Analysis of metabolic stability under environmental perturbations using a kinetic model

Abstract

Cells robustly maintain metabolic functions despite environmental fluctuations that broadly alter reaction kinetics. However, kinetic models of cellular metabolism often exhibit fragility, losing stability with minor parameter changes. This discrepancy suggests that cells possess metabolic regulations that preserve the stable state under environmental perturbations. To understand the principles of metabolic robustness, we investigated the effects of temperature changes on a kinetic model of Escherichia coli central metabolism. We found that a gradual temperature decrease destabilized the metabolic state, leading to an abrupt shift to a new state in which glycolytic and tricarboxylic acid cycle (TCA cycle) fluxes decreased, and ATP production efficiency dropped. This shift was triggered by an elevated ATP/ADP ratio, which created a bottleneck in the glycolytic pathway. To assess the relationship between the destabilization and the ATP/ADP ratio, we introduced a rapid ATP–ADP exchange reaction and prevented this surge in the ATP/ADP ratio. Under the ATP/ADP ratio homeostasis, ATP production efficiency remained high across temperatures. Furthermore, we demonstrated that the destabilization was also avoided by altering enzyme abundances through sampling multiple stable steady states under cold conditions. The predicted enzyme regulation to maintain high ATP production efficiency was consistent with experimental observations of E. coli at low temperatures. Our findings indicate that balancing key cofactors, particularly the ATP/ADP ratio, is crucial for preserving metabolic stability under environmental perturbations.

Significance

Cells maintain stable metabolic states across multiple environments. However, theoretical studies suggest that complex systems such as metabolism can be destabilized by minor parameter changes, indicating that cells possess regulatory systems to preserve metabolic stability under fluctuating conditions. We found that a gradual decrease in temperature induced a sudden shift in metabolic state, reducing ATP production efficiency. This shift was characterized by an increase in the ATP/ADP ratio, and sustaining the cofactor balance prevented the destabilization and maintained high ATP production efficiency. Our findings indicate that maintaining cofactor balance plays a crucial role in the robustness of cellular metabolism.

Introduction

Cellular metabolism comprises thousands of reactions that together generate energy, build biomolecules, and recycle cellular components. As external conditions fluctuate, cells adjust their metabolic state to maintain function. For example, bacteria switch between fermentative and respiratory modes, reroute carbon through the pentose phosphate pathway (PPP) under oxidative stress, and upregulate anaplerotic reactions when nutrients are scarce [1–3]. This plasticity arises from regulatory mechanisms such as enzyme reallocation and allosteric modulation, enabling cells to occupy distinct, environment-specific metabolic states while sustaining growth [4,5].

Mathematical models of metabolic dynamics have been widely used to probe this plasticity [6–8]. Such models allow the analysis of how steady states shift as parameters vary. An important finding is that steady-state stability is often sensitive to parameter changes. For instance, a kinetic model of E. coli core metabolism was stable for only a small fraction of fluxes and concentrations consistent with its kinetic parameters [9]. Similarly, Steuer et al. (2006) sampled thousands of parameter sets for a kinetic model of yeast glycolysis and found that over 80% of the resulting steady states were unstable [10]. These narrow stability ranges suggest that environmental perturbations can readily destabilize cellular metabolism.

Yet despite the fragility predicted by models, real cells maintain stable metabolic states when environmental conditions such as temperature and nutrient availability change [11–14]. This discrepancy implies the existence of mechanisms that stabilize metabolism. Indeed, cells adjust enzyme levels and pathway activities in response to environmental changes [15–18]. However, how such regulation preserves dynamic stability remains unclear.

Among environmental perturbations, temperature shifts directly and broadly affect metabolism [19,20]. Temperature modulates the rate constants of all intracellular reactions, with magnitudes set by each reaction’s activation free energy [21,22]. Because forward and backward reactions do not respond identically, temperature changes can alter not only rates but also reversibilities, reshaping flux throughout the network. Such non-uniform effects may destabilize the steady state of the metabolic system. Clarifying how cells prevent this outcome would reveal general principles of metabolic robustness.

Here, we investigated mechanisms that keep complex metabolic systems stable under temperature-induced parameter perturbations. Using a kinetic model of E. coli central metabolism constructed from experimental data, we varied the model parameters continuously as a function of temperature and observed a bifurcation: the system transitioned discontinuously to a new steady state. The post-bifurcation state exhibited an elevated ATP/ADP ratio and reduced ATP production efficiency. Imposing constraints that maintain the ATP/ADP ratio during the temperature shift eliminated the bifurcation and maintained a high ATP production efficiency even at low temperatures. Moreover, adjusting enzyme abundances to increase ATP-utilizing reaction rates also stabilized the metabolic system during temperature decreases. These results indicate that maintaining the balance of key cofactors is critical for stabilizing the metabolic system under parameter perturbations caused by environmental change.

Materials and methods

Model description

We used a kinetic model based on Himeoka et al. (2025), which is a modified version of the Khodayari et al. (2014) model [23,24]. The Khodayari model was constructed using flux data from the wild-type and single-gene knockout E. coli strains reported by Ishii et al. [25]. Reaction parameters were determined using the ensemble modeling method [26] combined with a genetic algorithm, ensuring that the model reproduced the flux distributions observed in both the wild-type and knockout strains. The resulting parameters were validated by comparison with experimentally reported values. However, the steady state reported in the Khodayari model was found to be unstable, which was inconsistent with the experimental observations. To address this issue, the Himeoka model removed the competitive inhibition of phosphoenolpyruvate carboxylase to ensure that the steady state reported in Khodayari’s paper was stable.

