  Variation and selection are the core principles of Darwinian evolution, yet
quantitatively relating the diversity of a population to its capacity to
respond to selection is challenging. Here, we examine this problem at a
molecular level in the context of populations of partially randomized proteins
selected for binding to well-defined targets. We built several minimal protein
libraries, screened them in vitro by phage display and analyzed their response
to selection by high-throughput sequencing. A statistical analysis of the
results reveals two main findings: first, libraries with same sequence
diversity but built around different "frameworks" typically have vastly
different responses, second, the distribution of responses within a library
follows a simple scaling law. We show how an elementary probabilistic model
based on extreme value theory rationalizes these findings. Our results have
implications for designing synthetic protein libraries, for estimating the
density of functional biomolecules in sequence space, for characterizing
diversity in natural populations and for experimentally investigating the
concept of evolvability, or potential for future evolution.
