Biology · Intermediate · 30 min · Chapter 27 of 28

Populations and genetics

Exponential and logistic growth, and the Hardy–Weinberg accounting of alleles in a population that is not evolving.

Read first: Functions, exponentials, and logarithms, Probability and statistics, Differential equations

The idea

A population growing by a constant per-capita rate r satisfies dN/dt = rN, with solution N = N₀ e^{rt}. That is the growth of bacteria in fresh medium, and of any population far below the resources that will eventually limit it. The logistic equation puts the limit in: dN/dt = rN(1 − N/K). Growth is fastest at N = K/2 and stops at the carrying capacity K. Real populations overshoot, oscillate, and get eaten; the logistic is the first honest correction to pure exponential growth, not a portrait of an ecosystem.

Hardy–Weinberg starts from two alleles, A and a, with frequencies p and q, and p + q = 1. If mating is random and no selection, mutation, migration, or drift is changing the frequencies, the next generation is p² of AA, 2pq of Aa, and q² of aa. The useful move is usually backwards. You observe the recessive phenotype frequency q², take a square root to get q, and then the carrier frequency is 2pq. For a rare recessive, q is small, p is near 1, and carriers outnumber affected individuals by about 2/q.

The assumptions are the point of the model. A statistically significant excess of homozygotes suggests inbreeding or a hidden population split (the Wahlund effect). A persistent excess of heterozygotes suggests balancing selection. Genetic drift is the sampling error of finite populations; its strength is about 1/(2N) per generation for the variance in allele frequency, which is why small populations lose variation. None of these tests require you to believe that a natural population is actually at equilibrium. They require you to know what equilibrium would have looked like.

Hold onto this

Unchecked growth is exponential. Limits turn it into a logistic curve that levels off at a carrying capacity.
Hardy–Weinberg is a null model: if nothing interesting is happening, genotype frequencies are p², 2pq, and q².
Departure from those frequencies is evidence that selection, drift, mating, mutation, or migration is at work.
Recessive diseases hide in heterozygotes. The carrier frequency is much larger than the disease frequency.

Formulas

Exponential growth

N = N_{0} e^{r t}

Constant per-capita growth. Doubling time is ln 2 / r.

N₀: Population at t = 0
r: Per-capita growth rate

Logistic growth

\frac{d N}{d t} = r N (1 - \frac{N}{K})

Per-capita growth shrinks linearly as N approaches K.

K: Carrying capacity

Hardy–Weinberg

p + q = 1, p^{2} + 2 pq + q^{2} = 1

Allele frequencies, and the genotype frequencies AA, Aa, aa under the null assumptions.

p, q: Frequencies of alleles A and a

Carrier frequency

2 pq = 2 q (1 - q)

Frequency of heterozygotes. For a rare recessive, about 2√(disease frequency).

q²: Frequency of the recessive genotype

Worked examples

1. Where logistic growth is fastest

A population has r = 0.4 per year and K = 1000. It is currently 200. What is dN/dt, and at what size would growth be fastest?

dN/dt = 0.4 × 200 × (1 − 0.2) = 80 × 0.8 = 64 individuals per year.
The factor N(1 − N/K) is a parabola peaking at N = K/2.
Fastest growth at 500 individuals.

64 per year now. The maximum rate occurs at N = 500, not when the population is largest.

2. A rare recessive

A recessive condition appears in 1 in 10,000 births. Estimate the carrier frequency, assuming Hardy–Weinberg.

q² = 1/10000, so q = 1/100 = 0.01.
p = 0.99.
2pq = 2(0.99)(0.01) = 0.0198, about 1 in 50.

About 2% of people are carriers, roughly 200 times the frequency of affected births.

Try these

A population of 500 grows exponentially at r = 0.1 per year. Estimate the population in 10 years.

Allele frequencies are p = 0.7 and q = 0.3. What genotype frequencies does Hardy–Weinberg predict?

Common mistakes

q² is the frequency of the recessive genotype, not of the allele. Taking the square root is the step that recovers q.
Hardy–Weinberg is the null. Fitting it is not evidence that selection is absent until you have the sample size to have noticed a departure.
The logistic K is not a target the population “wants.” It is the point where the modeled per-capita growth hits zero.