The value structure of metabolic states
To improve their metabolic performance, cells have to realise good compromises between large production fluxes, low enzyme investments, and well-adapted metabolite levels. In models, this idea can be formulated in the form of optimality principles that trade a high metabolic benefit against low enzyme cost. However, different modelling approaches are often incompatible. I propose a unified theory, called metabolic economics, to bridge the gap between different optimality-based cell models that exploits hidden equivalences between these approaches. Metabolic economics introduces new variables on the network, called economic variables, which represent the cost and benefit of metabolites, fluxes, and enzymes, and can be defined by Lagrange multipliers, auxiliary variables that are commonly used to handle constraints in optimality problems. Metabolic economics translates optimality conditions into local balance equations between these variables. The economic potentials and loads describe the value of metabolite production and metabolite concentrations. As "proxy variables", they can describe indirect fitness effects, arising elsewhere in the network, as if they arose locally in a reaction of interest. Here I derive these variables and their balance equations for three types of optimality problems: for direct optimisation of enzyme levels in kinetic models; for flux cost minimisation (FCM), a minimisation of enzyme cost, with flux and metabolite profiles as the variables to be optimised; and for optimal protein allocation in whole-cell models, where growth rate or other whole-cell objectives are maximised. The economic balance equations add a new layer of description to mechanistic models, a description in terms of beneficial cell functions and associated costs, and can seen as economic laws of metabolism. Metabolic economics provides concepts for comparing and combining metabolic optimality problems, employing different modelling paradigms or different levels of detail, which can be useful for semi-automatic or modular modelling.
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