# Structural Thermokinetic Modelling

## Abstract

**:**

## 1. Introduction

_{r}G, but dimensionless and with a flipped sign [44]. The quantity $A=-{\Delta}_{\mathrm{r}}G=RT\theta $, still dependent on Boltzmann’s gas constant R and absolute temperature T, is called reaction affinity. The thermodynamic force $\theta $ depends on the chemical potentials and therefore on substrate and product concentrations, and can range between two extremes. In equilibrium reactions, the force vanishes, the one-way rates ${v}_{+}$ and ${v}_{-}$ are equal, and the net flux v vanishes too. Strongly driven reactions, in contrast, have large thermodynamic forces, a negligible backward flux ${v}_{-}$, and a net flux close to the forward flux ${v}_{+}$. By tuning the one-way fluxes (${v}_{+}=\frac{{\mathrm{e}}^{\theta}}{{\mathrm{e}}^{\theta}-1}v,\frac{1}{{\mathrm{e}}^{\theta}-1}{v}_{-}=v$), the thermodynamic force shapes elasticities: if a force is large, the backward rate will be negligible, the net rate does not depend on the product concentrations, and the product elasticities vanish. Since the forces themselves are subject to Wegscheider conditions [45] (they must sum to zero over loops in the network) and elasticities depend on them, elasticities may be interdependent across the entire network [46]. Ignoring this fact, SKM leads to thermodynamically inconsistent models [46] (for an example, see Supplementary Section S2.2).

## 2. Results

#### 2.1. Structural Thermokinetic Modelling

^{M}values and metabolite concentrations); for ensemble modelling, we may sample them in the range $]0,1[$ or around plausible values. In the model, small-molecule regulation of enzymes (such as allosteric inhibition) was ignored, but it could easily be included.

#### 2.2. Metabolic Effects of Gene Expression Changes

#### 2.3. Enzyme Synergisms and Epistasis

#### 2.4. Uncertain States and Metabolic Fluctuations

^{cat}= 10 s

^{−1}) catalyses around ${10}^{4}$ net reaction events per second. If we assume strongly driven reactions (i.e., neglect backward rates), count the reaction events within one-second time intervals, and describe the count numbers by a Poisson distribution, we expect ${10}^{4}$ events per second on average with a standard deviation of $\sqrt{{10}^{4}}={10}^{2}$, i.e., a relative standard deviation of one percent. Over larger time intervals, the relative standard deviation is smaller because fluctuations average out. How does this chemical noise translate into metabolite and flux fluctuations? In metabolism, noise from different reactions propagates through the network, adds up, becomes damped or sometimes amplified, and leads to correlated fluctuations of metabolite concentrations and fluxes. To model this, we can add a white noise term to each rate law, which leads to a chemical Langevin equation [64]. The white noise spectrum contains all frequencies with uniform amplitudes, but the linearised model acts as a linear filter that translates the white spectrum into a coloured noise spectrum of the resulting metabolite fluctuations (see Section 4.5).

#### 2.5. Network Structure and Thermodynamic Forces Shape Metabolic Dynamics

**N**and elasticity matrix

**E**

_{c}, which is shaped itself by thermodynamics and enzyme saturation as considered in STM. The matrix product

**N**

**E**

_{c}yields the Jacobian matrix, which is local and sparse. The inverse of this Jacobian matrix leads to the non-sparse control and response matrices that describe network-wide changes of steady states. Another link between (local) network structure and (network-wide) metabolic behaviour can be made by the summation and connectivity theorems of MCT [18,20]: the control coefficients along a stationary flux distribution (which depend on network structure, but not on kinetics) must have a fixed sum (summation theorem), while the control coefficients around a metabolite are constrained by local elasticities (connectivity theorem). Hence, the control coefficients are constrained by ths network structure, but their precise values also depend on the elasticities—which can be explored by STM.

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Constructing Kinetic Metabolic Models

#### 4.2. Elasticities and Their Dependence on Thermodynamic Forces

**N**has full row rank.

^{M}instead of ${K}_{\mathrm{M}}$, the superscripts are used for convenience to leave space for lower reaction and metabolite indices, for example ${k}_{li}^{\mathrm{M}}$. In modular rate laws, which rely on a quasi-equilibrium approximation [46], the ${k}^{\mathrm{X}}$ are dissociation constants and half-saturation concentrations at the same time. Saturation values range between 0 and 1, and from saturation values and metabolite concentrations, we can reconstruct the dissociation constants ${k}_{li}^{\mathrm{X}}$.

