The Gibbs free energy landscape based on liver metabolome revealed thermodynamic robustness against fasting and obesity

Summary

Mammalian liver metabolism shifts substantially during fasting, yet the Gibbs free energy change of reaction (Δ_rG′), a fundamental thermodynamic quantity that defines reaction driving force and constrains metabolic directionality, remains poorly characterized in intact organs. Here, we mapped the Δ_rG′ landscape of mouse liver glucose metabolism during fasting using absolute metabolite concentrations and GLEAM (Gibbs free energy of reaction landscape estimation from metabolome assisted by variance-covariance matrix), a computational method developed here. Despite metabolite fluctuations, Δ_rG′ landscapes remained robust, changing only modestly, even for reactions reversing direction between glycolysis and gluconeogenesis. Thermodynamically inferred potential rate-limiting steps were maintained during fasting. Enzyme-cost analysis further suggested a trade-off between thermodynamic robustness and enzyme requirements during glycolytic switching. Obese mouse liver also showed thermodynamic robustness despite metabolic disruption. Our framework provides a novel thermodynamic perspective on intact organ metabolism, revealing preservation of key thermodynamic characteristics across fasting and obesity.

Graphical abstract

Highlights

• •GLEAM estimates the reaction Gibbs free energy landscapes from metabolomic data
• •Liver free-energy landscapes remain robust during fasting despite metabolite shifts
• •Potential rate-limiting steps persist; robustness trades off with enzyme cost
• •Obese liver retains WT-like free energy landscape despite metabolic disruption

Biological sciences; Enzymology; Human metabolism; Cancer systems biology

Introduction

Blood glucose homeostasis is crucial in mammals for providing stable energy sources to organs and maintaining health. During fasting, the liver plays a central role in maintaining blood glucose homeostasis by shifting its metabolism from glycolysis to gluconeogenesis.^1,2,3,4,5 This metabolic shift is a highly regulated process that involves the directional reversal of metabolic reactions and fluctuations in the concentrations of numerous metabolites.^6,7,8 Obesity induces metabolic dysregulation in the liver,^{7,8,9,10,11,12} leading to impaired glucose homeostasis^{13,14,15,16,17} and the development of further diseases, including type 2 diabetes.^9,18,19,20

Metabolic processes are governed by the laws of thermodynamics. According to the second law of thermodynamics, in the cellular environment, metabolic reactions proceed in the direction where the Gibbs free energy change of reaction (Δ_rG′) is negative, and Δ_rG′ is zero when the reaction reaches equilibrium. Therefore, Δ_rG′ indicates the reaction distance from equilibrium. In biochemical thermodynamic notation, Δ_rG′ denotes the transformed Gibbs energy change of reaction under fixed pH, pMg, and ionic strength.²¹ These pH and pMg are reflected by treating molecules in different binding states of H⁺ and Mg²⁺ as a single equilibrium mixture under specified pH and pMg (e.g., ATP is the equilibrium mixture of ATP⁴⁻, HATP³⁻, H₂ATP²⁻, MgATP²⁻, MgHATP⁻, and Mg₂ATP). The effect of ionic strength is incorporated through the estimation of activity coefficients by the extended Debye–Hückel equation,²² though this equation does not explicitly capture effects of macromolecular crowding or other intracellular non-idealities. Δ_rG′ can be calculated as the difference between the Gibbs free energy of formation (Δ_fG′) of the product and substrate metabolites:

$${\Delta }_r{G}^{\prime }=\sum _i{S}_i{\Delta }_f{G}_i^{\prime }$$ (Equation 1)

where, S_i is the stoichiometric coefficient of metabolite i in the reaction. Δ_fG′ depends on the metabolite activity, which is practically substituted by the following concentration:

$${\Delta }_f{G}^{\prime }={\Delta }_f{G}^{\prime \circ }+RT\ln c$$ (Equation 2)

where ${\Delta }_f{G}^{\prime \circ }$ is the standard Gibbs free energy of formation representing Δ_fG′ at 1 M of metabolite concentration, R is the gas constant, T is temperature, and c is metabolite concentration. Although calculation of Δ_rG′ requires absolute metabolite concentrations, mass spectrometry (MS)-based metabolomics analyses often suffer from ion suppression, a phenomenon where co-eluting compounds compromise quantitative accuracy by reducing analyte ionization efficiency.²³ Nevertheless, previous studies have successfully achieved absolute quantification of intracellular metabolites, primarily by employing isotope-labeled internal standards.^24,25 These techniques have enabled the investigation of absolute metabolite concentrations and subsequent thermodynamic analyses, such as Δ_rG′ landscapes, in microorganisms and cultured cells.^26,27,28,29 Despite these advancements, thermodynamic profiles have rarely been investigated at in vivo mammalian organ levels. Furthermore, all prior thermodynamic studies have focused on glycolysis-dominant conditions where glucose is the primary nutrient, leaving a large gap in our understanding of metabolism during fasting, a condition dominated by gluconeogenesis.

Mathematical metabolic modeling is an essential framework for a quantitative system-level understanding of metabolism. The thermodynamic principle regarding reaction direction has been widely applied as a constraint to increase the reliability of metabolic models.^{30,31,32,33,34,35,36,37} Furthermore, the significance of Δ_rG′ extends beyond physical aspects such as distance from equilibrium and reaction directionality to several physiological meanings, including pathway rate-limiting steps and enzyme demands for carrying metabolic flux.^38,39,40,41 Metabolic control analysis (MCA) is a mathematical framework for quantifying how perturbations in individual reactions affect pathway flux.^42,43 The degree of flux control exerted by each reaction is described by its flux control coefficient (FCC),^42,44,45 which can be affected by reaction thermodynamics, including Δ_rG′. However, these thermodynamic frameworks have hardly been applied to experimental data, especially those from multicellular organisms.

In this study, we explored the Δ_rG′ landscape in the liver metabolism of mice at different fasting levels using time-series absolute quantitative metabolomic data^8,46 measured by capillary electrophoresis-MS (CE-MS), which can reduce the effect of ion suppression,^47,48 without using isotopically labeled standards. Our results are the first in vivo identification of large-scale Δ_rG′ landscapes in mammalian organisms, and also the first elucidation of Δ_rG′ behavior during fasting in any cell type. Since metabolomic data contain uncertainties, we developed GLEAM (Gibbs free energy of reaction landscape estimation from metabolome assisted by variance-covariance matrix) to estimate Δ_rG′ with thermodynamically consistent metabolite concentrations from experimental metabolomic data. The estimated Δ_rG′ landscape during fasting showed that the central carbon metabolism in mouse liver globally exhibits thermodynamic robustness, as reflected by small changes in Δ_rG′ values despite large fluctuations in metabolite concentrations. We further integrated the Δ_rG′ and enzyme kinetics to investigate the physiological significance of the Δ_rG′ landscape in terms of rate-limiting steps and efficiency of metabolic switching. In addition to the thermodynamic robustness of mouse liver against fasting, we also demonstrated the robustness against obesity using leptin-deficient obese (ob/ob) mice.^49,50 Thus, our study provides novel insights into glucose metabolism in in vivo mammalian liver from a thermodynamic standpoint. These findings extend our understanding of mammalian metabolism beyond individual metabolite concentrations.

Results

The Δ_rG′ landscape is robust against fasting despite metabolite concentration fluctuations

In this study, we developed GLEAM to estimate Δ_rG′ from time-series metabolom and ${\Delta }_f{G}^{\prime \circ }$. One of the challenges in estimating Δ_rG′ in an entire reaction pathway is inherent uncertainties and missing values in metabolome and ${\Delta }_f{G}^{\prime \circ }$ data. To address this issue, GLEAM utilizes thermodynamic constraints and variances and covariances of metabolome and ${\Delta }_f{G}^{\prime \circ }$ data, and estimates a complete set of thermodynamically consistent metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$, and the subsequent Δ_rG′ (see STAR Methods). Briefly, GLEAM fits metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$ to data from the experiment or database within a range in which every reaction holds Δ_rG′ < 0 by the weighted least-squares method. The weights of the residuals are determined by variances and covariances of metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$ data. To consider uncertainties from metabolomic measurements, the Δ_rG′ estimation also provides confidence intervals based on parametric bootstrap sampling, which incorporates the standard deviations of the experimental metabolite concentrations and the residuals in GLEAM.

By applying GLEAM to time-series metabolomic data from ad libitum feeding to fasting in mice^8,46 and ${\Delta }_f{G}^{\prime \circ }$ data from the eQuilibrator database⁵¹ (Table S1), we estimated the Δ_rG′ landscape (Figures 1A and 1B and Table S2) in glycolysis, gluconeogenesis, and the TCA cycle (Table S3) in the liver. At ad libitum feeding (0-h fasting), when glycolysis is dominant, most of the irreversible reactions exhibited Δ_rG′ < −10 kJ/mol, which were defined as far-from-equilibrium reactions, such as Δ_rG′ = −21.56 kJ/mol in PFK and Δ_rG′ = −34.34 kJ/mol in PDH. Among the 11 irreversible reactions, only three reactions (ACO:IDH, PC, and PCK) exhibited Δ_rG′ > −10 kJ/mol, and PC and PCK were exceptionally close to equilibrium, with Δ_rG′ = −3.48 kJ/mol and −3.70 kJ/mol, respectively. In contrast, most of the reversible reactions exhibited Δ_rG′ > −5 kJ/mol, which were defined as near-equilibrium reactions, at ad libitum feeding, such as Δ_rG′ = −0.24 kJ/mol in GPI and Δ_rG′ = −0.03 kJ/mol in SDH. Four reversible reactions (ALDO, LSC, MDH, and ME) were exceptions that exhibited Δ_rG′ < −5 kJ/mol. Among those reactions, MDH was largely away from equilibrium in the opposite direction of the oxidative TCA cycle, with Δ_rG′ = −18.13 kJ/mol.

Figure 1.

Δ_rG′ landscape is robust against fasting despite metabolite concentration fluctuations

(A) Overview of the application of GLEAM to estimate Δ_rG′ landscape in the mouse liver. Metabolomic data were acquired from the liver, skeletal muscle, and plasma of WT and ob/ob mice during fasting (Table S1).^8,46n = 5 biological replicates were used per group. The estimated metabolite concentrations, ${\Delta }_f{G}^{\prime \circ }$, and Δ_rG′ values by GLEAM are shown in Table S2.

(B) Δ_rG′ landscape of glucose metabolism in mouse liver at ad libitum feeding and after 16-h fasting. Δ_rG′ is depicted by arrow colors. Shaded texts represent reaction names; boxed reactions are irreversible, while unboxed reactions are reversible. Blue shadings indicate near-equilibrium reactions with Δ_rG′ > −5 kJ/mol, and gray shadings indicate the other reactions with Δ_rG′ ≤ −5 kJ/mol. Unshaded texts represent metabolites. [h], [m], and [b] indicate hepatic, myocytic, and blood metabolites, respectively. Abbreviations of reactions and metabolites are defined in the glossary and Table S3.

(C) Comparison of Δ_rG′ in WT mouse liver at ad libitum feeding and after 16-h fasting. The forward direction (Δ_rG′ < 0) is defined as the reaction direction at ad libitum feeding. Error bars show 95% confidence intervals.

(D) Comparison of metabolite concentrations in WT mouse liver at ad libitum feeding and after 16-h fasting. Plotted metabolite concentrations represent those normalized by the mean value at all seven time points and two genotypes (14 conditions in total) to eliminate scale differences in concentrations among metabolites. Error bars show 95% confidence intervals.

(E) Comparison of Δ_rG′ in glycolysis across cell types and organisms. The data in iBMK cell, yeast, and E. coli were measured using LC-MS and isotope tracers.²⁸ Standard Gibbs free energy was calculated by treating all metabolite concentrations as 1 M. The vertical axis represents cumulative Δ_rG′ from GPI. Error bars show 95% confidence intervals.

