491612.fig.001a
(a)
491612.fig.001b
(b)
491612.fig.001c
(c)
491612.fig.001d
(d)
491612.fig.001e
(e)
Figure 1: (a) Phage library can be described by multisets made of and . Any change to the library can be described as function/operator acting on the . (b) Relevant functions are calculations of total sequences (sum) and unique sequences (uni). (c) Any transformation of library to another library is an operator acting on . Sampling of libraries to yield a sublibrary is the most important operator. (d) It can be described as matrix. Specifically, is a diagonal matrix of values derived from random distribution. Rounding function is necessary to ensure the physical meaning of the sampling results. acting on the same vector yields one of many vectors that have the same number of total elements. As a consequence, is nonlinear, nondistributive, and noncommutative operator. Average of many operators is a scalar (dilution factor). (e) Any screen of any library can be described as operators acting on the copy number vectors of the naïve (or theoretical) library. Copy number vectors cannot be observed directly. They have to be measured through sequencing. As sequencing contains sampling process ( operator), the result of sequencing is nondeterministic. Sequencing yields one of many possible observed copy number vectors, none of which are equal to the real copy number vector.