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

Catalytic particles are spatially organized in a number of biological systems across different length scales, from enzyme complexes to metabolically coupled cells. Despite operating on different scales, these systems all feature localized reactions involving partially hindered diffusive transport, which is 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 the depletion of substrate, and the second between steric confinement of intermediate products and the 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 well-known Thomson problem of electrostatics.