Protein Engineering Protocols - Mycobacteriology research center

Protein Library Design and Screening 141Table 2Ratio n/m = fold λ=expected Expectedlibrary number not percentage P (at leastn m representation picked not picked 1 missing) (%)1000 500 0.5 606 60.6 1001000 1 368 36.8 1002000 2 135 13.5 10010,000 10 0.05 0.00 4.4220,000 20 0.00 0.00 0.001 × 10 6 5 × 10 5 0.5 6.07 × 10 5 60.7 1001 × 10 6 1 3.68 × 10 5 36.8 1002 × 10 6 2 1.35 × 10 5 13.5 1001 × 10 7 10 45.4 0.00 1002 × 10 7 20 0.00 0.00 0.211 × 10 10 5 × 10 9 0.5 6.07 × 10 9 60.7 1001 × 10 10 1 3.68 × 10 9 36.8 1002 × 10 10 2 1.35 × 10 9 13.5 1001 × 10 11 10 4.54 × 10 5 0.00 1002 × 10 11 20 20.6 0.00 100where λ is defined as in Eq. 1 (see also Notes 1 and 3). From there, the probabilitythat at least one theoretical variant is missing is approximately 1 – e –λ(see Note 4). The probability that at most K of the theoretical variants are missingis obtained by summing the values of the probability in Eq. 2 for k = 0, 1,2, … up to K.To resolve problem B, we need to first compute the value λ as in Subheading3.1.1.1. For our example given in problem A, in which we sample 1 × 10 7 timesfrom a theoretical pool of 1 × 10 6 variants, we computed λ ≈45.4. Then, theprobability that at least one theoretical variant has not been sampled is computedas 1 – e –45.4 1.00, or 100%. Therefore, it is essentially certain (100%) that atleast some of the theoretical variants are missing in the sample of m variants. Inpractice, what does this mean? This result (and the additional examples providedin Table 2) indicates that if one screens a sample size m that is 10 times greaterthan the theoretical library size n, for n greater or equal to 10 5 , it is highly probable(almost 100%) that the theoretical library n has not been completely sampled.For smaller libraries (n = 1000 and less), there is only a weak probabilitythat sampling with m = 10 × n gives an incomplete coverage of the theoreticallibrary (for example, the probability is ~5% for n = 1000, see Table 2).By way of contrast, if we sample twice as many, i.e., m = 2 × 10 7 items sampledfrom the same theoretical pool of n = 1 × 10 6 variants, λ=0.00206 and the