  The emerging field of high-throughput compartmentalized in vitro evolution is
a promising new approach to protein engineering. In these experiments,
libraries of mutant genotypes are randomly distributed and expressed in
microscopic compartments - droplets of an emulsion. The selection of desirable
variants is performed according to the phenotype of each compartment. The
random partitioning leads to a fraction of compartments receiving more than one
genotype making the whole process a lab implementation of the group selection.
From a practical point of view (where efficient selection is typically sought),
it is important to know the impact of the increase in the mean occupancy of
compartments on the selection efficiency. We carried out a theoretical
investigation of this problem in the context of selection dynamics for an
infinite non-mutating subdivided population that randomly colonizes an infinite
number of patches (compartments) at each reproduction cycle. We derive here an
update equation for any distribution of phenotypes and any value of the mean
occupancy. Using this result, we demonstrate that, for the linear additive
fitness, the best genotype is still selected regardless of the mean occupancy.
Furthermore, the 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. Our conclusions may also
apply to natural genetic compartmentalized replicators, such as viruses or
early trans-acting RNA replicators.
