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Original URL path: /TsunamiIdWeb.htm (2016-02-12)- CEB1 Evolutionary Model

4 Determining the model parameters From the data of observed mutation rates for 23 progenitor alleles of various sizes we developed a model in stages 4 a mutation probability m s as a function of progenitor size s figure 2a A linear regression m s as b gives b 1 75 but there is no evidence p 0 45 from the data that b not 0 If b 0 then the linear fit becomes m s s 521 with considerable random variation R 2 0 42 A quadratic fit m s s 303 s 2 50000 explains the data only a little better R 2 0 52 at the expense of an extra parameter It is illogical anyway for it says that the mutation rate eventually declines above 82 repeats and even becomes negative above 165 repeats Therefore we choose the simple linear model m s s 521 4 b proportion d s of mutations that are contractions expansions figure 2b Contractions and expansions approach equal frequency for large sizes suggesting a model d s 0 5 k s for contractions and e s 0 5 k s for expansions A linear regression d s 0 5 as a function of 1 s gives a constant term of only 1 130 and the p value of 0 8 as well as the attraction of symmetry further suggest constraining the extra constant to be 0 Therefore we obtain d s 0 5 3 28 s R 2 0 6 For s 4 c contraction amount assuming contraction occurs D s figure 2c A linear regression gives D s 0 67 s 15 4 for which R 2 0 71 However the proposition that the constant term is non zero is only weakly supported by the data p 0 17 and D s s 13 explains the data nearly as well R 2 0 67 Therefore we choose the simpler formula 4 d expansion amount assuming expantion occurs E s figure 2c We choose to model E s by the linear regression E s 3 54 s 50 It may be p 0 12 that the linear term is unnecessary the constant fit E s 4 2 fits the data only a little less well 4 e distribution of contraction amounts From the data the mutation amount a s j tends to be small This suggests a model of the form P a repeat unit contraction proportional to k s a 1 0 k 1 However such a model doesn t account for the occasional contraction by a large amount therefore we chose a more elaborate model wherein decay of mutation probabilities with a 1 is slower than exponential One way to accomplish this is to replace a 1 by a 1 beta for some beta 1 For any given beta for each s the value of k s is constrained by the condition that the mean contraction amount must be D s and thus an expected number of contractions S s

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