Of the three stages of natural selection outlined before, the only one that occurs purely by chance is the first one, that of the occurrence of mutations. I discussed how although the chances of producing a favorable mutation by changes in any individual site in the DNA (called ‘point mutations’) on an individual member of the species is very small, when the number of individuals in a species and the long times available for the changes to occur are factored into the calculation, the result is that such mutations are not only likely, they are almost inevitable to occur and furthermore are likely to occur many times.

The Hardy-Weinberg law showed that if natural selection was not at work (along with some other conditions), populations settled into stable equilibrium values after just one generation of random mating. The next question to be addressed is to see how the populations change when natural selection is allowed to act. How likely is it that favorable mutations produced in the set of genes (the genotype) that characterize the organism (the phenotype) will end up with that actual organism predominating in the species? (Recall that natural selection acts on the phenotype and not the genes directly, while mutations act only on the genes and inheritance is passed on purely via the genes. A genetic change that has no effect on the phenotype will not influence the fitness of the organism.)

It is not the case that this happens every time. Most mutations are deleterious and do not spread and even favorable changes usually disappear by chance before they can spread to become a significant number in the population. But when the favorable mutations do take hold, that particular variety becomes widespread and dominates the population.

There are many such cases that have occurred in nature. The most famous and widely quoted example of this kind of growth of a favored phenotype is the case of the peppered moths in industrial areas of England and America. As a result of the pollution that created dark backgrounds on the lichen covering the trees where the moths rested, the darker varieties of the moths were camouflaged better from predators than the lighter ones and thus had a significant survival advantage. From 1848 to 1896, the darker forms grew to as much as 98% of the population. Subsequently, with the advent of pollution control measures that cleaned up the environment and reduced the soot pollution, the dark moth population decreased to as low as 10%. In The Making of the Fittest (2006, p. 52-53), Sean B. Carroll points out that peppered moths are not the only such examples, that many similar changes in coloration due to selection pressures have occurred in land snails, ladybird beetles, desert mice, and other species.

The way that even a very small natural selection advantage can result in that variety dominating a species can be appreciated using the more familiar example of compound interest. Suppose a parent gives each of two children $1,000 at the same time. One of the children invests in a bank that offers an interest rate of 5.0% while the other, being slightly more thrifty, shops around and invests in a different bank at 5.1%. Although they start out with the total money being split 50-50, in 7,000 years the second child (or rather that child’s descendents) will have 99.9% of the total money, thanks to that very small advantage in the annual rate of return.

It is exactly this kind of differential survival rate that plays such an important role in natural selection. Even minute differences in fitness can result, over the long term, in the runaway domination of a preferred variety. To see how fast this can happen, population geneticists have carried out calculations.

Suppose one variety has a mutated gene that has a very slight fitness advantage over the existing gene. ‘Fitness advantage’ can be quantified by defining the fitness w as the measure of the individual’s ability to survive and reproduce. (The concept of fitness is a combination of the organism’s ability to survive for any length of time (at least until its reproducing age is over) and its fecundity in terms of the number of offspring it produces.) Suppose the original gene has fitness w=1 and the new mutation has fitness w=1+s, where s is the selection advantage.

The selection advantage is a measure of how much more likely it is that that particular variety will propagate itself in future generations when compared with the standard type. So if, on average, the new mutated variety produces 101 fertile adult descendents while the same number of the standard organism produces 100, then s=0.01.

When this selection advantage is included in the calculation, the number of generations T it will take for a mutation to increase its frequency in the population from an initial value of f to a final value of F is given by the formula T=(1/s)ln[F(1-f)/f(1-F)], where ‘ln’ stands for the natural logarithm. (Molecular Evolution, Wen-Hsiung Li, 1997, p. 39)

So if we start with a trait that is present in just 0.1% of the population (i.e., f=0.001), and if this has a small selection advantage of size s=0.01, this variety will grow to 99.9% (F=0.999) of the population in just under 1,400 generations, which is a very short time on the geological scale.

But in a population of one million, an initial fraction of f=0.001 means that we are starting with about 1000 organisms having the favorable mutation. In reality, we are likely to begin with just one mutation in one organism. How does that affect the calculation?

That will be discussed in the next post in the series.

POST SCRIPT: Lil George and evolution

This clip from a cartoon show I had never heard of (probably because it is on cable) is pretty funny.