Probability and Allele Frequency
The mechanisms of genetic drift can be illustrated with a simplified example. Consider a very large colony of bacteria isolated in a drop of solution. The bacteria are genetically identical except for a single gene with two alleles labeled A and B. Half the bacteria have allele A and the other half have allele B. Thus both A and B have allele frequency 1/2.
A and B are neutral alleles—meaning they do not affect the bacteria's ability to survive and reproduce. This being the case, all bacteria in this colony are equally likely to survive and reproduce. The drop of solution then shrinks until it has only enough food to sustain four bacteria. All the others die without reproducing. Among the four who survive, there are sixteen possible combinations for the A and B alleles:
(A-A-A-A), (B-A-A-A), (A-B-A-A), (B-B-A-A),
(A-A-B-A), (B-A-B-A), (A-B-B-A), (B-B-B-A),
(A-A-A-B), (B-A-A-B), (A-B-A-B), (B-B-A-B),
(A-A-B-B), (B-A-B-B), (A-B-B-B), (B-B-B-B).
If each of the combinations with the same number of A and B respectively are counted, we get the following table. The probabilities are calculated with the slightly faulty premise that the peak population size was infinite.
| A | B | Combinations | Probability |
| 4 | 0 | 1 | 1/16 |
| 3 | 1 | 4 | 4/16 |
| 2 | 2 | 6 | 6/16 |
| 1 | 3 | 4 | 4/16 |
| 0 | 4 | 1 | 1/16 |
The probability of any one possible combination is
where 1/2 (the probability of the A or B allele for each surviving bacterium) is multiplied four times (the total sample size, which in this example is the total number of surviving bacteria).
As seen in the table, the total number of possible combinations to have an equal (conserved) number of A and B alleles is six, and its probability is 6/16. The total number of possible alternative combinations is ten, and the probability of unequal number of A and B alleles is 10/16.
The total number of possible combinations can be represented as binomial coefficients and they can be derived from Pascal's triangle. The probability for any one of the possible combinations can be calculated with the formula
where N is the number of bacteria and k is the number of A (or B) alleles in the combination. The function '' signifies the binomial coefficient and can be expressed as "N choose k". Using the formula to calculate the probability that between them the surviving four bacteria have two A alleles and two B alleles.
Genetic drift occurs when a population's allele frequencies change due to random events. In this example the population contracted to just four random survivors, a phenomenon known as population bottleneck. The original colony began with an equal distribution of A and B alleles but chances are that the remaining population of four members has an unequal distribution. The probability that this surviving population will undergo drift (10/16) is higher than the probability that it will remain the same (6/16).
Read more about this topic: Genetic Drift
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