Optimal spatial allocation of enzymes as an investment problem

Given a limited number of molecular components, cells face various allocation problems demanding decisions on how to distribute their resources. For instance, cells decide which enzymes to produce at what quantity, but also where to position them. Here we focus on the spatial allocation problem of how to distribute enzymes such as to maximize the total reaction flux produced by them in a system with given geometry and boundary conditions. So far, such distributions have been studied by computational optimization, but a deeper theoretical understanding was lacking. We derive an optimal allocation principle, which demands that the available enzymes are distributed such that the marginal flux returns at each occupied position are equal. This homogeneous marginal returns criterion (HMR criterion) corresponds to a portfolio optimization criterion in a scenario where each investment globally feeds back onto all payoffs. The HMR criterion allows us to analytically understand and characterize a localization-delocalization transition in the optimal enzyme distribution that was previously observed numerically. In particular, our analysis reveals the generality of the transition, and produces a practical test for the optimality of enzyme localization by comparing the reaction flux to the influx of substrate. Based on these results, we devise an additive construction algorithm, which builds up optimal enzyme arrangements systematically rather than by trial and error. Taken together, our results reveal a common principle in allocation problems from biology and economics, which can also serve as a design principle for synthetic biomolecular systems.