Population and Evolutionary Genetics

Evolution, at its most basic, is a change in allele frequencies in a population over time. Population genetics measures those frequencies, predicts what happens to them when nothing disturbs them, and then calculates how mutation, migration, chance and natural selection push them around. The same few equations let you estimate how many people carry a recessive disease allele, why small populations lose diversity, how a harmful allele can persist for generations, and how new species arise.

Section 1Allele and genotype frequencies

Populations and gene pools

Transmission genetics follows alleles through individual families. Population genetics steps back and asks about all the alleles in a group of interbreeding individuals. A population in this sense is a group of individuals of the same species that live in the same area and can mate with one another, sometimes called a Mendelian population. The complete set of alleles carried by all its members at a given time is its gene pool. For a diploid population of N individuals, the gene pool at an autosomal locus contains 2N alleles.

Two kinds of frequencies describe a gene pool, and the distinction is fundamental:

• A genotype frequency is the proportion of individuals in the population with a particular genotype. For a locus with alleles A and a, the three genotype frequencies are often written f(AA), f(Aa) and f(aa), or P, H and Q. They must add to 1.
• An allele frequency (sometimes called a gene frequency) is the proportion of all copies of the gene in the population that are a particular allele. By convention, p is the frequency of A and q the frequency of a, and p + q = 1.

Genotype frequencies are what we observe when we classify individuals; allele frequencies are what is transmitted from one generation to the next, because individuals pass on alleles, not genotypes. Evolution, in the population genetic definition, is a change in allele frequencies across generations.

Calculating allele frequencies from counts

There are two equivalent ways to calculate allele frequencies at an autosomal locus with codominant alleles, where every genotype can be recognized.

The counting method. Each homozygote carries two copies of its allele and each heterozygote carries one copy of each. Count all copies and divide by the total number of alleles, 2N:

p = (2 × number of AA + number of Aa) / 2N

The genotype frequency method. Half of a heterozygote's alleles are A, so

p = f(AA) + ½ f(Aa), and q = f(aa) + ½ f(Aa).

Both methods give the same answer, and you can use whichever is more convenient. Both require that you can identify heterozygotes. When one allele is completely dominant, AA and Aa look alike, and you cannot count alleles directly; you need the Hardy Weinberg relationships of Section 2 to estimate them.

Multiple alleles

The same logic extends to loci with more than two alleles. With three alleles A¹, A² and A³ there are six genotypes: three homozygotes and three heterozygotes. The frequency of A¹ is the frequency of A¹A¹ plus half the frequency of each heterozygote containing A¹:

p₁ = f(A¹A¹) + ½f(A¹A²) + ½f(A¹A³)

In general, a locus with k alleles has k(k + 1)/2 genotypes, of which k are homozygous and k(k − 1)/2 are heterozygous. A locus with 10 alleles therefore has 55 genotypes. This is why highly polymorphic loci, such as forensic short tandem repeats and HLA genes, distinguish individuals so well.

X-linked loci

For genes on the X chromosome, females (XX) carry two copies and males (XY) carry one. Male genotype frequencies equal allele frequencies directly, because each male has exactly one allele. To find the population allele frequency, count every X chromosome: two per female and one per male.

Measuring genetic variation

Allele frequencies are the raw data for describing how much variation a population carries. Population geneticists commonly use three summary measures:

Note one important point: allele frequencies alone do not determine genotype frequencies. Two populations can both have p = 0.5, one with 50 percent AA and 50 percent aa (no heterozygotes), and another with 25 percent AA, 50 percent Aa and 25 percent aa. Something must connect allele frequencies to genotype frequencies, and under random mating that connection is the Hardy Weinberg principle.

Section 2Hardy Weinberg equilibrium

The principle

The mathematician G. H. Hardy and the physician Wilhelm Weinberg independently showed that dominant alleles do not automatically spread. Their result, the Hardy Weinberg principle, states that in a large, randomly mating population with no mutation, migration or selection:

1. Allele frequencies do not change from generation to generation.
2. After one generation of random mating, genotype frequencies are given by p² (AA), 2pq (Aa) and q² (aa), and they stay at those values.