In the present study, the parameters of inflow and outflow reactions were further modified so that glucose served as the sole carbon source in the temperature range used for the simulations. The model includes glycolysis, the TCA cycle, the pentose phosphate pathway (PPP), glutamate metabolism, the electron transport chain (ETC), and the Entner-Doudoroff pathway. In addition to metabolite concentrations, the model includes the concentrations of free enzymes and substrate-enzyme complexes as variables. The metabolites and reactions are summarized in Supplementary Tables S1–S3.

In our model, all reactions are decomposed into elementary reactions that represent substrate-enzyme binding and dissociation. This decomposition can be illustrated using a simple reversible reaction as follows:

When a reversible reaction A+B ⇌ C catalyzed by an enzyme E is included in the model, it is decomposed into the following elementary reactions:

$$\begin{array}{c}\text{1. A + E ⇌ EA}\hfill \\ \text{2. EA + B ⇌ EAB}\hfill \\ \text{3. EAB ⇌ EC}\hfill \\ \text{4. EC ⇌ E +C.}\hfill \end{array}$$

All steps are reversible, and the forward and backward directions are treated as distinct reactions. The total concentration of free and complex enzymes does not change over time:

$$\begin{array}{c}[E]+[EA]+[EAB]+[EC]=\mathrm{const}\text{.}\hfill \end{array}$$ (1)

In our model, the dynamics of concentrations are described by an ODE system

$$\begin{array}{c}\frac{d\mathit{x}(t)}{dt}=S\mathit{r}(\mathit{x}(t);\mathit{k}(T))\text{,}\hfill \end{array}$$ (2)

where $\mathit{x}$ is the concentration vector of metabolites, free enzymes, and substrate-enzyme complexes; $S$ is the stoichiometric matrix; $\mathit{k}(T)$ is the vector of rate constants at temperature $T$; and $\mathit{r}(\mathit{x};\mathit{k})$ is the vector of reaction rates. Note that in our model, metabolite concentrations are dimensionless because they are normalized to 1.0 at the steady state as a standard practice in ensemble modeling [24,26]. All reactions are decomposed into elementary reactions, and each reaction rate is modeled by the law of mass action

$$\begin{array}{c}{r}_i(x)=k_i(T)\prod _{j=1}^n{x}_j^{{\nu }_{ji}}\text{,}\hfill \end{array}$$ (3)

with exponents ${\nu }_{ji}\in \left\{0\text{,}1\right\}$ determined by the stoichiometry, and $k_i(T)$ is the temperature-dependent rate constant (specified in the next section).

ODE simulations were performed using the ode15s function in MATLAB [27].

Temperature-dependent rate constants

In this analysis, rate constants were adjusted using the Eyring–Polanyi equation

$$\begin{array}{c}k(T)=\frac{k_{B}T}{h}\exp \left(-\frac{\mathrm{\Delta }{G}^{‡}}{RT}\right)\hfill \end{array}$$ (4)

for each temperature. Here, $k$ is the rate constant, $h$ is the Planck constant, $k_B$ is the Boltzmann constant, $R$ is the gas constant, and $T$ is the absolute temperature $(K)$. The activation free energy $\mathrm{\Delta }{G}^{‡}$ for each reaction was back-calculated from parameter values reported by Khodayari et al. [24] using the following equation:

$$\begin{array}{c}\mathrm{\Delta }{G}^{‡}=-R{T}_{ref}\ln \left(k^{*}\frac{h}{k_B{T}_{ref}}\right)\text{,}\hfill \end{array}$$ (5)

with $T_{ref}=310.15$, and $k^{*}$ is the rate constant at $T_{ref}$. Based on this value, reaction rate constants at altered temperatures were calculated.

Maintaining the ATP/ADP ratio

To maintain the ATP/ADP ratio during temperature decrease, we added a new reversible reaction to the model that exchanges ATP and ADP. The rate constant for this exchange was set to ${10}^6$, a value larger than the median of all rate constants at 37°C. During low-temperature simulations, this constant was not adjusted for temperature dependence to ensure that it maintained the ATP/ADP ratio at the value corresponding to the 37°C condition of the original model (without the ATP–ADP exchange reaction) across the entire temperature range investigated. This design allowed us to directly test the hypothesis that the bifurcation disappears when this specific cofactor ratio is kept constant. The concentrations of metabolites at the steady state were normalized to unity in the ensemble modeling procedure carried out for the model construction in Khodayari et al. (2014). Thus, the ATP/ADP ratio in the original model was 1:1 in the scale of normalized concentrations. Therefore, forcing the ATP/ADP ratio to unity corresponded to the recovery of the ATP/ADP ratio in the original model.

Perturbed initial condition sampling procedure

To identify multiple stable steady states in the system, we performed numerical integrations starting from perturbed initial conditions at specified temperatures. These initial conditions were generated by multiplying random values with the concentrations of the original steady state, while ensuring that the conserved quantities remained unchanged in the following procedure.

