Free-energy transduction mechanisms shape the flux space of metabolic networks

Abstract

The transduction of free energy in metabolic networks represents a thermodynamic mechanism by which the free energy derived from nutrients is converted to drive nonspontaneous, energy-requiring metabolic reactions. This transduction is typically observed in processes that generate energy-rich molecules such as ATP and NAD(P)H, which, in turn, power specific reactions, particularly biosynthetic reactions. This property establishes a pivotal connection between the intricate topology of metabolic networks and their ability to redirect energy for functional purposes. The present study proposes a dedicated framework aimed at exploring the relationship between free-energy dissipation, network topology, and metabolic objectives. The starting point is that, regardless of the network topology, nonequilibrium chemostatting conditions impose stringent thermodynamic constraints on the feasible flux steady states to satisfy energy and entropy balances. An analysis of randomly sampled reaction networks shows that the network topology imposes additional constraints that restrict the accessible flux solution space, depending on key structural features such as the reaction’s molecularity, reaction cycles, and conservation laws. Notably, topologies featuring multimolecular reactions that implement free-energy transduction mechanisms tend to extend the accessible flux domains, facilitating the achievement of metabolic objectives such as anabolic flux maximization or flux rerouting capacity. This approach is applied to a coarse-grained model of carbohydrate metabolism, highlighting the structural requirements for optimal biomass yield.

Significance

The thermodynamic properties of metabolic pathways are commonly employed to narrow down the solution space in underdetermined stoichiometric or kinetic models. Thermodynamic considerations can also be leveraged to decipher the functional patterns of Gibbs free-energy dissipation. Within the framework of chemostatted chemical reaction networks, the combination of sampling approaches and coarse-grained models enables the exploration of the extent to which the repertoire of flux states is constrained by the network topology. The results underscore the crucial role of bimolecular reaction schemes in governing free-energy transduction, essential for achieving various metabolic objectives. This reveals inherent metabolic trade-offs between the establishment and utilization of thermodynamic forces.

Introduction

Cellular metabolism is a nonequilibrium process in which the free energy derived from the environment is converted into chemical work that sustains many cellular processes (1,2). The nature and availability of free-energy sources therefore impose fundamental energetic constraints that require efficient partitioning and allocation of such limited resources (3). The landscape of free-energy dissipation throughout the metabolic network tends to be inhomogeneous and context dependent (4,5,6,7). On the one hand, metabolic reactions operating far from equilibrium enable an efficient enzyme usage with a high chemical motive force per enzyme and flexible flux control via enzyme regulation (5,8). On the other hand, reactions operating near equilibrium enable flexible flux control by substrate and product concentrations, for instance along the glycolytic pathway (6,9). Free energies of reaction are therefore expected to subtly contribute to objective trade-offs involving metabolic and enzyme efficiencies and determining metabolic fluxes in various conditions (10,11,12,13). Network topology itself plays an important role in determining free energies over reactions, particularly through the key roles of energy and electron/proton carriers (14,15,16,17,18) in coupling favorable and unfavorable reactions such as those in catabolic and anabolic pathways. The intricate interplay between network topology, flux objectives, and thermodynamic constraints in metabolic pathways necessitates dedicated modeling frameworks.

From a purely thermodynamic perspective, metabolic networks can be described as open chemical reaction networks (CRNs) where reactions are reversible and driven by chemical motive forces (19,20) and nutrient sources and solvents can be considered as chemostat reservoirs, imposing nonequilibrium boundary conditions (21). CRN theory has been developed to explore the link between the nonequilibrium properties of flux and concentration states and the structural properties of reaction pathways (21,22). Some general relations can be established based on the breakage of conservation laws through chemostatting or the decomposition of steady-state entropy production in terms of reaction cycles (21). This framework is thus valuable for specifically addressing both the mechanisms and roles of free-energy transduction in metabolic networks. This issue can be approached at the level of specific reaction pathways (23,24), as well as through a more systematic sampling approach of CRNs. The solution spaces of dynamic models are often sampled by varying enzyme kinetic parameters, assuming the implicit regulation of enzyme activities (25,26,27), but the topologies of metabolic reaction networks can also be sampled to capture the qualitative differences between distinct classes of topologies (28).

The present study aims to unveil the relationships between the topology, nonequilibrium thermodynamics, and functional objectives in metabolic networks. The materials and methods section first outlines the general framework of chemostatted CRNs, describing nonequilibrium flux states in terms of the free-energy dissipation rate across reactions and highlighting the importance of reaction cycles and conservation laws. These core thermodynamic and topological properties motivate a comprehensive approach to computing the feasible flux space, constrained primarily by chemostatting boundary conditions and subsequently by network topology. Random and systematic sampling of both kinetic and topological properties of CRNs is employed to comprehensively characterize the metabolic state space under various chemostatting conditions, addressing the significance of several topological features. First, a systematic comparison of the flux solution domains in CRNs composed of unimolecular versus bimolecular reactions highlights qualitative differences in the accessible repertoire of nonequilibrium flux states and the fulfillment of metabolic objectives related to anabolic or adaptive functions. Secondly, the minimal topologies of bimolecular CRNs that reach these metabolic objectives are identified, emphasizing the key role of free-energy transduction mechanisms and other topological requirements. Finally, the relationship between free-energy transduction mechanisms, metabolic solution space, and biomass yield objective is investigated in coarse-grained and thermodynamics-informed models of carbohydrate metabolism.

Materials and methods

Nonequilibrium steady states in open CRNs

We consider an open CRN comprising $n_s$ internal species and $n_c$ chemostatted external species (also termed chemostat species) whose concentrations are fixed, following a framework developed by Esposito and colleagues (21,22) (Fig. 1 A). The species interact through $n_r$ reversible chemical reactions, including a set of exchange reactions with the chemostat reservoirs. The topology of the CRN is represented by the stoichiometric matrix S, which contains the stoichiometric coefficient for the molecular species i involved in the reactions j. This matrix can be decomposed in various ways to analyze different aspects of network’s structure:

$$\mathbb{S}\mathbb{\equiv }\left(\begin{array}{l}{\mathbb{S}}^x\\ {\mathbb{S}}^y\end{array}\right)\equiv {\mathbb{S}}^{-}-{\mathbb{S}}^{+}\text{,}$$ (1)