The fasting-induced metabolic shift in the liver from glycolysis to gluconeogenesis may lead to changes in thermodynamic characteristics of metabolic reactions, including the sign reversal of Δ_rG′ of some reactions, i.e., reversal of their direction, by fluctuations in several metabolite concentrations. After 16-h fasting when gluconeogenesis is dominant, Δ_rG′ values remained similar in the liver (Figure 1C), with variation in 24/29 reactions being less than 5 kJ/mol. As a result of this thermodynamic robustness, as reflected by small changes in Δ_rG′ values, all irreversible reactions exhibited Δ_rG′ < −10 kJ/mol after 16-h fasting, indicating that the overall trend of irreversible reactions being far-from-equilibrium reactions was maintained across ad libitum feeding and after fasting. Although PC and PCK had been near-equilibrium at ad libitum feeding, these reactions moved away from equilibrium to exhibit Δ_rG′ = −9.07 kJ/mol and −5.25 kJ/mol after fasting, respectively. As for reversible reactions, despite reversal of some directions to switch the liver’s metabolism from glycolysis to gluconeogenesis, all the reversible reactions with Δ_rG′ > −5 kJ/mol at ad libitum feeding remained Δ_rG′ > −5 kJ/mol, indicating that the overall trend of reversible reactions being near-equilibrium reactions was maintained across ad libitum feeding and after fasting. In contrast to Δ_rG′, individual metabolite concentrations widely fluctuated during 16-h fasting (Figure 1D), with log2 fold change exceeding 0.5 for over 50% of metabolites. This result suggests that the thermodynamic architecture in glucose metabolism was robust against fasting while metabolite concentrations were reorganized. Note that the robustness of Δ_rG′ and metabolite concentrations cannot be directly compared by correlation coefficients for Figures 1C and 1D because Δ_rG′ values were not normalized while metabolite concentrations were normalized, leading to a much higher correlation coefficient for Δ_rG′. The reason why Δ_rG′ values were not normalized in Figure 1C is that Δ_rG′ does not represent abundance and thus its robustness should be evaluated not by fold changes but by difference values.

To assess the dependence of Δ_rG′ estimation on model constraints, we performed sensitivity analyses using GLEAM without direction constraints for reversible reactions or with relaxed bounds on metabolite concentrations and cofactor ratios. Removing direction constraints had little effect on overall Δ_rG′ values (Figure S1A), indicating that Δ_rG′ estimation was not strongly dependent on direction constraints. Although some reversible reactions were predicted to proceed in the glycolytic direction after 16-h fasting without these constraints, Δ_rG′ differences from the original model were small except for ALDO. Relaxed concentration and cofactor bounds altered Δ_rG′ values for several reactions, including CS and PC (Figure S1B), but the classification of most near- and far-from-equilibrium reactions and main downstream conclusions remained unchanged. As an additional control, substituting raw measured metabolite concentrations for GLEAM-estimated values produced similar Δ_rG′ values, suggesting that the estimated thermodynamic landscape was not driven by concentration adjustment alone (Figure S2).

To examine whether the thermodynamic characteristics of metabolic reactions observed in mouse liver are conserved across species, we compared the estimated Δ_rG′ in glycolytic reactions in mouse liver with those in other biological systems. The previous study²⁸ determined Δ_rG′ in glycolysis of immortalized baby mouse kidney epithelial (iBMK) cells, yeast, and Escherichia coli, reporting that in all these cell types, similar to mouse liver, reversible reactions were closer to equilibrium compared to the irreversible reaction, PFK (Figure 1E). This result suggests that the thermodynamic characteristics in glycolysis are widely conserved across species. An exception is PKLR, whose Δ_rG′ value in mouse liver was more than 20 kJ/mol smaller than those in the other cell types. Among the metabolite concentrations (or ratios) involved in PKLR, pyruvate concentration and ATP/ADP ratio were smaller in mouse liver than in mammalian iBMK cells, yeast, and E. coli (Table S4). In particular, pyruvate concentrations were two orders of magnitude smaller in mouse liver (3.77× 10⁻² mM) compared to the other three cell types (5.88 mM in iBMK cells, 9.40 mM in yeast, and 3.66 mM in E. coli).

Δ_rG′ maintenance during fasting is associated with metabolite concentration fluctuations in different ways among pathways

To characterize the metabolic reactions exhibiting large thermodynamic changes accompanying the fasting-induced metabolic shift, we linked the attributes of reactions including reversibility and pathway to ${\Delta }_r{G}_{16\mathrm{h}}^{\prime }-{\Delta }_r{G}_{0\mathrm{h}}^{\prime }$, defined as Δ_rG′ variations during fasting (Figure 2A). Among the seven irreversible reactions of glycolysis and gluconeogenesis, four reactions (GCK, PFK, PKLR, and PC) exhibited an absolute Δ_rG′ variation during fasting greater than 5 kJ/mol. Note that the identification of these large Δ_rG′ variations focuses solely on Δ_rG′ values and does not necessarily correspond to biological significance, such as reaction direction and flux. Contrary to the irreversible reactions, only one reversible reaction (ALDO) out of 11 showed an absolute Δ_rG′ variation during fasting exceeding 5 kJ/mol, despite the reversal of reversible reaction directions between glycolysis and gluconeogenesis under these conditions. This result suggests that reversible reactions maintain Δ_rG′ more strictly than irreversible reactions, even though the sign of Δ_rG′ is reversed. In the TCA cycle, the absolute Δ_rG′ variations during fasting were less than 3 kJ/mol for all reactions, indicating that the TCA cycle is thermodynamically stable in response to fasting.

Figure 2.

Δ_rG′ maintenance during fasting is associated with metabolite concentration fluctuations in different ways among pathways

(A) Absolute Δ_rG′ variation = $\left|{\Delta }_r{G}_{16\mathrm{h}}^{\prime }-{\Delta }_r{G}_{0\mathrm{h}}^{\prime }\right|$. Bars depict irreversible reactions in glycolysis and gluconeogenesis (orange), reversible reactions in glycolysis and gluconeogenesis (blue), irreversible reactions in the TCA cycle (yellow), reversible reactions in the TCA cycle (purple), and the other reactions in the metabolic network (gray), respectively. Error bars show 95% confidence intervals.

(B) Relationship between absolute Δ_rG′ variation and reaction distance from equilibrium represented by absolute Δ_rG′ at ad libitum feeding. Error bars show 95% confidence intervals. The reaction PC is highlighted and discussed in the results section.

(C) Comparison of absolute Δ_rG′ variation of reactions and total absolute Δ_fG′ variation of metabolites. Absolute Δ_fG′ variation is dependent on concentrations of metabolites involved in the reaction, and defined by $\sum \left|S\left({\Delta }_f{G}_{16h}^{\prime }-{\Delta }_f{G}_{0h}^{\prime }\right)\right|$. When Δ_fG′ variation = $S\left({\Delta }_f{G}_{16h}^{\prime }-{\Delta }_f{G}_{0h}^{\prime }\right)$ of the metabolites involved in the reaction are canceled out, absolute Δ_rG′ variation is smaller than total absolute Δ_fG′ variation, resulting in the reaction being plotted in the lower right. Plotted reactions for glycolysis and gluconeogenesis, and the TCA cycle are the same as Figure 2B. Error bars show 95% confidence intervals. The reactions GPD, GAPDH:PGK, and GCK are highlighted and discussed in the results section.

(D) Time courses of Δ_rG′ variation and Δ_fG′ variation for representative reactions: GPD (small Δ_fG′ variations), GAPDH:PGK (large Δ_fG′ variations which are canceled out), and GCK (large Δ_fG′ variations which are not canceled out). Red and gray lines depict Δ_rG′ variation and Δ_fG′ variation, respectively. Error bars show 95% confidence intervals.

As Δ_rG′ variations during fasting differed between irreversible and reversible reactions in glycolysis and gluconeogenesis, we further examined the relationship between the distance from equilibrium (|Δ_rG′|) at ad libitum feeding and the absolute Δ_rG′ variation (Figure 2B). In glycolysis and gluconeogenesis, the absolute Δ_rG′ variations during fasting were less than 3.5 kJ/mol for near-equilibrium reactions with |Δ_rG′| < 5 kJ/mol at ad libitum feeding except for PC. PC moved away from equilibrium only when gluconeogenesis was dominant in response to fasting. In contrast, the absolute Δ_rG′ variations of the reactions with |Δ_rG′| ≥ 5 kJ/mol at ad libitum feeding ranged between 0 and 10 kJ/mol, and were significantly larger than those of the near-equilibrium reactions (p value = 0.037 < 0.05 calculated by two-tailed Welch’s t test, n = 12 for near-equilibrium reactions with |Δ_rG′| < 5 and n = 6 for the other reactions with |Δ_rG′| ≥ 5 kJ/mol). This result suggests that Δ_rG′ of near-equilibrium reactions in glycolysis and gluconeogenesis tended to be strictly maintained, while this tendency was lost among other reactions. In the TCA cycle, Δ_rG′ was strictly maintained regardless of the distance from equilibrium. The statistical test was not performed for the TCA cycle due to a lack of near-equilibrium reactions.

To investigate the reason for maintaining Δ_rG′ values during fasting, we compared the absolute Δ_rG′ variation with the total absolute Δ_fG′ variation of metabolites involved in the reaction (Figure 2C). The total absolute Δ_fG′ variation of metabolites is defined as $\sum \left|S\left({\Delta }_f{G}_{16\mathrm{h}}^{\prime }-{\Delta }_f{G}_{0\mathrm{h}}^{\prime }\right)\right|$, where S is the stoichiometric coefficient of a metabolite involved in the reaction. The Δ_rG′ variation is associated in various ways with fluctuations in the Δ_fG′ of substrates and products, which depend on the metabolite concentrations. For example, when substrate and product concentrations remain relatively constant and changes in their Δ_fG′ remain small, the Δ_rG′ variation is small (e.g., GPD in Figure 2D). Even when substrate and product concentrations largely fluctuate and changes in their Δ_fG′ are large, the Δ_rG′ variation can remain small when the changes in Δ_fG′ of substrates and products are canceled out. This can happen, for instance, when both the substrate and product Δ_fG′ values increase simultaneously (e.g., GAPDH:PGK in Figure 2D). In contrast, when the changes in Δ_fG′ of substrates and products are not canceled out, the Δ_rG′ variation becomes large (e.g., GCK in Figure 2D). In glycolysis and gluconeogenesis, reactions with small absolute Δ_rG′ variation due to small fluctuations in metabolite concentrations, such as GPD, and due to the cancellation of Δ_fG′ variations such as GAPDH:PGK, were both observed (Figure 2C). For the latter type of reaction, the total absolute Δ_fG′ variations often corresponded to more than 5 kJ/mol, while the resulting Δ_rG′ variations were several kJ/mol smaller. This result indicates that the modest Δ_rG′ variation cannot be explained solely by logarithmic compression of fold changes, but also reflects cancellation among coordinated substrate and product concentration changes. In the TCA cycle, the overall contribution of metabolite concentration changes to Δ_rG′ was small, and accordingly, the absolute Δ_rG′ variations were also small.

Thermodynamic rate-limiting candidates were maintained during fasting

One of the physiological consequences of reaction distances from equilibrium is the control of steady-state pathway fluxes. The extent to which perturbation of each reaction can control pathway steady-state flux is quantified by the FCC in MCA.^42,44 To infer thermodynamic effects on the liver’s flux control, we analytically calculated FCCs under both arbitrary Δ_rG′ landscapes and those observed in mouse liver within glycolysis (Figure 3A), gluconeogenesis (Figure 3B), and the TCA cycle (Figure 3C) from an enzymatic rate law using the theorems of MCA. To incorporate reaction distance from equilibrium, we used a reversible kinetics called the direct binding modular rate law⁵³ as the enzymatic rate law. Under this framework, a flux v can be decomposed into the effect of thermodynamics η_rev (Figure 3D) and substrate saturation η_sat,s⁵⁴:

$$v=E{k}_{\text{cat}}^{+}{\eta }_{\text{rev}}{\eta }_{\text{sat},\mathrm{s}}\text{,}$$ (Equation 3)

$${\eta }_{\text{rev}}=1-\exp \left(\frac{{\Delta }_r{G}^{\prime }}{RT}\right)\text{,}$$ (Equation 4)

$${\eta }_{\text{sat},\mathrm{s}}=\frac{\prod _{i\in M^{\mathrm{s}}}{\left(\frac{c_i}{K_{\mathrm{m},i}}\right)}^{\left|S_i\right|}}{1+\prod _{i\in M^{\mathrm{s}}}{\left(\frac{c_i}{K_{\mathrm{m},i}}\right)}^{\left|S_i\right|}+\prod _{i\in M^{\mathrm{p}}}{\left(\frac{c_i}{K_{\mathrm{m},i}}\right)}^{\left|S_i\right|}}$$ (Equation 5)

where E is an enzyme concentration, $k_{\text{cat}}^{+}$ is the forward turnover number, K_m is the Michaelis-Menten constant, M^s is substrates, and M^p is products. Using this reaction rate law, we calculated FCCs based on the connectivity theorem and summation theorem in MCA⁵⁵ (see STAR Methods). This FCC calculation depends on metabolite concentrations, ${\Delta }_f{G}^{\prime \circ }$, and K_m. For ${\Delta }_f{G}^{\prime \circ }$, we used sampled values for the confidence interval calculation in GLEAM. K_m values were sampled from distributions based on the BRENDA database data.⁵² Metabolite concentrations were obtained by the following two approaches.

Figure 3.

TRCs were maintained during fasting

(A) Glycolysis pathway where FCCs were calculated.

