Natural selection in compartmentalized environment with reshuffling

The emerging field of compartmentalized in vitro evolution, where selection is carried out by differential reproduction in each compartment, is a promising new approach to protein engineering. From practical point of view, it is important to know the effect of the increase in the average number of genotype bearing agents per compartment. This effect is also interesting on its own in the context of primordial evolution in the hypothetical RNA world. The question is important as genotypes with different phenotypes in the same compartment share their fitness (the number of produced copies) rendering the selection frequency-dependent. I will show the results of a theoretical investigation of this problem in the context of selection dynamics for a simple model with an infinite population that is periodically redistributed among infinite number of identical compartments, inside which all molecules are copied without distinction with the success rate as a function of the total genomic composition in the compartment. Surprisingly, with a linear selection function, the selection process is slowed down only approximately inversely proportional to the average number of individuals per compartment. I will also demonstrate exact forms of the governing equations for some nonlinear selection functions. Finally, I will expose an intriguing open problem of an apparent phase transition for an exponential selection function seen in numerical experiments, which is missed by the current infinite population theory.