If a vector $\mathit{d}$ satisfies

$$\begin{array}{c}{\mathit{d}}^{T}S=0\text{,}\hfill \end{array}$$ (6)

then the inner product of $\mathit{d}$ and a concentration vector $\mathit{x}$ does not change over time, since

$$\begin{array}{c}\frac{d}{dt}\left({\mathit{d}}^T\mathit{x}\right)=0\hfill \end{array}$$ (7)

If random perturbations were applied directly to the concentrations of the original steady state ${\mathit{x}}_{org}$, the resulting concentration vector ${\mathit{x}}_{rand}$ generally had conserved quantities different from those of ${\mathit{x}}_{org}.$ To obtain perturbed concentrations that preserve the conserved quantities of ${\mathit{x}}_{org}$, we searched for the point closest to ${\mathit{x}}_{rand}$ that satisfies the conservation constraints.

This was formulated as the following optimization problem:

$$\begin{array}{c}\begin{array}{c}\hfill minimize{\parallel \begin{array}{c}\hfill \mathit{x}-{\mathit{x}}_{rand}\hfill \end{array}\parallel }_1\hfill \\ \hfill subjectto\hfill \\ \hfill {\mathit{d}}_i^T\mathit{x}={\mathit{d}}_i^T{\mathit{x}}_{org}\text{,}foralli.\hfill \end{array}\hfill \end{array}$$ (8)

where ${\left\{{\mathit{d}}_i\right\}}_i^{\dim \ker S^T}$ form a basis of $\ker S^T$. This optimization was performed using Gurobi Optimizer [28].

Steady state sampling procedure

To identify enzyme abundances that yield stable metabolic states with low-temperature parameters, we sampled steady states for each temperature-specific parameter set and computed their enzyme abundances. In our model, each reaction is decomposed into elementary reactions whose reactants are (i) substrate and free enzyme, (ii) substrate and substrate-enzyme complex, or (iii) substrate-enzyme complex only. Therefore, each reaction rate function is a first- or second-order monomial in its variables.

Fixing concentrations of substrates (metabolites) at their values near the original steady state turns every rate into a linear function of the free enzyme or substrate-enzyme complex concentrations. ${\mathit{x}}_{enzyme}$ is a vector of the remaining variables, i.e., the concentrations of free enzymes and substrate-enzyme complexes. Using a matrix $V$, whose non-zero values are determined by the temperature-specific rate constants and the fixed substrate concentrations, we have

$$\begin{array}{c}\mathit{r}=V{\mathit{x}}_{enzyme}\hfill \end{array}$$ (9)

where $\mathit{r}$ is the vector of reaction rates.

The matrix $V$ is constructed as follows. For example, if the system has only one reaction A+B ⇌ C catalyzed by an enzyme E, the reaction is decomposed into four reversible elementary reactions as described in the “Model description” section. The forward and backward rates of each reaction are calculated as

$$\begin{array}{c}\mathit{r}=\left(\begin{array}{c}\begin{array}{c}\hfill k_{1f}[A][E]\hfill \\ \hfill k_{1b}[EA]\hfill \end{array}\hfill \\ k_{2f}[EA][B]\hfill \\ k_{2b}[EAB]\hfill \\ k_{3f}[EAB]\hfill \\ k_{3b}[EC]\hfill \\ k_{4f}[EC]\hfill \\ k_{4b}[E][C]\hfill \end{array}\right)\text{,}\hfill \end{array}$$ (10)

where $k_{if}$ and $k_{ib}$ are the forward and backward rate constants of the $i$-th elementary reactions, respectively. When the substrate concentrations are fixed at ${([A{]}^{*}\text{,}[B{]}^{*}\text{,}[C{]}^{*})}^T$, the rates are obtained as

$$\begin{array}{c}\mathit{r}=\left(\begin{array}{c}\begin{array}{cccc}\hfill k_{1f}[A{]}^{*}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill k_{1b}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill k_{2f}[B{]}^{*}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill k_{2b}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill k_{3f}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill k_{3b}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill k_{4f}\hfill \\ \hfill k_{4b}[C{]}^{*}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\hfill \end{array}\right)\left(\begin{array}{c}[E]\hfill \\ [EA]\hfill \\ [EAB]\hfill \\ [EC]\hfill \end{array}\right)\hfill \end{array}$$ (11)

The matrix that appears on the right-hand side of Eq. (11) is the matrix $V$ in Eq. (9).

Using a submatrix of the stoichiometric matrix corresponding to the rows of free enzyme and substrate-enzyme complex concentrations, denoted as $S_{enzyme}$, the time-course change of ${\mathit{x}}_{enzyme}$ is written as

$$\begin{array}{c}\frac{d{\mathit{x}}_{enzyme}}{dt}=S_{enzyme}V{\mathit{x}}_{enzyme}.\hfill \end{array}$$ (12)

Therefore, steady-state solutions can be obtained by solving

$$\begin{array}{c}{S}_{enzyme}V{\mathit{x}}_{enzyme}=0.\hfill \end{array}$$ (13)

Eq. (13) is an undetermined linear equation when $S_{enzyme}V$ is not full-rank. In that case, Eq. (13) has infinitely many solutions. Among such solutions of Eq. (13), we selected the one closest to a reference point $\left({\mathit{x}}_{ref}\right)$, which was randomly sampled around the enzyme abundances of the original steady state. We also required that steady states satisfied:

1. Glucose served as the sole carbon source, and transport reactions of other metabolites acted as efflux.

2. The total enzyme abundance of all reactions was equal to that of the original steady state.

Steady states satisfying these conditions were obtained by solving the following optimization problem:

$$\begin{array}{c}\begin{array}{c}\hfill minimize{\parallel \begin{array}{c}\hfill {\mathit{x}}_{enzyme}-{\mathit{x}}_{ref}\hfill \end{array}\parallel }_1\hfill \\ \hfill subjectto\hfill \\ \hfill S_{enzyme}V{\mathit{x}}_{enzyme}=0\hfill \\ \hfill \sum _i{\mathit{x}}_{enzyme\text{,}i}=const.\hfill \\ \hfill r_{GlucoseInput}>0\hfill \\ \hfill r_{OtherTransports}<0\hfill \end{array}\hfill \end{array}$$ (14)

The solution ${\mathit{x}}_{enzyme}$ obtained from this optimization, together with the fixed metabolite concentrations near the 37°C steady state, yielded a steady state of the full system. This optimization was performed using Gurobi Optimizer [28].