(B) Gluconeogenesis pathway where FCCs were calculated.

(C) The TCA cycle where FCCs were calculated. MDH was excluded from this TCA cycle because it proceeded in the opposite direction to the oxidative TCA cycle.

(D) Correspondence between Δ_rG′ and η_rev.

(E–G) Relationship between η_rev and FCC in the reactions thermodynamically capable of achieving a relatively large FCC in glycolysis (E), gluconeogenesis (F), and the TCA cycle (G) in the mouse liver. η_rev and FCC values of each point were calculated from each sampled set of metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$ values for the calculation of confidence intervals and sampled K_m values based on the BRENDA database⁵² (see STAR Methods). Metabolite concentrations at ad libitum feeding were used for glycolysis (E) and the TCA cycle (G), and those after 16-h fasting were used for gluconeogenesis (F). Vertical variation is attributed to differences in K_m values and condition of the other reactions, while horizontal variation reflects differences in metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$.

(H) Alterations of TRC status based on transitions in η_rev for reactions in glycolysis and gluconeogenesis before and after starvation. Red, blue, and gray lines depict reactions remaining TRC (5 reactions), reactions remaining non-TRC (10 reactions), and reactions altering their TRC status (either TRC or not) during fasting (3 reactions), respectively. Black dashed lines indicate the boundary between TRC and non-TRC where η_rev = 0.86, i.e., Δ_rG′ = -5 kJ/mol. Error bars show 95% confidence intervals.

We investigated FCCs for metabolite concentrations in mouse liver using sampled values for confidence interval calculations in GLEAM, and FCCs for arbitrary Δ_rG′ landscapes to identify thermodynamic conditions for a large FCC. FCCs for arbitrary Δ_rG′ landscapes showed that an FCC value larger than 0.5, which is greater than the sum of FCCs of the other reactions in the pathway, required η_rev ≈ 1 except for reactions located upstream in a pathway (Figures S3A–S3D). In mouse liver, reactions with η_rev ≈ 1 at ad libitum feeding included irreversible reactions GCK, PFK, PKLR, and the reversible reaction ALDO in glycolysis. Among them, reactions that could achieve an FCC ≈1 were GCK and PFK, while the other reactions could achieve an FCC of between 0.5 and 0.7 (Figures 3E and S4A). We also found that GPI, which is a reversible reaction located upstream in the pathway, could achieve an FCC = 0.42 at η_rev ≈ 0.70. In gluconeogenesis, reactions with η_rev ≈ 1 after 16-h fasting were the irreversible reactions PC, PCK, FBP, and G6PC, all of which could achieve an FCC ≈1 (Figures 3F and S4B). All the other reactions exhibited only small FCC values. In the TCA cycle, similar FCC distributions were observed at ad libitum feeding and after fasting (Figures 3G, S4C, and S4D). Reactions with η_rev ≈ 1 were irreversible reactions CS, ACO:IDH, OGDH, and a reversible reaction LSC. Among these reactions, CS and OGDH could achieve an FCC ≈1, while ACO:IDH and LSC could achieve an FCC slightly below 0.7. Generally, reversible reactions have not been considered rate-limiting reactions because they have been recognized to be qualitatively closer to equilibrium than irreversible reactions. However, our quantitative analysis demonstrated that the reversible reactions GPI, ALDO, and LSC could achieve an FCC of between 0.4 and 0.7, comparable to several irreversible reactions. Considering the results of arbitrary Δ_rG′ landscapes, these large FCC values of the reversible reactions can be attributed to GPI being located relatively upstream in glycolysis, and ALDO and LSC being sufficiently away from equilibrium to exhibit η_rev ≈ 1 despite being closer to equilibrium than irreversible reactions.

As described above, a reaction with an FCC larger than 0.5 seems to be either away enough from equilibrium to exhibit η_rev ≈ 1, or located upstream in a pathway. Because the location of a reaction in a pathway depends on the definition of the pathway, a reaction being away enough from equilibrium to exhibit η_rev ≈ 1 is a key determinant for the reaction to have a large FCC, sometimes being referred to as the rate-limiting step. Therefore, we introduced thermodynamic rate-limiting candidate (TRC) as reactions with η_rev ≥ 0.86 corresponding to Δ_rG′ ≤ −5 kJ/mol, and examined which reactions are thermodynamically capable of being a rate-limiting step for any pathway definition at ad libitum feeding (Figure 3H, 0 h) and after fasting (Figure 3H, 16 h). Since this analysis is based on the reaction rate law described by Equations 3, 4, and 5, TRC status reflects the thermodynamic driving force and is distinct from activation energy and substrate saturation. TRC reactions both at ad libitum feeding and after fasting included GCK, PFK, CS, and seven other reactions, indicating that these reactions can be rate-limiting steps regardless of fasting levels. The parameter η_rev and TRC status (either TRC or not) are sensitive to Δ_rG′ variations at a near-equilibrium region (Δ_rG′ > −5 kJ/mol) while they are insensitive at a far-from-equilibrium region (Δ_rG′ < −10 kJ/mol) (Figure 3D). Due to this property, 15 out of 18 reactions remained either TRC (red line in Figure 3H) or non-TRC (blue line in Figure 3H) across ad libitum feeding and after fasting in glycolysis and gluconeogenesis, where only Δ_rG′ of near-equilibrium reactions were strictly maintained. This result suggests that fasting-induced fluctuations in metabolite concentrations occur within a range that maintains the TRC status for most reactions. The three reactions of ALDO, PC, and PCK were the exceptions that did not remain TRC or non-TRC (gray line in Figure 3H), and whether these reactions are rate-limiting or not would depend on fasting levels. The reactions maintaining their TRC status described above remain stable across the threshold from η_rev = 0.73 to 0.99.

Glycolysis and gluconeogenesis achieve efficient directional switching by requiring large enzyme costs

A fundamental question in biology is to understand what biological systems are optimized for. While objective functions optimized in microbial metabolism—such as maximizing growth or ATP production⁵⁶ and minimizing enzyme demands⁴¹—have been reported, the objective function for in vivo mammalian metabolism remains completely unknown. Therefore, we focused on the optimality of mouse liver metabolism, rather than on its exact regulatory mechanisms, by analyzing metabolite concentrations and Δ_rG′ landscape obtained from GLEAM as a model-based inference.

By solving the equation of the direct binding modular rate law (Equation 3) with respect to the enzyme amount, we obtain

$$E=\frac{v}{k_{\text{cat}}^{+}}{\eta }_{\text{rev}}^{-1}{\eta }_{\text{sat},\mathrm{s}}^{-1}\text{.}$$ (Equation 6)

This equation shows that the closer a reaction is to equilibrium (the smaller η_rev is), the greater the enzyme amount required to achieve the same flux v. Here, we defined enzyme cost (EC) as the product of enzyme amount and enzyme molecular weight, and EC minimum (ECM) as the theoretical state that minimizes the total EC within the entire pathway (see STAR Methods). By comparing ECs and metabolite concentrations in mouse liver to those in ECM, we investigated whether mouse liver metabolism is optimized to reduce enzyme demand. As with the FCC calculation, we calculated ECs using sampled values of metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$ used in the confidence interval calculation in GLEAM. For K_m values, we obtained them as geometric means of data in the BRENDA database.⁵²

We calculated ECs and ECM for the reactions in glycolysis at ad libitum feeding, gluconeogenesis after 16-h fasting, and the TCA cycle under both these conditions (Table S5). The liver showed a larger total EC compared to ECM, with a difference between pathways. The liver/ECM ratio of total EC was on the order of 10⁴ for glycolysis and gluconeogenesis, but only 10¹ for the TCA cycle (Figure 4A). Additionally, we compared ECM-predicted metabolite concentrations with liver metabolite concentrations in each pathway, and found typical fold errors of 10^RMSE = 29.79 (RMSE: root-mean-square error on a log₁₀-scale) for glycolysis and gluconeogenesis and 14.40 for the TCA cycle (Figure 4B). These findings show that enzyme and metabolite concentrations in the liver are not optimized for EC reduction, and this is more evident in glycolysis and gluconeogenesis than in the TCA cycle.

Figure 4.

Glycolysis and gluconeogenesis achieve efficient switching by requiring large EC

(A) Ratio of total enzyme cost under metabolite concentrations in the mouse liver and ECM (liver/ECM). The total EC was calculated for glycolysis, gluconeogenesis, and the TCA cycle, which are the same as the pathways used for the FCC calculation (Figures 3A–3C). Metabolite concentrations at ad libitum feeding and after 16-h fasting were used for glycolysis and gluconeogenesis, respectively. K_m values for the EC calculation were the geometric mean of data in the BRENDA database⁵² (Table S5). Error bars show 95% confidence intervals. ECs in mouse liver, and metabolite concentrations and ECs at ECM are shown in Table S5.

(B) Comparison of metabolite concentrations in the mouse liver and ECM. Plotted data are for the measured metabolites, and the scatterplot for the TCA cycle includes the data both at ad libitum feeding and after 16-h fasting. RMSE is the root-mean-square error on a log₁₀-scale of ECM’s prediction compared to metabolite concentrations in the mouse liver, and typical fold error (typical FE) = 10^RMSE. Error bars show 95% confidence intervals. Plotted data are for the measured metabolites.

(C) EC of each reaction in glycolysis, gluconeogenesis, and the TCA cycle. The left and right bars of each reaction are ECs in the mouse liver and ECM, respectively. Since the vertical axis represents a logarithmic scale, ECs can be depicted by the stacked bar of normalized enzyme molecular weight (green), the effect of thermodynamics, ${\eta }_{\text{rev}}^{-1}$ (red), and saturation, ${\eta }_{\text{sat},\mathrm{s}}^{-1}$ (blue). Note that although the total Gibbs free energy change across the entire pathway in ECM is fixed to that in the mouse liver (see STAR Methods), the sum of ${\eta }_{\text{rev}}^{-1}$ differs due to the non-linear increase of η_rev over Δ_rG′ (Figure 3D). Error bars show 95% confidence intervals.

(D) Comparison of metabolite concentrations in the liver or ECM, and those for MEG calculation, where the reactions required to reverse for metabolic switching are at equilibrium. Plotted data are for the measured metabolites. Error bars show 95% confidence intervals. Metabolite concentrations where the reactions required to reverse for metabolic switching are at equilibrium are shown in Table S6.

To identify a key reaction contributing to the large total EC of glycolysis and gluconeogenesis in mouse liver, we broke down the total EC of pathways into ECs of individual reactions (Table S5). The GAPDH:PGK reaction in mouse liver exhibited the highest EC of 10⁶ both in glycolysis and gluconeogenesis (Figure 4C), while ECs of reactions in the TCA cycle (e.g., SDH, ACO:IDH, and FH) were at most on the order of 10². The EC values of GAPDH:PGK are 99% and 89% of the total EC of glycolysis and gluconeogenesis, respectively, suggesting that GAPDH:PGK outstandingly elevates the total EC of glycolysis and gluconeogenesis. The high EC for GAPDH:PGK both in glycolysis and gluconeogenesis is attributed to two factors. One factor is low substrate saturation (Figure 4C, blue), which is given by the lower substrate metabolite concentrations (3.4 × 10⁻³ mM of GAP at ad libitum feeding and 1.3 × 10⁻² mM of 3 PG after fasting) than their respective K_m values (1.9 × 10⁻¹ mM for GAP and 3.5 × 10⁻¹ mM for 3 PG), resulting in large ${\eta }_{\text{sat},\mathrm{s}}^{-1}$. From Equation 6, a large ${\eta }_{\text{sat},\mathrm{s}}^{-1}$, indicating low substrate saturation, causes an increase in the EC to carry a certain flux. Under this condition, an increase in flux from higher enzyme levels is mitigated by fluctuations in metabolite concentrations. The other factor of the high EC for GAPDH:PGK is liver thermodynamics (Figure 4C, red), where the reaction was markedly close to equilibrium (Δ_rG′ = −0.01 kJ/mol both at ad libitum feeding and after fasting), leading to large ${\eta }_{\text{rev}}^{-1}$. When the reaction is close to equilibrium, increasing the enzyme abundance enhances both the forward and reverse fluxes, thus limiting the net flux increase. Thus, the elevated EC for this reaction is a result of both low substrate saturation and high reaction reversibility.

