## Abstract

To improve their metabolic performance, cells need to find compromises between high metabolic fluxes, low enzyme investments, and well-adapted metabolite concentrations. In mathematical models, such compromises can be described by optimality problems that trade metabolic benefit against enzyme cost. While many such modelling frameworks exist, they are often hard to compare and combine. To unify these modelling approaches, I propose a theory that characterises metabolic systems by a value structure, that is, a pattern of local costs and benefits assigned to all elements in the network. The economic values of metabolites, fluxes, and enzymes are interlinked by local balance equations. Formally defined as shadow values, the economic variables serve as local proxies for benefits that arise anywhere in the network, but are represented as local costs or benefits in the reaction of interest. Here I derive economic variables and their balance equations for kinetic, stoichiometric, and cell models. Metabolic value theory provides a new perspective on biochemical networks, defines concepts for comparing and combining metabolic optimality problems, and is useful for semi-automatic, layered, and modular modelling.

### Competing Interest Statement

The authors have declared no competing interest.

## Footnotes

Many changes in the text

^{1}In our metabolic “standard models”, the vector**e**contains enzyme concentrations. However, in other models it may also contain other control variables, such as mRNA levels, that appear as prefactors in rate laws.^{2}Derivation: where**v***/***e**stands for the enzyme elasticity matrix**E**_{e}= Dg(**v**)Dg(**e**)^{−1}.^{3}If*x*is (directly) constrained by bounds*x*^{min}≤*x*≤*x*^{max}or by a predefined value*x*=*x*^{fix}, this leads to a shadow price, which plays a similar role as*f*_{x}and can be added to it.^{4}In a linear pathway with thermodynamic forces*θ*_{l}and*K*_{M}values set to 1, the substrate elasticities are larger than the product elasticities [14]. According to the balance equation, around a metabolite, the enzyme investment in the producing reaction should thus be larger than in the consuming reaction. Since this holds for each metabolite, the enzyme investments must decrease along the chain.^{5}A nonlinear benefit function with gradient*b*_{v}=*∂b/∂***v**would lead similar optimality conditions. Here we consider linear functions just for simplicity.^{6}In models with moiety conservation (e.g. if ATP and ADP appear in reactions only as cofactor pairs, and therefore [ATP]+[ADP]=const.), the concentrations of conserved moieties can be treated as model parameters.^{7}In dynamically unstable states or bifurcation points, our theory does not apply. In a metabolic system, the enzyme profiles will not always uniquely determine the metabolic state. On the one hand, the state may also depend on conserved moiety concentrations (which follow from the initial conditions). On the other hand, in cases of multistability, we need to select one of the possible steady states.^{8}For simplicity we assume bounds on individual metabolites. Density constraints for a sum of metabolite concentrations can be treated similarly and will be discussed below.^{9}In a dynamical reaction system, dilution alone would lead to an exponential concentration decrease. Sine-wave oscillations of enzyme and metabolite concentrations can be described by complex-valued exponential functions. This resembles an example in physics: an overdamped pendulum shows an exponentially decreasing motion, which is mathematically related to the motion of oscillating pendulum because sine and cosine functions are exponential functions with an imaginary function argument.^{10}Here we consider deterministic, non-spatial network models in steady state. Other possible variables (which can be included in the theory) are for instance compartment sizes, electric potentials, or temperature.^{11}In some models such terms play a role. External concentration prices may matter in modular models, in which models are coupled (with external metabolites between them) and an optimal compromise (for the external metabolite concentrations) needs to be found. For example, one submodel produces ATP and the other submodel consumes it; the first submodel would run more efficiently at a lower ATP concentration, while the second submodel would run more efficiently at a higher ATP concentration. In the optimal state, the (positive) ATP price from the first model and the (negative) ATP price from the second model must exactly cancel out.Similarly, internal production gains matter in models with dilution or could be needed to penalise loss by diffusion (in models in which diffusion reactions are not explicitly described).^{12}Dual variables also play a role in physics. In thermodynamics, the chemical potentialsare defined as derivatives of the Gibbs free energy with respect to substance amounts (in moles), and are dual variables of the substance amounts (the physical variables). According to the second law of thermodynamics, thermodynamic systems tend towards states of minimal Gibbs free energy. The chemical potentials are a useful concept to describe this. The chemical potential*µ*_{i}measures the Gibbs free energy contained in one mole of substance*i*. Accordingly, a change in the mole number*δn*_{l}leads to an energy change*δG*=*µ**·***n**. Knowing the chemical potentials, we also know which processes would dissipate Gibbs free energy (by changing the substances’ mole numbers), and may therefore happen spontaneously.^{13}Whenever a variable hits a bound, the corresponding shadow value acts as a gain or price, creating an extra term in the value balance, and therefore an additional value “inflow” or “outflow”.^{14}Formally, like in the case of variables hitting a bound, this term can also be seen as virtual value inflow or outflow; together with this term, value conservation would hold again at least formally.^{15}Interestingly, variational principles in physics used are often not maximisation principles but extremality principles, i.e. they may be satisfied by maximal, minimal, or saddle point solutions. In contrast, if we apply variational principles to living systems, we typically know whether a target should be minimised or maximised. We assume that cells maximise, for example, a benefit or minimise a cost on a set of possible cell states. Optimising a simple objective can be seen as an approximation of more complicated multi-objective problems, which would be a more realistic description of cells.^{16}Metabolic value Theory also applies to non-optimal states. In optimal states, investments and value production are balanced. However, real cells are never exactly optimal. If an objective function has been defined, deviations from optimality can be expressed by “stress” terms in the cost-benefit balance equations.

## Abbreviations

- FBA
- Flux balance analysis,
- LP
- Linear programming,
- MCT
- Metabolic control theory,
- MVT
- Metabolic value theory.