## Wednesday, March 27, 2013

### The Hardy-Weinberg Equilibrium

It's important to understand modern evolutionary theory and that means it's important to understand the Hardy-Weinberg Equation and what it means.

The significance is explained in all the leading textbooks on genetics and evolution. I've chosen the explanation given by Carl Zimmer and Douglas Emlen because I know that Carl has spent a good deal of time getting it right in his new book Evolution: Making Sense of Life.

Imagine that you have a population with two alleles, A and a, at a single locus. The frequency of the first allele is f(A) to which we assign the value p. The frequency of the second allele is f(a)=q. In a randomly mating sexual population the probability of an A sperm being produced is p and the probability of an a sperm being produced is q. Similarly, the probability of an A egg cell is p and the probability of an a egg cell is q. These probabilities, p and q, do not have to be equal.

We can calculate the probabilities of all possible combinations or sperm and eggs in the population from a the following diagram (Punnett square). This one is from Wikipedia.

Since the total probability has to equal one, we have ....

p² + 2pq + q² = 1
This is the Hardy-Weinberg equation or the Hardy-Weinberg Equilbrium. What does it mean? Let's quote Zimmer and Emlen (page 156).
Hardy and Weinberg demonstrated that in the absence of outside forces (which we describe later), the allele frequencies of the population will not change from one generation to the next. As we'll see below, this theorem is a powerful tool for population geneticists looking for evidence of evolution in populations. But it's important to bear in mind that it rests upon some assumptions.

One assumption of the model is that a population is infinitely large. If a population is finite, allele frequencies can drift randomly from generation to generation simply due to chance variation, in which alleles happen to be passed on to the next generation. (We will explore genetic draft in detail later in the chapter.) While no real population is infinite, of course, very large ones behave quite similarly to the model. That's because variation due to chance is inconsequential, and the allele frequencies will not change very much from generation to generation.

The Hardy-Weinberg theorem also requires all of the genotypes of the locusts are equally likely to survive and reproduce. If individuals with certain genotypes produced twice as many offspring as individuals with other genotypes, for example, then the alleles that these certain individuals carry will comprise a greater proportion of the total in the offspring generation than would be expected given the Hardy-Weinberg theorem. In other words, selection for or against particular genotypes may cause the relative frequencies of alleles to change and results in evolution.

Yet another assumption of the Hardy-Weinberg theorem is that no alleles enter or leave a population through migration. This assumption can be violated in a population if some individuals disperse out of it or if new individuals arrive. The model also assumes that there is no mutation in the population, because it would lead to new alleles

In each of these four cases, the offspring genotype frequencies will differ from the equilibrium predictions of the Hardy-Weinberg theorem. That is, because they alter allele frequencies from one generation to the next, selection, migration, and mutation are all possible mechanisms of evolution.

The Hardy-Weinberg theorem is useful because it provides mathematical proof that evolution will not occur in the absence of selection, drift, migration, or mutation. By explicitly delineating the conditions under which allele frequencies do not change, the theorem serves as a useful null model for studying ways of allele frequencies do change. The Hardy-Weinberg theorem helps us understand explicitly how and why populations evolve. By studying how populations deviate from the Hardy-Weinberg equilibrium, we can learn about the mechanisms of evolution.
There you have it. The Hardy-Weinberg describes the situation where evolution DOES NOT HAPPEN and thus serves as the null hypothesis for testing whether evolution is happening. Every undergraduate knows this.

Let's see if the Intelligent Design Creationists know this. I'm quoting "niwrad" from a post on one of the leading ID websites, Uncommon Descent: The equations of evolution.
For the Darwinists “evolution” by natural selection is what created all the species. Since they are used to say that evolution is well scientifically established as gravity, and given that Newton’s mechanics and Einstein’s relativity theory, which deal with gravitation, are plenty of mathematical equations whose calculations pretty well match with the data, one could wonder how many equations there are in evolutionary theory, and how well they compute the biological data related to the Darwinian creation.

....

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 creation of entirely new organisms. To claim that the Hardy-Weinberg law explains evolution is as to say that in mechanics a principle of statics (immobility) explains dynamics (movement and the forces causing it).

....

So the initial question, how well math support Darwinian evolution, has the short answer: it doesn’t support evolution at all. Despite of the pretension of evolution to be a scientific theory with the mathematical certitude of the hard sciences, properly the equations of evolution do not exist.
As you can see, the Intelligent Design Creationists interpret the "Hardy-Weinberg law" very differently, I wonder who is right?

