in the parents.
Linkage disequilibrium (D) is a measure of whether an allele at one locus tends to be found more often with an allele at another locus.
http://www.zoology.ubc.ca/~otto/PopG...ture3/Fig1.jpghttp://www.zoology.ubc.ca/~otto/Pictures/hr.jpgTwo-Locus Selection Models

http://www.zoology.ubc.ca/~otto/Pictures/hr.jpgWe will consider a diploid life cycle as follows:http://www.zoology.ubc.ca/~otto/PopG...e3/Diploid.jpgwhere the survival of a diploid individual depends on its genotype:
http://www.zoology.ubc.ca/~otto/PopG.../Fitnesses.jpghttp://www.zoology.ubc.ca/~otto/Pictures/hr.jpgTwo-Locus Selection Models

http://www.zoology.ubc.ca/~otto/Pictures/hr.jpgWe census the gamete frequencies at the beginning of a generation (x_i).These gametes unite at random, creating all possible combinations according to their frequencies. Selection then acts on these diploid genotypes producing a new generation of adults. These adults undergo meiosis to produce the next generation of gametes.
These processes are illustrated in the following mating table:
http://www.zoology.ubc.ca/~otto/PopG...re3/Mating.jpghttp://www.zoology.ubc.ca/~otto/Pictures/hr.jpgTwo-Locus Selection Models

http://www.zoology.ubc.ca/~otto/Pictures/hr.jpgTo calculate gamete frequencies in the next generation, multiply the column "Frequency after selection" (= adult frequencies) by the appropriate gamete column and sum down the column.For instance,
http://www.zoology.ubc.ca/~otto/PopG...ture3/Eqn1.jpgwhere we define
http://www.zoology.ubc.ca/~otto/PopG...ture3/Eqn2.jpgas the marginal fitness of allele i (the fitness that the allele experiences averaged over all genetics backgrounds) and where
http://www.zoology.ubc.ca/~otto/PopG...ture3/Eqn3.jpgis the mean fitness of all members of the current population.
http://www.zoology.ubc.ca/~otto/Pictures/hr.jpgTwo-Locus Selection Models

http://www.zoology.ubc.ca/~otto/Pictures/hr.jpgAssuming that fitness depends only on the alleles carried regardless of whether they're on maternal or paternal chromosomes (i.e. w_ij=w_ji and w₁₄=w₂₃), the equations describing evolution in the two-locus diploid model are:http://www.zoology.ubc.ca/~otto/PopG...ture3/Eqn4.jpgNotice that recombination only enters into these equations when multiplied by the linkage disequilibrium D. (Why is recombination irrelevant if D=0?)
We will not analyse this model in detail, but will simply illustrate some interesting results and how they are obtained.
http://www.zoology.ubc.ca/~otto/Pictures/hr.jpgNo Selection

http://www.zoology.ubc.ca/~otto/Pictures/hr.jpgIf all genotypes are equally fit, the recursions become:http://www.zoology.ubc.ca/~otto/PopG...ture3/Eqn5.jpgWhat is p'_A₁ = x'₁+x'₂ ?
http://www.zoology.ubc.ca/~otto/Pictures/RArrow.jpg In the absence of selection, allele frequencies remain constant.
What is D' = x'₁x'₄ - x'₂x'₃ ?
http://www.zoology.ubc.ca/~otto/Pictures/RArrow.jpg Linkage disequilibrium decays at a rate r every generation.
After an amount of time t, the expected amount of disequilibrium is D[t]=(1-r)^t D[0].
Therefore, after enough time has passed, we expect to see little linkage disequilibrium between two neutral (=not selected) loci unless they are very tightly linked.
http://www.zoology.ubc.ca/~otto/Pictures/hr.jpgExample

http://www.zoology.ubc.ca/~otto/Pictures/hr.jpg
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:
http://www.zoology.ubc.ca/~otto/PopG...3/DrosoDis.jpghttp://www.zoology.ubc.ca/~otto/Pictures/RArrow.jpg Alleles at most pairs of loci are not significantly associated.
http://www.zoology.ubc.ca/~otto/Pictures/hr.jpgOne Selected Locus, One Neutral Locus

http://www.zoology.ubc.ca/~otto/Pictures/hr.jpg

Assume that the A locus is unselected (neutral), while the B locus experiences selection with fitnesses for B₁ B₁, B₁ B₂, and B₂ B₂ equal to 1, 1+ h s, and 1+s, respectively. http://www.zoology.ubc.ca/~otto/PopG...Fitnesses2.jpg

http://www.zoology.ubc.ca/~otto/PopG...ure3/Eqn5a.jpgHow does the selectively favored allele, B₂, change over time?
http://www.zoology.ubc.ca/~otto/PopG...ure3/Eqn5b.jpgThis is identical to the recursion for B₂ from a simple one-locus model.
How does the neutral allele, A₂, change over time?
http://www.zoology.ubc.ca/~otto/PopG...ure3/Eqn5c.jpgOnly when D=0 will the frequency of A₂ remain constant over time, as expected from the one-locus model.
Otherwise, the A₂ allele will change in frequency because of its association (linkage disequilibrium) with another selected locus: Genetic Hitchhiking.
http://www.zoology.ubc.ca/~otto/Pictures/RArrow.jpg When s is positive (B₂ favored) and D is positive (A₁B₁ and A₂B₂ are more common than expected), the A₂ allele will increase in frequency over time.
http://www.zoology.ubc.ca/~otto/Pictures/RArrow.jpg When s is positive (B₂ favored) and D is negative (A₁B₂ and A₂B₁ are more common than expected), the A₂ allele will decrease in frequency over time.
If recombination is loose (r > s), disequilibrium decays rapidly and the A₂ allele will change little in frequency.
If recombination is tight (r < s), the extent of hitchhiking can be great.
Implications?
http://www.zoology.ubc.ca/~otto/Pictures/hr.jpgTwo Selected Loci

