Design principles for the optimal arrangement of consecutive enzymes

The spatial organization of enzymes is important in the regulation and the efficiency of metabolic pathways. In previous studies, we showed how to optimize the production rate of the last metabolite of a cascade of enzymatic reactions. We observed a transition in the optimal enzyme arrangement from a clustered configuration into a more extended profile. This transition resulted to be independent of the reaction kinetics, spatial dimensions and loss mechanisms considered. However this generality was not fully understood. Here we derive a condition defining the boundary between the two different regimes. The condition compares two measurable quantities: the reaction flux at the cluster and the diffusive flux away from the cluster. Further, we describe the optimal arrangements as the result of a diversification strategy for enzymes investment. We analyze the marginal returns of the product flux per extra enzyme added at different positions in the system and we find that, for the optimal profiles, different positions generate the same marginal returns. Starting from this strategy, we derive an algorithm for building the optimal arrangements as the amount of enzymes in the system increases, thereby suggesting possible instructions for the realization of optimal bio-reactors.