Over at Uncommon Descent, “niwrad” has argued that the equations of theoretical population genetics show that evolution is unlikely. niwrad says that the equations of theoretical population genetics
consist basically in two main equations: the Hardy-Weinberg law and the Price equation.
Furthermore niwrad says that
The Hardy-Weinberg law mathematically describes how a population is in equilibrium both for the frequency of alleles and for the frequency of genotypes. Indeed because this law is a fundamental principle of genetic equilibrium, it doesn’t support Darwinism, which means exactly the contrary, the breaking of equilibrium toward the increase of organization and the creation of entirely new organisms.
I just finished teaching my course in theoretical population genetics (with lots of equations, but actually not the Price Equation, as it happens). And I can say that the statement about the Hardy-Weinberg law shows niwrad to be mixed up about the import of Hardy-Weinberg proportions. Let me explain …
If we have a gene with two alleles whose gene frequencies are p and q, random mating in effect brings together a pair of randomly chosen copies in each individual. Thus the probability of genotype AA is p2. The probability of the heterozygote is 2pq, and the probability of aa is q2. The genotype frequencies depend on the gene frequencies in the previous generation.
These proportions are sometimes called Hardy-Weinberg “equilibrium”, because if you change the genotype frequencies in a population in a way which does not alter the gene frequencies random mating will bring the genotypic composition of the population back to the same Hardy-Weinberg proportions. Thus if you have a gene frequency of 0.2 for A, the Hardy-Weinberg proportions will be 0.04 : 0.32 : 0.64. Now if we take half of the heterozygotes and replace them by a 50:50 mixture of the two homozygotes, we would have genotype frequencies of 0.12 : 0.16 : 0.64. In that population the gene frequency of A is unchanged, still 0.2. Now if these individuals mate randomly and have offspring, we once again get the Hardy-Weinberg proportions 0.04 : 0.32 : 0.64.
But what happens if we change the gene frequency? For example if we kill off all the AA individuals, we then have a population with genotype frequencies 0 : 0.32/0.96 : 0.64/0.96 which is 0 : 0.3333 : 0.6667. In that population of survivors, the gene frequency of A is now 0.16667 instead of 0.2. After they mate randomly and have offspring, the genotypes are in Hardy-Weinberg proportions at the new gene frequency, and their genotype frequencies are 0.027778 : 0.27778 : 0.694444. The gene frequency is now still 0.16667, and has not gone back to 0.2. So evolutionary forces that change gene frequencies are not counterposed by Hardy-Weinberg “equilibrium”. The gene frequency has no tendency to return to previous values. Thus continued natural selection can gradually change the gene frequencies.
In my (free, online) textbook Theoretical Evolutionary Genetics this is covered on page 7. Chapter II is devoted to the (many) equations for change of gene frequencies under natural selection. I hope that niwrad will taken a look at that chapter, and let us know whether it shows that the Hardy-Weinberg “equilibrium” is a force preventing response to natural selection.