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6 3.6 Exercise 3.5, if done repeatedly, will eventually reveal that the best distance measure (between distributions) is the symmetrized relative entropy (case (iii)). Notice that this means that when comparing two distributions we quantify their difference not by a difference on Shannon entropies, case (i). In other words, we choose:Difference(X, Y) = MI(X, Y) = H(X) + H(Y) – H(X, Y),Not Difference = ∣ H(X) – H(Y)∣.The latter case satisfies the metric properties, including triangle inequality, in order to be a “distance” measure, is this true for the mutual information difference as well?

7 3.7 Go to genbank (https://www.ncbi.nlm.nih.gov/genbank) and select the genome of the K‐12 strain of E. coli. (The K‐12 strain was obtained from the stool sample of a diphtheria patient in Palo Alto, CA, in 1922, so that seems like a good one.) Reproduce the MI codon discovery described in ssss1.

8 3.8 Using the E. coli genome (the one described above) and using the codon counter code, get the frequency of occurrence of the 64 different codons genome‐wide (without even restricting to coding regions or to a particular “framing,” these are still unknowns, initially, in an ab initio analysis). This should reveal oddly low counts for what will turn out to be the “stop” codons.