where ${\mathbb{S}}^{x,y}$ contains, respectively, the stoichiometric coefficients for internal versus chemostat species, with concentrations denoted by $\overrightarrow{x}$ and $\overrightarrow{y}$, while ${\mathbb{S}}^{+,-}$ contains, respectively, the nonnegative coefficients for forward versus backward reactions. Similarly, the net flux of a reversible reaction can be decomposed into directed forward and backward fluxes, $J_r=J_r^{+}-J_r^{-}$, whose ratio is determined by the local detailed balance, $J_r^{+}/J_r^{-}=\exp \left(-\Delta G_r/RT\right)$ (20). The chemical potential per mole of internal and chemostat species is given by ${\mu }_i={\mu }_i^{\circ }+RT\ln \left(x_i/Z\right)$ and ${\mu }_i^y={\mu }_i^{\circ ,y}+RT\ln \left(y_i/Z\right)$, where R denotes the gas constant ($RT\approx 2.5\text{kJ}/\text{mol}$ at $T=300\mathrm{K}$) and Z the standard state concentration set to $1\mathrm{M}$. The Gibbs free energy of reaction r is given by $\Delta G_r^{(\circ )}={\overrightarrow{\mu }}^{(\circ )}·{\mathbb{S}}_{:,r}$. The directed fluxes following the mass-action law are given by $J_r^{+/-}=k_r^{+/-}\prod _i{x}_i^{{\mathbb{S}}_{i,r}^{+/-}}$. Using a single overall rate constant $k_r=k_r^{+}{e}^{\Delta G_r^{\circ }/2RT}=k_r^{-}{e}^{-\Delta G_r^{\circ }/2RT}$, the net flux can be expressed as a combination of thermodynamic and kinetic parameters:

$$J_r=k_r\left(\prod _i{x}_i^{{\mathbb{S}}_{i,r}^{+}}{e}^{-\Delta G_r^{\circ }/2RT}-\prod _i{x}_i^{{\mathbb{S}}_{i,r}^{-}}{e}^{\Delta G_r^{\circ }/2RT}\right)\text{.}$$ (2)

Figure 1.

Feasible and restricted flux spaces in chemostatted chemical reaction networks. (A) Schematic representation of an open chemostatted chemical reaction network involving $n_c$ chemostat species (bold red outlined circle) and $n_s$ internal species (circles) of standard chemical potential ${\mu }^{\circ }$. The network includes internal and exchanged reactions (lines) with kinetic rate constants $k_r$ and is described by a stoichiometric matrix S. Steady-state variables of interest include the species’ chemical potentials ${\mu }_i$, chemostat fluxes $J^y$, and total entropy production rate σ or, equivalently, chemostat free-energy dissipation rate $T\sigma$, which can be decomposed into contributions from exchanged and internal reactions: $\sigma ={\sigma }^E+{\sigma }^I$. (B) The chemostatting conditions (${\overrightarrow{\mu }}^y$, $k_{r=1,n_c}$) and the conservation laws ($\overrightarrow{\mathcal{l}}$) define a thermodynamically feasible flux solution domain in the space of exchange fluxes bounded by a manifold where ${\sigma }^I=0$ (Eqs. 8a and 8b). (C) The stoichiometric matrices of diverse network topologies exhibit different kinetically sampled chemostat flux solution domains (Eq. 3).

Accordingly, the steady-state flux pattern $\overrightarrow{J}$ and chemical potential pattern $\overrightarrow{\mu }$ of a CRN satisfies

$${\mathbb{S}}^x\overrightarrow{J}\left(\overrightarrow{\mu }\right)=0\text{.}$$ (3)

The nonequilibrium nature of the steady-state solution of Eq. 3 is enforced by the chemostatting condition where both the extracellular environment and the intracellular solvent can be treated as chemostat reservoirs (21). Specifically, it is assumed that $n_c$ internal species are exchanged with reservoirs using Fick’s law of diffusion of exchanged fluxes:

$$J_r={\tilde{k}}_r\left(e^{{\mu }_r^y/RT}-e^{{\mu }_r/RT}\right)\text{for}r=1,.,n_c\text{,}$$ (4)

where ${\tilde{k}}_r=k_{r}Z{e}^{-{\mu }_r^{\circ }/RT}$ and ${\mu }_{r=1,.,n_c}$ denotes the chemical potential of exchanged internal species. To keep the concentrations of chemostat species fixed, chemostat fluxes must satisfy ${\overrightarrow{J}}^y=-{\mathbb{S}}^y\overrightarrow{J}$ and are thus equal to exchange fluxes $J_r^y=J_r$ for $r=1,.,n_c$. An important macroscopic observable of the steady state is the overall dissipation rate of free energy, that is, the sum of dissipation rate for all reactions, and can be thus decomposed into the contributions of exchange versus internal reactions:

$$T\sigma =\stackrel{T{\sigma }^E}{\overbrace{-\sum _{r=1}^{n_c}{J}_r\Delta G_r}}\stackrel{T{\sigma }^I}{\overbrace{-\sum _{r=n_c+1}^{n_r}{J}_r\Delta G_r}}\text{,}$$ (5)

where σ is the (molar) entropy production rate. It has been demonstrated that the steady dissipation rate of a chemostatted CRN can be conveniently expressed as a function of chemostat fluxes and potentials as $T\sigma =-{\overrightarrow{\mu }}^y·{\overrightarrow{J}}^y$ (21). By treating exchanged internal species as effectively chemostatted species, a similar expression can be used for the internal dissipation rate:

$$T{\sigma }^I=-\sum _{i=1,n_c}{\mu }_i{J}_i\text{.}$$ (6)

The role of chemostatting in shaping the steady-state dissipation pattern is influenced by the topological properties related to reaction cycles and conservation laws (21). The basis vectors of the right null space (kernel) of the internal stoichiometric matrix ${\mathbb{S}}^x$ define a set of reaction cycles $\overrightarrow{j}$, which are similar to elementary flux vectors (29). Following the terminology proposed in (21), these cycles are termed emergent cycles ($j^{ec}$), as they do not belong to the reaction cycles of the entire nonchemostatted CRN ($\mathbb{S}{\overrightarrow{j}}^{ec}\ne 0$), thus emerging from chemostatting some species. Reaction cycles of both the internal CRN and the entire nonchemostatted CRN are termed closed cycles ($\mathbb{S}{\overrightarrow{j}}^{cc}={\mathbb{S}}^x{\overrightarrow{j}}^{cc}=0$), as they do not involve exchange reactions with chemostats. A key functional distinction is that only emergent cycles carry a flux that dissipates chemostat free energy. The basis vectors of the left null space (cokernel) of S define conservation vectors denoted $\overrightarrow{\mathcal{l}}=\left({\overrightarrow{\mathcal{l}}}^x,{\overrightarrow{\mathcal{l}}}^y\right)$, where ${\overrightarrow{\mathcal{l}}}^x.\overrightarrow{x}+{\overrightarrow{\mathcal{l}}}^y.\overrightarrow{y}$ remains conserved over time. Chemostatting some species leads to broken conservation laws when ${\overrightarrow{\mathcal{l}}}^y\ne 0$, meaning that conserved quantities, such as mass or atom nuclei, are exchanged with reservoirs, thereby imposing a linear constraint on chemostat fluxes (see the supporting material for details):

$${\overrightarrow{\mathcal{l}}}^y.{\overrightarrow{J}}^y=0\text{.}$$ (7)

Importantly, the respective numbers of boundary conservation vectors $n_{bc}$ and emergent cycles $n_{ec}$ are linked and bounded by the relation $n_{bc}+n_{ec}=n_c$. Another topological feature of interest is the molecularity of a reaction $n_m$, i.e., number of reactants involved in an elementary reaction. Two main classes of network topologies can be distinguished based on whether internal reactions are exclusively unimolecular, $n_m=1$, or bimolecular, $n_m=2$, both characterized with the same mass conservation vector $\overrightarrow{\mathcal{l}}=\overrightarrow{1}$.