Stability assessment of steady states

The stability of each steady state found by the optimization was determined by computing the eigenvalues of the system’s Jacobian matrix at the steady state. The model contains 98 conserved quantities, so the full 643×643 Jacobian matrix has 98 zero eigenvalues. The zero eigenvalues were difficult to distinguish from the very small, non-zero eigenvalues due to numerical errors. Thus, we avoided obtaining zero eigenvalues by solving the dependent variables using the conserved quantities. Note that one conserved quantity leads to a single linear equation. For each conserved quantity, we chose a single chemical species and solved its concentration as a function of the remaining species’ concentrations. From this pre-solved ODE system, we obtained the reduced 545×545 Jacobian matrix. A steady state was classified as unstable if the largest real part of any eigenvalue of the reduced Jacobian matrix was positive. All eigenvalues were computed with interval-arithmetic routines in INTLAB, ensuring that numerical errors did not flip the sign of any eigenvalue [29].

Results

Temperature decrease induces a bifurcation

To examine how a low-temperature shift affects the metabolic state of E. coli central metabolism, we simulated metabolic dynamics using rate constants adjusted for different temperatures. Figure 1 shows the reaction network of the kinetic model used in this study (see Materials and methods for details). In our simulations, rate constants at each temperature were calculated using the Eyring–Polanyi equation. We focused on temperature downshifts because heating can cause irreversible enzyme inactivation, which is not captured by the standard Eyring–Polanyi formulation [19,30]. Activation free energies $\left(\mathrm{\Delta }{G}^{‡}\right)$ for individual reactions were obtained from parameter values fitted to the in vivo metabolic state at 37°C reported by Khodayari et al. (2014) (see Materials and methods for details). Without a temperature shift, the system converged to a stable steady state in which glucose serves as the sole carbon source and ATP was produced via glycolysis, the TCA cycle, and the electron transport chain (ETC). We refer to this as the “original steady state” throughout the paper.

Figure 1.

Reaction network of the kinetic model used in our simulations. Although each reaction was decomposed into elementary steps for the simulations, the network is shown here with the pre-decomposition reactions for clarity.

When the temperature was lowered from 37°C to 0°C, each rate constant decreased according to its activation free energy. As a result, the distribution of rate constants shifted, and the median value decreased about ten-fold (Figure 2A). We simulated system dynamics using the rate constants at 0°C, starting from the original steady state. Numerical simulations showed that metabolite concentrations converged to a new stable steady state (Figure 2B). Compared with the original state, metabolite concentrations in this new steady state changed by factors ranging from ${10}^{-9}$ to ${10}^4$—far greater than the fold change observed in the median of the rate constant.

Figure 2.

(A) Distribution of reaction rate constants in the system. (B) Time courses of relative metabolite concentrations with rate constants at 0°C and initial concentrations set to the original steady state; each line represents an individual metabolite. (C) Steady-state concentration of D-glycerate 1,3-diphosphate at each temperature. (D) Relative reaction rates at steady state for each temperature, normalized to the rates at the original steady state. (E, F) Steady-state ATP yield (E) and ATP/ADP ratio (F) at each temperature.

To determine the temperature at which this substantial shift occurred, we tracked the steady state as the temperature gradually decreased from 37°C to 0°C. At each step, system dynamics were integrated up to $t={10}^{10}s$ to ensure convergence, and the resulting concentrations were used as initial conditions for the next step, with temperature reduced by 0.1°C. At 14.2°C, the steady state shifted discontinuously, with abrupt changes in the concentrations of all metabolites (Figure 2C shows one example; Supplementary Figure S1 presents all metabolite concentrations at the converged state across temperatures). Reaction rates also changed discontinuously, and relative changes compared with the original steady state are shown in Figure 2D.

This discontinuous shift could arise from either the disappearance or destabilization of the original steady state, or from a switch between coexisting attractors. To discriminate between these possibilities, we enumerated multiple stable steady states across temperatures including the transition point. We generated 10,000 random initial conditions by scaling each chemical concentration of the original steady state by ${10}^{U(-3.0\text{,}3.0)}$. Here $U(a\text{,}b)$ denotes a uniform distribution ranging from $a$ to $b$. See Materials and methods for details of the sampling procedure. Starting from each of these points, we numerically integrated the system dynamics to $t={10}^{10}s$ and checked convergence for all cases.

This procedure revealed three coexisting stable steady states above 14.2°C and two below it. These results indicate that the discontinuous shift was induced by a bifurcation, in which the physiologically relevant steady state above 14.2°C was destabilized as temperature decreased. Bifurcation diagrams for all metabolites are provided in Supplementary Figure S1. Notably, the steady state that disappeared due to the bifurcation was characterized by aerobic respiration and ATP production from glucose.

