In the Wright-Fisher model, the 2Ncopies of a locus in consideration could come eitherfromNdiploidindividuals (who have two copies of their genetic material in each cell) or2Nhaploidindividuals (who have one copy of their genetic material in each cell). ConsiderMoran modelwithonetype of mutation. We think ofn= 2N haploid individuals, anda genetic locus with two allelesAandathat have the same fitness. This model evolvesacoording to the following rules:
(i) At each time step, an individualxis chosen uniformly at random out ofnindividuals.We want to replacexwith a clone.
(ii) For this, we “clone” an individual chosen at random from the set of allnindividuals,includingxitself. Then, we replacexwith the clone.
(iii) A specific mutation may occur during cloning. If we are cloning alleleA, then it maymutate to alleleawith probability 0
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iii) A specific mutation may occur during cloning. If we are cloning alleleA, then it maymutate to alleleawith probability 0< ρ <1. Cloning alleleaalways happens withoutmutation.Problem 1.LetS={0,1,2,…,n}be the sample space andXtbe the number of allelesAat timet. Write down the transition probabilities for the Markov chainXtover the statespaceS. Next, properly define thefixation timeTf.Problem 2.Compute the mean fixation time condition on the event that the Markov chainbegins withX0=jallelesA, i.e. findE[Tf|X0=j].Is there a differential equations approach?