Feasible flux space constrained by chemostatting and network topology

A steady chemostat flux state ${\overrightarrow{J}}^y$ is considered thermodynamically feasible when it satisfies the weak criterion of nonnegative internal entropy production ${\sigma }^I\ge 0$ (Fig. 1 B). The values ${\overrightarrow{J}}^y$ that correspond to the boundary ${\sigma }^I=0$ of this thermodynamically feasible flux space can be determined by combining Eqs. 4 and 6, in addition to the conservation constraint given by Eq. 7:

$$\sum _{i=1,n_c}{J}_i^y\log \left(e^{{\mu }_i^y/RT}-J_i^y/{\tilde{k}}_i\right)=0$$ (8a)

$${\overrightarrow{\mathcal{l}}}^{y,i}.{\overrightarrow{J}}^y=0\text{for}i=1,n_{bc}\text{.}$$ (8b)

The boundary manifold is of dimension $n_c-n_{bc}-1\equiv n_{ec}-1$ and fully determined by the chemostatting condition: the chemical potentials of chemostat species ${\overrightarrow{\mu }}^y$, the exchange rate constants with chemostat reservoirs $k_{r=1,n_c}$, and the chemostat part of broken conservation laws ${\overrightarrow{\mathcal{l}}}^{y,i=1,n_{bc}}$. The largest feasible flux space is thus obtained when there is a single broken conservation vector associated with mass conservation. Adding conservation constraints associated, for instance, with the atomic composition of chemosttated species would restrict the feasible flux space to a subspace of lowest dimensionality.

A CRN of given topological and thermodynamic structure (defined by ${\mathbb{S}}^x$ and ${\overrightarrow{\mu }}^{\circ }$) covers a subdomain of such feasible space but can approach the boundary manifold in some asymptotic limit in the kinetic space, such that $\Delta G_{i=n_c+1,n_r}\to 0$ and ${\sigma }^I\to 0$. For such a quasi-equilibrium state, species chemical potentials are a linear combination of conservation vectors (22) with ${\mu }_{i=1,n_c}=\sum _{j=1,n_{bc}}{\alpha }_j{\mathcal{l}}_i^{y,j}$ given that ${\mathcal{l}}_i^x={\mathcal{l}}_i^y$. Eq. 8a can be rewritten as a function of ${\mu }_{i=1,n_c}$ instead of $J_i^y$ to finally obtain

$$\sum _{i=1,n_c}{\tilde{k}}_i\left(\sum _{j=1,n_{bc}}{\alpha }_j{\mathcal{l}}_i^{y,j}\right)\left(e^{{\mu }_i^y/RT}-\prod _{j=1,n_{bc}}{e}^{{\alpha }_j{\mathcal{l}}_i^{y,j}/RT}\right)=0\text{.}$$ (9)

For a general class of chemostatted CRNs with a single broken mass conservation vector ($n_{bc}=1$ and $\overrightarrow{\mathcal{l}}=\overrightarrow{1}$), $\overrightarrow{\mu }={\mu }^{\ast }\overrightarrow{1}$ is incorporated into Eq. 9 to obtain

$$\sum _{i=1,n_c}{\tilde{k}}_i\left(e^{{\mu }_i^y/RT}-e^{{\mu }^{\ast }/RT}\right)=0\text{.}$$ (10)

The unique solution ${\mu }^{\ast }$ of Eq. 10 is replaced into Eq. 4 to obtain the chemostat flux solutions ${\overrightarrow{J}}^y$ and the overall dissipation rate given by Eq. 5:

$$T{\sigma }^{\ast }=\sum _{i=1,n_c}{\mu }_i^y{\stackrel{\sim }{k}}_i\left(e^{{\mu }_i^y/RT}-\frac{{\sum }_{i=1,n_c}{\stackrel{\sim }{k}}_i{e}^{{\mu }_i^y/RT}}{{\sum }_{i=1,n_c}{\stackrel{\sim }{k}}_i}\right)\text{.}$$ (11)

For $n_{bc}=1$ and $\overrightarrow{\mathcal{l}}=\overrightarrow{1}$, the only nonequilibrium steady state of vanishing internal entropy production is featured with homogeneous chemical potentials ${\mu }_i={\mu }^{\ast }$ and maximum entropy production rate $\sigma ={\sigma }^{\ast }$. This state can be naively illustrated in the simplest chemostatted CRN $y_1\leftrightarrow x_1\leftrightarrow x_2\leftrightarrow y_2$. In the limit $k_3\gg k_{1,2}$, the steady-state concentrations of $x_1$ and $x_2$ are dominated by the backward and forward fluxes through $r_3$, resutling in quasi-equilibrium ($\Delta G_3\to 0$ and ${\mu }_1\approx {\mu }_2$) and maximal entropy production rate $\sigma \to {\sigma }^{\ast }$ (Eq. 11). In the opposite limit $k_3\ll k_{1,2}$, the exchange reactions are in quasi-equilibrium (${\mu }_1\approx {\mu }_1^y$ and ${\mu }_2\approx {\mu }_2^y$), such that $J_{1,2,3}\approx 0$ and $\sigma \to 0$. $T{\sigma }^I=-J_3\Delta G_3$ therefore vanishes in these two limits and reaches a maximum between.

