Natural selection in compartmentalized environment with reshuffling

The emerging field of high-throughput compartmentalized in vitro evolution, a promising new approach to protein engineering, can be described as a lab-scale mimicking of natural selection processes. Libraries of mutants genotypes are randomly distributed in emulsified droplets, and the selection of desirable variants is achieved by differential amplification according to the phenotype in each microcompartment. The random partitionning leads to a fraction of compartments receiving more than one molecule, which increase with the mean occupancy—i.e. the average number of genotype-bearing agents per compartment. In these compartments, genotypes with different phenotypes will pool their activity but also share the total number of produced copies, in effect rendering the selection frequency-dependent. From a practical point of view (where efficient selections are typically sought), it is then important to know the impact of an increase in mean occupancy. We carried out a theoretical investigation of this problem in the context of selection dynamics for a simple model with an infinite, non-mutating population that is periodically redistributed among an infinite number of identical compartments. We derive here an update equation for any distribution of phenotypical activities and any value of the mean occupancy. It can be interpreted as a generalization of Price’s formula to the case of random clustering of the individuals. Using this result, we demonstrate that, for the linear additive fitness function, the best genotype is still selected regardless of the mean occupancy. Furthermore, the “natural” selection process is remarkably resilient to the presence of multiple genotypes per compartments, and slows down approximately inversely proportional to the mean occupancy at high values. We extend out results to more general expressions that cover nonadditive and non-linear fitnesses, as well non-Poissonian distribution among compartments. While the discussion uses the terminology of high-throughput directed evolution methods, our conclusions may also apply to natural genetic compartmentalized replicators, such as viruses or early transacting RNA replicators. ∗e-mail: [email protected] †e-mail: [email protected] 1 ar X iv :1 70 7. 07 46 1v 1 [ qbi o. PE ] 2 4 Ju l 2 01 7