Since glycolysis and gluconeogenesis in mouse liver required much larger ECs than ECM, especially in the midstream reversible reactions due to large ${\eta }_{\text{rev}}^{-1}$ values, we hypothesized that the reason for this may be that the enzyme costs are required for enabling switching from glycolysis to gluconeogenesis during fasting. To assess the metabolic hurdle for directional switching from glycolysis to gluconeogenesis, we introduced the metabolite equilibrium gap (MEG) as the typical fold change in metabolite concentration required for the directional switching (Table S6) (see STAR Methods). In glycolysis and gluconeogenesis, mouse liver exhibited MEG = 1.35 while ECM exhibited MEG = 16.01 (Figure 4D), suggesting that the liver achieves directional switching from glycolysis to gluconeogenesis with much smaller metabolite concentration changes than ECM. Unlike glycolysis and gluconeogenesis, the TCA cycle, which does not undergo directional switching during fasting, exhibited a larger MEG (6.85) in the liver than that of glycolysis and gluconeogenesis. The mouse liver requires a much larger enzyme cost in glycolysis and gluconeogenesis than the theoretical minimum, allowing it to achieve directional switching with smaller changes in metabolite concentration. Conversely, a smaller enzyme cost is required for the TCA cycle—which does not undergo directional switching—than for glycolysis and gluconeogenesis.

Robustness of Δ_rG′ in the liver against obesity

Obesity disrupts liver metabolism, leading to impaired homeostasis in response to fasting^7,8,10 and the subsequent development of various diseases.^9,18,19,20 Recent studies have reported that the obesity-associated dysregulation of liver metabolism occurs even at ad libitum feeding.^11,12 To reveal the impact of obesity-induced metabolic disruption on the thermodynamic characteristics of reactions, we calculated Δ_rG′ in the ob/ob mouse liver during fasting.

The Δ_rG′ landscape in ob/ob mouse liver (Figure 5A and Table S2) showed that the difference in Δ_rG′ from WT mouse liver was less than 5 kJ/mol for most reactions both at ad libitum feeding and after 16-h fasting (Figures 5B and 5C). Consequently, the overall trend of irreversible reactions being far from equilibrium (Δ_rG′<-10 kJ/mol) and reversible reactions being near equilibrium (Δ_rG′>-5 kJ/mol) was preserved even under obesity. The similar Δ_rG′ values between WT and ob/ob mice indicate that the concentration ratios of substrates and products for the reactions are comparable between the genotypes. Individual metabolite concentrations (Table S2), however, showed discrepancies between WT and ob/ob mouse livers (Figure 5D), with log2 fold change exceeding 0.5 for 16/31 and 13/31 metabolite concentrations at ad libitum feeding and after 16-h fasting, respectively. This result suggests that the thermodynamic characteristics of reactions remain robust against obesity despite the changes in individual concentrations. Particularly, in most reversible reactions of glycolysis and gluconeogenesis, the differences in Δ_fG′ between WT and ob/ob mice were not reflected in the differences in Δ_rG′ (Figure 5E). This result prominently demonstrates that the reaction thermodynamic characteristics are robust to obesity while the individual metabolite concentrations are sensitive.

Figure 5.

Robustness of Δ_rG′ in the liver against obesity

(A) Δ_rG′ landscape of glucose metabolism in ob/ob mouse liver at ad libitum feeding and after 16-h fasting. Δ_rG′ is depicted by arrow colors. Shaded texts represent reaction names; boxed reactions are irreversible, while unboxed reactions are reversible. Blue shadings indicate near-equilibrium reactions with Δ_rG′ > −5 kJ/mol, and gray shadings indicate the other reactions with Δ_rG′ ≤ −5 kJ/mol. Unshaded texts represent metabolites. The estimated metabolite concentrations, ${\Delta }_f{G}^{\prime \circ }$, and Δ_rG′ values by GLEAM are shown in Table S2.

(B) Comparison of Δ_rG′ in WT and ob/ob mouse liver. The forward direction (Δ_rG′ < 0) is defined as the reaction direction in WT at ad libitum feeding. Error bars show 95% confidence intervals.

(C) Difference in Δ_rG′ between WT and ob/ob mouse livers (${\Delta }_r{G}_{ob/ob}^{\prime }-{\Delta }_r{G}_{\text{WT}}^{\prime }$). Reactions are arranged from left to right in order of decreasing absolute difference in Δ_rG′. Error bars show 95% confidence intervals.

(D) Comparison of metabolite concentrations in WT and ob/ob mouse liver. Plotted data represent the concentrations normalized by the mean value at all seven time points and two genotypes (i.e., 14 conditions) to eliminate scale differences in concentrations among metabolites. Error bars show 95% confidence intervals.

(E) Comparison of absolute difference in Δ_rG′ variation and total absolute difference in Δ_fG′ = $\sum \left|S\left({\Delta }_f{G}_{ob/ob}^{\prime }-{\Delta }_f{G}_{\text{WT}}^{\prime }\right)\right|$, between WT and ob/ob mouse livers. Plotted data include those at ad libitum feeding and after 16-h fasting. The reactions for glycolysis and gluconeogenesis, and the TCA cycle are the same as in Figure 2B. Error bars show 95% confidence intervals.

Discussion

We revealed the Δ_rG′ landscape of glucose metabolism in in vivo mouse liver for the first time from fasting time-series absolute quantitative metabolomic data using the newly developed GLEAM. In mouse liver, the overall trend of the Δ_rG′ landscape was maintained regardless of fasting level (Figures 1B and 1C). The Δ_rG′ maintenance was markedly strict in near-equilibrium reactions in glycolysis and gluconeogenesis, such as GAPDH:PGK (Figure 2B), despite large fluctuations in metabolite concentrations (Figures 1D and 2C), while this strictness was not true for the other reactions, such as GCK. In the TCA cycle, overall Δ_rG′ variations were small due to minimal metabolite concentration fluctuations during fasting (Figures 2B and 2C). This thermodynamic robustness against fasting led to maintenance of the reactions that could have a large FCC, i.e., TRCs, during fasting (Figure 3H) because FCC values are sensitive to Δ_rG′ variation only in the near-equilibrium region. The thermodynamic landscape in the liver could also contribute to efficient switching from glycolysis to gluconeogenesis by requiring larger ECs than that theoretically minimized (Figures 4A–4D). The results regarding FCC and EC were obtained by applying inferences based on a model using the assumed kinetic parameters to the Δ_rG′ values in mouse liver. We also found similar Δ_rG′ landscapes between WT and ob/ob mice (Figures 5A–5C), while metabolite concentrations were quite different (Figure 5D). Our results revealed that the liver robustly maintains Δ_rG′ landscapes favorable for metabolic control in contrast to fluctuations in individual metabolite concentrations caused by fasting and obesity.

Prior reports on Δ_rG′ landscapes have been limited to microorganisms and cultured cells under conditions where glycolysis is dominant.^26,27,28,29 In contrast, we targeted mouse liver during fasting and revealed the Δ_rG′ landscape at the in vivo mammalian organ level not only for glycolysis but also for gluconeogenesis. Our findings, therefore, extend the thermodynamic understanding of metabolic networks to higher organisms and different metabolic modes. Moreover, this study also introduces a novel computational framework for analyzing Δ_rG′ (Table S7). Methods for estimating thermodynamically consistent metabolite concentrations, ${\Delta }_f{G}^{\prime \circ }$, and Δ_rG′ from absolute quantitative metabolomic data have been proposed in previous studies.^28,29,57 In these methods, ${\Delta }_f{G}^{\prime \circ }$ values were fixed to the inputs of the model, or estimated without considering the covariance. However, metabolites sharing common chemical groups, such as ADP and ATP, tend to exhibit similar ${\Delta }_f{G}^{\prime \circ }$ values, because ${\Delta }_f{G}^{\prime \circ }$ depends on the structures of metabolites. Incorporating covariance information into the estimation thus better reflects the underlying chemical relationships between metabolites and allows for narrower confidence intervals in the estimated values.⁵⁸ By leveraging the covariance matrix of ${\Delta }_f{G}^{\prime \circ }$ from eQuilibrator,⁵¹ GLEAM achieves more precise and consistent estimation of thermodynamic characteristics throughout the entire metabolic network.

During fasting, hepatic enzyme activities are changed by hormonal signaling, enzyme expression, and allosteric regulation,^8,59 leading to redistribution of metabolic fluxes and subsequent changes in metabolite concentrations. As a result of these changes, the signs of Δ_rG′ values for the reversible reactions change, allowing flux reversal and thereby contributing to transition between glycolysis and gluconeogenesis. Therefore, changes in Δ_rG′ values should be interpreted primarily as a consequence of hormonal signaling, enzyme expression, and allosteric regulation, rather than as a direct target of regulation. Nevertheless, for the reversible reactions in glycolysis and gluconeogenesis, our results showed that the Δ_rG′ varied only modestly despite large changes in the metabolite concentrations (Figure 2C). This finding suggests that the Δ_rG′ landscape is more robust than the underlying metabolite concentrations during fasting-induced switching between glycolysis and gluconeogenesis. Whether such thermodynamic robustness also holds for other metabolic pathways, organs, and physiological or pathological conditions remains an important question for future research.

Using the obtained Δ_rG′ landscape in mouse liver, our study narrowed down the reactions that could achieve a large FCC in the pathways, i.e., candidates for a rate-limiting step (Figures 3E–3G and S4). A previous study experimentally measured FCCs in glycolysis in iBMK cells, and reactions exhibiting large FCCs were hexokinase (HK), which catalyzes the same reaction as GCK, and PFK.⁶⁰ Another previous study showed that PC had a dominant FCC in gluconeogenesis in rat liver.^61,62 Although no studies have measured FCCs in the TCA cycle to our knowledge, CS, IDH, and OGDH have conventionally been regarded as rate-limiting steps. All these reactions are included in the TRCs in this study, supporting the validity of our Δ_rG′ estimation even though this study focuses on mouse liver metabolism.

Our analyses showed that TRCs were maintained in mouse liver during fasting (Figure 3H). The maintenance of TRCs across metabolic states may have physiological advantages. Because TRCs represent reactions that can potentially exert strong control over pathway flux, their conservation during metabolic transitions suggests that mouse liver metabolism may rely on a stable set of reactions that are thermodynamically capable of controlling pathway flux under different physiological conditions. Such stability could allow metabolic fluxes to be regulated efficiently without requiring major reorganization of enzyme-expression or allosteric regulatory programs. This view is compatible with a previous standard-state analysis showing that reactions away from equilibrium share topological features with driver nodes of metabolic networks.⁶³ However, whether stable TRCs provide a functional advantage for metabolic regulation remains to be experimentally examined.

When calculating ECs, we made two assumptions: having common turnover number, k_cat values for all reactions and setting saturation (c/K_m) to the geometric mean of the pathway when K_m could not be obtained from the BRENDA database.⁵² These assumptions were required because of the difficulty of obtaining kinetic parameters in mammals. The former assumption may cause inaccurate weighting of individual enzymes in the total EC minimization, and the latter may lead to inaccurate ${\eta }_{\text{sat},\mathrm{s}}^{-1}$. However, the metabolite concentrations close to ECM were reported in a model in which k_cat and K_m information was lacking and only the influence of ${\eta }_{\text{rev}}^{-1}$ was considered.⁴¹ This means that the Δ_rG′ landscape in ECM would be similar under any k_cat and K_m values. Therefore, our conclusion of mouse liver metabolism not being optimized for enzyme cost reduction would be reasonable regardless of k_cat and K_m assumptions.

According to Equations 23, 24, 30, 31, 32, 33, 34, 35, and 38, the design of Δ_rG′ landscape faces a trade-off between FCC concentration and enzyme cost minimization. If only a few reactions in a metabolic pathway are far from equilibrium and exhibit η_rev close to 1, large FCCs are likely to be concentrated in these reactions. In this case, η_rev of other reactions are close to zero, leading to a large EC. This trade-off can be rephrased as a trade-off between flexibility in responding to multiple conditions and survival cost reduction under a single condition because the pathway flux can be efficiently regulated through a minimal number of enzyme activity changes when FCCs are concentrated in limited reactions. As a result, in mouse liver, particularly in glycolysis and gluconeogenesis, reactions that could achieve a large FCC were concentrated in a few reactions (Figures 3E–3G and S4), while there were large deviations in a total EC and metabolite concentrations between mouse liver and ECM (Figures 4A and 4B). Although a large EC does not directly indicate large enzyme expression, these results suggest that metabolic regulation in mouse liver adopts a strategy of flexibly responding to changes in environmental and physiological conditions rather than optimizing for a certain condition. In contrast, the previous study in E. coli showed that both enzyme concentrations and metabolite concentrations are well predicted by ECM within a typical fold error range of 10⁰⁴¹ (Figure S5). Taken together, these facts suggest differences in metabolic regulation strategies between cell types; mouse liver expends costs to perform flexible metabolic regulation, while E. coli optimized survival cost in a certain condition. Furthermore, our study also showed that this difference in metabolic regulation strategy may also exist between pathways by demonstrating the large differences in TRC proportions and the total EC optimality between glycolysis and gluconeogenesis, and the TCA cycle. Note that although glycolysis in mouse liver and E. coli shared a common thermodynamic tendency of reversible reactions being close to equilibrium with Δ_rG′ > −10 kJ/mol (Figure 1E), ECs in them can be greatly different. As shown in Equations 38 and 39, thermodynamic characteristics of reactions affect the EC through ${\eta }_{\text{rev}}^{-1}$, which indicates v^total/v^net, where v^total is the total flux of forward and backward reactions, and v^net is the net flux. In E. coli, most reversible reactions exhibited Δ_rG′ < −1 kJ/mol, contributing to reducing the EC with ${\eta }_{\text{rev}}^{-1}$ on the order of 10° (Figure S6, orange).^28,64 In contrast, most reversible reactions in mouse liver exhibited Δ_rG′ > −0.1 kJ/mol, where ${\eta }_{\text{rev}}^{-1}$ rises sharply (Figure S6, green), leading to a much larger EC than E. coli.

