Q. 3(Genetic drift).We have a population ofNpeople. Each person has one of two mutationsAandBof a certain gene. How are these mutations inherited by successive generations? The simplest model of this type of evolution is theWright-Fisher model. In this model, the population sizeNis assumed to be the same in every generation. Given a population ofNpeople (some of whom have mutationAand some of whom have mutationB), we form the next generation ofNpeople as follows. First, select one individual from the parent population uniformly at random (that is, each individual has probability1/Nof being selected). That person gives birth to a child who inherits her parent's mutation of the gene. Then, independently, select another individual from the parent population uniformly at random, and that person gives birth to a child who inherits her parent's mutation. This is repeatedNtimes, so that we obtain a new population ofNchildren who form the next generation. (Note that in this random sampling process, some parents may be selected multiple times and thus have multiple children, while some parents may never be selected and thus are childless.) The next generation reproduces in the same manner, and so on.

LetXnbe the number of people in generationnthat have gene mutationA.

a. What possible values canXntake?

b. Show that{Xn}is a Markov chain with the following transition probabilities:NijiNj

Pij=j N1N.1

c. What states are recurrent and what states are transient?

d. What happens to the numberXnof mutations of typeAafter a long timen ? Can you give a biological interpretation?