A population with these genotype frequencies is in Hardy Weinberg equilibrium. Dominance has nothing to do with it: a dominant allele at a frequency of 0.01 stays at 0.01. Dominance affects which phenotype a genotype shows, not how often alleles are transmitted.

Why it works

Random mating of individuals is equivalent to random union of gametes drawn from the gene pool. A fraction p of all gametes carries A and a fraction q carries a. The probability that an egg and a sperm both carry A is p × p = p²; both a, q²; and one of each, pq + qp = 2pq (an A egg with an a sperm or an a egg with an A sperm). These terms are the expansion of (p + q)² = p² + 2pq + q² = 1.

Now check that allele frequencies are unchanged. In the offspring, the frequency of A is f(AA) + ½f(Aa) = p² + ½(2pq) = p² + pq = p(p + q) = p. Allele frequencies are preserved, so the same genotype frequencies are produced in every later generation. Meiosis and fertilization, by themselves, do not change the gene pool.

The assumptions

The principle holds exactly only under idealized conditions. Each assumption corresponds to a force that can change the population, and much of the rest of this chapter is about those forces.

The surprising lesson is how robust the principle is in practice. Many loci in large human populations fit Hardy Weinberg proportions closely, because mutation and migration are weak over a single generation, and selection on most variants is weak. Even when a population departs from equilibrium, one generation of random mating restores Hardy Weinberg proportions for an autosomal locus, provided allele frequencies are the same in both sexes. For X-linked loci the approach is gradual: if male and female allele frequencies differ, the difference is halved and changes sign each generation, so the population oscillates toward equilibrium.

Uses of the Hardy Weinberg equation

The most common practical use is estimating allele and carrier frequencies for a recessive trait, where heterozygotes cannot be recognized. If the population is assumed to be in equilibrium, the frequency of affected individuals equals q², so q = √(frequency of affected). The carrier frequency is then 2pq. When q is small, p is close to 1 and 2pq ≈ 2q.

That last observation follows from the ratio of heterozygotes to recessive homozygotes, 2pq/q² = 2p/q. When q = 0.02, the ratio is 2(0.98)/0.02 = 98. This has an important consequence for selection: eliminating affected individuals barely changes the frequency of a rare recessive allele, because almost all copies are in carriers who are not selected against.

For X-linked recessive traits, the frequency of affected males equals q, and the frequency of affected females is q². About 8 percent of males of northern European ancestry have red green color vision deficiency, so q ≈ 0.08, the expected frequency of affected females is 0.08² = 0.0064 (about 0.6 percent), and carrier females are 2(0.92)(0.08) = 0.147, about 15 percent. This is why X-linked recessive conditions are much more common in males: a male needs one copy, a female two.

Multiple alleles: the ABO system

With three alleles at frequencies p, q and r (p + q + r = 1), Hardy Weinberg genotype frequencies come from (p + q + r)²: p², q², r², 2pq, 2pr and 2qr. For ABO blood groups, with I^A (p), I^B (q) and i (r), the phenotypes are type A = p² + 2pr, type B = q² + 2qr, type AB = 2pq and type O = r². Because type O is homozygous recessive, r = √f(O) is the natural starting point. For example, if 49 percent of a population is type O and 32 percent type A, then r = 0.7; solving p² + 2(0.7)p = 0.32 gives p = 0.2; and q = 1 − 0.7 − 0.2 = 0.1. The model then predicts type B = 0.01 + 0.14 = 0.15 and type AB = 0.04, which sum with the others to 1.00.

Testing for equilibrium with chi square

To ask whether a population fits Hardy Weinberg proportions, calculate allele frequencies from the observed data, use them to calculate expected genotype numbers, and compare observed and expected with a chi square test: χ² = Σ(O − E)²/E. The degrees of freedom are the number of genotype classes minus 1, minus the number of independent allele frequencies estimated from the data. For two alleles and three genotypes, df = 3 − 1 − 1 = 1, not 2. This is one of the most common errors on population genetics problems. With df = 1, the critical value at P = 0.05 is 3.84.

