Problem
The Jukes-Cantor distance is an estimate of the number of mutations that occurred per site over the course of one sequence evolving from another. A simpler estimate for this number is just p, the proportion of sites that have changed from the initial to final sequence.
1) Explain why multiple mutations at the same site would cause p to be less reliable. Does it give an overestimate or underestimate of the true amount of mutation?
2) Give an intuitive explanation of why, if p is relatively small, so that the sequences have few differences, this simpler estimate might be reasonable anyway.
3) Explain why part (b) is consistent with the Jukes-Cantor model. That is, explain why for small p, -3/4ln(1-4/3p)=p by using the approximation for ln(1 + x) valid for small x.
4) It has been claimed that, if p is less than .1, it can be used as a reasonable approximation of the Jukes-Cantor distance. Do you agree? Illustrate by graphing both and d = p for 0 ≤ p < 3/4.