Mutation, gene flow, genetic drift and Hardy–Weinberg

Natural selection is only one of the forces that change populations. Mutation creates new genetic variation, gene flow moves it between populations, and genetic drift changes allele frequencies by pure chance. To see any of these forces at work, biologists need a baseline: what would a population look like if nothing were changing it? That baseline is the Hardy–Weinberg principle. This lesson introduces the vocabulary of population genetics, the Hardy–Weinberg model and the four main evolutionary forces. BIOL-322 develops the mathematics much further.

Genes, alleles and gene pools

A gene is a stretch of DNA that codes for a product such as a protein. Different versions of a gene are called alleles. A diploid organism such as a human, a pea plant or a fruit fly carries two copies of each gene, one from each parent, and so has two alleles at each locus (gene position). If the two alleles are the same, the individual is homozygous; if they differ, it is heterozygous. The combination is the genotype; the observable trait it produces is the phenotype.

Population geneticists think in terms of the gene pool: all the alleles at a locus across all the individuals in a population. The allele frequency is the proportion of all copies that are a given allele. If a population of 100 diploid moths has 200 copies of a colour gene and 120 of them are allele A, then the frequency of A, written \(p\), is 0.6 and the frequency of the other allele a, written \(q\), is 0.4. Because there are only two alleles here, \(p + q = 1\). Evolution, in the modern synthesis definition, is a change in allele frequencies over generations.

The Hardy–Weinberg principle

In 1908 the English mathematician G. H. Hardy and the German physician Wilhelm Weinberg independently showed that, in a large, randomly mating population with no evolutionary forces acting, allele frequencies stay constant from generation to generation and genotype frequencies settle at predictable values after one generation of random mating:

Here \(p^2\) is the frequency of AA homozygotes, \(2pq\) the frequency of Aa heterozygotes and \(q^2\) the frequency of aa homozygotes. The logic is simply probability: if eggs and sperm combine at random, the chance of an offspring receiving A from both parents is \(p \times p\), and the chance of one A and one a is \(2pq\) because it can happen two ways.

Hardy–Weinberg equilibrium holds only when five conditions are met. Each condition corresponds to one evolutionary force that, if present, will change allele or genotype frequencies.

Real populations never meet all five conditions perfectly, so why is the model useful? Because it is a null hypothesis. If observed genotype frequencies match Hardy–Weinberg expectations, no strong force is acting on that locus, at least not detectably. If they do not match, something is happening, and the pattern of the mismatch hints at what.

Testing for equilibrium

To test whether a sample fits Hardy–Weinberg proportions, biologists count genotypes, estimate allele frequencies from the counts, calculate the expected genotype numbers and compare observed with expected using a chi-square test. For one gene with two alleles there is one degree of freedom (three genotype classes, minus one for the total, minus one for the estimated allele frequency).

from scipy.stats import chi2
# Genotype counts at a two-allele locus in a sample of 500 deer (invented data)
obs = {"AA": 205, "Aa": 210, "aa": 85}
n = sum(obs.values())
p = (2 * obs["AA"] + obs["Aa"]) / (2 * n)
q = 1 - p
exp = {"AA": p * p * n, "Aa": 2 * p * q * n, "aa": q * q * n}
stat = sum((obs[g] - exp[g]) ** 2 / exp[g] for g in obs)
pval = chi2.sf(stat, df=1)
print(f"p = {p:.3f}, q = {q:.3f}")
for g in obs:
    print(f"{g}: observed {obs[g]:3d}, expected {exp[g]:6.1f}")
print(f"chi-square = {stat:.2f}, df = 1, P = {pval:.4f}")

p = 0.620, q = 0.380
AA: observed 205, expected  192.2
Aa: observed 210, expected  235.6
aa: observed  85, expected   72.2
chi-square = 5.90, df = 1, P = 0.0151

Here there are fewer heterozygotes than expected and more of both homozygotes. A deficit of heterozygotes is the classic signature of inbreeding, or of a sample that accidentally mixed two populations with different allele frequencies (called the Wahlund effect). A small P value tells us the population is not in equilibrium at this locus; the biology tells us which explanation to investigate.