Section 3Inbreeding and nonrandom mating

Kinds of nonrandom mating

Random mating means that the chance two individuals mate does not depend on their genotypes or their relationship. Real populations often break this rule in one of three ways:

• Inbreeding is mating between relatives, more often than expected by chance. It affects every locus in the genome, because relatives share alleles everywhere.
• Positive assortative mating is mating between individuals with similar phenotypes. Humans mate assortatively for height, educational attainment and, notably, deafness and blindness. It affects only the loci that influence the trait in question (and loci linked to them).
• Negative assortative mating (disassortative mating) is mating between individuals with dissimilar phenotypes. Self incompatibility systems in plants enforce it, and in some animals, mate choice favors partners with different immune (MHC) genotypes.

The central point about all three is that nonrandom mating changes genotype frequencies but not allele frequencies. Inbreeding and positive assortative mating increase homozygosity and decrease heterozygosity; negative assortative mating does the reverse. On its own, nonrandom mating is therefore not an evolutionary force in the strict sense, but by exposing recessive alleles in homozygotes it changes how selection acts.

The inbreeding coefficient

Two alleles at a locus can be identical in two different senses. They can be identical by state, meaning they have the same DNA sequence, or identical by descent (IBD), meaning they are both copies of a single allele in a common ancestor. The inbreeding coefficient (F) is the probability that the two alleles at a locus in an individual are identical by descent. An individual whose two alleles are identical by descent is called autozygous. F ranges from 0 (no inbreeding beyond the reference population) to 1 (complete inbreeding, as in a fully inbred line).

With inbreeding coefficient F, genotype frequencies in a population become:

• f(AA) = p² + Fpq
• f(Aa) = 2pq(1 − F)
• f(aa) = q² + Fpq

Heterozygotes are reduced by the fraction F, and the missing heterozygotes are added equally to the two homozygote classes. Allele frequencies stay the same: f(AA) + ½f(Aa) = p² + Fpq + pq − Fpq = p. Rearranging the heterozygote equation gives a way to estimate F from data: F = 1 − H_o/H_e, where H_o is the observed heterozygosity and H_e = 2pq is the heterozygosity expected under random mating.

Calculating F from a pedigree

F for an individual can be calculated by path counting. Find every common ancestor of the individual's two parents. For each common ancestor, trace the path that runs from one parent up to the common ancestor and back down to the other parent, and count n, the number of individuals in that path including both parents and the common ancestor (but not the inbred individual). Each step in a path is a 1/2 chance of passing on a particular allele, and the common ancestor's two alleles are themselves identical by descent with probability F_A. Then

F = Σ (1/2)ⁿ (1 + F_A)

summed over all paths. If the common ancestor is not itself inbred, F_A = 0 and each path contributes (1/2)ⁿ.

Repeated inbreeding and inbred lines

Continued inbreeding pushes F toward 1. Self fertilization is the extreme case: each generation of selfing halves the remaining heterozygosity, so after t generations H_t = H₀(1/2)^t and F = 1 − (1/2)^t. Starting from a fully heterozygous plant, five generations of selfing leave only (1/2)⁵ = 1/32, about 3 percent, of the original heterozygosity (F ≈ 0.97). Repeated full sibling mating approaches homozygosity more slowly, reaching F ≈ 0.99 after about 20 generations. This is how inbred strains of mice, rats and many crop lines are created, and it is why every member of such a strain is essentially genetically identical, a property exploited in Section 3 of Chapter 11 to estimate environmental variance.

Inbreeding depression and hybrid vigor

Inbreeding depression is the decline in fitness and in trait values such as growth, fertility and survival that follows inbreeding. It has two main genetic explanations. The more important is the dominance hypothesis: populations carry many rare, partially recessive deleterious alleles, mostly hidden in heterozygotes; inbreeding makes them homozygous and exposes them. The overdominance hypothesis proposes that heterozygotes at some loci are intrinsically superior, so losing heterozygosity lowers fitness. The reverse effect, heterosis or hybrid vigor, appears when two different inbred lines are crossed: the F1 is heterozygous at many loci, masking each line's recessive defects, and is often more vigorous than either parent. Hybrid corn, which dominates modern corn production, exploits heterosis, and farmers buy new hybrid seed every year because the F2 loses much of the advantage as segregation restores homozygosity.

Section 4Mutation, migration, drift and selection

Four processes change allele frequencies: mutation, migration, genetic drift and natural selection. They differ in speed, in direction and in whether they increase or decrease variation. The table below gives the overview; the rest of the section works through each force.