Examination of the steady state after the bifurcation revealed substantial changes in pathway fluxes and ATP production efficiency. The discontinuous shift caused glycolytic and TCA cycle fluxes to decrease by factors of 100–1,000, while flux through the pentose phosphate pathway (PPP) slightly increased (Figure 2D). As a result, the ATP yield—defined as the sum of ATP-producing fluxes divided by the glucose uptake rate—declined following the temperature-induced bifurcation (Figure 2E).

To investigate the cause of the decreased ATP yield, we examined concentrations of metabolites involved in glycolysis and the TCA cycle. While most metabolite concentrations decreased after the temperature-induced bifurcation, the concentration of D-glycerate 1,3-diphosphate (13dpg) increased by two orders of magnitude compared with its pre-bifurcation level (Figure 2C; Supplementary Figure S1). Despite this elevation, the reaction degrading 13dpg—catalyzed by phosphoglycerate kinase (PGK), which converts 13dpg and ADP to 3-phosphoglycerate (3pg) and ATP in glycolysis (Figure 1)—showed a reduced rate. This reduction was caused by an approximately 40-fold increase in the ATP/ADP ratio, driven by elevated ATP and decreased ADP concentrations (Figure 2F; Supplementary Figure S1). Because the PGK reaction lies near the center of glycolysis, its slowdown created a bottleneck that decreased flux through both glycolysis and the downstream TCA cycle.

Fixing the ATP/ADP ratio prevents the temperature-induced bifurcation

The above results suggest that a sudden rise in the ATP/ADP ratio can trigger the temperature-induced bifurcation. To investigate the relationship between the ATP/ADP ratio and the temperature-induced bifurcation, we maintained the ATP/ADP ratio nearly constant by introducing a rapid, reversible ATP–ADP exchange reaction into the system. Because the ATP/ADP ratio at the steady state before the bifurcation was approximately 1:1, the forward and backward rate constants were set to ${10}^6$, far exceeding the median values of other reactions’ rate constants in the model. The detailed procedures for fixing the ATP/ADP ratio during temperature decrease are provided in Materials and methods.

The modified model with the ATP–ADP exchange reaction maintained a single, stable steady state under the 37°C parameter set. The resulting concentrations of 13dpg, the ATP/ADP ratio, and the ATP yield were similar to those of the original steady state (Figure 3A–C). With the exchange reaction, lowering the temperature from 37°C to 0°C no longer caused a sharp rise in 13dpg concentration or discontinuous changes in pathway fluxes (Figure 3A, D). Consistently, the ATP/ADP ratio remained near unity, and the ATP yield showed no abrupt decline across the temperature range (Figure 3B, C). To test multiple attractors at each temperature, we generated 10,000 random initial points by scaling each chemical concentration of the original steady state by ${10}^{U(-3.0\text{,}3.0)}$ and simulated the dynamics until convergence. All trajectories converged to a single steady state at each temperature. See Supplementary Figures S2A–C for the time-course changes of chemical concentrations starting from perturbed initial conditions at 37°C. These results indicate that, if the ATP/ADP ratio is fixed at approximately 1.0, the system becomes monostable across the examined temperatures and does not undergo the bifurcation.

Figure 3.

Changes in steady states of the modified model with ATP/ADP ratio homeostasis. (A) Steady-state concentration of D-glycerate 1,3-diphosphate at each temperature. (B, C) Steady-state ATP/ADP ratio (B) and ATP yield (C) at each temperature. (D) Relative changes in reaction rates at steady state for each temperature, normalized to the rates at the original steady state.

We showed that introducing the ATP–ADP exchange reaction with a large rate constant $\left({10}^6\right)$ rendered the system monostable and eliminated the bifurcation during cooling. However, whether bifurcation occurs and how many stable steady states exist depend on the rate constant of the added exchange reaction. We examined how the bifurcation depended on the rate constant of the hypothetical ATP–ADP exchange reaction by plotting the ATP yield as a function of both the reaction rate constant and temperature (Supplementary Figure S3A). Within the range we investigated, the high-ATP-yield state disappeared when the rate constant exceeded approximately ${10}^{0.5}$ and the temperature fell below roughly 14°C.

Furthermore, we investigated the multistability of the system across the parameter range shown in Supplementary Figure S3A (Supplementary Figures S3B–E). We found that the system exhibited multistability when the rate constant of the ATP–ADP exchange reaction was lower than approximately ${10}^2$. For example, when the rate constant was ${10}^{0.5}$, multiple stable steady states coexisted across almost all examined temperatures (Supplementary Figure S3C). These results indicate that the number of stable steady states varies in a complex manner depending on the rate constant of the hypothetical ATP–ADP exchange reaction.

The ATP/ADP ratio emerged as the critical factor for the temperature-induced bifurcation we observed. However, other cofactor ratios, NAD⁺/NADH and NADP⁺/NADPH, can also play fundamental roles in metabolic regulation. Therefore, we analyzed whether fixing the NAD⁺/NADH or NADP⁺/NADPH ratio would suppress the temperature-induced bifurcation. The steady states for each temperature were obtained while the NAD⁺/NADH or NADP⁺/NADPH ratio was fixed by adding an exchange reaction with a rate constant of ${10}^6$.