Mutation: the source of new variation

A mutation is a change in DNA sequence. Mutations arise from copying errors during DNA replication, from chemical damage and radiation, and from mobile genetic elements that jump around the genome. They range from single base changes (point mutations) to insertions, deletions, duplications of whole genes and rearrangements of chromosomes. Gene duplications are especially important in evolution, because the spare copy is free to accumulate changes and take on new functions; the globin genes that carry oxygen in our blood arose this way.

Mutation rates per gene are low. In humans, each newborn carries roughly 60 to 70 new point mutations across the whole genome that neither parent had, out of about 3 billion base pairs per genome copy. Most mutations are neutral, having no effect on fitness; many of the rest are harmful; a few are beneficial in a particular environment. Crucially, mutations are random with respect to need: an environment does not cause the mutations that would help in it. Mutation alone changes allele frequencies very slowly, but it supplies the raw material for every other force.

Gene flow

Gene flow is the movement of alleles between populations through migrating individuals or dispersing gametes, such as pollen carried by wind or bees. Gene flow tends to make populations more similar to each other, counteracting the differences produced by selection and drift. It can also introduce useful alleles: populations of plants on old mine sites receive pollen from nearby populations, and the balance between gene flow and local selection for metal tolerance can be measured across just a few metres of ground.

A simple way to see gene flow's effect: if a fraction \(m\) of a population each generation is made up of migrants from a source with allele frequency \(p_m\), the local frequency moves toward the source value: \(p' = (1-m)\,p + m\,p_m\). Even a few migrants per generation is enough to prevent populations from diverging very far by drift alone.

Genetic drift

Genetic drift is change in allele frequencies due to chance: which individuals happen to survive, mate and pass on which of their two alleles. In a large population these chance effects average out. In a small population they do not, and allele frequencies wander randomly from generation to generation until, eventually, one allele is lost and the other is fixed (reaches 100 per cent). Drift does not favour helpful alleles; it is blind.

The Wright–Fisher model, named after Sewall Wright and Ronald Fisher, simulates drift: each new generation of \(N\) diploid individuals draws its \(2N\) allele copies at random from the previous generation's gene pool. The simulation below runs it for a small and a large population.

import numpy as np
# Wright-Fisher genetic drift: 2N allele copies drawn at random each generation
rng = np.random.default_rng(7)

def drift(N, p0=0.5, gens=100, reps=200):
    p = np.full(reps, p0)
    for _ in range(gens):
        p = rng.binomial(2 * N, p) / (2 * N)
    return p

for N in (10, 100, 1000):
    p = drift(N)
    lost_or_fixed = np.mean((p == 0) | (p == 1))
    print(f"N = {N:4d}: after 100 generations, {lost_or_fixed:5.1%} of 200 populations "
          f"have lost or fixed the allele; spread of p (SD) = {p.std():.3f}")

N =   10: after 100 generations, 100.0% of 200 populations have lost or fixed the allele; spread of p (SD) = 0.500
N =  100: after 100 generations, 13.0% of 200 populations have lost or fixed the allele; spread of p (SD) = 0.317
N = 1000: after 100 generations,  0.0% of 200 populations have lost or fixed the allele; spread of p (SD) = 0.113

With only 10 individuals, every one of the 200 simulated populations has lost one allele within 100 generations; with 100 individuals, about one population in eight has; with 1,000, none has and frequencies have spread only modestly around 0.5. Drift also erodes genetic diversity. A useful measure is heterozygosity, the proportion of individuals that are heterozygous. Under drift alone it declines each generation by a factor of \(1 - \frac{1}{2N}\), so a population of 25 loses about 2 per cent of its heterozygosity every generation, and about 87 per cent over 100 generations.

Two special cases of drift matter for conservation and human history:

Putting the forces together

In real populations all these forces act at once. Mutation slowly adds variation; drift removes it at random, quickly in small populations; gene flow homogenizes populations; selection pushes allele frequencies toward whatever works best locally; non-random mating rearranges genotypes. Whether selection or drift wins depends on population size: roughly, if the selection coefficient is much smaller than \(1/(2N)\), drift dominates and the allele behaves as if it were neutral. This is why small, isolated populations of endangered species can accumulate harmful alleles, and why conservation programs work hard to keep populations large and connected (lesson 12).