http://www.zoology.ubc.ca/~otto/Pictures/hr.jpgWhen selection acts on both loci, the equations cannot be analysed completely to predict where the population will be at any future point in time.There are, however, some general and counter-intuitive results.
Result 1
Mean fitness can go down.
For example, Karlin and Carmelli (1975) found that fitness went down for some period of time in most of their simulations of the two-locus model with fitnesses drawn at random:
http://www.zoology.ubc.ca/~otto/PopG...3/Fitness1.jpghttp://www.zoology.ubc.ca/~otto/PopG...3/Fitness2.jpg
http://www.zoology.ubc.ca/~otto/PopG...3/Fitness3.jpg
http://www.zoology.ubc.ca/~otto/PopG...3/Fitness4.jpg
Result 2
Linkage disequilibrium does not always decay to zero but can be maintained indefinitely within a population.
In fact, very special conditions are required for linkage equilibrium (D=0) to be maintained when both loci are polymorphic and under selection.
Result 3
Continuous cycling can occur (Hastings 1981): "apparent cases of directional selection may be due to stable cycling".
Result 4
Even when the double heterozygote has the highest fitness, there may be no stable equilibrium maintaining polymorphisms at both loci!
Claim 1
If there is pure directional selection favoring a genotype, say A₂B₂ (such that w_i,j+1 > w_i,j and w_i+1,j > w_i,j), it has been claimed but not proven that selection will lead eventually to the fixation of A₂B₂.
Nordborg (pers. comm.) ran simulations with millions of fitness sets, which all confirmed this claim, but it has not been shown analytically.
More results are known for special cases of the two-locus model:
Special Case 1: r=0
When r=0, the system is dynamically the same as a one-locus four allele model.
http://www.zoology.ubc.ca/~otto/Pictures/RArrow.jpg Mean fitness never decreases.
Special Case 2: Additive fitnesses
If the fitnesses of the A and B loci add together to give two-locus fitnesses, then:

There is only one equilibrium with both loci polymorphc for r>0, which has x₁ = p_A₁ p_B₁, x₂ = p_A₁ p_B₂, x₃ = p_A₂ p_B₁, and x₄ = p_A₂ p_B₂, where p_A₁ and p_B₁ equal the allele frequencies found at equilibrium in the one-locus model with heterozygote advantage.
This "internal" equilibrium is globally stable if there is heterozygote advantage at both loci.
D=0 at equilibrium.
Mean fitness never decreases (Ewens 1969).
Away from equilibrium, selection WILL generate linkage disequilibrium within a population that is initially in linkage equilibrium.

Special Case 3: Multiplicative fitnesses
If the fitnesses of the A and B loci multiply together to give two-locus fitnesses, then:

The same internal polymorphism exists with x₁ = p_A₁ p_B₁, x₂ = p_A₁ p_B₂, x₃ = p_A₂ p_B₁, and x₄ = p_A₂ p_B₂.
This "internal" equilibrium is only stable if there is heterozygote advantage at both loci AND if r is large enough.
For tight linkage, the population goes to one of two stable equilibria at which D ≠ 0http://www.zoology.ubc.ca/~otto/PopG...ure3/DisEq.jpg
Disequilibrium may or may not be present at equilibrium.
Away from equilibrium, selection WILL NOT generate linkage disequilibrium within a population that is initially in linkage equilibrium.
Mean fitness can decrease over time (Ewens 1969).
At equilibrium, mean fitness is highest when r=0, at which point the population beomes fixed on only two chromosomes (A₁B₁ and A₂B₂ OR A₂B₁ and A₁B₂). Recombination generates whichever chromosomes are missing and reduces the mean fitness.

[EXERCISE: Let the fitnesses at both loci A and B be: 0.8 for homozygotes and 1.0 for the heterozygote. Let the two-locus fitnesses be multiplicative. Calculate the mean fitness when only chromosomes A₁B₁ and A₂B₂ are present (at frequency 0.5 each because the homozygotes all have equal fitness). Compare this to the mean fitness when x₁ = p_A₁ p_B₁, x₂ = p_A₁ p_B₂, x₃ = p_A₂ p_B₁, and x₄ = p_A₂ p_B₂, where p_A₁ and p_B₁ equal the allele frequencies found at equilibrium in the one-locus model with heterozygote advantage.]IMPORTANT CONCEPTS TO REMEMBER:

Linkage disequilibrium measures associations among alleles at different loci.
Linkage disequilibrium decays over time in the absence of selection.
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 neutral loci to change in frequency (hitchhiking).
Mean fitness need not increase.

SOURCES:

Summary of two-locus results: Karlin (1975) TPB, 7:364-398.
Cycling in two-locus model: Hastings (1981) PNAS 78(11):7224-5.
Elegant proof that mean fitness does not decrease in additive model: Ewens (1969) Nature 221: 1076.
Mean fitness can decrease under non-additive fitness regimes: Karlin and Carmelli (1975) TPB, 7:399-421.
Disequilibrium at equilibrium in multiplicative model: Hastings (1981) JTB 89:69-81.