With the NAD⁺/NADH ratio fixed, a bifurcation occurred at 12.8°C, and with the NADP⁺/NADPH ratio fixed, a bifurcation occurred at 4.7°C. In both cases, a discontinuous drop was observed in the ATP yield during the temperature shift (Supplementary Figures S4A and B). We also compared the ATP/ADP ratio changes associated with these bifurcations. In both models, as in the original model, the ATP/ADP ratio increased upon bifurcations (Supplementary Figures S4C and D). These results indicate that among cofactor balances in the system, the ATP/ADP ratio specifically plays an important role in maintaining a stable steady state under temperature change.

Maintaining the ATP yield by adjusting enzyme abundances

The analysis in the previous section demonstrated that the ATP yield can be maintained at a high level by fixing the ATP/ADP ratio using a rapid, reversible ATP–ADP exchange reaction. However, cells do not naturally possess such a reaction, and precise control of the ATP/ADP ratio in this manner is biologically implausible. Instead, cells adjust intracellular enzyme abundances in response to environmental perturbations, including temperature fluctuations, osmotic stress, oxidative conditions, and nutrient limitations [15–18]. We therefore investigated whether the ATP yield could be maintained under decreasing temperatures by modulating enzyme abundances. To this end, we sampled steady states with varied enzyme levels under low-temperature conditions and identified enzyme abundances that supported a high ATP yield.

A naïve approach of randomly varying enzyme levels and integrating the system dynamics until convergence for each set would require computationally intensive simulations. To overcome this challenge, we developed a faster method to sample numerous low-temperature steady states. Because every reaction in our model was decomposed into elementary reactions, each rate function depended on one or two variables and followed mass-action kinetics. Therefore, the rate functions were at most quadratic monomials, with the second variable, if present, corresponding to the concentration of a substrate. Based on this model feature, the rate functions became linear when the concentrations of metabolites, which served as substrates for reactions, were fixed near their values in the original steady state. Consequently, the system of differential equations was transformed into a linear one. Steady states of this linear system could then be obtained simply by solving a linear equation (Eq. (13) in Materials and methods).

The linearized system had infinitely many steady states. To sample one, we first selected random reference concentrations of free enzymes and enzyme-substrate complexes, denoted as ${\mathit{x}}_{ref}$, near those of the original steady state as follows. Each concentration in the original steady state was multiplied by a value drawn from a log-normal distribution over the interval $(1/3\text{,}3.0)$. Broadening this range reduced the fraction of sampled states that were stable, so we chose the widest interval that still yielded stable solutions with reasonable frequency. We then solved a linear optimization problem to identify the steady state closest to ${\mathit{x}}_{ref}$, subject to additional constraints: (1) the glucose transport reaction must proceed inward while other transport reactions act outward, and (2) the total enzyme abundance must equal that of the original steady state. These constraints were imposed to ensure the biological plausibility of the sampled steady states. The second constraint reflected the limited proteome budget, because total enzyme amount varies only modestly across conditions in E. coli [16]. A mathematical formulation and detailed description of the sampling procedure are provided in Materials and methods. This approach sampled steady states more efficiently than the naïve method. Because stability was not enforced during sampling, we assessed it by calculating the eigenvalues of the Jacobian matrix at each steady state. Materials and methods also describe how steady states were classified as stable based on the Jacobian matrix.

Using this method, we sampled 5,000 steady states with rate constants corresponding to 0°C and visualized the 94-dimensional enzyme abundances in two dimensions via principal component analysis (PCA; Figure 4A, Supplementary Figures S5A and B). The contributions from the fourth and higher principal components were minor, and therefore we focused on the first three principal components in this analysis (Supplementary Figure S5C). Among all computed solutions, the stable steady states were enriched in a specific region, where the total TCA cycle enzyme levels were elevated. Figure 4B shows the sum of TCA enzyme abundances at each steady state in the PCA plot. This trend was also observed in the individual TCA enzyme abundances as shown in Supplementary Figures S5D–O.

Figure 4.

(A) PCA plot of 94-dimensional enzyme abundances at steady states obtained with parameters at 0°C. Explained variance: PC1, 16.9%; PC2, 4.7%. Stability of each steady state is indicated by color: blue, stable; gray, unstable. (B) TCA cycle enzyme abundances (color-coded) at steady states under low-temperature conditions. (C) Box plot of the ATP yield for stable steady states at each temperature.

The above results demonstrate that adjusting enzyme abundances allows the system to achieve stable steady states near the original steady state, even at low-temperature conditions. However, the ATP yields at these stable states are not necessarily high. To evaluate the ATP yield at low temperatures, we sampled stable steady states across temperatures from 37°C to 0°C using the same method and assessed their ATP production rate. The ATP yield remained high even at low temperatures. Figure 4C shows box plots of ATP yields for stable steady states at 37°C, 19°C, and 0°C. These findings indicate that cells can maintain a high-ATP-yield state during a temperature decrease by appropriately modulating enzyme levels. It should be noted that the stability of the sampled steady states does not imply stability of the original steady state, as their enzyme abundances differ.

In practice, cells may not experience a sudden drop in temperature but rather undergo gradual cooling. We therefore investigated which enzyme reallocations would allow the system to maintain a high ATP yield as temperature decreased from 37°C to 0°C. To do this, we fixed the target concentrations of free enzymes and substrate-enzyme complexes $\left({\mathit{x}}_{ref}\right)$, and computed the steady-state enzyme abundances at each temperature. During continuous cooling, we assumed that enzyme reallocation was completed before the subsequent temperature change. Therefore, we considered that the cooling was slow enough for enzyme reallocation to be completed. In E. coli, proteome composition changes typically occur over hours [31]. Thus, our continuation procedure corresponds to a situation in which the temperature changes over the course of several days to weeks. Along this path, enzyme levels at stable steady states changed with temperature: TCA cycle enzymes increased, whereas ETC enzyme levels decreased. Figures 5A and 5B show the summed abundances of TCA cycle and ETC enzymes at stable steady states across temperatures. This trend is consistent with experimental observations that cold-adapted E. coli cells upregulate TCA cycle enzymes and downregulate ATP synthase subunits [12,32].

