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

Abstract

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.

Keywords: diminishing returns; electrical circuit; metabolic flux; robustness.

Figure 1.

Two basic types of electrical circuits. (a) A circuit with resistors exclusively in series and in parallel. (b) A Wheatstone bridge. (c) Relationship between the equivalent conductance ${\sigma }_{\text{E}}$ and the individual conductances in circuit (a) (same colour code as in (a)). For each curve, one conductance increased from 0 to 20, the other conductances being fixed. The fixed conductance values are $1/R_1=1$, $1/R_2=10$ and $1/R_3=1.25$. (d) Relationship between the equivalent conductance ${\sigma }_{\text{E}}$ and the individual conductances in the Wheatstone bridge in (b). The fixed conductance values are $1/R_1=0.43$, $1/R_2=1$, $1/R_3=1.25$, $1/R_4=1$ and $1/R_5=5$.