Trade-offs and design principles in the spatial organization of catalytic particles

Spatial organization of catalytic particles is ubiquitous in biological systems across different length scales, from enzyme complexes to metabolically coupled cells. Despite the different scales, these systems share common features of localized reactions with partially hindered diffusive transport, determined by the collective arrangement of the catalysts. Yet it remains largely unexplored how different arrangements affect the interplay between the reaction and transport dynamics, which ultimately determines the flux through the reaction pathway. Here we show that two fundamental trade-offs arise, the first between efficient inter-catalyst transport and depletion of substrate, and the second between steric confinement of intermediate products and accessibility of catalysts to substrate. We use a model reaction pathway to characterize the general design principles for the arrangement of catalysts that emerge from the interplay of these trade-offs. We find that the question of optimal catalyst arrangements generalizes the famous Thomson problem of electrostatics.