Mutation

Mutation is the ultimate source of all new alleles, but as a force changing allele frequencies it is extremely weak. Let A mutate to a at rate u per generation (forward mutation) and a mutate back to A at rate v. The change in the frequency of A in one generation is

Δp = vq − up

At equilibrium Δp = 0, so up = vq, which gives q̂ = u/(u + v) and p̂ = v/(u + v). Typical mutation rates are around 10⁻⁵ to 10⁻⁶ per gene per generation, so the approach is glacial. With forward mutation alone at u = 10⁻⁵, it takes about ln 2/u ≈ 0.693/10⁻⁵ ≈ 69,000 generations to halve p. Mutation supplies variation; other forces determine what happens to it.

Migration

Migration, or gene flow, is the movement of alleles between populations when individuals or gametes (such as pollen) move and reproduce. In the simplest model, a fraction m of a recipient population each generation consists of migrants from a source population with allele frequency q_m. The new frequency is a weighted average:

q' = (1 − m)q + m q_m, so Δq = m(q_m − q)

Migration changes allele frequencies fastest when the populations differ most, and it makes populations more similar. Over t generations the difference shrinks geometrically: (q_t − q_m) = (q₀ − q_m)(1 − m)^t. Even a modest amount of migration, on the order of one migrant per generation, is enough to prevent drift from making populations diverge completely at neutral loci.

Genetic drift

Genetic drift is random change in allele frequencies caused by sampling error. Each generation, the gametes that form the next generation are a finite sample of the parents' gene pool, and like any sample they rarely match it exactly. Flip a coin 10 times and you often get 7 heads; flip it 10,000 times and you get very close to half. In the same way, drift is strong in small populations and weak in large ones. The variance of the change in allele frequency in one generation is pq/2N.

Drift has four key consequences:

• Allele frequencies wander randomly, with no tendency to return to their starting point.
• Eventually an allele is either fixed (frequency 1) or lost (frequency 0). For a neutral allele, the probability of fixation equals its current frequency. A new mutation, present as one copy among 2N, is fixed with probability 1/(2N).
• Heterozygosity declines by a fraction 1/(2N) each generation: H_t = H₀(1 − 1/2N)^t.
• Replicate populations diverge from one another, each drifting in its own direction.

Effective population size

The N in drift equations is not simply a census count. It is the effective population size (N_e), the size of an idealized population that would drift at the same rate as the real one. N_e is usually much smaller than the census size for three reasons. Unequal numbers of breeding males and females: N_e = 4N_mN_f/(N_m + N_f). Fluctuating population size: N_e over several generations is the harmonic mean of the sizes, which is dominated by the smallest generation. And variation in family size, since a few parents may leave most of the offspring.

Founder effects and bottlenecks

Two special cases of drift have great practical importance. The founder effect occurs when a new population is started by a small number of individuals, whose alleles are a chance sample of the source population. A bottleneck occurs when an existing population is drastically reduced in size for one or more generations. In both cases, rare alleles are often lost, others rise by chance to high frequency, and the resulting population can differ markedly from its ancestors. Many human populations show founder effects: Ellis van Creveld syndrome (a form of dwarfism with extra fingers) is unusually common among the Old Order Amish of Pennsylvania, traced to a single founding couple, and several recessive disorders, such as Tay Sachs disease, are more frequent among Ashkenazi Jewish populations, partly because of historic bottlenecks. Northern elephant seals, hunted to perhaps 20 to 30 animals, have recovered to large numbers but retain extremely low genetic diversity.

Natural selection

Natural selection occurs when genotypes differ in their contribution to the next generation. That contribution, combining survival and reproduction, is fitness. In population genetics we use relative fitness (w), with the best genotype usually set to 1. The selection coefficient s = 1 − w measures how strongly a genotype is selected against. A lethal genotype that never reproduces has w = 0 and s = 1.

The general method for any selection problem is always the same, and it is worth memorizing as a procedure:

1. Write the Hardy Weinberg genotype frequencies before selection: p², 2pq, q².
2. Multiply each by its relative fitness.
3. Add the products to get the mean fitness, w̄.
4. Divide each product by w̄ to get genotype frequencies after selection.
5. Calculate the new allele frequency: q' = f(aa) + ½f(Aa) after selection.