Let's check with Joe Felsenstein. He's an expert on population genetics so he should know. Read his decision at: Evolution disproven — by Hardy and Weinberg?.

1. I don't recall ever seeing Hardy-Weinberg called a law. And I've never seen it used as some kind of enforced straight-jacket that prevents evolution. Oddly enough, it would prevent microevolution, which most IDiots claim really happens, rather than macroevolution, which they claim doesn't.

I've seen that phenomenon -- of IDiots reflexively rejecting microevolution -- many times, though, as in claims that the Biston betularia example is faked, fitness is tautological, and so on.

1. On this note, Professor Moran says: "The Hardy-Weinberg describes the situation where evolution DOES NOT HAPPEN and thus serves as the null hypothesis for testing whether evolution is happening. Every undergraduate knows this." If one defines evolution strictly in terms of phenotype, this is perhaps correct. But DNA is constantly changing and so-called "sibling species" with no apparent phenotypic differences, HAVE CHANGED (presumably following DNA changes) to the extent that their members are reproductively isolated from each other. We should at least pause before declaring that evolution has not happened.

2. (presumably following DNA changes)

The Hardy-Weinberg model assumes, among other things, that there are no mutations. Mutations, whether phenotypically visible or not, disturb the HW equilibrium of allele frequencies and they "make evolution happen" (even if only by shifting the point of equilibrium).

3. If one defines evolution strictly in terms of phenotype, this is perhaps correct.

Is this a typo or error? HW is quite clearly just measuring allele frequencies(ie genotype).

4. @Donald Forsdyke,

Don, your comment doesn't make sense. Can you re-phrase it?

5. Gasiorowski gets near to sorting it out for alleles. Any mutation within an allele will disturb the equilibrium, whether it impacts phenotype or not, so, as TheOtherJim explains, HW concerns genotype. But even assuming there are no changes between alleles, surely there will be ongoing non-allelic changes - genotypic changes - elsewhere in a genome, that may be relevent for macroevolution? In this respect, it may be tidy for the mathematics, but one cannot really assume that evolution does not happen.

2. I should be surprised no longer by their stupidity.

3. They confuse a neutral equilibrium with a stable one. Little wonder, if Niwrad the Toidi thinks a trivial binomial expansion of (p + q)² is some sort of advanced maths. He can't have seen the inside of any handbook of population genetics, or he wouldn't be asking dumb questions like "How many equations are there in evolutionary theory?"

4. The phrase "Hardy-Weinberg law" gets 36,100 hits on Google and that includes the title of a Britannica Online encyclopedia entry. It is unfortunate,but common. In my book I call them Hardy-Weinberg Proportions. The fact that altering the genotype frequencies in such a way that the gene frequency isn't altered does not change the future composition of the population is mildly interesting, but the phrase "Hardy-Weinberg Equilibrium" for that is misleading. Almost any alteration of genotype frequencies you do will not leave the gene frequency unchanged.

5. When I first read the post, I misunderstood and thought Larry was quoting one of the comments from UD, which would be unfair. So I was surprised to click thru and find that this was actually a feature article from the site. I don't know why I should be surprised, but for some reason I keep underestimating the depths of stupidity to which these people will sink.

The response to Joe's article is particularly interesting, in that they (predictably) say "Yeah, but that's just microevolution", without realizing (or at least without admitting) that this is already enough to refute niwrad's entire thesis.

IDiots, the lot of them.

6. Apparently I have led a sheltered life with regard to "law". But "equilibrium" seems perfectly appropriate, since any frequency combination of genotypes will reach HW frequencies in a single generation, given the HW conditions. OK, the least little change in allele frequencies will destroy the supposed equilibrium, but at least it's robust to changes in genotype frequencies that don't effect allele frequencies. (How you would achieve that is unclear; but usually it's just the initial condition out of whack.) That should count for something.

7. John, if you take the heterozygotes and replace them by a 50:50 mixture of the two corresponding homozygotes, you will get a change of genotype frequencies which keeps the gene frequencies the same. It even works for multiple alleles. That is does work is obvious once you consider what that does to the number of copies of each allele -- it keeps the allele numbers the same.

1. I meant "how you would achieve that in a real population", rather than in say, some fruit flies in a bottle.

2. Oh. Reality. I was talking about models ...

8. To be pedantic...

If we are talking about a "randomly mating sexual population" that produces sperm and eggs then the following is possible:

If females all have alleles A (p = 1.0) and males all have allele a (q = 1.0) then you need two generations of random mating--not a single generation--to achieve the equilibrium.