Next: General properties of genetic Up: Genetic Drift Previous: Introduction

# A simple example

To understand in more detail what happens when there is genetic drift, let's consider the simplest possible example: a haploid population consisting of 2 individuals.1 Suppose that there are initially two alleles in this population and . This implies that , but we'll ignore that numerical fact and simply imagine that the frequency of the allele is .

We imagine the following scenario:

• Each individual in the population produces a very large number of offspring.

• Each offspring is an identical copy of its parent, i.e., begets and begets . In other words, there's no mutation.

• The next generation is constructed by picking two offspring at random from the very large number of offspring produced by these two individuals.

Then it's not too hard to see that

Of course if both offspring sampled are , if one is and one is , and if both are , so that set of equations is equivalent to this one:
 (1) (2) (3)

In other words, we can no longer predict with certainty what allele frequencies in the next generation will be. We can only assign probabilities to each of the three possible outcomes. Of course, in a larger population the amount of uncertainty about the allele frequencies will be smaller,2 but there will be some uncertainty associated with the predicted allele frequencies unless the population is infinite.

The probability of ending up in any of the three possible states obviously depends on the current allele frequency. In probability theory we express this dependence by writing equations (1)-(3) as conditional probabilities:

 (4) (5) (6)

I've introduced the subscripts so that we can distinguish among various generations in the process. Why? Because if we can write equations (4)-(6), we can also write the following equations:3

Now if we stare at those a little while, we4 begin to see some interesting possibilities. Namely,

It takes more algebra than I care to show,5 but these equations can be extended to an arbitrary number of generations.

Why do I bother to show you these equations?6 Because you can see pretty quickly that as gets big, i.e., the longer our population evolves, the smaller the probability that becomes. In fact, it's not hard to verify two facts about genetic drift in this simple situation:

1. One of the two alleles originally present in the population is certain to be lost eventually.

2. The probability that is fixed is equal to its initial frequency, , and the probability that is fixed is equal to its initial frequency, .

Both of these properties are true in general for any finite population and any number of alleles.

1. Genetic drift will eventually lead to loss of all alleles in the population except one.7

2. The probability that any allele will eventually become fixed in the population is equal to its current frequency.

Next: General properties of genetic Up: Genetic Drift Previous: Introduction
Kent Holsinger 2012-09-23