Sampling approach

A sampling approach is designed to investigate how the solution domain in the space of exchanged fluxes depends on the network topology (Fig. 1 C). It involves computing solutions to the steady-state equation Eq. 3 while randomly sampling the rate constant parameters ${\overrightarrow{k}}_r$, the standard chemical potentials ${\overrightarrow{\mu }}^{\circ }$, and the stoichiometric matrices S associated with a given class of topology. The sampling procedure is detailed in section 1.2 in the supporting material and Table S1 and summarized below. The sample size ranges between ${10}^4$ and ${10}^6$, which is set on a case-by-case basis to avoid both undersampling and oversampling. The sampled range of thermodynamic parameters is ${\mu }_{i=1,n_s}^{\circ }\in \left[-6,6\right]\text{kJ}/\text{mol}$ (unless otherwise specified). The sampled range of kinetic rate constants is $k_{r=n_c+1,n_r}\in \left[{10}^{-2},10\right]$ in units of ${\mathrm{s}}^{-1}$ or ${\mathrm{s}}^{-1}.{\mathrm{M}}^{-1}$, respectively, for unimolecular or bimolecular reactions, given a normalized time unit with $k_{r=1,n_c}=1{\mathrm{s}}^{-1}$. Random stoichiometric matrices ${\mathbb{S}}^x$ are generated by randomly choosing substrates and products for all reactions, with certain restrictions: 1) an equal number of reactants and products, 2) the same species cannot be both a substrate and a product (excluding autocatalytic reactions), and 3) the exclusion of duplicated reaction schemes. Some topological hyperparameters are fixed, such as the size of the CRN set to $n_s=8$ and $n_r=12$ (unless otherwise specified), and the same molecularity for all internal reactions set to $n_m=1$ or $n_m=2$. This specific sampling approach is complemented by control simulations to ensure that the qualitative results are not sensitive to network size or sampled parameter ranges (Figs. S2 and S3). The volume of the sampled solution space relative to the volume of the feasible flux space is quantified by

$$\mathcal{V}=\frac{{\int }_{-\infty }^{+\infty }{f}_2\left({\overrightarrow{J}}^y,\mathbb{S},{\overrightarrow{\mu }}^{\circ }\right)d{J}_1^y\cdots d{J}_{n_c-1}^y}{{\int }_{-\infty }^{+\infty }{f}_1\left({\overrightarrow{J}}^y\right)d{J}_1^y\cdots d{J}_{n_c-1}^y}\text{,}$$ (12)

where $f_1\left({\overrightarrow{J}}^y\right)=1$ when the criteria of thermodynamic feasibility (Eqs. 8a and 8b) are met and 0 otherwise and $f_2\left({\overrightarrow{J}}^y,.\right)=1$ when a steady-state solution of Eq. 3 can be found by sampling k values in a CRN of given topology.

Results

Nonequilibrium patterns of free-energy dissipation in random CRNs

A CRN coupled to two chemostats is the most basic framework for performing a random sampling analysis of nonequilibrium properties, such as the overall versus internal dissipation rates (Eqs. 5 and 6; Fig. 2 A). First, the overall dissipation rates across a large set of CRNs with random topologies and kinetic rates (see materials and methods) are broadly distributed between 0 and the maximum value ${\sigma }^{\ast }$ computed with Eq. 11 (top, Fig. 2 B). Characteristics of the distribution weakly depend on the network size and reaction molecularity (middle, Fig. 2 B), while it significantly depends on the averaged kinetic constant of internal reactions $k_{r=n_c+1,n_r}$ (bottom, Fig. 2 B). Faster (resp. slower) internal reactions are correlated with higher dissipation rates, consistent with the predicted asymptotic behaviors where ${\sigma }^I\to 0$ and $\sigma \to {\sigma }^{\ast }$ for $k_{r=n_c+1,n_r}\gg k_{r=1,n_c}$. In this limit, the internal concentration states of the CRN are dominated by internal reaction fluxes such that the system converges to a quasi-equilibrium state.

Figure 2.

Distribution and bounds of thermodynamic variables in random CRNs. (A) CRN coupled to two chemostats (${\overrightarrow{\mu }}^y=\left[5,-5\right]$, $\overrightarrow{y}=\overrightarrow{1}$, and $k_{1,2}=1$) characterized by a maximum dissipation rate of $T{\sigma }^{\ast }\approx 9.64$ computed using Eq. 11. (B) Probability distribution functions of $T\sigma$ (Eq. 5) for a set of ${10}^5$ random CRNs with random ${\mathbb{S}}^x$, $k_{r=n_c+1,n_r}$, and ${\mu }_i^{\circ }$. Top: default values for $n_s$, $n_r$, and $n_m$. Middle: example of varied values for $n_m$ or $n_s$ and $n_r$. Bottom: same as at the top but with fixed kinetic rate constants $k_{r=n_c+1,n_r}=0.01$, 0.1, 1, or 10 instead of randomly sampled constants. (C) Distribution of thermodynamic variables associated with reactions and species as a function of the total free-energy dissipation rate $T\sigma$ in the CRN for a set of ${10}^4$ random CRNs with unimolecular (left) or bimolecular (right) reactions. The violet line represents $T{\sigma }^I$ computed from Eq. 7, and the blue line represents ${\mu }_1-{\mu }_2$, bounding the reaction free energy only for $n_m=1$. All quantities are expressed in units listed in Table S1.

In these random CRNs, the pattern of free energy, flux and entropy production over reactions exhibits a typical dependency on the overall dissipation rate $T\sigma$ (Fig. 2 C). Regardless of whether reactions are unimolecular or bimolecular, the sampled states of maximum entropy production rate $\sigma \lesssim {\sigma }^{\ast }$ are characterized with $\Delta G\to 0$ (near-equilibrium internal reactions) and ${\mu }_i\to {\mu }^{\ast }$ given by Eq. 10 (homogeneous chemical potentials). More interestingly, an increased dissipation rate in random CRNs coincides with increased maximum reaction fluxes and decreased maximum reaction free energies or species chemical energies. Such opposing trends manifest themselves with a maximal value of internal dissipation rate $T{\sigma }^I$ for a nonmaximum level of total dissipation rate $T\sigma$, highlighting a trade-off where the free-energy dissipation through internal reactions necessitates the dissipation of chemostat free energy to power some internal flux but also dissipate chemostat free energy, which is no longer available to power fluxes through exchanged reactions. The quantitative characteristics of such trade-off, however, depend on the chemostatting parameters, such as their chemical potentials (Fig. S4).

The main difference between unimolecular and bimolecular reaction networks lies in the strict bounding of thermodynamic quantities (compare the left and right images of Fig. 2 C). In unimolecular CRNs, the chemical potentials of internal species range between the chemical potentials of the two exchanged internal species with ${\mu }_2^y\le {\mu }_2\le {\mu }_i\le {\mu }_1\le {\mu }_1^y$, imposing an upper bound on the free energies of all reactions with $\left|\Delta G_r\right|\le {\mu }_1-{\mu }_2\le {\mu }_1^y-{\mu }_2^y$. In contrast, bimolecular CRNs exhibit a much broader range of nonequilibrium thermodynamic states, often violating the aforementioned bounds. This behavior can be attributed to both the increased free energy from additional metabolites involved in the reactions and the greater degrees of freedom resulting from the multivariate dependencies of the reaction rates. Such broader steady-state solution domains in the space of chemical potentials and free energies can lead to qualitative functional differences in the high-dimensional flux state space, an issue that is explored in the following sections.