The speed of selection depends on dominance and on allele frequency. Selection against a dominant allele is fast, because every copy is exposed. Selection against a recessive allele slows dramatically as the allele becomes rare, because nearly all copies hide in heterozygotes. For a recessive lethal (s = 1), q_t = q₀/(1 + tq₀), so the number of generations to go from q₀ to q_t is t = 1/q_t − 1/q₀. Reducing q from 0.02 to 0.01 takes 100 − 50 = 50 generations, and from 0.01 to 0.005 another 100. This is why eugenic proposals to eliminate recessive disorders by preventing affected people from reproducing were not only unethical but biologically futile.

Modes of selection

• Directional selection favors one extreme or one allele and moves the population toward it, as in the spread of insecticide resistance or antibiotic resistance.
• Stabilizing selection favors intermediate phenotypes, as for human birth weight, where both very small and very large babies historically had higher mortality.
• Disruptive selection favors both extremes over the middle and can contribute to divergence.
• Balancing selection maintains two or more alleles indefinitely. It includes heterozygote advantage (overdominance) and frequency dependent selection, in which rare alleles have an advantage.

Heterozygote advantage and the sickle cell allele

Let the fitnesses be AA = 1 − s, Aa = 1 and aa = 1 − t. Selection then removes both alleles from homozygotes and keeps them in heterozygotes, and the population reaches a stable equilibrium at

q̂ = s/(s + t), where q is the frequency of a.

The classic example is the sickle cell allele, Hb^S, in regions with endemic falciparum malaria. Hb^AHb^A individuals are more susceptible to severe malaria, Hb^SHb^S individuals have sickle cell disease, and Hb^AHb^S carriers are partly protected from malaria while rarely having symptoms. With illustrative values s = 0.12 (malaria cost to Hb^AHb^A) and t = 0.80 (cost of sickle cell disease), q̂ = 0.12/(0.12 + 0.80) = 0.12/0.92 = 0.13, similar to the Hb^S frequencies of 0.1 or more observed in parts of sub-Saharan Africa. Where malaria is absent, the advantage disappears and the allele slowly declines.

Mutation selection balance

Deleterious alleles persist because mutation keeps recreating them while selection removes them. At equilibrium the two rates match. For a recessive deleterious allele, q̂ ≈ √(u/s); for a dominant one, q̂ ≈ u/s. With u = 10⁻⁶ and s = 1, a recessive lethal settles at q ≈ √10⁻⁶ = 0.001, but a dominant lethal at only 10⁻⁶, essentially just the new mutations each generation. Recessive deleterious alleles can therefore be a thousand times more common than dominant ones with the same mutation rate.

Section 5Molecular evolution and speciation

The neutral theory

Sequencing revealed far more molecular variation than selection alone seemed able to explain. Motoo Kimura's neutral theory of molecular evolution proposed that most of the differences we see at the DNA level are selectively neutral, or nearly so, and were fixed by genetic drift rather than by positive selection. The theory does not deny that selection shapes adaptations. It says that most mutations are either harmful, and quickly removed by purifying selection, or neutral, and left to drift; only a small fraction are advantageous.

The theory makes a striking prediction. In a diploid population of size N, about 2Nu new neutral mutations arise at a locus each generation, and each has a probability of 1/(2N) of eventually being fixed by drift. The rate of substitution is the product:

k = 2Nu × 1/(2N) = u

The rate at which neutral substitutions accumulate equals the neutral mutation rate, regardless of population size. Large populations produce more mutations, but each has a smaller chance of fixation, and the two effects cancel exactly.

The molecular clock

If neutral substitutions accumulate at a roughly constant rate, then the number of sequence differences between two species should be roughly proportional to the time since they shared a common ancestor. This is the molecular clock. Because both lineages accumulate changes independently after they split, the divergence between them is

K = 2μT

where K is the number of substitutions per site, μ the substitution rate per site per year, and T the time since divergence. A clock must be calibrated with fossils or dated geological events, and it ticks at different rates in different genes and lineages (for example, faster in rodents, with short generation times, than in primates), so modern methods use "relaxed" clocks that allow rates to vary.

