entropy test

After the last post, you might wonder what the big deal is about these information theoretic measures of linkage. After all, we’ve got lots of other measures of linkage to choose in population genetics, with many years of theory behind them. The basic conclusion about genetic drift was that it adds mutual information to samples over short regions, but that recombination over longer areas washes it out. If the net effect is no linkage, why would we bother to come up with some non-standard linkage measure?

One answer: If the existing linkage measures were so great for testing neutrality, then we might expect some of the recent genome-wide selection scans to have used them. But they didn’t – instead we have several partially incompatible methods, all of which eschew the usual measures of linkage.

We tend to think of genetic drift as a random process. Random processes operating repeatedly over time are called “stochastic,” and changes in gene frequency under genetic drift are certainly that.

Since entropy is a measure of uncertainty, it might seem natural to think that stochastic changes in gene frequency would increase the entropy in a population. After all, the gene frequency in a population under genetic drift will be more and more uncertain over time. So, considering the frequency of a single allele as the system, genetic drift appears to increase entropy over time.

But even this simple system isn’t quite so simple as it might appear. Sure if you start out knowing the allele frequency, then genetic drift will increase your uncertainty over time. You will become less and less able to say that it lies in any given interval. But what if you don’t start out knowing? What if all you know is that the locus has been subjected to t generations of genetic drift?

As t increases, the probability of fixation of the locus also increases. The net effect is to reduce the entropy in the system – going from uncertainty about the allele frequency to more and more certainty that it will be either one or zero. The only thing that will stop this process is some other evolutionary force – mutation, migration from other populations, balancing selection. Each of these will have its own distinctive effects on the entropy of the single-locus system.

This is the second in a series on information theory and tests for recent selection. The first entry, "Information theory: a short introduction" reviewed the basic concepts of information measures and their background.

The International HapMap is a massive project to determine the genotypes for up to 3 million single nucleotide polymorphisms (SNPs) in samples of people from 11 population samples around the world. The current data release (Phase 3) includes genotypes for a subset of over 1.5 million SNPs in 1,115 people. The 11 population samples include people of African ancestry from the US Southwest, Utah residents of Northern and Western European ancestry, Han Chinese from Beijing, people of Chinese ancestry from Denver, people in the Houston Gujarati Indian community, Japanese people from Tokyo, Luhya and Maasai people from Kenya, people of Mexican ancestry from Los Angeles, Italians in Tuscany, and Yoruba from Ibadan, Nigeria.

As impressive as this effort is, we may wonder why exactly SNP genotyping of so many people is a valuable enterprise in itself. The project’s homepage includes this short statement:

There are theoretical and practical objections to this simple explanation (as I discussed here last month). However, what no one involved with the project seems to have expected is the extent to which the data would demonstrate the importance of recent adaptive evolution in human populations.

Here, I am describing some of the ways that we can test hypotheses about natural selection by using the SNP genotypes from the HapMap. This is a theory-centric description, with some digression into practical aspects of handling the genotype data. First, I consider how we might use information theoretic concepts to test the hypothesis of independence between two genetic loci.

I lectured this week in my Biology of Mind course about information theory, and in particular the concept of Shannon entropy. I’ve typed up a few notes for my students, and I’m cross-posting them on my own blog because they are relevant to another topic I’ll be writing about: discovery and testing of natural selection in the human genome. You see, the kind of data that are presently being collected as part of the International HapMap , single nucleotide polymorphisms (SNPs), are naturally treated by information theoretic measures. So first, it may help to define the essential concepts of information theory.