#### 4.3. Model Construction by STM

^{M}values can be converted into saturation values. These values can then be inserted directly, or saturation values can be sampled around them (uniformly or following a beta distribution) to account for uncertainties or missing information [87]. Known k

^{cat}values can be used similarly: in the dependence schema, the k

^{V}values, together with equilibrium constants and fluxes, determine the turnover rates ${k}_{+}^{\mathrm{cat}}$ and ${k}_{-}^{\mathrm{cat}}$ and the enzyme concentrations e for the same reaction. To match them to data, we can first choose k

^{V}values at random and then modify them to obtain a good fit of forward k

^{cat}value, enzyme concentration, and flux; or we use a different schema with forward k

^{cat}values instead of k

^{V}as basic variables.

- Cell growth. To model metabolism in growing cells, the formulae must be adapted. For balanced growth at a cell growth rate $\lambda $, all cellular compounds must be reproduced continuously, so the fluxes follow a mass balance condition $\mathbf{N}\mathbf{v}-\lambda \mathbf{c}=0$ with an extra dilution term. Metabolite concentrations and fluxes are tightly coupled by this equation, and in model construction they must be chosen together. In the kinetic model, a dilution term $-\lambda {c}_{i}$ must be added to the ODE of each metabolite, and the Jacobian matrix (which also appears in formulae for the control matrices) contains an extra term $-\lambda \mathbf{I}$. Moreover, conserved moieties in such models must always vanish because otherwise they would be diluted, thus preventing a steady state.
- Kinetic constants as basic variables. Instead of choosing saturation values and concentrations and then computing the constants ${k}_{li}^{\mathrm{X}}$, we may treat the constants ${k}_{li}^{\mathrm{X}}$ as basic variables and compute the saturation values from them. The distributions for ratios $c/{k}^{\mathrm{M}}$ and for saturation values $\beta =\frac{c/{k}^{\mathrm{M}}}{1+c/{k}^{\mathrm{M}}}=\frac{c}{{k}^{\mathrm{M}}+c}$ are obviously related. If $\beta $ is uniformly distributed in $]0,1[$, the ratio $c/{k}^{\mathrm{M}}$ shows a probability density function $\mathrm{prob}(c/{k}^{\mathrm{M}})=\frac{1}{{(1+c/{k}^{\mathrm{M}})}^{2}}$, i.e., $ln(c/{k}^{\mathrm{M}})$ follows a logistic distribution with location parameter 0 and scale parameter 1 (see Supplementary Section S5.1). In practice, we may conveniently sample saturation values from uniform or beta distributions, while kinetic constants may be sampled from sufficiently narrow gamma distributions or from similar log-normal distributions (for the choice of probability distributions, see Supplementary Text 3.1). By sampling saturation values not within $]0,1[$ but in a smaller range, one may avoid full saturation, and one may sample ${k}^{\mathrm{M}}$, ${k}^{\mathrm{A}}$, and ${k}^{\mathrm{I}}$ values around known experimental values.
- Multiple steady states. To build a model with multiple steady states, in the metabolic state phase we choose one set of equilibrium constants, but several sets of concentrations and fluxes for the different steady states; in the kinetics phase the constants ${k}_{li}^{\mathrm{M}},{k}_{li}^{\mathrm{A}}$, and ${k}_{li}^{\mathrm{I}}$ and velocity constants ${k}_{l}^{\mathrm{V}}$ (geometric means of forward and backward catalytic constants) are chosen or sampled, for example, by parameter balancing [37,52,54]. Finally, enzyme concentrations for each state are computed by matching reaction rates from the rate laws to predefined fluxes.

