The Equilibrium That Measures Evolution: The Hardy-Weinberg Principle

A null hypothesis for genes

The most useful idea in population genetics is, at first glance, a statement about nothing happening. The Hardy-Weinberg principle describes what a population’s genetics look like when no evolution is occurring at all. That sounds like a strange thing to bother proving, but it is exactly what makes it powerful. To detect a force, you first need to know what the world looks like without it. Hardy-Weinberg is the genetic equivalent of a level floor: once you know what flat looks like, any tilt becomes visible.

The principle was published independently in 1908 by an English mathematician, G. H. Hardy, and a German physician, Wilhelm Weinberg. Hardy worked it out almost dismissively, in a short letter, because a biologist had been confused about whether a dominant trait would automatically spread through a population just by being dominant. Hardy showed it would not. In the absence of disturbing forces, the proportions of genes simply stay put.

The algebra

Consider a single gene with two versions, called alleles, which we can label A and a. Let the frequency of A in the population be p and the frequency of a be q. Since those are the only two options, they must add to one:

$$p + q = 1$$

Now, if individuals pair up at random, the genotypes of the next generation are just the ways of drawing two alleles from this pool. The chance of drawing A twice is p times p; the chance of drawing a twice is q times q; and the chance of one of each is two times p times q, because it can happen in either order. So the three genotype frequencies are:

$$p^2\;(\text{AA}) \;+\; 2pq\;(\text{Aa}) \;+\; q^2\;(\text{aa}) \;=\; 1$$

That is the whole principle. If nothing perturbs the population, these proportions stay constant generation after generation, and the allele frequencies p and q never change. The population is said to be in Hardy-Weinberg equilibrium.

A worked example

The equation earns its keep when you use it to find numbers you cannot measure directly. Suppose a recessive genetic disorder, which only appears in individuals carrying two copies of the recessive allele, affects one person in ten thousand. The affected individuals are the aa genotype, so:

$$q^2 = \frac{1}{10000} \quad\Rightarrow\quad q = 0.01$$

Then p, the frequency of the normal allele, is 1 minus 0.01, or 0.99. Now you can calculate something clinically important that is otherwise invisible: the number of healthy carriers, the Aa individuals who carry one copy but do not have the disease.

$$2pq = 2\,(0.99)(0.01) = 0.0198$$

So roughly one person in fifty is a silent carrier of an allele that shows up as disease in only one person in ten thousand. The carriers vastly outnumber the affected, which is a general and counterintuitive feature of recessive conditions, and the Hardy-Weinberg equation pulls that hidden number straight out of the visible one.

What the assumptions are really for

The equilibrium holds only under a specific set of conditions, and the list is worth knowing because the conditions are exactly the forces of evolution. For frequencies to stay constant, the population must be large, so that chance fluctuations average out; mating must be random with respect to the gene; and there must be no mutation creating new alleles, no migration adding or removing them, and no natural selection favoring one genotype over another.

In the real world at least one of these is almost always violated, which is the point. Because Hardy-Weinberg tells you what a non-evolving population should look like, a population that does not match it is telling you that something is acting on it. If the observed genotype frequencies depart from p squared, two pq, and q squared, you go looking for the cause: selection against a genotype, perhaps, or non-random mating, or migration between populations with different allele frequencies. The equilibrium is not interesting because populations sit in it. It is interesting because the ways they fall out of it are the signatures of evolution at work. Hardy handed biology a ruler, and every deviation from it is a measurement.