Formalizing the law of diminishing returns in metabolic networks using an electrical analogy

The way biological systems respond to changes in parameter values caused by mutations is a key issue in evolution and quantitative genetics, as it affects fundamental aspects such as adaptation, selective neutrality, robustness, optimality, evolutionary equilibria, etc. We address this question using the enzyme-flux relationship in a metabolic network as a model of the genotype-phenotype relationship. The lack of a suitable mathematical tool from biochemical theory to investigate this relationship led us to use an analogy between electrical circuits and metabolic networks with uni-uni reactions. We show that a behaviour of diminishing returns, which is commonly observed at various phenotypic levels, is inevitable, irrespective of the complexity of the system. We also present a possible generalization to metabolic networks with both uni-uni and bi-bi reactions.