In glycolysis, MEG for the switching to gluconeogenesis is expected to be smaller as the reversible reactions are closer to equilibrium. The sum of Δ_rG′ values of the reversible reactions in each cell type shown in Figure 1E is −8.06 kJ/mol for mouse liver, −21.14 kJ/mol for iBMK cells, −24.41 kJ/mol for yeast, and −13.03 kJ/mol for E. coli. Therefore, if the correlation between the small value of MEG and reversible reactions being close to equilibrium holds, the metabolism in mouse liver is advantageous for the transition from glycolysis to gluconeogenesis in terms of changes in metabolite concentrations. This speculation is consistent with the physiological function of the liver, which is the maintenance of systemic glucose homeostasis rather than proliferation, supporting a specific link between MEG and the switching flexibility of metabolic modes.

The functional interpretation of the thermodynamically robust Δ_rG′ landscape should also consider hepatic energetic and redox requirements. Stoichiometrically, glycolysis from one glucose molecule to two pyruvate molecules generates a net of two ATP and two NADH molecules, whereas gluconeogenesis from two pyruvate molecules to one glucose molecule consumes four ATP, two GTP, and two NADH molecules. Therefore, the fasting-induced transition from glycolysis to gluconeogenesis involves not only changes in reaction directions and Δ_rG′ values, but also increased requirements for high-energy phosphate compounds and reducing equivalents. The ATP/ADP ratio increased after 16-h fasting in this study (Table S2), suggesting that hepatic energy status is maintained despite the energetic demand of gluconeogenesis. In particular, PGK, represented in our model as part of the GAPDH:PGK reaction, remains close to equilibrium after 16-h fasting, suggesting that the increased ATP/ADP ratio may help provide the thermodynamic condition required for gluconeogenic directionality. Thus, ATP production during fasting, mainly supported by mitochondrial metabolism including the TCA cycle and β-oxidation, likely contributes to sustaining gluconeogenesis while maintaining the thermodynamic architecture of glucose metabolism.

In conclusion, we developed GLEAM to estimate Δ_rG′ for the glucose metabolism in mouse liver and revealed the thermodynamic robustness against fasting and obesity. Furthermore, we suggested that these thermodynamic characteristics in the liver contribute to efficient metabolic control in two ways: the maintenance of rate-limiting step candidates and the switching between glycolysis and gluconeogenesis with minimal metabolite fluctuations by requiring a large enzyme cost. This study thus provides crucial insights into the principles underlying the highly flexible and robust metabolic regulation observed in mammalian liver, paving the way for future investigations into therapeutic targets for metabolic diseases.

Limitations of the study

One limitation of this study is that we treated hepatocytes as a single compartment and calculated Δ_rG′ from metabolite concentrations averaged within the cell. Some metabolites, including Cit, AKG, and OAA, are distributed between the cytosol and mitochondria, and Δ_rG′ values for reactions involving these metabolites may therefore be less accurate when calculated from whole-cell average concentrations. In addition, NAD⁺/NADH and NADP⁺/NADPH ratios are known to differ between the cytosol and mitochondria,⁶⁵ which may affect the Δ_rG′ values of redox reactions such as GAPDH, GPD, and PDH. Furthermore, localized concentrations can be affected by substrate channeling.⁶⁶ Substrate channeling is a phenomenon in which enzymes of consecutive reactions form complexes, allowing the product of the former enzyme to be directly transferred to the active site of the latter enzyme without being released into the solvent. For reactions involved in substrate channeling, Δ_rG′ is dependent on the channeled metabolite concentration in a limited space among the enzymes. Although substrate channeling has not yet been confirmed in living organisms, the channeling of oxaloacetate between MDH and CS has been indirectly supported by experiments,^67,68 and ALDO, TPI, and GAPDH have also been suggested to form complexes.^69,70

Another limitation is that the FCC calculations were based solely on stoichiometric relationships and did not account for allosteric regulation because of the severe lack of K_i in mammals and the simplification of the calculation. Inclusion of allosteric regulation would alter FCC values, allowing for further narrowing down of the reactions that can have a large FCC.

Lastly, due to a lack of flux measurements, functional interpretation drawn from the Δ_rG′ landscape is limited. Although Δ_rG′ constrains the direction of the reaction and reflects the thermodynamic driving force, it does not determine the metabolic flux by itself. Fluxes are also governed by enzyme abundance, turnover number, metabolite saturation, and allosteric regulation. Therefore, the thermodynamic robustness is not equivalent to functional robustness. To reveal the connection between thermodynamic and functional robustness, the integration of flux measurements is an important future direction.

Resource availability

Lead contact

Requests for further information and resources should be directed to and will be fulfilled by the lead contact, Shinya Kuroda (skuroda@bs.s.u-tokyo.ac.jp).

Materials availability

This study did not generate new unique reagents.

Data and code availability

• •Metabolomic data from WT and ob/ob mice in our previous studies^8,46 were used in this study. The kinetic-parameter distributions (Michaelis–Menten constants and turnover numbers) used for the FCC calculations and the FCC values generated in this study have been deposited on GitHub (https://github.com/tabekawa/Thermodynamics_MouseLiver). All other data needed to evaluate the conclusions in this study are provided in the study or the supplemental information.
• •The MATLAB code for GLEAM and the FCC and EC analyses has been deposited on GitHub and is publicly available at https://github.com/tabekawa/Thermodynamics_MouseLiver.
• •Any additional information required to reanalyze the data reported in this study is available from the lead contact upon request.

Acknowledgments

We thank our laboratory members for critically reading this manuscript. The computational analysis of this work was performed in part with the support of the supercomputer system of the National Institute of Genetics (NIG), Research Organization of Information and Systems (ROIS).

This study was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI (grant numbers JP17H06300, JP17H06299, JP18H03979, JP21H04759, JP22K17992, JP23H04946, JP23H04939) (S.K.); the Japan Science and Technology Agency (JST) as part of CREST (JPMJCR2123, JPMJCR25T3) and as part of ASPIRE (JPMJAP24B1) (S.K.); The Uehara Memorial Foundation (S.K.); AMED grant number JP21zf0127001 (T.S.); JST, CREST grant number JPMJCR2123 (T.S.); MEXT KAKENHI grant number JP23H04946 (T.S.); World Premier International Research Center Initiative (WPI), Human Biology-Microbiome-Quantum Research Center (Bio2Q), MEXT, Japan (T.S.); JSPS KAKENHI JP24K11417 (A.H.); and the FOREST Program of the Japan Science and Technology Agency (JPMJFR2052) (A.H.); JSPS KAKENHI JP21K14467 (S.O.).

Author contributions

Conceptualization, T.A., S.O., and S.K.; formal analysis, T.A.; software, T.A.; methodology, T.A., S.O., A.H., and T.S.; data curation, A.H. and T.S.; visualization, T.A.; funding acquisition, S.O., A.H., T.S., and S.K.; project administration, S.K.; supervision, S.K.; writing – original draft, T.A., S.O., and S.K.; writing – review and editing, T.A., S.O., A.H., T.S., and S.K.

Declaration of interests

The authors declare no competing interests.

Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work, the authors used ChatGPT, DeepL, and Claude in order to improve the readability and language of the manuscript. After using these tools, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Deposited data
Metabolomic data in the liver and blood of WT and ob/ob mice during fasting (n = 5)	Morita et al.8	N/A
Metabolomic data in the liver of WT and ob/ob mice during fasting (n = 5)	Li et al.46	N/A
Software and algorithms
MATLAB R2021b	MathWorks	https://jp.mathworks.com/products/matlab.html
Gurobi 10.0.3	Gurobi Optimization	https://www.gurobi.com/product
Python 3.10	Python Software Foundation	https://www.python.org/downloads/
Other
GLEAM (Gibbs free energy of reaction Landscape Estimation from metabolome Assisted by variance-covariance Matrix)	This study	https://github.com/tabekawa/Thermodynamics_MouseLiver

Experimental model and study participant details

Metabolomic data from WT and ob/ob mice in our previous studies^8,46 were used in this study.

Method details

Metabolic network for glucose metabolism in mice

A metabolic network for glucose metabolism in mice was constructed to calculate Δ_rG′. The network consists of a total of 55 reactions and 80 metabolites in the liver, muscle, and blood (the same reactions and metabolites in different compartments were counted as distinct ones), and involves glycolysis, gluconeogenesis, the TCA cycle, glycogen metabolism, alanine metabolism, and transporters (Figure S7 and Table S3). Among the 55 reactions, we defined 23 reversible reactions and 32 irreversible reactions based on the tissue-specific genome-scale metabolic network.⁷¹ All the transporters were defined as reversible reactions.

Reversible reactions GAPDH and PGK were combined into a single reversible reaction GAPDH:PGK because 1,3-bisphosphoglycerate was not measured in the metabolomic data.⁸ According to the experimentally measured metabolite concentrations, the reversible reaction ACO proceeded in the reverse direction of the oxidative TCA cycle with a little thermodynamic driving force at all time points in WT and ob/ob mice, breaking a sequential pathway from CS to FH. Since this long sequential pathway is necessary for the downstream analysis, such as the FCC and EC calculations, we summed ACO and irreversible reaction, IDH, which have a large thermodynamic driving force in the direction of the oxidative TCA cycle, as a single irreversible reaction, ACO:IDH.

In the skeletal muscle, FBP, G6PC, and PCK in gluconeogenesis were not included in the metabolic network. Although PC is also involved in gluconeogenesis, PC is included in the metabolic network of skeletal muscle because we could not exclude the possibility of its expression in the skeletal muscle.^72,73

In this study, we assumed the liver and skeletal muscle as single compartments of hepatocytes and myocytes, respectively, and cytosolic and mitochondrial compartments in these cells were not considered for simplification of the modeling. Therefore, the Δ_rG′ values calculated in this study represent the average values within the cells.

Algorithm of GLEAM

We developed GLEAM as a method to estimate a complete set of thermodynamically consistent metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$ values, from metabolomic data on a logarithmic scale, $\mathbf{ln}\mathit{c}\in {\mathbb{R}}^{m\times h\times g}$, and ${\Delta }_f{G}^{\prime \circ }$ data calculated by the component contribution method,⁷⁴ ${\mathbf{\Delta }}_{\mathit{f}}{\mathit{G}}^{\prime \circ }\in {\mathbb{R}}^m$, where m, h, and g are the number of metabolites, time points, and genotypes, respectively. Since metabolomic and ${\Delta }_f{G}^{\prime \circ }$ data contain uncertainties and missing values, directly substituting them into the definition equations of Δ_rG′ (Equations 1 and 2) often violates the second law of thermodynamics. To address this issue, GLEAM corrects for the uncertainties and missing values by calculating the most plausible values based on measured metabolomic data within a thermodynamically feasible space.