Requirement for multimolecular reactions and free-energy transduction to achieve anabolic function

To investigate more thoroughly how the accessible flux space is determined by the network topology, we consider CRNs coupled to a third chemostat reservoir with a chemical potential hierarchy ${\mu }_3^y>{\mu }_1^y>{\mu }_2^y$ (Fig. 3 A). A negative exchange flux $J_3<0$ thus mimics an anabolic or biosynthetic process that produces energy-rich compounds, here assumed to be exported from the CRN. Given such chemostatting conditions, the two-dimensional feasible flux space can be computed using Eqs. 8a and 8b, showing that maximum entropy production $\sigma ={\sigma }^{\ast }$ would coincide with the maximum influx of energy-rich compounds ($J_3>0$), whereas maximum efflux of energy-rich species $-J_3=J^{\ast }$ occurs for a much lower entropy production rate (black lines and violet circles in Fig. 3, B and C). The capacity of CRNs to achieve negative $J_3$ values is investigated by computing the steady flux solutions of Eq. 3 for a large set of CRNs of randomly varied topologies and kinetics, distinguishing two main classes of topologies: unimolecular and bimolecular reaction networks. Unimolecular reaction networks fail to achieve an anabolic flux mode with $J_3<0$ (Fig. 3 B), which can be naively explained by the allowed range of chemical potentials ${\left({\mu }^y\right)}^{\min }<{\mu }_i<{\left({\mu }^y\right)}^{\max }$, precluding $J_3<0$, as ${\mu }_3$ cannot be larger than the highest chemostat potential ${\mu }_3^y$. Relieved from this constraint, bimolecular reaction networks cover the entire feasible flux domain, including the anabolic mode where $J_3<0$ and approaching the maximum feasible value $-J_3=J^{\ast }$ (Fig. 3 C).

Figure 3.

Anabolic capacity in bimolecular reaction networks. (A) CRN coupled to three chemostats (${\overrightarrow{\mu }}^y=\left[5,-5,5.77\right]$, $\overrightarrow{y}=\left[1,1,0.5\right]$, and $k_{1,.,n_c}=1$), focusing on the maximization of an anabolic-like flux $-J_3$. (B) Thermodynamically feasible solution domains (black line computed by solving Eqs. 8a and 8b) and sampled solutions of ${10}^4$ random unimolecular CRNs (dots, where color denotes the internal contribution to entropy production ${\sigma }^I/\sigma$) plotted in the space of the exchange flux $J_3$ and the normalized entropy production rate $\sigma /{\sigma }^{\ast }$. (C) Similar to (B) but for $5\times {10}^4$ random bimolecular CRNs ($n_m=2$). (D) Examples of three CRN topologies of specific stoichiometric matrix ${\mathbb{S}}^x$ (red = $+1$; blue: $-1$; white: 0) and their corresponding kinetically sampled solution domains (gray dots on the bottom). These topologies differ qualitatively in their maximum ${\left(-J_3\right)}^{\mathit{max}}$ values. (E) Probability distribution of ${\left(-J_3\right)}^{\mathit{max}}$ for a set of $2\times {10}^3$ random network topologies (each sampled with ${10}^3$ random kinetic parameters). Topologies are sampled based on a defined topological class characterized by the number broken conservation law $n_{bc}$ and the reaction molecularity $n_m$. (F) CRN of minimal topological complexity ($n_s=5$, $n_r=5$, $n_{bc}=2$) characterized by a maximum anabolic exchange flux $-J_3\simeq J^{\ast }$ (for $k_{4,5}={10}^2$). (G) Flux solution domain of the minimal CRN sampled with ${10}^3$ random kinetic parameters (red), with respect to flux solution domains of ${10}^5$ random CRNs with $n_{bc}=2$ and nonzero fluxes (gray). All quantities are expressed in the units listed in Table S1.

Given that bimolecular reaction networks can virtually cover the entire feasible flux domain, the subsequent inquiry revolves around characterizing the topological features required to reach the anabolic objective of maximizing $-J_3$. The flux solution domain of diverse bimolecular CRN topologies can be kinetically sampled and shows a very distinct capacity of maximizing $-J_3$, ranging from ${\left(-J_3\right)}^{\mathit{max}}$ being positive or slightly negative to close to $J^{\ast }$ (Fig. 3 D). The distribution of the maximum anabolic flux ${\left(-J_3\right)}^{\mathit{max}}$ across randomly sampled CRN topologies demonstrates indeed that only a small fraction of multimolecular network topologies can display an anabolic flux, where the small subset having two broken conservation laws ($n_{bc}=2$) is more prone to achieving the maximum anabolic flux, as depicted by the shape of the sampled distribution (Fig. 3 E). The key topological features required for reaching anabolic objectives can be identified by seeking the reaction network of minimal topological complexity that achieves $-J_3\simeq J^{\ast }$. The minimal CRN with optimal anabolic capacity comprises a simple set of $n_r=5$ reactions involving $n_s=5$ species, including two nonexchanged internal species whose interconversions are coupled to the reactions interconverting exchanged internal species (Fig. 3 F). The substantial chemical energy difference between these two nonexchanged internal species, where ${\mu }_4\gg {\mu }_5$, drives the transduction of free energy released in the conversion from the nutrient species ${\mu }_1$ into the waste species ${\mu }_2$ to power the conversion from nutrient species to anabolic species ${\mu }_3$. This free-energy transduction process supporting anabolic function is implemented at the level of the stoichiometric matrix through the existence of one emergent cycle and, importantly also, two broken and one unbroken conservation vectors (mass conservation: ${\overrightarrow{\mathcal{l}}}^{x,1}=\overrightarrow{1}$; structural conservation: ${\overrightarrow{\mathcal{l}}}^{x,2}=\left[2,0,4,1,-1\right]$; energy-carrier conservation: ${\overrightarrow{\mathcal{l}}}^{x,3}=\left[0,0,0,1,1\right]$). Accordingly, the kinetically sampled flux solution domain of this minimal model coincides with a line along a single elementary flux vector approaching the optimal state $J_3=J^{\ast }$ for fast-enough internal reaction constants $k_4$ and $k_5$ (Fig. 3 G). In relation to Eq. 9, this quasi-equilibrium state is indeed decomposable into conservation laws $\overrightarrow{\mu }=\sum _{i=1,3}{\alpha }_i{\overrightarrow{\mathcal{l}}}^{x,i}$, where $\overrightarrow{\alpha }\approx \left[-4.2,2.84,2.84\right]$. Fig. 3 G also shows that this one-dimensional flux domain associated with maximum $-J_3$ is one of many flux line domains associated with CRNs containing two nontrivial conservation laws. It is remarkable that those CRNs cover only domains that are inaccessible to unimolecular networks (compare Fig. 3, G and B). Although two broken conservation laws are neither sufficient nor necessary to exhibit states associated with $J_3<0$, the role of this topological feature in maximizing $-J_3$, regardless of network size, demonstrates the requirement for specific structural constraints to optimally redirect free energy for anabolic purposes.

