Protein Engineering Protocols - Mycobacteriology research center

138 Denault and Pelletierwe analyze the significance of results, either when assessing library quality or whenscreening a library. In both subheadings, formulae are presented and numericalexamples are provided, without full theoretical explanation, and are illustrated withan example. The full theoretical motivation of the examples is provided in the Notes(Subheading 4.).3.1. Library RepresentationWe design a library containing n possible, different, theoretical variants. Wesample m times, randomly, from this theoretical pool; we, thus, form a sampleof m variants. The problems we will address are the following:A. How many of the n theoretical variants do we expect not to appear among the mvariants chosen?B. What is the probability that at least one of the n theoretical variants has not beensampled, among the m variants chosen? What is the probability that at most a certainnumber of the theoretical variants have not been sampled?C. How many times can we expect a specific variant i to appear in the sample? Moregenerally, what is the probability that it appears r times?These problems are presented as a group because their resolution relies on thesame general theory, as described in Subheading 3.1.1. and 3.1.2. We shalldenote by p ithe probability that variant i will come up any time we sample once,randomly, from the theoretical pool.We make a distinction between two cases, and treat them in separate subheadingsbelow:1. In Subheading 3.1.1., the case of equiprobable outcomes, in which each of the nvariants has an equal probability of being sampled. Here, p i= 1/n; this case istreated in full generality.2. In Subheading 3.1.2., the case in which outcomes occur with unequal probabilities,i.e., some of the variants have a better chance of being sampled than others.By definition, it is difficult to generalize this case, because the answers to problemsA, B, and C depend on the probabilities themselves. As an example, we treatone case of unequal probabilities in detail.Equiprobable outcomes occur if there is no bias either in the expected frequencyof occurrence of a given sequence nor in the encoded characteristic ofinterest. In the case of library selection, if the characteristic of interest is independentof other parameters, or if the impact of other parameters is negligible,we consider the outcomes to be equiprobable. Outcomes may have unequalprobabilities if either considering biased codons or codon degeneracy, or else ifthe outcome affects the system. An example of the outcome affecting the systemis the following: when screening in vivo for an increase in the catalyticactivity of enzyme X, the increase in reaction product may be toxic toward the