In the previous post, we saw that 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 will grow to 99.9% (F=0.999) 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. But it could be argued that new mutations usually start with just a single new kind of organism being produced in one single organism. How does that affect the calculation?

Suppose that you have a population of organisms of size N and they all start out having the same gene at a particular position (called the ‘locus’) on one of the chromosomes that make up the DNA. Now suppose a random mutation occurs in just one organism, the way that it was described in an earlier post in this series describing the shift from violet to UV sensitive sight in some birds. Most of the time, even a favorable mutation will disappear because of random chance because (say) that mutated organism died before it produced any offspring or it did produce a few and that particular gene was not inherited. But on occasion that mutation will spread. How likely is it that such a single mutation will spread to every single organism (i.e., become ‘fixed’ in the population)?

When one is not dealing with deterministic systems involving smoothly varying numbers (as was done in the previous case), a different kind of calculation (based on probabilities and known as ‘stochastic’) has to be done, and in this case the expectation value for the number of generations T taken for the single new mutation to spread all over and become fixed in the population (i.e. to spread to 100% of the organisms) is given by T=(2/s)ln(2N) generations, where ‘ln’ stands for natural logarithms. (Molecular Evolution, Wen-Hsiung Li, 1997, p. 49)

Even if s is taken as a very small advantage of size 0.01, for a population of N=one million, the average time taken for just a single mutation to become fixed is just 2,900 generations. So we see that mutations occurring in a single organism can become universal in a very short period on the geological time scale.

There are two important points that need to be emphasized.

There first is that even a very small selection advantage is sufficient to have that mutation dominate the species. This means that the advantage may not be even visible in the organism itself, which may look like every other organism in the species. For example, an eye mutation that works better by just a tiny bit may look like every other eye. Thus we should not think in terms of big changes for natural selection to work.

The second point is that even starting from a single mutation, as long as it takes hold (which has a probability of 2s of happening) and does not disappear and has an selection advantage however small, the mutation can spread surprisingly rapidly in the population and become universal and form the basis for future mutations.

It is interesting that even if there is no survival advantage to the new gene (i.e., s=0 and the mutation is said to be ‘neutral’), the mutation can on occasion still spread and become fixed, except that now the average time taken is much longer and given by T=4N generations. So that for a population of one million it would take on average about 4 million generations for a neutral mutation to spread everywhere, as compared to just 2,900 generations for a selection advantage of 0.01.

Darwin did not have access to this kind of mathematical analysis, which came long after his death. It is a tribute to his genius that he intuitively sensed the power of cumulative change over long time scales.

So far, I have shown how the first two items in the three components of natural selection, although seeming to have small probabilities of occurring, actually are quite likely. The third aspect of natural selection that has to be looked at is how the cumulative effects of small changes lead to big changes.

POST SCRIPT: Some real fact-checking

In yesterday’s post, I spoke about how the media does almost no fact-checking on Bush. Well, except for Jon Stewart who catches Bushies making up stuff about Iraq.