Requirement for closed cycles and free-energy transduction to achieve metabolic flexibility

In addition to biosynthetic objectives, the ability to flexibly reroute fluxes through regulated kinetics is a crucial aspect of metabolic function. To address this issue, we consider a slightly different chemostatting protocol involving two nutrient species with equally high chemical potential and one waste species with much lower chemical potential (Fig. 4 A). The goal is to explore whether certain topologies exhibit a greater capacity to manifest a broad range of functional flux states, such as coconsumption, selective consumption, or export of nutrient species. Similar to the previous chemostatting condition investigated in Fig. 3, only bimolecular reaction networks demonstrate the ability to fully access the whole thermodynamically feasible flux space, including flux states characterized by the export of nutrient-like species $J_{1,2}<0$ (compare Fig. 4, B and C).

Figure 4.

Metabolic flexibility in bimolecular reaction networks. (A) CRN coupled to three chemostats (${\overrightarrow{\mu }}^y=\left[2.5,2.5,-2.5\right]$, $\overrightarrow{y}=\overrightarrow{1}$, and $k_{1,.,n_c}=1$), focusing on the flexible utilization of two nutrient sources. (B and C) Thermodynamically feasible flux domain (black line computed using Eqs. 8a and 8b) and sampled solutions of ${10}^4$ random unimolecular reaction networks (dots, with color denoting the internal contribution to entropy production ${\sigma }^I/\sigma$), plotted in the space of exchange fluxes $J_1$ and $J_2$. (C) Similar to (B) but for $5\times {10}^4$ random bimolecular reaction networks ($n_m=2$). (D) Examples of three CRN topologies with specific stoichiometric matrices ${\mathbb{S}}^x$ (red = $+1$; blue: $-1$; white: 0) and their corresponding kinetically sampled solution domains (gray dots on the bottom). These topologies differ qualitatively in their flux domain size $\mathcal{V}$ (gray area normalized to the bounded area). (E) Probability density of $\mathcal{V}$ for a set of $2\times {10}^3$ random topologies (each sampled with ${10}^3$ random kinetic parameters). Topologies are sampled based on a defined topological class characterized by the number of closed cycles $n_{cc}$ and the reaction molecularity $n_m$. (F) CRN of minimal topological complexity ($n_s=5$, $n_r=7$, $n_{ec}=2$, $n_{uc}=1$, $n_{bc}=1$, $n_{cc}=1$) characterized with maximum domain size $\mathcal{V}\approx 0.82$. (G) Flux solution domain of the minimal CRN, with color codes representing the energy charge ${\mu }_4-{\mu }_5$ as an indication of free-energy transduction. All quantities are expressed in units listed in Table S1.

The subsequent question is whether and which specific CRN topologies are more prone to display the broadest flux solution domain, including states that maximize exchange fluxes. Again, the flux solution domains of diverse bimolecular CRN topologies can be kinetically sampled and show very distinct flux domain size quantified by a normalized volume $\mathcal{V}$ using Eq. 12 (Fig. 4 D). Distribution of $\mathcal{V}$ across a random set of CRN topologies broadly ranges between 0 and 1, but different topologies are more or less prone to display a broad flux solution space. Unimolecular reaction networks show two peaks around 0.6 associated with the capacity to fill the whole sampled space shown in Fig. 4 B, and half this value is associated with topologies prone to import preferentially one of the two nutrient species. In contrast, bimolecular reaction networks show a broader range of flux domain size, including smaller and larger flux domains than unimolecular reaction networks. Interestingly, only bimolecular reaction networks involving closed reaction cycles are statistically much more prone to display highest values for $\mathcal{V}$ approaching one (Fig. 4 E). It is worth noting that, for the same number of species, the number of closed cycles increases with the number of reactions or the number of unbroken conservation laws. The key topological features required for displaying the broadest flux solution domain can be again identified by seeking the CRN of minimal complexity that maximizes $\mathcal{V}$. Such minimal topology is shown to comprise $n_r=7$ reactions between $n_s=5$ species, including one unbroken conservation law where $x_4+x_5=L$ and one closed cycle involving all internal reactions (Fig. 4 F). Kinetic sampling of this minimal CRN highlights that steady-state flux solutions cover nearly the entire feasible space, including the possibility of reaching the boundary of the feasible flux space at several locations corresponding to the modes of coconsumption, selective consumption, and selective export of each nutrient (Fig. 4 G). The presence of a closed cycle confers the possibility to implement various simple CRN topologies capable of achieving a given optimal flux configuration. For instance, $k_{4,5}=0$ or $k_{6,7}=0$ would generate minimal network topologies similar to the one depicted in Fig. 3 F: each of these topologies is characterized by two broken conservation laws and the ability to maximize the export flux of either nutrient species. It is also worth noting that the sampled flux states are all associated with ${\mu }_4>{\mu }_5$, indicating that the same energy-carrier species is exploited to transduce free energy to sustain a diversity of functional flux modes.

In summary, the functional capacity to exhibit a very diverse range of flux states is enhanced in network topologies comprising closed cycles, which provide flexibility, and multimolecular reactions implementing free-energy transduction mechanisms, enabling an expanded coverage of the thermodynamically feasible flux space.

Case study of a coarse-grained model of carbohydrate metabolism

Multimolecular reactions and free-energy transduction mechanisms are common in cellular metabolic pathways, notably through reaction coupling involving energy carriers such as ATP or NAD(P)H. The aim of this section is to apply the proposed framework to a coarse-grained model of carbohydrate metabolism and investigate whether the relations between free-energy transduction mechanisms and the geometry of the metabolic solution space hold true. Consistent with the assumption of a chemostatted CRN, some internal species are exchanged with chemostat reservoirs (i.e., glucose, acetate, water, carbon dioxide, oxygen, protons, biomass), while others are not (acetyl-coenzyme A, ADP, and ATP). Besides ATP, other important energy carriers, such as NAD(P)H, are implicitly considered for through conversion factors, although strict conservation laws for associated moieties are not enforced. The metabolic pathways that interconvert exchanged species with internal species (glycolysis, fermentation, aerobic respiration, biomass synthesis, and basal ATP consumption) are described as single effective reactions (Fig. 5 A). The set of simplifying assumptions and thermodynamic properties of this coarse-grained model are detailed in the supporting material and Table S2.

Figure 5.

