Protein Engineering Protocols - Mycobacteriology research center

Protein Library Design and Screening 139system, resulting in indirect adverse effects toward the detected catalytic activity.Such parameters are generally too complex to be systematically accounted for.However, we can take into account the use of biased codons, in which aminoacids will occur with unequal probabilities, as well as other parameters which,although being unequal, can be quantified.3.1.1. The Case of Equiprobable OutcomesThis is the simplest case, in which all outcomes are considered to have anequal probability: after sampling any one library member, all n variants have thesame chance of being chosen. Formally, p i= 1/n.3.1.1.1. PROBLEM AHow many of the n theoretical variants do we expect not to appear amongthe m variants chosen?An easy, approximate answer is given by the parameter λ (see Notes 1 and 2),defined as:⎛ ⎞λ = n⎝1−1n⎠m(1)For example, suppose we sample m = 1 × 10 7 times from a theoretical poolof n = 1 × 10 6 variants. The probability of any variant of coming up on any onedraw is:1 1pi = =n 1×10 6The expected number of missing theoretical variants is given by λ calculatedas follows:6 ⎛ 1 ⎞λ = 1× 10⎝⎜1−610 . × 10 ⎠⎟1×10 7≅ 45.4Note that the expected number of missing variants need not be an integer;here, the expected number of missing variants is between 45 and 46 variants outof the 1 × 10 6 theoretical variants.If instead we sample m = 2 × 10 7 times from the same theoretical pool ofn = 1 × 10 6 variants, λ=0.00206. Thus, the expected number of missing variantsis between 0 and 1, although much closer to 0 than 1. In fact, in this example,the probabilities that, respectively, 0 or 1 variants are missing are the onlyprobabilities that are not very small, with 0 being much more likely (>99%, ascalculated in Subheading 3.1.1.2.).We can conclude that sampling a 20-fold excess of items relative to thetheoretical library size (m = 20n) results in essentially complete sampling of thetheoretical library, as illustrated in Fig. 5. Sampling a 10-fold excess results