We first formulate the quadratic programming problem to minimize the departures of estimated metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$ from the experimental and database values under thermodynamic and physiological constraints:

$$\begin{array}{c}\underset{y\in \left\{\ln c,{\Delta }_f{G}^{\prime \circ }\right\},a}{\min }{Z}_{\text{QP}}=\sum _y\frac{1}{\#y^{\exp }}\left\{{\left(\mathit{y}+\mathit{a}-{\mathit{y}}^{\mathbf{exp}}\right)}^{\mathrm{T}}{\mathbf{\Sigma }}_{\left(\mathit{y}\right)}^{-1}\left(\mathit{y}+\mathit{a}-{\mathit{y}}^{\mathbf{exp}}\right)+\sum _i\lambda \left|\frac{{\left({\mathit{P}}_{\mathit{i}}\right)}^{\mathrm{T}}\mathit{a}}{\sqrt{D_{ii}}}\right|\right\}\\ \begin{array}{ccc}\mathrm{s}.\mathrm{t}.& {\Delta }_r{G}_j^{\prime }<0& \forall i\in M,\forall j\in N\end{array}\\ \begin{array}{cc}{\Delta }_r{G}_j^{\prime }=\sum _{i\in M}{S}_{i,j}\left({\Delta }_f{G}_i^{\prime \circ }+RT\ln c_i\right)& \forall j\in N^{\text{cell}}\end{array}\\ {\Delta }_r{G}_{\text{GLUT}}^{\prime }=RT\ln \frac{\left[\text{Glu}\left[\mathrm{h}\right]\right]}{\left[\text{Glu}\left[\mathrm{b}\right]\right]}\\ {\Delta }_r{G}_{\text{MCT}}^{\prime }=RT\ln \frac{\left[{\mathrm{H}}^{+}\left[\mathrm{b}\right]\right]\left[\text{Lac}\left[\mathrm{b}\right]\right]}{\left[{\mathrm{H}}^{+}\left[\mathrm{h}\right]\right]\left[\text{Lac}\left[\mathrm{h}\right]\right]}\\ \begin{array}{cc}{lb}_i\le \ln c_i\le u{b}_i& \forall i\in M\end{array}\\ \begin{array}{cc}{lb}_i\le \ln \frac{c_{i_1}}{c_{i_2}}\le u{b}_i& \forall \left(i_1,i_2\right)\in M^{\text{cofactor}}\end{array}\end{array}$$ (Equation 7)

where c_i is concentration of metabolite i, S_ij is stoichiometric coefficient of metabolite i in reaction j, R is the gas constant, and T is temperature. M, N, and N^cell are the sets of metabolites, reactions, and intracellular reactions, respectively. M^cofactor is the set of cofactor pairs whose the ratios were constrained. Σ is a variance-covariance matrix whose eigendecomposition is Σ=PDP^T. a is systematic error, and λ is regularization parameter. The superscript exp means measured or database value. The first term in curly brackets of the objective function represents the variance and covariance-weighted residuals of estimated metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$, which considers systematic errors from experimental operations, while the second term represents a regularization term for systematic errors to avoid overfitting. By using the covariances in ${\Delta }_f{G}^{\prime \circ }$ for the weight of residuals, the estimation reflects the correlations in ${\Delta }_f{G}^{\prime \circ }$ due to shared chemical groups. In contrast, all covariances between metabolite concentrations were set to zero, making Σ a diagonal matrix, since no prior correlation in metabolite concentrations is expected.

The solution of the optimization problem (7) may not provide unique concentrations, especially for unmeasured metabolites, which can lead to undetermined Δ_rG′ of the reactions involving the unmeasured metabolites. To estimate unique Δ_rG′, two terms with variance of unmeasured metabolite concentrations were added to the objective function:

$$\begin{array}{cc}{Z}_{\text{QP}\_\text{unique}}\left(y,a\right)& \\ & =\sum _y\frac{1}{\#y^{\exp }}\left\{{\left(\mathit{y}+\mathit{a}-{\mathit{y}}^{\mathbf{exp}}\right)}^{\mathrm{T}}{\mathbf{\Sigma }}_{\left(\mathit{y}\right)}^{-1}\left(\mathit{y}+\mathit{a}-{\mathit{y}}^{\mathbf{exp}}\right)+\sum _i\lambda \left|\frac{{\left({\mathit{P}}_{\mathit{i}}\right)}^{\mathrm{T}}\mathit{a}}{\sqrt{D_{ii}}}\right|\right\}\\ & +{ϵ}_1\sum _{\left(h_1,h_2\right)\in H^{\text{adj}}}\frac{1}{\left(h_2-h_1\right)}\sum _{i\in M^{\text{unmeasured}}}{\left(\ln c_{i,h_2}-\ln c_{i,h_1}\right)}^2\\ & +{ϵ}_2{\sum }_{i\in M^{\text{unmeasured}}}{\left(\ln c_i-\widehat{\ln c_i}\right)}^2\end{array}$$ (Equation 8)

where H^adj is the set of pairs of adjacent time points, M^unmeasured is the set of unmeasured metabolites, $\widehat{\ln c}$ is the median of upper and lower bounds of metabolite concentration, and ϵ₁ is a sufficiently small value to prioritize the first term over the second term. Similarly, ϵ₂ (ϵ₁≫ϵ₂) allows the second term to take precedence over the third term. The second term of the objective function (the first additional term) reduces the sum of concentration changes of unmeasured metabolites across time points, making smooth concentration changes more likely to be estimated than unnatural repetition of increases and decreases. The third term of the objective function (the second additional term) reduces the departures of the estimated unmeasured metabolite concentrations from the median of their upper and lower bounds. This term can be justified by the trade-off in increasing concentrations between elevation of enzyme efficiency and cellular burden from osmotic pressure and toxicity.^39,75 Since the objective function includes terms with greatly different orders of magnitude due to ϵ∈{ϵ₁,ϵ₂}, it has difficulty in being optimized at once. Therefore, in implementation, we first solve the optimization problem only for the first term, then, with the measured metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$ fixed to the obtained solutions, and then solve the optimization problem for the second and third terms.

Metabolic networks can include reversible reactions, and their directions are unknown under a condition of interest. By introducing binary variables representing reaction directions, the GLEAM model is finally formalized by the following mixed-integer quadratic programming problem:

$$\begin{array}{c}\underset{y\in \left\{\ln c,{\Delta }_f{G}^{\prime \circ }\right\},a,v,d}{\min }{Z}_{\text{MIQP}}=\sum _y\frac{1}{\#y^{\exp }}\left\{{\left(\mathit{y}+\mathit{a}-{\mathit{y}}^{\mathbf{exp}}\right)}^{\mathrm{T}}{\mathbf{\Sigma }}_{\left(\mathit{y}\right)}^{-1}\left(\mathit{y}+\mathit{a}-{\mathit{y}}^{\mathbf{exp}}\right)+\sum _i\lambda \left|\frac{{\left({\mathit{P}}_{\mathit{i}}\right)}^{\mathrm{T}}\mathit{a}}{\sqrt{D_{ii}}}\right|\right\}+{ϵ}_1\sum _{\left(h_1,h_2\right)\in H^{\text{adj}}}\frac{1}{\left(h_2-h_1\right)}\sum _{i\in M^{\text{unmeasured}}}{\left(\ln c_{i,h_2}-\ln c_{i,h_1}\right)}^2+{ϵ}_2{\sum }_{i\in M^{\text{unmeasured}}}{\left(\ln c_i-\widehat{\ln c_i}\right)}^2\\ \begin{array}{ccc}\mathrm{s}.\mathrm{t}.& d_j·{\Delta }_r{G}_j^{\prime }<0& \forall i\in M,\forall j\in N\end{array}\\ \begin{array}{cc}{\Delta }_r{G}_j^{\prime }=\sum _{i\in M}{S}_{i,j}\left({\Delta }_f{G}_i^{\prime \circ }+RT\ln c_i\right)<0& \forall i\in M,\forall j\in N^{\text{cell}}\end{array}\\ {\Delta }_r{G}_{\text{GLUT}}^{\prime }=RT\ln \frac{\left[\text{Glu}\left[\mathrm{h}\right]\right]}{\left[\text{Glu}\left[\mathrm{b}\right]\right]}\\ {\Delta }_r{G}_{\text{MCT}}^{\prime }=RT\ln \frac{\left[{\mathrm{H}}^{+}\left[\mathrm{b}\right]\right]\left[\text{Lac}\left[\mathrm{b}\right]\right]}{\left[{\mathrm{H}}^{+}\left[\mathrm{h}\right]\right]\left[\text{Lac}\left[\mathrm{h}\right]\right]}\\ \begin{array}{cc}{d}_j\in \left\{-1,1\right\}& \forall j\in N^{\text{rev}}\end{array}\\ \begin{array}{cc}{d}_j\in \left\{1\right\}& \forall j\in N^{\text{irrev}\_\mathrm{f}}\end{array}\\ \begin{array}{cc}{d}_j\in \left\{-1\right\}& \forall j\in N^{\text{irrev}\_\mathrm{b}}\end{array}\\ \begin{array}{cc}{d}_j·v_j>0& j\in N\end{array}\\ \begin{array}{cc}\sum _{j\in N}{S}_{i,j}{v}_j=0& i\in M^{\text{steady}}\end{array}\\ \begin{array}{cc}l{b}_i\le \ln c_i\le u{b}_i& \forall i\in M\end{array}\\ \begin{array}{cc}l{b}_i\le \ln \frac{c_{i_1}}{c_{i_2}}\le u{b}_i& \forall \left(i_1,i_2\right)\in M^{\text{cofactor}}\end{array}\end{array}$$ (Equation 9)

where v_j is metabolic flux through reaction j, d_j={1,-1} is a binary variable representing direction of reaction j, and N^rev is the set of reversible reactions. $N^{\text{irrev}\_\mathrm{f}}$ and $N^{\text{irrev}\_\mathrm{b}}$ are the sets of irreversible reactions whose directions are fixed in the forward and backward with respect to the definition of metabolic network, respectively. M^steady is the set of metabolites assumed to be in steady state. To solve this mixed integer quadratic programming problem efficiently, GLEAM treats it as a bilevel optimization problem, where the optimization of binary variables is the upper-level problem and the optimization of continuous variables is the lower-level problem:

$$\begin{array}{cc}\underset{d}{\min }& Z_{\text{MIQP}}\left(y^{\ast },a^{\ast },v^{\ast },d\right)\\ \mathrm{s}.\mathrm{t}\text{.}& y^{\ast },a^{\ast },v^{\ast }\in \arg \underset{y,a,v}{\min }{Z}_{\text{MIQP}}\left(y,a,v,d\right)\\ & y,a,v,d\in {\mathrm{\Omega }}_{\text{GLEAM}}\end{array}$$ (Equation 10)

where Ω_GLEAM is the solution space defined in the problem (9). In the bilevel optimization problem, binary variables are optimized by a genetic algorithm, while continuous variables are optimized by the interior-point method under the individuals of binary variables (Figure S8). The genetic algorithm was performed by the MATLAB function ga with a termination condition of FunctionTolerance = 10^-9 and a population size of PopulationSize = 500. The lower-level quadratic programming problem was solved using Gurobi⁷⁶ MATLAB API with dual feasibility tolerance of OptimalityTol = 10^-9 and primal feasibility tolerance of FeasibilityTol = 10^-9. This bilevel optimization was performed 10 times from random initial values, and the solution with the minimum objective function value was selected as the optimal solution.

Calculation of confidence intervals

The confidence intervals of the estimation by GLEAM were calculated using parametric bootstrap sampling of metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$. These data were first resampled by adding perturbations based on the variance and residual of each metabolite concentration and ${\Delta }_f{G}^{\prime \circ }$ in the Δ_rG′ estimation by Equation 9 using the experimental and database values.^29,77 To generate ${\Delta }_f{G}^{\prime \circ }$ samples reflecting their covariance, the perturbations were applied after the basis of ${\Delta }_f{G}^{\prime \circ }$ vector was transformed by transposed matrix of the eigenvectors of the variance-covariance matrix:

$$\ln c_i^b=\ln c_i^{\exp }+{\sigma }_i{\delta }_i^b,\forall b\in B\text{,}$$ (Equation 11)

$${\left({\mathit{P}}_{\left({\Delta }_{\mathit{f}}{\mathit{G}}^{\prime \circ }\right)\mathit{i}}\right)}^{\mathrm{T}}{\mathbf{\Delta }}_{\mathit{f}}{\mathit{G}}^{\prime \mathit{o},\mathit{b}}={\left({\mathit{P}}_{\left({\Delta }_{\mathit{f}}{\mathit{G}}^{\prime \circ }\right)\mathit{i}}\right)}^{\mathrm{T}}{\mathbf{\Delta }}_{\mathit{f}}{\mathit{G}}^{\prime \mathit{o},\mathbf{exp}}+\sqrt{D_{\left({\Delta }_f{G}^{\prime o}\right)ii}}{\delta }_i^b\text{,}$$ (Equation 12)

$${\mathbf{\Sigma }}_{\left(\mathit{y}\right)}^{\mathit{b}}={\mathbf{\Sigma }}_{\left(\mathit{y}\right)}\text{,}$$ (Equation 13)

$${\delta }_i^b\sim \mathcal{N}\left(0,{\left(s{d}^{\ast }\right)}^2\right)\text{,}$$ (Equation 14)

$${sd}^{\ast }=\sqrt{\frac{1}{\#y^{\exp }-1}\sum _i{\left(\frac{{\left({\mathit{P}}_{\left(\mathit{y}\right)\mathit{i}}\right)}^{\mathrm{T}}{\mathit{y}}^{\ast }+{\left({\mathit{P}}_{\left(\mathit{y}\right)\mathit{i}}\right)}^{\mathrm{T}}{\mathit{a}}^{\ast }-{\left({\mathit{P}}_{\left(\mathit{y}\right)\mathit{i}}\right)}^{\mathrm{T}}{\mathit{y}}^{\mathbf{exp}}}{\sqrt{D_{\left(y\right)ii}}}-{err}^{\ast }\right)}^2}\text{,}$$ (Equation 15)

$${err}^{\ast }=\frac{1}{\#y^{\exp }}{\left({\mathit{y}}^{\ast }+{\mathit{a}}^{\ast }-{\mathit{y}}^{\mathbf{exp}}\right)}^{\mathrm{T}}{\sum }_{\left(y\right)}^{-1}\left({\mathit{y}}^{\ast }+{\mathit{a}}^{\ast }-{\mathit{y}}^{\mathbf{exp}}\right)$$ (Equation 16)

where b is an index of bootstrap samples B, ∗ represents the optimized values by Equation 9 using the experimental and database values. δ is Gaussian noise generated from the normal distribution with mean 0 and standard deviation sd∗, which represents the standard deviation of metabolite concentration and ${\Delta }_f{G}^{\prime \circ }$ residuals in the Δ_rG′ estimation from the experimental and database values. Using the resampled metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$ values, we conducted the estimation of thermodynamically consistent metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$ in the same manner as the estimation from the experimental and database values. In this optimization, the directions of reversible reactions were fixed to those estimated from the experimental and database values. This resampling and optimization were performed 2000 times. By applying the percentile method to the resulting 2000 sets of the estimated values, we obtained the 95% confidence intervals.

