Protein Engineering Protocols - Mycobacteriology research center

150 Denault and Pelletieressentially states that if all of the λs, defined in Eq. 9 and Eq. 14, are small, thenthe number of variants that do not occur in the sample follows approximately aPoisson distribution with parameter λ = ∑ iλ (see Chapter 10 of ref. 15, particularlyExample 10.2(B) regarding the multinomial distribution, for the mathemat-n= 1 iical details).The Poisson approximation should be “very successful if n ≥ 20 and λ≤10,”according to (16), page 252.2. The number of missing variants approximately follows a Poisson distribution withnthe parameter λ, defined as λ= ∑ 1λ i , and where λ iis defined as:λ i= P(variant i is missing from the sample) = (1 – p i) m (9)By the property of Poisson distributions, the expected value of the number of missingvariants is the parameter λ itself.In the case of equiprobable outcomes, because pi = 1 , it is easy to compute λ iandnλ as:and:i=m ⎛λ i= ( 1− p i) = 1−1 ⎞⎝⎜n⎠⎟mn⎛λ= 1 1 ⎞− 1 1⎝⎜⎠⎟ = ⎛⎝⎜ − ⎞∑ n n n⎠⎟i=1mm(10)Note that, given the special form of λ 1, the value λ can be computed directly withoutcomputing the λ i.3. Computations of Poisson distributions with Excel can stall when the distribution’sparameter λ is too large. In such cases, one can simply use the normal distributionwith the mean equal to λ and the variance equal to λ, i.e., N(λ,λ) instead. This trickis illustrated in Fig. 2.4. By virtue of the Poisson distribution:–e λ kλPk ( variants are missing) ≅ , k ∈ { 01 , ,…,n}k!(11)where λ is defined in Eq. 1, or, identically, in Eq. 10.The probability that none is missing is then approximately:−λe0!= eso that the probability that at least one is missing is approximately:λ 0−λ1− e –λ(12)