Tuesday, December 20, 2005

Mean Fitness, Genetic Load, and the Misapplication of Population Genetics Metrics

I will probably deal with some of these concepts in my Detecting Natural Selection series, but a couple of letters in the American Scientist (hat tip: Deinonychus antirrhopus) have inspired me to post my opinion. The letters -- one a reply to an article by Paul E. Turner, and the other a reply by Turner to the first letter -- deal with how natural selection shapes the fitness of individuals and populations. In the first letter, Dmitri E. Kourennyi complains:

"In the last paragraph of the 'Cheaters Sometimes Prosper' section, the author mentions an apparent conflict between the prediction of evolutionary game theory in the prisoner's dilemma and the theory of evolution. Cheaters can lower the average fitness of the population. On the other hand, paraphrasing the author's statement, Darwin's theory of evolution suggests that the population becomes better adapted to its environment over time.

"As far as I know, evolutionary theory does not claim that populations are driven toward higher fitness. Evolutionary pressure acts at the level of individuals. As a result, the average fitness of the population usually increases, but it can decrease in some cases, such as when cheaters have an advantage over cooperators and can take over the population."

To which Turner replies:

"In my opinion, Dr. Kourennyi is misreading this sentence in at least two ways. First, his literal reading is that 'steers' is equivalent to 'guides,' where the population is purposefully taken in the direction of increased fitness. As he correctly indicates, natural selection is blind and the process does not drive populations to increased fitness through time. As for Dr. Kourennyi's conclusion, I wrote that '[Prisoner's dilemma] is somewhat counter to Darwin's theory by natural selection.' Selection can favor takeover by cheaters, leading to a surprising (in a Darwinian sense) decrease in mean fitness of the population."

I think they’re both pissing into the wind on this one, as there is no meaningful way to measure the fitness of a population (someone can correct me on this, but please read below for my argument). The two measures I usually equate with population fitness are mean fitness and genetic load. Mean fitness, however, does not have any meaning outside of the context for which it was derived -- as a factor used in determining the expected change in allele frequencies under natural selection. Genetic load, which is calculated using the meaningless statistic mean fitness, is therefore a useless metric.

Let us first examine mean fitness. Under Hardy-Weinberg equilibrium allele frequencies remain constant from generation to generation and can be used to predict the genotype frequencies. If we have a one locus, two allele system (alleles A and a), where the frequency of A is given by p and the frequency of a is given by q (and q = 1 - p), we get the following genotype frequencies:

  • Freq(AA) = p2
  • Freq(Aa) = 2pq
  • Freq(aa) = q2

Now, imagine that one genotype is more fit than the others (leaves more offspring per individual carrying that genotype), so we get the following fitness measurements for each genotype: WAA, WAa, Waa. The fitness of a genotype can be thought of as the number of offspring left per individual carrying that genotype, such that WAA is the number of offspring from an AA individual. To standardize for differences in the number of individuals carrying each of the three genotypes and the fitness of the different genotypes relative to each other, we use the mean fitness:

Wbar = p2WAA + 2pqWAa + q2Waa

The frequency of each genotype after selection can be calculated using the frequency of each genotype before selection, the fitness of that genotype, and the mean fitness:

  • Freq(AA after selection) = p2WAA / Wbar
  • Freq(Aa after selection) = 2pqWAa / Wbar
  • Freq(aa after selection) = q2Waa / Wbar

By favoring particular genotypes, natural selection leads to changes in allele frequencies between generations. The frequency of the A allele can be calculated using the frequency of the genotypes carrying that allele:

p = Freq(AA) + ½Freq(Aa)

If we represent the frequency of the A allele before selection as p and after selection as p', we get the following:

p' = (p2WAA / Wbar) + ½(2pqWAa / Wbar)

This model can be applied to any locus regardless of the number of alleles segregating at that locus (I’ll leave it up to the reader to perform this derivation).

As you can see, Wbar is simply a metric used to account for the fitness of each genotype relative to each other, and is used when we would like to calculate changes in allele and genotype frequencies due to natural selection. It is called “mean fitness”, but it does not measure any meaningful aspect of the population. This did not stop J.B.S. Haldane from using Wbar in his equation for genetic load:

L = (Wmax - Wbar) / Wmax

where Wmax is the theorertical maximum of mean fitness. Haldane decided that mean fitness actually measures something, and that a population with higher mean fitness is, well, more fit. This goes contra to the paradigm of natural selection, in which fitness is measured for individuals not populations.

Kourennyi makes a valid point in his letter when he points out that evolutionary theory says nothing about the fitness of populations, but he slips a bit when he claims that cheaters affect the fitness of a population. The fitness of a population does not measure the health or vitality of a population; it simply standardizes an equation for calculating changes in allele frequencies.


At 3:39 AM, Blogger Steve said...

Hey, cool post. I was just telling somebody the other day that biology is becoming more and more mathematical and here I find a post like this.

At 10:55 AM, Blogger RPM said...

Biology may be becoming more mathematical, but these ideas are half a century old.

At 1:12 PM, Blogger Mike the Mad Biologist said...

Following up on RPM, didn't Bruce Wallace make a similar argument in Fifty Years of Genetic Load?

At 3:26 PM, Blogger Timothy Chase said...

Yes -- Bruce Wallace made a similar argument. Incidentally, just got done reading "Fifty Years." Great book! Going to be reading through it again. (Love his style of writing, too!)

At 4:00 PM, Blogger RPM said...

Alright, I'm gonna have to check out 50yrs. In my defense, I wasn't familiar with anything from Wallace other than his journal articles, and we appeared to have reached the same conclusion. I guess it comes from having the same academic pedigree.

At 3:32 AM, Anonymous Anonymous said...

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