Framework applied to a coarse-grained carbohydrate metabolic model. (A) A coarse-grained metabolic network model involving $n_c=7$ chemostat species, $n_s=10$ internal species, and $n_r=12$ reactions (the $H^{+}$ species is not shown for readability). (B) Workflow to compare feasible and sampled spaces for a chemostatted metabolic network model. (C) Feasible solution domains and (gray lines computed by solving Eqs. 8a and 8b) and kinetically sampled solutions of $2\times {10}^4$ reaction networks, plotted in the two-dimensional spaces of biomass exchange flux as a function of other exchange fluxes. The sampled flux solution polyhedron displays vertices (black geometrical shapes) representing elementary flux modes. Maximum carbon-based biomass yield $\beta =-J_{bex}/J_{gex}$ is represented for the sampled solution space ($\beta \approx 0.5$), the feasible space ($\beta \approx 0.66$), and the case where all glycolytic carbons are converted into biomass carbons ($\beta =1$). Chemostatting parameters are $y_{Glu}=0.02\mathrm{M}$, $y_{Bio}=y_{C{O}_2}=y_{Ace}=0.001\mathrm{M}$, and $y_{O_2}=y_{H_{2}O}=y_{H^{+}}=1\mathrm{M}$, and thermodynamic and structural parameters are given in Table S2.

The assumption of steady chemostatting conditions allows for a strict application of the framework designed to depict and compare the thermodynamically feasible and kinetically sampled flux spaces (Fig. 5 B). On the one hand, the chemostatting properties and the conservation laws associated with atom and other moieties (see Table S2) define the thermodynamically feasible domain in the space of exchange fluxes using Eqs. 8a and 8b. On the other hand, kinetic sampling of the steady-state flux solutions using Eqs. 2 and 3 determine the accessible flux subdomain, highlighting whether the topology is well-suited for maximizing a given flux objective. This comparative approach can be applied to any chemostatting condition of interest, here associated with glucose and oxygen availability and lack of acetate, providing several insights into the anabolic state of highest biomass yield (Fig. 5 C). Firstly, the sampled flux solution domain is polyhedron-shaped with vertices corresponding to elementary flux modes (Table S2) or their combinations, associated with a full fermentation mode, full aerobic respiration mode, and maximal biomass production through either fermentation only or respiration only. Similar solution polyhedra and their vertices are obtained with a constraint-based stoichiometric modeling approach, with the additional constraint imposed by the steady solutions in the chemical potential (or concentration) space. Secondly, the sampled flux solution domain covers a broad, but not complete, part of the thermodynamically feasible flux space, revealing topological constraints on thermodynamic efficiency. Accordingly, the optimal biomass yield obtained from kinetic sampling is lower than the maximum biomass yield constrained by conservation laws and thermodynamics and much lower than the maximum biomass yield considering carbon conservation only, where all glycolytic carbons are converted into biomass carbons (dashed lines in Fig. 5 C). Lastly, the paradoxical result shown in Fig. 3 where maximum anabolic flux coincides with nonmaximum nutrient import, is still valid but not observable, as $J_{gex}\lesssim J_{gex}^{\mathit{max}}$ (left, Fig. 5 C), due to higher chemical potential differences between chemostat species.

An important result obtained from the thermodynamic and sampling analysis of random CRNs is that free-energy transduction mechanisms, implemented by specific topologies, are crucial for efficient anabolic functions and biomass yield. In the carbohydrate metabolism model, the sampled solution domain in the space of biomass flux versus total dissipation rates shows that a higher optimal biomass yield correlates with lower dissipation rates (i.e., $T{\sigma }^{opt}\ll T{\sigma }^{\ast }\approx 60\text{kJ}/\mathrm{l}/\mathrm{h}$) (Fig. 6 A). The characteristics ${\beta }^{\mathit{max}}$, ${\beta }^{opt}$, or $T{\sigma }^{opt}$ of these optimal biomass-yield states depend on model parameters (Fig. S5), such as the standard chemical potential of the effective biomass species, chemostatting parameters like oxygen availability, and network topological properties such as ATP demand and yield per reaction. It is noteworthy that solutions seem to approach the feasible space boundary where ${\sigma }^I=0$, but a closer examination reveals that asymptotically reaching this quasi-equilibrium state is challenging due to the high standard free energies of many reactions (Fig. S4). The role of free-energy transduction in determining ${\beta }^{opt}$ is investigated by modulating the ATP yield of both glycolytic and respiratory reactions (${\mathbb{S}}_{ATP,gly/ares}$). A higher ATP yield in both reactions results in a broader flux solution domain and an increased optimal biomass yield (Fig. 6 B). This effect is mediated by changes in free-energy efficiency through ATP-producing reactions, defined as the ratio of the output over the input of free energy in a coupled process (24):

$${\eta }_j=-\frac{S_{ATP,j}\left({\mu }_{ATP}+{\mu }_{H_{2}O}-{\mu }_{AD{P}^{\prime }}\right)}{{\mu }_{prod,j}-{\mu }_{subs,j}}\text{.}$$ (13)

This efficiency varies between 0 (uncoupling) and 1 (maximum efficiency at equilibrium of the coupled reaction). For a given topology, free-energy transduction efficiencies can significantly vary depending on kinetically sampled flux states, but they also increase strongly in stoichiometric matrices with higher ATP yield per reactions (Fig. 6 C). A more systematic computation of optimal biomass yield ${\beta }^{opt}$ as a function of ATP yield per reaction ${\mathbb{S}}_{ATP,gly/ares}$ shows that biomass yield can approach its maximum feasible value ${\beta }^{\mathit{max}}$ (Fig. 6 D). Interestingly, the optimal biomass yield drops after a certain level of ATP yield due to the thermodynamically unfavorable glycolytic and respiratory reactions, given constraints on ADP and ATP concentrations. In summary, the structural properties and feasible space geometry of the carbohydrate metabolic model coincide with those of the small subset of sampled random multimolecular CRNs that achieve anabolic objectives, though there are notable differences related to higher standard free energies in the coarse-grained metabolic pathways.

Figure 6.

Relation between biomass yield and free-energy transduction. (A) $J_{bex}$ as a function of the total dissipation rate, highlighting that maximum feasible $\beta ={\beta }^{\mathit{max}}$ and optimal sampled $\beta ={\beta }^{opt}$ occurs for low dissipation rates. (B) Effect of ATP yield of glycolytic and aerobic respiratory reactions on the sampled flux solution space (bottom/red: ${\mathbb{S}}_{ATP,gly}={\mathbb{S}}_{ATP,ares}=1$; center/orange: $S_{ATP,gly}=S_{ATP,ares}=10$; top/blue: ${\mathbb{S}}_{ATP,gly}=15$, $S_{ATP,ares}=20$). Circles represent the states of optimal biomass yield ${\beta }^{opt}$ for each case. (C) Representation of the sampled solutions in (B) as a function of free-energy transduction efficiencies in glycolytic and respiratory reactions. (D) Effect of $S_{ATP,gly}$ and $S_{ATP,ares}$ on the optimal biomass yield ${\beta }^{opt}$. Circles represent parameters used in (B). Dashed line represents the limit case where ${\beta }^{opt}={\beta }^{\mathit{max}}$. Chemostatting parameters are $y_{Glu}=0.02\mathrm{M}$, $y_{Bio}=y_{C{O}_2}=y_{Ace}=0.001\mathrm{M}$, and $y_{O_2}=y_{H_{2}O}=y_{H^{+}}=1\mathrm{M}$, and thermodynamic and structural parameters are given in Table S2.

