Protein Engineering Protocols - Mycobacteriology research center

Protein Library Design and Screening 145where Met and Trp are encoded by 1 out of 64 codons each; Phe, Tyr, His, Gln,Asn, Lys, Asp, Glu, and Cys are encoded by 2 out of 64 codons each; and soon. Thus, there are, respectively, 2, 9, 2, 5, and 3 amino acids or stop codons,respectively, corresponding to each of the five probabilities given.3.1.2.2. PROBLEM A{ }1 2 3 4 6, , , , ,64 64 64 64 64How many of the n theoretical variants do we expect not to appear amongthe m variants chosen?To resolve this problem, the principal factor to consider is the codon degeneracy.We consider that each unique decapeptide has the same probability ofoccurrence as any other decapeptide of the same amino acid composition (and,thus, of the same degeneracy), irrespective of the order of the amino acids. Theanswer is the parameter λ (see Note 7), computed as:λ≅∑( n1, n2, n3, n4, n5)⎛ 10!2 9 2 5 3⎝⎜ nnnnn ! ! ! ! !1 2 3 4 5⎞e⎠n1 n2 n3 n4 n5⎟ − pn ( 1, n 2, n 3, n 4, n 5)m(6)where the summation is over all ordered series of five numbers (n 1,n 2,n 3,n 4,n 5),which represent the five probabilities of codon degeneracy given. Each of thefive numbers is taken from the set {0,1,…,10} to represent how many aminoacids within the decapeptide are of each of the five degeneracies. Thus,5∑ n i= 10 because we consider 10 amino acids. For example, (n 1,n 2,n 3,n 4,n 5) =i=1(8,0,0,1,1) means that 8 positions of the decapeptide are encoded by a nondegenerateamino acid (1/64), no positions contain amino acids of degeneracy2/64 or 3/64, and one position each is encoded by an amino acid with degeneracy4/64 and 6/64.p(n 1,n 2,n 3,n 4,n 5) is computed as:1pn (1, n2, n3, n4, n5)= ⎛ ⎞⎝ 64⎠n1 n2⎛ 2 ⎞⎝ 64⎠⎛ 3 ⎞⎝ 64⎠n3 n4 n5⎛ 4 ⎞⎝ 64⎠⎛ 6 ⎞⎝ 64⎠(7)This p(n 1,n 2,n 3,n 4,n 5) is a new name for what was introduced as simply p iatthe outset of Subheading 3.1., the probability that a certain variant comes upany time we sample once, randomly, from the theoretical pool.Unfortunately, there seems to be no simpler expression for λ; the summationinvolved in the computation of λ must be programmed on a computer.For example, we sample m = 10 14 decapeptides, randomly, from the theoreticalpool of n = 21 10 (i.e., ~1.67 × 10 13 ; see Fig. 2).