Selection of regularization parameter

The regularization parameter λ for systematic errors in GLEAM was selected by the reduced chi-square (${\chi }_{\text{red}}^2$). ${\chi }_{\text{red}}^2$ was calculated using the estimated metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$ from resampled values^29,77:

$${\chi }_{\text{red}}^2=\frac{\#y^{\exp }EPE}{df}\text{,}$$ (Equation 17)

$$EPE=\frac{1}{\#B}\sum _{b\in B}\frac{1}{\#y^{\exp }}{\left({\mathit{y}}^{\mathit{b}\ast }+{\mathit{a}}^{\mathit{b}\ast }-{\mathit{y}}^{\mathbf{exp}}\right)}^{\mathrm{T}}{\sum }_{\left(y\right)}^{-1}\left({\mathit{y}}^{\mathit{b}\ast }+{\mathit{a}}^{\mathit{b}\ast }-{\mathit{y}}^{\mathbf{exp}}\right)\text{,}$$ (Equation 18)

$$df=\frac{\#y^{\exp }}{{2sd}^{\ast }}\left(EPE-{err}^{\ast }\right)$$ (Equation 19)

where b∗ represents the optimized values calculated from the resampled values. The model is interpreted as fitting the data ideally when ${\chi }_{\text{red}}^2$ = 1, as overfitting when ${\chi }_{\text{red}}^2$ < 1, and as poorly fitting when ${\chi }_{\text{red}}^2$ > 1. In this study, ${\chi }_{\text{red}}^2$ = 1.03 when λ = 1.51, and therefore it was selected as the best regularization parameter.

Application of GLEAM to mice data

We applied GLEAM to absolute quantitative metabolomic data in the liver, skeletal muscle, and plasma from WT and ob/ob mice (n = 5) after 0, 2, 4, 6, 8, 12, and 16-hour fasting (Table S1), which were measured in our previous studies.^8,46 We used the logarithmic mean and variance of the metabolites, which were successfully measured in three or more samples per condition, and metabolites with two or fewer measured samples were treated as unmeasured metabolites. The cofactors ATP, ADP, GTP, GDP, NAD⁺, NADH, FAD, FADH2, NADP⁺, and NADPH were treated as unmeasured metabolites regardless of the number of measured samples due to sample degradation, and instead, their ratios were constrained by previous reports (Table S8). Intracellular lactate was also treated as an unmeasured metabolite in the same manner as the cofactors because it might leach out of the container during measurement. The upper and lower bounds for many metabolites in GLEAM were set to a physiologically typical range of intracellular metabolite concentration (lower bound: 10^-7 M, upper bound: 10^-2 M),²⁸ while the bounds for some unmeasured metabolites were individually defined based on biological knowledge (Table S8).

${\Delta }_f{G}^{\prime \circ }$ data calculated by the component contribution method⁷⁴ were retrieved from the eQuilibrator database⁵¹ (Table S1). We used the identical ${\Delta }_f{G}^{\prime \circ }$ data for both liver and skeletal muscle under the condition in eQuilibrator of pH = 7.2, pMg = 3.0, and ionic strength = 0.15 M.²⁸ We also set the condition in blood to pH = 7.4⁷⁸ because Δ_rG′ for MCT depends on pH in the cells and blood. Temperature was set to T = 310.15 K for all the compartments.

In the liver, glycolysis was assumed to be dominant at the 0-hour fasting state (ad libitum feeding) while gluconeogenesis was assumed to take over by 16-hour fasting, both in WT^{79,80,81,82,83} and ob/ob mice.⁸⁴ At these two time points, the directions of reversible reactions involved in glycolysis and gluconeogenesis (GPI, ALDO, TPI, GAPDH:PGK, PGAM, ENO, PGM, GPD, LDH) were fixed and treated as those of the irreversible reactions in the calculation of GLEAM.

Since the reaction rates are sufficiently faster than the temporal changes in metabolite and enzyme levels during starvation, our metabolic model was assumed to be in a pseudo-steady state at each measured time point. This assumption leads to mass balance equilibrium for the intermediate metabolites (G6P, F6P, F1,6BP, DHAP, GAP, 3PG, 2PG, PEP, Pyruvate, AcCoA, Cit, Isocit, AKG, SucCoA, Suc, Fum, Mal, OAA) within the metabolic network.

Sensitivity analysis

We conducted sensitivity analyses to assess the dependency of Δ_rG′ estimation by GLEAM on the assumption of reversible reaction directions and bounds of metabolite concentrations and cofactor ratios. For the former analysis, the constraints of reversible reaction directions, which force reactions to operate in glycolysis or gluconeogenesis, were removed from the GLEAM model to estimate Δ_rG′. For the latter alnalysis, we multiplied the lower and upper bounds shown in Table S8 by 0.1 and 10, respectively, and conducted the estimation by GLEAM. However, when the value obtained by multiplying the upper bound by 10 is larger than 0.1 M, the relaxed upper bound for the sensitivity analysis was set to 0.1 M to keep the estimated metabolite concentrations within a physiologically plausible range.

Derivation of FCC

We derived flux control coefficients (FCCs) within glycolysis (Figure 3A), gluconeogenesis (Figure 3B), and the TCA cycle (Figure 3C) from the direct binding modular rate law,⁵³ which is equivalent to the reversible Michaelis-Menten kinetics⁸⁵ for simple reversible uni-uni reactions, using theorems of Metabolic Control Analysis (MCA).^42,43 In the direct binding modular rate law, flux can be described by the effect of the maximal rate, the thermodynamics, and the saturation of substrate⁵⁴:

$$v=E{k}_{\text{cat}}^{+}{\eta }_{\text{rev}}{\eta }_{\text{sat},\mathrm{s}}\text{,}$$ (Equation 20)

$${\eta }_{\text{rev}}=1-\exp \left(\frac{{\Delta }_r{G}^{\prime }}{RT}\right)\text{,}$$ (Equation 21)

$${\eta }_{\text{sat},\mathrm{s}}=\frac{\prod _{i\in M^{\mathrm{s}}}{\left(\frac{c_i}{K_{\mathrm{m},i}}\right)}^{\left|S_i\right|}}{1+\prod _{i\in M^{\mathrm{s}}}{\left(\frac{c_i}{K_{\mathrm{m},i}}\right)}^{\left|S_i\right|}+\prod _{i\in M^{\mathrm{p}}}{\left(\frac{c_i}{K_{\mathrm{m},i}}\right)}^{\left|S_i\right|}}$$ (Equation 22)

where M^s is the set of substrates, M^p is the set of products. From Equation 20, we calculated the elasticity of substrate ${ϵ}_{\mathrm{s},i}^v$ and product ${ϵ}_{\mathrm{p},i}^v$:

$${ϵ}_{\mathrm{s},i}^v=\frac{\partial v}{\partial c_i}\frac{c_i}{v}=\left|S_i\right|\left({\eta }_{\text{rev}}^{-1}-{\eta }_{\text{sat},\mathrm{s}}\right)\text{,}$$ (Equation 23)

$${ϵ}_{\mathrm{p},i}^v=\frac{\partial v}{\partial c_i}\frac{c_{\mathrm{i}}}{v}=\left|S_i\right|\left(1-{\eta }_{\text{rev}}^{-1}-{\eta }_{\text{sat},\mathrm{p}}\right)\text{,}$$ (Equation 24)

$${\eta }_{\text{sat},\mathrm{p}}=\frac{\prod _{i\in M^{\mathrm{p}}}{\left(\frac{c_i}{K_{\mathrm{m},i}}\right)}^{\left|S_i\right|}}{1+\prod _{i\in M^{\mathrm{s}}}{\left(\frac{c_i}{K_{\mathrm{m},i}}\right)}^{\left|S_i\right|}+\prod _{i\in M^{\mathrm{p}}}{\left(\frac{c_i}{K_{\mathrm{m},i}}\right)}^{\left|S_i\right|}}\text{.}$$ (Equation 25)

The stoichiometric matrices of the pathways we analyzed here are full rank, and each of their null spaces is one-dimensional, which means the degree of freedom of steady-state flux is one. For these pathways, the following Equations 26 and 27 hold from the connectivity theorem and the summation theorem in MCA, respectively⁵⁵:

$${\mathbf{C}}_{\mathbf{v}}^{\mathbf{J}}{\mathbf{ϵ}}_{\mathbf{c}}^{\mathbf{v}}=\mathbf{0}\text{,}$$ (Equation 26)

$$\sum _j{C}_{v_j}^J=1\text{,}$$ (Equation 27)

$${\mathbf{C}}_{\mathbf{v}}^{\mathbf{J}}=\left(\begin{array}{ccc}{C}_{v_1}^J& \cdots & C_{v_n}^J\\ ⋮& \ddots & ⋮\\ C_{v_1}^J& \cdots & C_{v_n}^J\end{array}\right)\text{,}$$ (Equation 28)

$${\mathbf{ϵ}}_{\mathbf{c}}^{\mathbf{v}}=\left(\begin{array}{ccc}{ϵ}_{c_1}^{v_1}& \cdots & {ϵ}_{c_k}^{v_1}\\ ⋮& \ddots & ⋮\\ {ϵ}_{c_1}^{v_n}& \cdots & {ϵ}_{c_k}^{v_n}\end{array}\right)$$ (Equation 29)

where ${ϵ}_c^v=\left\{{ϵ}_s^v,{ϵ}_p^v\right\}$. Integrating Equations 23, 24, 26, and 27, we obtained FCC as:

$$C_{v_l}^J=\frac{{\alpha }_l}{\sum _j{\alpha }_j}\text{,}$$ (Equation 30)

where α for GAPDH:PGK in glycolysis is described as

$$\alpha =\prod _{j=0}^{\text{ALDO}}\left(-{ϵ}_{\mathrm{p}}^{v_j}\right)\left({ϵ}_{\mathrm{s}}^{v_{\text{TPI}}}-{ϵ}_{\mathrm{p}}^{v_{\text{TPI}}}\right)\prod _{j=l+1}^{n+1}{ϵ}_{\mathrm{s}}^{v_j}\text{,}$$ (Equation 31)

α for downstream reactions of GAPDH:PGK in glycolysis is described as

$$\alpha =\prod _{j=0}^{\text{ALDO}}\left(-{ϵ}_{\mathrm{p}}^{v_j}\right)\left({ϵ}_{\mathrm{s}}^{v_{\text{TPI}}}-{ϵ}_{\mathrm{p}}^{v_{\text{TPI}}}\right)\prod _{j=\text{GAPDH}:\text{PGK}}^{l-1}\left(-{ϵ}_{\mathrm{p}}^{v_l}\right)\prod _{j=l+1}^{n+1}{ϵ}_{\mathrm{s}}^{v_j}\text{,}$$ (Equation 32)

α for ALDO in gluconeogenesis is described as

$$\alpha =\prod _{j=0}^{\text{GAPDH}:\text{PGK}}\left(-{ϵ}_{\mathrm{p}}^{v_j}\right)\left({ϵ}_{\mathrm{s}}^{v_{\text{TPI}}}-{ϵ}_{\mathrm{p}}^{v_{\text{TPI}}}\right)\prod _{j=l+1}^{n+1}{ϵ}_{\mathrm{s}}^{v_j}\text{,}$$ (Equation 33)

α for downstream reactions of ALDO in gluconeogenesis is described as

$$\alpha =\prod _{j=0}^{\text{GAPDH}:\text{PGK}}\left(-{ϵ}_{\mathrm{p}}^{v_j}\right)\left({ϵ}_{\mathrm{s}}^{v_{\text{TPI}}}-{ϵ}_{\mathrm{p}}^{v_{\text{TPI}}}\right)\prod _{j=\text{ALDO}}^{l-1}\left(-{ϵ}_{\mathrm{p}}^{v_j}\right)\prod _{j=l+1}^{n+1}{ϵ}_{\mathrm{s}}^{v_j}\text{,}$$ (Equation 34)

α for other reactions is described as

$${\alpha }_l=\prod _{j=0}^{l-1}\left(-{ϵ}_{\mathrm{p}}^{v_j}\right)\prod _{j=l+1}^{n+1}{ϵ}_{\mathrm{s}}^{v_j}\text{.}$$ (Equation 35)

Note that reaction indices j∈{1, ...,N} are denoted such that 1 represents the most upstream reaction and N represents the most downstream reaction in the pathway, and ${ϵ}_{\mathrm{s}}^{v_0}={ϵ}_{\mathrm{p}}^{v_0}={ϵ}_{\mathrm{s}}^{v_{n+1}}={ϵ}_{\mathrm{p}}^{v_{n+1}}=1$.