Discussion

Feasibility constraints and optimality principles offer a powerful explanatory framework for the observed metabolic phenotypes with respect to high-dimensional metabolic solution space (11,12). Such a framework suitably matches with a nonequilibrium thermodynamics characterization of metabolic phenotypes, particularly in terms of their entropy production rates or, equivalently, free-energy dissipation rates, at both the reaction and system levels (7,30,31). Nonequilibrium chemostatting conditions are sufficient to draw a thermodynamically feasible domain in the space of exchanged fluxes bounded by a manifold of vanishing internal entropy production. The geometry of this feasible space is shaped by concentrations, standard chemical potential and exchange rates, and conservation laws of the chemostat species. This space also displays few singular states, characterized by null or maximum total entropy production, and those associated with maximum internal entropy production. Of particular interest is a subdomain within the feasible space that is accessible only through multimolecular networks that implement free-energy transduction mechanisms. This includes flux states associated with the export of energy-rich species, similar to biomass production fueled by the free energy derived from nutrients and transduced by energy carriers. Based on this analogy, the theory suggests that achieving the highest biomass yield necessitates maximizing free-energy transduction efficiencies through near-equilibrium reactions, which turns out to be challenged by experimental data. Free-energy transduction in energy-producing pathways, such as glycolysis, fermentation, the citric acid cycle, and the electron transport chain, is far from reaching the maximum efficiency of 1, with estimates ranging between 0.3 and 0.6 (24). This aligns with the observation of significant dissipation of free energy in specific reactions, such as those catalyzed by phosphofructokinase, pyruvate kinase, and ATP synthase (5,7,32). A plausible hypothesis for this discrepancy is that dissipative, far-from-equilibrium reactions facilitate more efficient enzyme usage and are therefore more effective targets for regulating pathway flux (5,33). It would be intriguing to address whether an optimal level for free-energy transduction efficiency could emerge from combining constraints about enzyme utilization, flux controllability, and thermodynamic efficiency. This issue intersects with the search for cellular-scale constraints, such as those associated with overflow metabolism (34), including limits on free-energy dissipation rates (7), constraints on protein allocation (35), or considerations of macromolecular crowding effects (36).

In constraint-based stoichiometric modeling approaches, feasible flux solution spaces generally correspond to flux polyhedra resulting from the intersection of the null space and a region defined by a set of inhomogeneous linear constraints, such as bounded flux capacity or thermodynamically unfeasible directionalities (29,37). In particular, thermodynamics-based metabolic flux analysis integrates constraints based on experimental ranges of metabolite concentrations to restrict the solution flux space and facilitate flux analysis (38). Although related through the concept of thermodynamically feasible extreme pathways, the present approach provides a different perspective based on network topology rather than flux estimation or flux prediction. In this respect, two distinct types of constraints are here considered: exchange fluxes are constrained by nonequilibrium chemostatting conditions and internal fluxes are constrained by thermodynamic forces imposed by steady-state concentrations. Distinguishing between external and internal constraints allows for the definition of chemostatting-dependent flux solution space and subsequent determination of how much this space is occupied by flux solutions of a given network topology. This approach identifies topological features that contribute to the efficient utilization of thermodynamic forces to achieve metabolic objectives, such as flux maximization or flux rerouting. In other words, optimal states, coinciding in constraint-based stoichiometric modeling with the vertices of the flux polyhedron (12,29), can potentially be improved or impaired by specific modifications of the network topology, be it through regulation (39) or truncation (40,41).

A key topological feature is the multimolecularity of chemical reactions, common in metabolic networks due to the chemical interactions between molecules of heterogeneous atomic composition (42) and between multiple reactants, among them energy and electron carriers (17). The ability of multimolecular reaction networks to effectively achieve diverse metabolic objectives can be interpreted from several related features that distinguish them from unimolecular reaction networks. Firstly, multimolecular mass-action reactions introduce nonlinearities that generate a greater variability in species chemical potentials and, consequently, a broader range of nonequilibrium concentration states (28). Secondly, multimolecular reactions structurally implement energy coupling and free-energy transduction mechanisms (24), not only between pathways but also more generally between elementary flux modes. Lastly, the complex stoichiometries of multimolecular networks commonly exhibit nontrivial conservation laws (43), enabling a variety of optimal flux configurations beyond those maximizing the entropy production rates. This particular property of multimolecular networks relates to their hypergraph structure, which has motivated graph-theoretical and geometrical approaches to nonequilibrium reaction networks (43).

The relevance of a nonequilibrium thermodynamic framework is particularly valuable for studying metabolic shifts associated with diverse chemostatting conditions, such as changes in nutrient sources (44,45) or the division of labor and metabolite exchanges in microbial communities (46,47). Nonequilibrium models do not necessarily strive for the scalability of constraint-based stoichiometric modeling approaches, given the challenges in providing a self-consistent thermodynamic description of genome-scale models. Coarse-grained or modular models (48,49) are better suited for including the main pathways involved in metabolic flexibility. Notably, pathway motifs, such as cycles, bypasses, and shunts, are likely to increase the number of reactions relative to species and provide flexible flux rerouting in a regulation-dependent or chemostat-dependent manner (41,50,51). In such models, considering various moieties (e.g., phosphate, nicotinamide, coenzyme A) is crucial for exploring the role of conservation constraints in energy and biosynthetic metabolism. Besides investigating the design logic of cellular metabolic networks, it is also appealing to refine the theoretical framework concerning the restrictive assumptions of steady-state and reversible mass-action kinetics. The analysis of stability, fluctuations, or responses to driven time-dependent chemostat conditions (22) would enable understanding dynamic behaviors such as metabolic shifts (51), jamming and flux limits due to conserved moieties (18,52), or instabilities arising from autocatalytic cycles (53). Furthermore, integrating saturated reaction kinetics, which also relies on thermodynamic considerations (54,55), represents a natural continuation in exploring the interplay between thermodynamic and concentration-dependent contributions in shaping the flux space of a metabolic network.

Data and code availability

All Fortran codes and data used in this study are available at https://github.com/bpfeuty/CRN_fluxspace.

Acknowledgments

This work has been supported by the LABEX CEMPI (ANR-11-LABX-0007).

Author contributions

The author designed the research, carried out all simulations, and wrote the manuscript.

Declaration of interests

The author declares no competing interests.

Editor: Margaret Johnson.

Supporting citations

References (56,57,58,59,60,61) appear in the supporting material.