# Probability in genetics pdf

Whether a gene will ultimately be lost or fixed is probability in genetics pdf on selection coefficients and chance fluctuations in allelic proportions. The earliest mention of gene fixation in published works was found in Kimura’s On Probability of Fixation of Mutant Genes in a Population.

This paper was published in 1962. In Kimura’s paper, he is able to use mathematical techniques to determine the probability of fixation of mutant genes in a population. He was able to determine that the probability of fixation is dependent on the initial frequency of the allele and the mean and variance of the gene frequency change per generation. Thus, the rate of fixation for a mutation not subject to selection is simply the rate of introduction of such mutations. Markov chain under the following assumptions.

The number of offspring of any one individual must follow a fixed distribution and is independently determined. This estimate can be obtained directly from Kimura’s 1962 work. Probability of fixation is also influenced by population size changes. For growing populations, selection coefficients are more effective. This means that beneficial alleles are more likely to become fixed, whereas deleterious alleles are more likely to be lost.

In populations that are shrinking in size, selection coefficients are not as effective. Thus, there is a higher probability of beneficial alleles being lost and deleterious alleles being fixed. This is because if a beneficial mutation is rare, it can be lost purely due to chance of that individual not having offspring, no matter the selection coefficient. In growing populations, the average individual has a higher expected number of offspring, whereas in shrinking populations the average individual has a lower number of expected offspring. Thus, in growing populations it is more likely that the beneficial allele will be passed on to more individuals in the next generation. This continues until the allele flourishes in the population, and is eventually fixed.

However, in a shrinking population it is more likely that the allele may not be passed on, simply because the parents produce no offspring. This would cause even a beneficial mutation to be lost. Additionally, research has been done into the average time it takes for a neutral mutation to become fixed. Usually the population statistic used to define effective population size is heterozygosity, but others can be used. Fixation rates can easily be modeled as well to see how long it takes for a gene to become fixed with varying population sizes and generations. Additionally, fixation rates can be modeled using coalescent trees. A coalescent tree traces the descent of alleles of a gene in a population.