#### 4.4. Metabolic Control and Synergy Effects

#### 4.5. Variability and Fluctuations in Cells

**Σ**, but are complex-valued (Hermitian instead of symmetric) and frequency-dependent. The spectral power density $\mathcal{S}\left(\omega \right)$ describes the “weight” of at frequency $\omega $ in the stochastic process. Correlated fluctuating variables are described by a matrix with diagonal elements describing the spectral power densities and off-diagonal matrix elements describing correlations. For uncorrelated white noise, this matrix is an identity matrix $\mathcal{S}\left(\omega \right)=\mathbf{I}$. If the spectral densities of the original parameter fluctuations are known, the resulting spectral densities of state variables can be computed in a linear approximation (see Supplementary Section S4.2). In linear metabolic models with a stable reference state, noise parameters ${p}_{j}$ with spectral density matrix ${\mathcal{S}}_{\mathrm{p}}$ will lead to fluctuating metabolite concentrations with a spectral density matrix ${\mathcal{S}}_{\mathbf{c}}\left(\omega \right){=}_{}^{}\left(\omega \right){\mathbf{R}}_{\mathbf{p}}^{\mathbf{c}}\left(\omega \right){\mathcal{S}}_{\mathbf{p}}\left(\omega \right){\mathbf{R}}_{\mathbf{p}}^{\mathbf{c}}{\left(\omega \right)}^{\top}$. Again, diagonal elements are real-valued and contain the spectral density of a concentration ${c}_{i}$, while correlations are described by complex-valued off-diagonal elements. Random distributions of steady-state fluxes are described similarly by a matrix ${\mathcal{S}}_{\mathbf{v}}\left(\omega \right){=}_{}^{}\left(\omega \right){\mathbf{R}}_{\mathbf{p}}^{\mathbf{v}}\left(\omega \right){\mathcal{S}}_{\mathbf{p}}\left(\omega \right){\mathbf{R}}_{\mathbf{p}}^{\mathbf{v}}{\left(\omega \right)}^{\top}$. The effects of parameter fluctuations depend on the frequency: slow fluctuations (that is, ${\mathcal{S}}_{}\left(\omega \right)\approx 0$ except for $\omega \approx 0$) have quasi-static effects, creating permanent differences between cells.

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SKM | Structural kinetic modelling |

MCT | Metabolic control theory |

FBA | Flux balance analysis |

MoMA | Minimisation of metabolic adjustment |

ECM | Enzyme Cost Minimisation |

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**Figure 1.**Metabolic rate laws and dynamics: elasticities, thermodynamic forces, and metabolic control. (

**a**) A reaction rate depends on metabolite concentrations as described by a rate law $v(e,\mathbf{c})$. The slope $\partial v/\partial c$ is called reaction elasticity. (

**b**) In strongly driven reactions, with a driving force $\theta =ln\frac{{v}_{+}}{{v}_{-}}\gg 1$, the net rate is dominated by the forward rate and the product elasticity is almost zero. In contrast, close to chemical equilibrium (with a driving force $\theta \approx 0$ and large forward and backward rates, but a small net rate), the scaled elasticities ${\hat{\mathbf{E}}}_{{c}_{i}}^{{v}_{l}}=\frac{{c}_{i}}{{v}_{l}}\frac{\partial {v}_{l}}{\partial {c}_{i}}$ are large. (

**c**) Metabolic control coefficients describe how perturbations in single reactions shape steady-state fluxes and metabolite concentrations across the network. If an enzyme is inhibited or repressed, upstream metabolites accumulate and downstream metabolites deplete. The details of this response depend on network structure, flux distribution, and reaction elasticities. Thermodynamic forces shape metabolic control via the elasticities: a strongly driven reaction, with its low product elasticity, is insensitive to downstream processes and deprives all downstream enzymes of their flux control.

**Figure 2.**Dependence schema and systematic model construction. (

**a**) Dependencies between model variables (kinetic constants and state variables) in kinetic metabolic models. A dependence schema describes physical or logical dependencies between variables and can serve as a blueprint for model construction. For simplicity, the schema is shown in four parts, corresponding to four steps of model construction. (1) Metabolic state phase. Fluxes and thermodynamic forces must have the same signs. Predefined flux directions define sign constraints on fluxes and thermodynamic forces, and fluxes and chemical potentials can be sampled under these constraints. (2) Kinetics phase. Saturation values ${\beta}_{li}^{\mathrm{M}}$, ${\beta}_{li}^{\mathrm{A}}$, and ${\beta}_{li}^{\mathrm{I}}$ can be chosen independently between 0 and 1. Together with metabolite concentrations and thermodynamic forces, they determine kinetic constants (${k}_{li}^{\mathrm{M}}$, ${k}_{li}^{\mathrm{A}}$, and ${k}_{li}^{\mathrm{I}}$) and reaction elasticities. The elasticities further determine control properties (3) as well as kinetic constants and enzyme concentrations (4), allowing us to reconstruct the entire kinetic model. In the graphics, some variables stem from previous steps (types of variables marked by colours). (

**b**) Model construction around a metabolic reference state. Based on a dependence schema, basic model variables can be freely chosen or sampled, while derived variables are computed from them.

