Protein Engineering Protocols - Mycobacteriology research center

Protein Library Design and Screening 1515. The answer to Problem C relies on the Poisson approximation of the binomial distributionfollowed by the random variable that counts the number of times the variantis drawn. Here, the Poisson approximation should be “very successful if m ≥ 20and λ ≤ 10,” still according to (16).6. The event of obtaining a specific variant i a certain number of times r in the samplefollows a binomial distribution of parameters m (sample size) and (the prob-1nability of picking variant i anytime a variant is picked randomly). This binomialdistribution is well approximated, and much easier to compute, as a Poisson distributionwith parameter λ= m n.7. See Note 2. The λ for this case is computed as follows. In opposition to theequiprobable case, the λ iare not identical here. At first glance, to compute λ, wewould need to compute all of n = 21 10 parameters λ i,which is enormous, butwhich contains many repetitions, a characteristic that we use to simplify the computations.Rather, let us identify any decapeptide with the 5-tuple (n 1,n 2,n 3,n 4,n 5)16that indicates the number of codons of, respectively, probability and in64 , 264 , 364 , 4645the said decapeptide. Clearly, ∑ n , otherwise there are not 10 codons ini i= 10= 1the decapeptide. Also, there are:64 ,10!2 9 1 5 3n ! n ! n ! n ! n !1 2 3 4 5n1 n2 n3 n4 n5(13)different decapeptides associated with the 5-tuple (n 1,n 2,n 3,n 4,n 5). This numberaccounts for all shufflings of the codons with the peptide, i.e., the order of the aminoacids within the peptide needs not be explicitly defined. It also accounts for the factthat more than one amino acid displays each probability (except for Ile, the onlyamino acid encoded by 3/64 codons). The point we make is that the 5-tuples containall of the information regarding the decapeptides required to answer our questions.We can now replace the computation of the λ i= (1 – p i) m ,which is specific todecapeptide i, by the computation of a λ(n 1,n 2,n 3,n 4,n 5), such that:λi= λ( n , n , n , n , n ) = [ 1− p( n , n , n , n , n )]1 2 3 4 5 1 2 3 4 5m(14)and which is specific to the 5-tuple (n 1,n 2,n 3,n 4,n 5) associated with decapeptide i.That is, of the 21 10 parameters λ i, we will compute only the different values thatoccur in the λ i. There are:⎛10 + 5 −1⎞141001⎝⎜ 5−1⎠⎟ = !10!! 4=(15)different values in the λ i; here, we use the binomial coefficient notation,⎛ ⎞rr!=⎝⎜ s⎠⎟ ( r− s ) ! s!