Jul 20 2007

Evolution-12: Population genetics and the Hardy-Weinberg law

(Please see here for previous posts in this series.)

In the previous post, I discussed the puzzle posed by a naïve understanding of Mendelian genetics, which was that one might expect that organisms that displayed recessive gene traits would slowly disappear in a population while those with dominant gene traits would grow in number. But if that were true that would prevent new mutations from gaining a foothold in the population and growing in number, if it happened to be a recessive trait.

The crucial work that formed the breakthrough that revived the theory of natural selection was done in 1908 by G. H. Hardy (a Cambridge University mathematician and author of a fascinating book A Mathematician’s Apology) and Wilhelm Weinberg (a German physician), working independently. What is nice is that the result is quite simple to derive, and surprising.

The main result is that whatever the distribution of gene pairs AA, Aa, and aa you start with in a population, after just one generation the number of people with those distributions will reach an equilibrium value that will never subsequently change. In other words, the numbers of the different types of genes in a population are stable. So traits, once they appear, do not disappear simply because of the accidents of random mating. This counters the ‘blending inheritance’ objections to Darwin’s theory.

The proof of this result assumes that certain conditions apply so that only mating effects are at play: that the total population is large enough (effectively infinite for statistical purposes) to avoid the phenomenon of genetic drift, whereby the ratio of a particular gene varies purely due to statistical fluctuations (i.e., say the population with a particular gene happens to breed disproportionately, thus causing that gene’s frequency to change), is diploid, that the population reproduces sexually and that mating within the population is totally random, that natural selection is not working to change the distributions of the genotypes, and that other factors like genetic mutations and migrations in or out of the population are not occurring (i.e., no gene flow).

Here’s the result. Suppose that you start with a population in which AA types occur with probability p, Aa types occur with probability 2q (where the 2 is inserted just to make the arithmetic a little simpler), and aa types account occur with probability r. Since the total population must add to 100%, this means that the total probabilities p+2q+r=1.

Under the conditions given above, the Hardy-Weinberg result says that: (1) after just one generation, the AA types occur with probability P where P=(p+q)2, Aa with probability 2Q where Q=(p+q)(q+r), and aa with probability R where R=(q+r)2; and (2) these new probabilities will remain unchanged with each succeeding generation.

As an example, if we started with the population of AA being 50% (p=0.5), Aa being 40% (2q=0.4), and aa being 10% (r=0.1), then after just one generation, the Hardy-Weinberg result predicts that the proportions will be P=0.49 or 49% for AA, 2Q=0.42 or 42% for Aa, and R=0.09 or 9% for aa, and remain fixed at these values forever afterwards.

The proof of this result is quite simple and elegant and here it is:

If there is random mating, then the probability of any particular mating combination is just the product of their individual probabilities.

The probability of an AA mating with another AA is p2. The offspring will get just one gene from each parent, and in this case the result will always be AA.

The probability of an AA mating with an Aa is 4pq (the extra factor of 2 comes from the fact that this mating combination can occur two ways, that either the father could be AA and the mother Aa, or the father could be Aa and the mother AA) and there is a 50% chance that the offspring will be an AA and 50% chance of being an Aa.

Similarly, the probability of an AA mating with an aa is 2pr. The offspring will get just one gene from each parent, and in this case the result will always be Aa.

The probability of an Aa mating with an Aa is 4q2 and there is a 25% chance that the offspring will be AA, 50% chance of being an Aa, and 25% of being an aa.

The probability of an Aa mating with an aa is 4qr and there is a 50% chance that the offspring will be Aa and 50% chance of being an aa.

The probability of an aa mating with another aa is r2. The offspring will get just one gene from each parent, and in this case the result will always be aa.

When you add all the probabilities for each type of offspring together, the probabilities of getting AA and Aa and aa are just the expressions for P, 2Q, and R given above.

What Hardy and Weinberg noticed was that if, by some chance, the starting values p, q, and r were such that they satisfied the equation q2=pr, then after one generation, P=p, Q=q, and R=r. In other words, if the starting values satisfied that particular relationship, the probabilities are unchanged from one generation to the next.

Of course, the values of p, q, and r we actually start with for a random population can have any value, as long as p+2q+r=1. But after the first generation of mixing, the values P, Q, and R actually do satisfy the relationship Q2=PR, irrespective of the starting values of p, q, and r.

Since the values of P, Q, R that are obtained after one generation become the starting values to calculate the distributions for the subsequent generation, and since P, Q, R satisfy the required relationship Q2=PR, these values will remain the same for every succeeding generation after the first.

What I found particularly surprising is that usually equilibrium conditions tend to be approached gradually and even asymptotically. Here, whatever the starting point, you get equilibrium after just one iteration.

The stability of population distributions under conditions of random mating is an important result. It implies that gene distributions do not change due to mating but only under some kind of pressure to do so..

From the year 1908 onwards, mathematical biologists proceeded to make rapid advances in the embryonic field now known as population genetics. The names of R. A. Fisher, Sewall Wright, and J. B. S. Haldane are the ones associated with the birth of this field and by the 1930s or so, their work had put Darwinian natural selection and Mendelian genetics on a firm scientific and mathematical footing (William B. Provine, The Origins of Theoretical Population Genetics, 2001).

In the next post in the series, I will look at how natural selection causes the population distributions to shift.

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