Rates differ with functional constraint

The neutral theory explains why different parts of the genome evolve at very different rates. The more a change is likely to harm function, the more often it is removed by purifying selection, and the slower the observed rate. Changes in sequences with no function accumulate at the full mutation rate.

Comparing the two kinds of coding sites provides one of the most useful tests of selection. Let dN be the number of nonsynonymous substitutions per nonsynonymous site and dS the number of synonymous substitutions per synonymous site. Their ratio, ω = dN/dS, reveals the kind of selection:

• ω < 1: purifying selection; most amino acid changes were harmful and removed. Most genes fall here, often with ω around 0.1 to 0.3.
• ω ≈ 1: neutral evolution, as expected for a pseudogene.
• ω > 1: positive selection; amino acid changes have been favored. Classic examples include the peptide binding region of MHC genes, viral surface proteins such as influenza hemagglutinin, and some genes involved in reproduction and immune defense.

Gene duplication and gene families

New genes most often arise by gene duplication. After a duplication, one copy can keep the original function while the other is free to accumulate changes. Most extra copies decay into pseudogenes, but some acquire new functions (neofunctionalization) or split the original functions between them (subfunctionalization). The globin family is the textbook case: duplications produced α and β globin clusters, myoglobin, and fetal and embryonic globins with different oxygen affinities. Genes related by duplication within a genome are paralogs; genes in different species descended from one gene in their common ancestor are orthologs.

What is a species?

Speciation connects population genetics to the diversity of life. Several species concepts are used. The biological species concept defines a species as a group of actually or potentially interbreeding natural populations reproductively isolated from other such groups. It captures why species stay distinct, but it cannot be applied to asexual organisms or fossils. The morphological species concept uses physical traits and is practical for fossils. The phylogenetic species concept defines a species as the smallest group of populations sharing a common ancestor, diagnosable by shared derived traits, and is widely used with DNA data.

Reproductive isolating mechanisms

Under the biological species concept, the key to speciation is the evolution of reproductive isolation. Barriers are classified by whether they act before or after the formation of a zygote.

Modes of speciation

Speciation is classified by geography. In allopatric speciation, the most common mode, a population is divided by a geographic barrier, such as a rising mountain range, a new river course or the formation of an isthmus, and the separated populations diverge through mutation, drift and selection until they can no longer interbreed. Peripatric speciation is a special case in which a small peripheral population, such as a few colonists on an island, diverges rapidly under strong drift and new selective pressures; the founder effect plays a large role. In parapatric speciation, populations along a continuous range diverge across an environmental gradient despite some gene flow at the boundary. In sympatric speciation, a new species arises within the range of its parent species with no geographic separation. It is rare in animals, though host shifts in insects such as the apple maggot fly and diversification of cichlid fishes in African lakes are candidate examples. In plants, sympatric speciation by polyploidy is common: a tetraploid formed from a diploid ancestor produces sterile triploid offspring when it crosses back, so it is reproductively isolated in a single generation. Bread wheat is an allopolyploid (hexaploid) combining three ancestral genomes.

The genetics of speciation

How can two populations evolve hybrid incompatibility if every intermediate step must remain fit? The Dobzhansky Muller model answers this. Suppose the ancestral genotype is aabb. In one isolated population, allele A arises and becomes fixed, giving AAbb, which is perfectly fit. In the other, allele B arises and becomes fixed, giving aaBB, also fit. A and B have never been tested together. When the populations meet and hybridize, the hybrid carries A and B, and if those alleles interact badly, the hybrid is inviable or sterile. No population ever passed through an unfit stage; the incompatibility appears only in the hybrid. Several such interacting gene pairs have been identified in fruit flies, mice and plants.

A second regularity, Haldane's rule, states that when only one sex of a hybrid is inviable or sterile, it is usually the heterogametic sex: males in mammals and fruit flies (XY), females in birds and butterflies (ZW). A leading explanation is that recessive incompatibility alleles on the X (or Z) chromosome are exposed in the hemizygous sex, much as X-linked recessive diseases appear in males. When hybrids have reduced fitness and the incipient species come back into contact, selection can favor individuals that avoid mating with the other population, strengthening prezygotic isolation, a process called reinforcement.

Words worth knowing