• Random mating, large population
• No migration
• No mutation
• No selection
• No genetic drift

Small subsamples of any large population are not representative of the population as a whole. This is why political pollsters use samples of 1000 or more people to estimate voter preferences.

Offspring are "genetic samples" of the parental population. If there are few offspring, the allele frequency distribution among them may be different from the distribution among the parents.

Example from Hartl, Primer of Population Genetics

Assume you have 2 plants, and every year you take just 2 seeds from them and plant the seeds.

If both parents are heterozygous (Aa), then the allele frequencies in the parents are p (A) = 0.5 and q (a) = 0.5. The genotype frequencies among the seeds are
AA = p² = 1/4,
Aa = 2pq = 1/2
aa = q² = 1/4.

If we pick 2 seeds at random, the chance of picking a AA seed on the first try is 1/4, and the chance of picking a AA seed on the second try is 1/4. Thus the probability of picking both seeds of type AA is 1/4 x 1/4 = 1/16. Thus in a single generation there is a 1/16 chance that the A allele will become fixed (be 100%) in this population. By similar logic, the chance of picking one aa seed on the first try is 1/4, and on the second try the chance of picking another aa seed is 1/4, so the probability of picking both seeds of type aa is 1/16. Hence there is a 1/16 chance that in one generation the A allele will be lost.

Therefore in a population of size 2, each generation there is a 6.25% chance of losing or fixing one allele in a 1-locus 2-allele system.

The intensity of this process of random genetic drift is dependent upon population size. Small populations drift a great deal from one generation to the next, and large populations drift little. This process occurs in subdivided populations and in situations where few individuals reproduce, such as conserved populations in zoos (example: Snow Leopards and other endangered species), and in populations that are stocked with the offspring of few parents (example: Striped Bass in South Carolina lakes).

The rate of loss of alleles is dependent upon population size, and the process is similar to radioactive decay.

Allele A --> a at rate "mu".
Allele a --> A at rate "nu". This is back mutation to original allele.

How do allele frequencies change over time? Build a model to estimate what happens.
If p_t is the frequency of "A" alleles in the population at time t, and p_t-1 is the frequency of "A" alleles in the previousgeneration,

p_t = p _t-1 ( 1 - mu) + (1- p_t-1) nu

the quantity (1-mu) represents the fraction of "A" alleles that did not mutate into "a" alleles : these alleles are still "A" at time t.

the quantity (1 - p_t-1) represents the fraction of alleles that were a in the previous generation (if p is the fraction that was "A", then 1 minus p is the fraction that was "a"). Of the individuals that were type "a", fration n mutated.

There is a balance between the rates of forward and backward mutation, resulting in an equilibrium allele frequency
p equilibrium = mu/(mu+nu)

Typical rates are mu = 10^-4 and nu = 10^-5

The approach to equilibrium occcurs at a rate (1 - mu - nu) per generation:

p_t = p_equilibrium + (p₀ - p_equilibrium)(1 - (mu - nu))^t
This is a very slow process.

This graph shows the approach to equilibrium from an initial allele frequency of 0.5, with mu=10^-4 and nu=10^-5 . Note the time scale in generations.
 
 

In a habitat at time (t-1) the frequency of "A" alleles was p_t-1. If m is the rate of migration, then

p_t = p_t-1 (1-m) + P m

the proportion of the population that was "A" in the last generation (p_t-1 remains "A" if individuals do not migrate). The quantity (1-m) is the fraction of individuals that do not migrate. If P is the proportion of A alleles in the surrounding habitat, then "A" alleles migrate into the local area each generation when fraction "m" of the surrounding population migrates in. "A" alleles in the local population result from "A" individuals that stay, and "A" individuals that migrate from outside.

p_t = (p₀ - P) (1-m)^t

Typical values of m are 0.1, so this is a rapid process.

This graph shows the approach to equilibrium from an initial allele frequency of 0.5, a migration rate of 0.1 and a surrounding habitat allele frequency P=0.3. Note the time scale is 20 generations.

This page last modified Wednesday, 11-Sep-2002 22:44:45 EDT