Problem H-701. During meiosis, two genes on the same chromosome are recombined if an odd number...

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Problem H-701. During meiosis, two genes on the same chromosome are recombined if an odd number of crossovers occurs in-between them. Let d be the distance in Morgans between two genes. (a) Show that the probability that the number of crossovers is odd equals e (b) Show that x2k+1 (2k + 1)! k=0 xm ex = m! m=0 -d d2k+1 (2k + 1)! k=0 X (which is called sinh(x)) by using the Taylor series (c) Combine (a) and (b) to derive a formula considered earlier this quarter: if genes A and B are a distance d apart (in Morgans) then the recombination rate between them is r = (1-exp(-2d)). (d) When d is "small, the distance d in Morgans is approximately the same as the recombina- tion rate as a fraction. Now we will see why. (i) Expand r as a Taylor series in d about d = 0 to at least three terms. (ii) The first nonzero term in the Taylor series is d. Plot r1 = d and r2 = (1-exp(-2d))/2 as functions of d on the same graph. You may copy it from your graphing calculator or use a printout from a graphing program on a computer. (iii) Examine the plots and give approximate values of d where r and r are almost equal. Problem H-701. During meiosis, two genes on the same chromosome are recombined if an odd number of crossovers occurs in-between them. Let d be the distance in Morgans between two genes. (a) Show that the probability that the number of crossovers is odd equals e (b) Show that x2k+1 (2k + 1)! k=0 xm ex = m! m=0 -d d2k+1 (2k + 1)! k=0 X (which is called sinh(x)) by using the Taylor series (c) Combine (a) and (b) to derive a formula considered earlier this quarter: if genes A and B are a distance d apart (in Morgans) then the recombination rate between them is r = (1-exp(-2d)). (d) When d is "small, the distance d in Morgans is approximately the same as the recombina- tion rate as a fraction. Now we will see why. (i) Expand r as a Taylor series in d about d = 0 to at least three terms. (ii) The first nonzero term in the Taylor series is d. Plot r1 = d and r2 = (1-exp(-2d))/2 as functions of d on the same graph. You may copy it from your graphing calculator or use a printout from a graphing program on a computer. (iii) Examine the plots and give approximate values of d where r and r are almost equal.