Figure 5.

(A, B) Changes in the summed abundances of TCA cycle enzymes (A) and ETC enzymes (B) at stable steady states across temperatures. Each line represents one ${\mathit{x}}_{\mathit{r}\mathit{e}\mathit{f}}$. (C, D) Changes in the summed absolute values of TCA cycle reaction rates (C) and ATP-utilizing reaction rates (D), both normalized to the glucose uptake rate, at stable steady states across temperatures. Each line represents one ${\mathit{x}}_{\mathit{r}\mathit{e}\mathit{f}}$.

Because rate constants also vary with temperature, changes in enzyme abundances do not directly result in proportional changes in flux. Therefore, to understand how enzyme adjustments maintain a high ATP yield during cooling, it is essential to examine the resulting metabolic fluxes rather than enzyme levels alone. We analyzed reaction rates normalized to the glucose uptake rate to investigate flux-level modulations arising from enzyme reallocation. The analysis revealed that, as temperature decreased, normalized flux through the TCA cycle increased, along with fluxes of reactions involved in ATP synthesis or consumption. Figures 5C and 5D show the summed reaction fluxes of the TCA cycle and ATP-utilizing reactions at stable steady states for each temperature. These findings indicate that the adjustment of enzyme abundances enhanced ATP–ADP turnover, analogous to the rapid ATP–ADP exchange artificially imposed in the previous section. Together, these results suggest that maintaining the ATP/ADP ratio near unity through increased cofactor turnover allows cells to sustain a high ATP yield even at low temperatures.

Discussion

In this study, we computationally investigated how perturbations in environmental parameters influence the stability of a metabolic state. By varying rate constants in a temperature-dependent manner, we found that lowering the temperature induced a bifurcation and a discontinuous drop in the ATP yield. This finding provides a model-based illustration of how complex metabolic systems can be destabilized by parameter perturbations [9,10,33].

One way to maintain a high ATP yield was to keep the ATP/ADP ratio constant through external control. Several studies have reported that coenzymes such as ATP and ADP exert major influences on the perturbation responses of kinetic models [23,34,35]. In our model, fixing this cofactor ratio preserved a stable metabolic state across the temperature range and maintained a high ATP yield, as shown in Figure 3C. We also demonstrated that the ATP yield can be maintained at a high level by adjusting enzyme abundances. Under such enzyme-tuning scenarios, the relative flux through ATP-utilizing reactions increased. Thus, regulation of enzyme abundances can, in principle, achieve the same effect as cofactor-ratio control through the ATP–ADP exchange reaction.

The analysis of enzyme abundances to maintain a high ATP yield revealed an increase in the TCA cycle enzymes and a decrease in ETC enzymes. This trend aligns with experimental reports that E. coli upregulates TCA cycle enzymes and downregulates ATP synthase subunits at low temperatures [12,32]. This suggests that cells may modulate enzyme abundances in vivo to maintain the ATP/ADP ratio and prevent an abrupt shift in their metabolic state. Nevertheless, these findings require experimental validation, such as comprehensive fluxomic analyses, to determine whether cells increase the fluxes of ATP-utilizing reactions at low temperatures.

Our analysis perturbed only temperature, starting from parameters fitted to the 37°C metabolic state. Therefore, the conclusions depend on that particular parameter set. Additionally, the model is limited to central metabolism and omits lipid pathways and many amino acid biosynthesis routes that are reported to be important for cold adaptation [14,36]. Network-structure modifications in kinetic models are known to exert strong influences on responses to parameter perturbations [37]. Future work can probe cellular responses to cooling with higher quantitative accuracy using a more detailed network and more precise temperature dependence of parameters.

At steady states under low temperatures, some relative concentrations fell outside the range in which our ODE model remains biophysically meaningful. For example, glutamate exhibited the largest decrease in relative concentration at 14.2°C, dropping to $7.19\times {10}^{-9}$. This value corresponds to a state in which the average number of molecules per cell is lower than one, making the application of ODE-based calculations inappropriate.

To evaluate how such extremely low metabolite concentrations affect system dynamics, we performed simulations in which the concentration of a metabolite was set to zero when it fell below a defined threshold value. The threshold concentration for each metabolite was set to correspond to fewer than ten molecules per cell, calculated using the intracellular metabolite concentrations of E. coli at 37°C [38]. The threshold values for all metabolites are provided in Supplementary Table S4. When any metabolites had steady-state relative concentrations below their thresholds, we recomputed the dynamics starting from that state while setting their concentrations and time derivatives to zero.

When the temperature was higher than the bifurcation point, no metabolite concentration fell below the defined threshold, indicating that the pre-bifurcation state was unaffected by this correction. In contrast, at 14.2°C, which is below the bifurcation point, the concentrations of several metabolites changed considerably when concentrations below the thresholds were fixed to zero (Supplementary Figure S6A). Nevertheless, the temperature-induced bifurcation and the associated decrease in ATP yield remained unchanged (Supplementary Figure S6B).

