Deviations from the null hypotheses: Finite populations sizes and genetic drift, mutation and gene flow
I. Genetic drift: the effect of population size alone on allele and genotype frequencies
A. Finite population size alone results in a change in allele frequency (which results in a decline in heterozygosity)
1. The probability of an individual being autozygous in a population of finite size N (by random sampling alone) is the probability of getting any gene copy as one of the alleles in a diploid genotype (=1) AND the probability of getting the same gene copy by random sampling (= 1/(2N)). That is, for any particular individual in a randomly mating population of finite size N,
F = 1/(2N).
2. The probability of an individual in this population being allozygous for these gene copies is thus:
1-F = 1 - (1/(2N))...
UNLESS these gene copies came from a population already inbred to an extent Ft-1, (where t is the present generation), in which case,
Ft = 1/(2N) + [1 - 1/(2N)]Ft-1
(i.e., the probability of either being autozygous in the present generation OR the probability that an individual comes from that fraction of the population that is already autozygous due to inbreeding in the previous generation).
3. Because Ft = (H0-Ht)/H0 and Ft-1 = (H0-Ht-1)/H0,
Ht = Ht-1(1 - 1/(2N))...
That is, in each successive generation (from t-1 to t), the heterozygosity (H) declines by 1/(2N).
4. Also, the smaller the population size (N), the faster the decline in heterozygosity.
5. This decline in heterozygosity is due to the increase in frequency of one of the alleles, which approaches fixation.
6. But only chance (i.e., random sampling) dictates which allele leaves more descendents and becomes fixed.
7. Note that this kind of loss in heterozygosity differs from that due to nonrandom mating, since a HW equilibrium is still approximately maintained in this finite population.
8. This stochastic change in allele frequency resulting simply from the finite size of a population is called "genetic drift".
II. Genetic drift can result in evolutionary divergence
1. Because of genetic drift, the variance between demes (small subpopulations) increases over time (i.e., demes will diverge, esp. if they become isolated).
2. Given enough time, allele A or a will become fixed (p = 1 or p = 0).
3. The allele that is already more frequent will have a higher probability of being fixed:
The probability of a ("neutral") allele is its frequency (= 1/(2N) for a single gene copy).
4. Thus, the main features of genetic drift are:
- A loss of genetic variation results within populations
- Genetic divergence results between populations
- Evolution results (i.e., allele frequencies change, until H = 0)
III. Mutations, their rate of fixation and Kimura's molecular "clock"
A. Mutations form new alleles which thus arise at an initial frequency of 1/(2N). Because the probability that an allele will be fixed is equal to its frequency, there is a small probability (1/(2N)) that it will become fixed (by chance alone, if it is neutral to selection).
B. Rate of fixation of new mutations
- In any generation, the number of possible copies at which a new mutation could originate is 2N (i.e., new mutants could exist at any of the gene copies in the population).
- The probability of getting a new mutant is thus 2Nu (i.e., the rate at which mutation occurs at any one gene copy, applied to all gene copies)
- Thus, the total number of mutations that will be fixed per generation is:
(2Nu) (1/(2N)) = u
(i.e., the probability of getting a new mutant in the population and then the probability that this mutant will be fixed)
- That is, the rate at which new (neutral) mutants are fixed is 1/u (i.e., the number of generations for a mutant to be fixed, which is the inverse of the number of mutants fixed per generation); this rate is INDEPENDENT of population size, N.