Protein Engineering Protocols - Mycobacteriology research center

Protein Library Design and Screening 143Fig. 6. Probability of obtaining “x” times a specific variant vs the number ofoccurrences of that variant. The example presented here uses the theoretical librarysize as n = 1 × 10 6 and m = 1 × 10 7 . Thus, there is a probability greater than 10% ofobtaining 8, 9, 10, or 11 times a specific variant, and a probability more than 50%of obtaining between 8 and 12 times a specific variant, using the specified values ofn and m.e−λr⋅ λ r!= e⎛ 110 × 7 ⎞−⎝⎜110 × 6 ⎠⎟7⎛ 1×10 ⎞⎝ 61×10 ⎠·10!10≅ 12.5%(5)This computation can be bypassed by using a built-in Poisson distributionfunction, available in several software programs (Excel, Matlab, Mathematica,and so on); refer to the accompanying Excel sheet (see Heading 2. and Fig. 1)for an example.As illustrated in Fig. 6, there is a reasonable likelihood (12.5%) of obtaining10 identical hits, and calculating for nearby values (i = 8, 9, 11, and 12) revealsthat it is also quite likely to obtain those numbers of identical hits (11.3, 12.5,11.4, and 9.5%, respectively). However, the probability that variant i appearsthree times is only:⎛ ⎝⎜e − 110 × 7 ⎞110 × 6 ⎠⎟7⎛ 1×10 ⎞⎝ 61×10 ⎠·3!3≅ 0.76%