Metabolite concentrations and ${\mathbf{\Delta }}_{\mathit{f}}{\mathit{G}}^{\prime \circ }$ values used in FCC calculation

We calculated FCCs under arbitrary Δ_rG′ landscapes as well as under the hepatic ones. For both conditions, we used ${\Delta }_f{G}^{\prime \circ }$ values sampled for confidence interval calculation in GLEAM. To generate the hepatic Δ_rG′ landscapes, we used metabolite concentrations sampled in the confidence interval calculation. To generate arbitrary Δ_rG′ landscapes, logarithmic metabolite concentrations were uniformly sampled from the physiologically reasonable solution space Ω_lnc:

$${\mathrm{\Omega }}_{\ln c}=\left\{\ln c|\begin{array}{c}{\Delta }_r{G}_j^{\prime }<0\\ {lb}_i\le \ln c_i\le u{b}_i\\ l{b}_i=u{b}_i=\ln c_i^{b\ast }i\in M^{\text{blood}}\end{array}\right\}$$ (Equation 36)

where M^blood represents blood metabolites. The upper and lower bounds of metabolite concentrations were the same as those used in GLEAM (Table S8). This sampling was performed 20 times for each ${\Delta }_f{G}^{\prime \circ }$ set using MATLAB function polySampler in the Volume-and-Sampling package.⁸⁶ Thus, we obtained 2000 × 20 = 40000 arbitrary Δ_rG′ landscapes in total.

K_m values used in FCC calculation

In the FCC calculation, we sampled Michaelis-Menten constants K_m for each Δ_rG′ landscape to calculate reaction saturation level. To sample K_m, we first retrieved K_m and the turnover number, $k_{\text{cat}}\in \left\{k_{\text{cat}}^{+},k_{\text{cat}}^{-}\right\}$ values for enzymes of Mus musculus, Rattus norvegicus, Homo sapiens, Bos taurus, Sus scrofa, Canis lupus familiaris, and Mammalia⁸⁷ from the BRENDA database.⁵² For K_m and k_cat values for which three or more data points were available without reports of mutant enzymes, we calculated their logarithmic means and standard deviations. For K_m and k_cat with fewer than three data points, we set the logarithmic mean of k_cat (/s) to 1 with a standard deviation of 1.5, and the logarithmic mean of K_m (mM) to −1 with a standard deviation of 1.⁸⁸ With the means and standard deviations of K_m and k_cat, we uniformly sampled logarithmic K_m, k_cat values for each enzyme from the solution space Ω_KP:

$${\mathrm{\Omega }}_{\text{KP},j}=\left\{\begin{array}{c}(\log K_{\mathrm{m}},\\ \log k_{\text{cat}})\end{array}|\begin{array}{c}{\log {\mu }^{K_{\mathrm{m},i,j}}-2\sigma }^{K_{\mathrm{m},i,j}}\le \log K_{\mathrm{m},i,j}\le {\log {\mu }^{K_{\mathrm{m},i,j}}+2\sigma }^{K_{\mathrm{m},i,j}}\\ {\log {\mu }^{k_{\text{cat},j}}-2\sigma }^{k_{\text{cat},j}}\le \log k_{\text{cat},j}\le {\log {\mu }^{k_{\text{cat},j}}+2\sigma }^{k_{\text{cat},j}}\\ \log k_{\text{cat},j}^{+}-\log k_{\text{cat},j}^{-}-\sum _{i\in M^{\mathrm{s}}}\log K_{\mathrm{m},i,j}^{\left|S_{i,j}\right|}+\sum _{i\in M^{\mathrm{p}}}\log K_{\mathrm{m},i,j}^{\left|S_{i,j}\right|}=\log K_{\text{eq},j}\\ K_{\text{eq},j}=\exp \left(-\frac{1}{RT}\sum _{i\in M}{S}_{i,j}{\Delta }_f{G}_i^{\prime \circ }\right)\end{array}\right\},\forall i\in M,\forall j\in N$$ (Equation 37)

where μ is mean, σ is standard deviation, and K_eq is equilibrium constant. The third constraint reflects the Haldane relationship.⁸⁵ These samplings were performed 20 times for each Δ_rG′ landscape using MATLAB function polySampler in the Volume-and-Sampling package.⁸⁶ Thus, we finally calculated FCCs under the arbitrary and hepatic Δ_rG′ landscapes with 20 × 20 × 2000 = 800000 and 20 × 2000 = 40000 sets of metabolite concentrations, ${\Delta }_f{G}^{\prime \circ }$ and K_m values, respectively.

Definitions of EC and total EC

By solving the equation of the direct binding modular rate law (Equation 20) with respect to the enzyme amount, we can obtain enzyme demand needed to carry a certain flux:

$$E=\frac{v}{k_{\text{cat}}^{+}}{\eta }_{\text{rev}}^{-1}{\eta }_{\text{sat},\mathrm{s}}^{-1}\text{.}$$ (Equation 38)

With this equation, we defined Enzyme Cost (EC) as the product of the enzyme demand and the enzyme molecular weight w:

$$\text{EC}=wE\text{.}$$ (Equation 39)

The enzyme molecular weights were retrieved from the UniProt database.⁸⁹ We also defined the pathway’s total EC as the sum of ECs in the pathway:

$$\text{total}\text{EC}=\sum _{j\in N^{\text{pathway}}}{\text{EC}}_j$$ (Equation 40)

where N^pathway is the set of reactions included in the pathway. We calculated ECs in glycolysis (Figure 3A), gluconeogenesis (Figure 3B), and the TCA cycle (Figure 3C), which were the same as those used in the FCC calculations, under the hepatic Δ_rG′ landscape using metabolite concentrations and ${\Delta }_f{G}^{\prime \circ }$ values sampled for confidence interval calculation in GLEAM, and geometric mean of K_m values from the BRENDA database⁵² obtained in the FCC calculation (Table S5). For K_m with fewer than three data found in the BRENDA database, we set their c/K_m values to the geometric mean of the pathway. $k_{\text{cat}}^{+}$ values were assumed to be the same within a pathway, and their effect can be ignored in this study because we only focused on the relative values of ECs.

Minimization of total EC

Since ECs are dependent on metabolite concentrations, we defined Enzyme Cost Minimum (ECM) as a theoretical condition in which the total EC is minimized with metabolite concentrations. We obtained ECM by solving the following non-linear optimization problem⁴¹:

$$\begin{array}{cc}\underset{\ln c}{\min }& \text{total}\text{EC}=\sum _{j\in N^{\text{pathway}}}{\text{EC}}_j\\ \mathrm{s}.\mathrm{t}\text{.}& {\text{EC}}_j=wv{\eta }_{\text{rev},j}^{-1}{\eta }_{\text{sat},\mathrm{s},j}^{-1},j\in N\\ & {\eta }_{\text{rev},j}=1-\exp \left(\frac{{\Delta }_r{G}_j^{\prime }}{RT}\right)\\ & {\eta }_{\text{sat},\mathrm{s},j}=\frac{\prod _{i\in M^{\mathrm{s}}}{\left(\frac{c_i}{K_{\mathrm{m},i,j}}\right)}^{\left|S_{i,j}\right|}}{1+\prod _{i\in M^{\mathrm{s}}}{\left(\frac{c_i}{K_{\mathrm{m},i,j}}\right)}^{\left|S_{i,j}\right|}+\prod _{i\in M^{\mathrm{p}}}{\left(\frac{c_i}{K_{\mathrm{m},i,j}}\right)}^{\left|S_{i,j}\right|}}\\ & {\Delta }_r{G}_j^{\prime }=\sum _{i\in M}{S}_{i,j}\left({\Delta }_f{G}_i^{\prime \circ ,\text{liver}}+RT\ln c_i\right)<0,j\in N\\ & l{b}_i\le \ln c_i\le u{b}_i,i\in M\\ & l{b}_i=u{b}_i=\ln c_i^{\text{mouse}},i\in M^{\text{blood}}\\ & l{b}_i=u{b}_i=\ln c_i^{\text{mouse}},i\in M^{\text{terminal}}\\ & \ln \frac{c_{i_1}}{c_{i_2}}={\left(\ln \frac{c_{i_1}}{c_{i_2}}\right)}^{\text{mouse}},\forall \left(i_1,i_2\right)\in M^{\text{cofactor}}\text{.}\end{array}$$ (Equation 41)

w and v are normalized by the minimum molecular weight and flux, respectively. The relative fluxes for each analyzed pathway were unique because the degree of freedom of these pathways’ steady-state flux is 1. lnc^mouse represents metabolite concentration in mouse, i.e., lnc∗ and lnc^b∗. M^blood represents the set of blood metabolites, and M^terminal represents the set of intracellular metabolites at the termini of each pathway: Glucose and Pyr for glycolysis and gluconeogenesis, and AcCoA, OAA, and Mal for the TCA cycle. By fixing these concentrations and cofactor ratios to those in mouse liver, the total Gibbs free energy change across the entire pathway in ECM is consistent with that in the mouse liver. ECM was calculated by MATLAB function fmincon with termination tolerance on the first-order optimality of OptimalityTolerance = 10^-7, tolerance on the constraint violation of ConstraintTolerance = 10^-9, maximum number of function evaluations of MaxFunctionEvaluations = 1000000, and maximum number of iterations of MaxIterations = 100000. The optimization was performed 100 times for a single problem, and the solution minimizing the objective function was selected as ECM.

Calculation of MEG

To quantify the efficiency of metabolic switching in terms of metabolite concentration changes, we defined the Metabolite Equilibrium Gap (MEG) as the minimum concentration changes required to reverse the direction of metabolism:

$$\begin{array}{cc}& \text{MEG}=\exp A\\ & A=\underset{\ln c}{\min }\sqrt{\frac{1}{\#M}\sum _{i\in M}{\left(\ln c_i-\ln c_i^{\text{sys}}\right)}^2}\\ \mathrm{s}.\mathrm{t}\text{.}& {\Delta }_r{G}_j^{\prime }=\sum _{i\in M}{S}_{i,j}\left({\Delta }_f{G}_i^{\prime \circ }+RT\ln c_i\right)=0,j\in N^{\text{shift}}\\ & {\Delta }_r{G}_j^{\prime }=\sum _{i\in M}{S}_{i,j}\left({\Delta }_f{G}_i^{\prime \circ }+RT\ln c_i\right)<0,j\in N\notin N^{\text{shift}}\\ & l{b}_i\le \ln c_i\le u{b}_i,j\in M\end{array}$$ (Equation 42)

where sys represents a value in a target system corresponding to the liver or ECM in this study. N^shift is the set of reactions that must proceed in the reverse direction when switching metabolic modes; it includes reversible reactions in glycolysis and gluconeogenesis, and all reactions in the TCA cycle. ${\Delta }_f{G}^{\prime \circ }$ values used in the MEG calculation were the same as those in the liver (and also the same as those in ECM). MEG was calculated using Gurobi⁷⁶ MATLAB API with dual feasibility tolerance of OptimalityTol = 10^-9 and primal feasibility tolerance of FeasibilityTol = 10^-9.

Quantification and statistical analysis

Statistical tests, optimizations, computational samplings, and calculations of Δ_rG′, FCC, and EC were performed using MATLAB 2021b (The MathWorks Inc.) and Gurobi (version 10.0.3).⁷⁶ Database handling was performed using Python (version 3.10). All details of statistical tests are described in the results section. The MATLAB package Plot Groups of Stacked Bars⁹⁰ was used to create the figures.