These results suggest that the key finding of this study—the destabilization of the high-ATP-yield state under low-temperature conditions—is robust to stochastic fluctuations in metabolite concentrations. At the same time, the results shown in Supplementary Figure S6A indicate that precise quantitative estimation of the metabolic state using deterministic dynamical system modeling becomes difficult when metabolite concentrations are extremely low, as stochastic effects can no longer be neglected.

We showed that the ATP/ADP ratio is a key variable in metabolic control; however, other factors may also contribute. To identify other important factors, we examined whether the temperature decrease still induced the bifurcation when the concentration of each small metabolite was fixed to its value in the original 37°C steady state. In doing so, we found that fixing certain metabolites rendered the original steady state unstable; these cases were excluded from subsequent analyses.

Among the metabolites tested, five cases did not exhibit the temperature-induced bifurcation: 6-phospho-D-glucono-1,5-lactone (6pgl), 6-phospho-D-gluconate (6pgc), D-ribulose 5-phosphate (ru5p), reduced nicotinamide adenine dinucleotide (NADH), and ubiquinone-8 (q8) (Supplementary Figure S7). Notably, fixing metabolite concentrations in the oxidative phase of the PPP eliminated the temperature-induced bifurcation, even though the reaction rates among these metabolites changed only modestly across the transition (Figure 2D). Although the mechanism remains unclear, these results suggest that the oxidative PPP may play an important role in maintaining the stability of the central metabolic system.

We showed that suppressing temperature-induced bifurcation through the ATP–ADP exchange reaction requires the introduced reaction to have a sufficiently large rate constant. Although this exchange reaction is not intended to represent a specific biological process, similar adaptive mechanisms might contribute to maintaining the stability of metabolic dynamics under low-temperature perturbations. For example, E. coli rapidly restores membrane fluidity after cooling by altering lipid composition [39]. Because respiratory efficiency depends on membrane fluidity, such an adaptation could restore the functionality of the electron transport chain and thereby limit ATP/ADP excursions. This adaptive response may thus have an effect similar to that of the ATP–ADP exchange reaction introduced in our model.

Some studies show that poikilotherms often maintain metabolic activity across temperatures via physiological compensation and acclimation, implemented by cellular and organ-level mechanisms and altered enzyme activities [40,41]. Our results identified a concrete intracellular basis for such compensation: maintaining the ATP/ADP ratio prevents a temperature-induced bifurcation and preserves the ATP yield across temperatures. Thus, cofactor homeostasis provides a mechanistic explanation for temperature compensation in bacterial metabolism.

The same computational framework can be applied to environmental perturbations beyond temperature. By allowing reaction parameters to vary with factors such as pH, osmotic pressure, or nutrient availability, we can reveal which metabolic functions become unstable and what regulations—such as cofactor balance control or targeted enzyme-tuning—can restore stability.

Overall, our results highlight the possibility of destabilization in complex metabolic systems under environmental perturbations and identify cofactor balance—specifically the ATP/ADP ratio—as a critical parameter for maintaining stability. Applying this approach to other external parameters may allow us to identify key variables that cells must regulate to maintain metabolic stability. Such insights would advance our understanding of the principles underlying biological robustness.

Conclusion

We showed that gradual cooling induced a bifurcation for a kinetic model of E. coli central metabolism near 14°C and reduced fluxes of glycolysis and the TCA cycle, thereby lowering the ATP yield. A surge in the ATP/ADP ratio was observed after the bifurcation that created a bottleneck in glycolysis. Constraining this ratio through an ATP–ADP exchange reaction with a large rate constant eliminated the bifurcation and maintained a high ATP yield. The regulation of enzyme abundances also contributed to stabilizing the metabolic system. Importantly, in the enzyme reallocation case, we observed an increase in the flux of ATP‐utilizing reactions, which may help maintain the ATP/ADP ratio.

These results highlight cofactor balance, particularly the ATP/ADP ratio, as an important factor for stabilizing the metabolic system under broad parameter perturbations, such as temperature decreases. The same computational framework can be extended to other environmental parameters (e.g., pH, osmolarity, or nutrient shifts) to identify which cofactors or reaction rates should be modulated to prevent the destabilization of the metabolic system.

Our analysis has limitations. All results depend on parameters fitted to a 37°C metabolic state of E. coli and focus only on central metabolism, omitting other pathways involved in cold adaptation. Future work should extend this approach with more comprehensive networks and refined temperature dependencies of reaction parameters. Moreover, our results require quantitative experiments to test the predicted enzyme regulations and cofactor homeostasis. Nonetheless, by linking destabilization of the metabolic system to a simple parameter, i.e., cofactor balance, this study provides important insight into how cells maintain metabolic stability under environmental perturbations.

Conflict of interest

The authors declare that no competing interests exist.

Author contributions

A.H., Y.H., and C.F. designed the research. A.H. conducted computational analysis. A.H., Y.H. and C.F. wrote the manuscript.

Data availability

All code used in this study is available at https://github.com/hishidagit/TemperatureResponse/. A preliminary version of this work was deposited in bioRxiv (https://doi.org/10.1101/2025.08.27.672763) on September 01, 2025.

Acknowledgements

This work was supported in part by JSPS KAKENHI (23KJ1324 to A. H.; 22K15069, 24H01118 and 25H01390 to Y. H.; 22K21344 and 23H02471 to C. F.), and GteX Program Japan Grant No. JPMJGX23B4.