**Figure 3.**Systematic model construction by STM. Model of central carbon metabolism in Escherichia coli (for model details, see Data Availability). Network structure and flux data (from cells respiring on glucose) were taken from [51]. The panels show different types of variables obtained during model construction following the schema in Figure 2 (circles: metabolites; arrows and squares: reactions). By repeatedly sampling the basic variables and computing the others, a model ensemble can be constructed. (

**a**) Thermodynamically feasible fluxes (red arrows) obtained by flux minimisation (data from [53]); (

**b**) metabolite log-concentrations. Metabolite concentrations and thermodynamic forces were determined by thermodynamic balancing [54] of metabolite and reaction Gibbs free energy data. (

**c**) chemical potentials; (

**d**) thermodynamic forces in units of RT; (

**e**) saturation values, set to standard values of $\frac{1}{2}$. Alternatively, the saturation values could be determined from data or be sampled at random between 0 and 1; (

**f**) scaled reaction elasticities ${\hat{\mathbf{E}}}_{{c}_{i}}^{{v}_{l}}$; (

**g**) scaled control coefficients $\widehat{(}{C)}_{{v}_{l}}^{{r}_{\mathrm{PFK}}}$ for the flux in upper glycolysis (phosphofructokinase reaction) as the target variable; (

**h**) enzyme synergies (given by scaled second-order control coefficients) for the glycolytic flux. Positive values are shown in blue, negative values in red, zero values in white. For clarity, only enzyme synergies in the outer 5 percent quantiles are shown.

**Figure 4.**Paradoxical flux control: a model of UTP rephosphorylation in human hepatocytes. Fluxes in central metabolism were obtained in [53] by FBA with UTP production as the objective. Here, flux control coefficients were obtained by STM with standard elasticities, i.e., saturation values of 0.5. Paradoxically, the UTP-regenerating enzyme NDK has a negative control over its own flux. (

**a**) Metabolic dynamics after an NDK upshift. A higher NDK activity first speeds up the reaction, but later the rate drops below its initial value. (

**b**) Dose–response curve between NDK amount and steady-state UTP production. In the reference state (vertical line), the curve slope (response coefficient) is negative: an enzyme increase decreases the flux. (

**c**) The response coefficient depends on thermodynamic forces. To see this, all thermodynamic forces were increased by a common factor (x-axis): with higher thermodynamic forces, the response (y-axis) becomes more negative.

**Figure 5.**Enzyme synergy effects in a linear pathway with alternative routes. The (scaled) synergisms towards a fitness-relevant objective (e.g., biomass production) can tell us about epistasis. (

**a**) network structure. A conversion from S${}_{1}$ to S${}_{3}$ can occur directly or via the intermediate S${}_{2}$. The following plots show synergy effects for double inhibitions predicted by FBA (

**b**) or MoMA (

**c**). To simulate enzyme inhibitions, a flux decrease to 90 percent of the original value was imposed by setting a bound at 0.9 times the unperturbed reaction flux. The double inhibition of an enzyme is simulated by applying the inhibition twice, i.e., leading to a relative flux decrease factor of 0.81. Synergy effects are shown by line colours (red: aggravating, blue: buffering) and as a matrix. Colour ranges span the observed synergy effects in each panel (red: negative; white: zero; blue: positive). Small values, below one percent of the maximal absolute value, are not shown. (

**d**) Synergy effects computed by Metabolic Control Theory, assuming common modular rate laws and half-saturated enzymes. A model ensemble with saturation values sampled from uniform distributions yields almost the same results on average. Also a different rate law (the simultaneous binding modular rate law) yields very similar results (see Supplementary Figure S4). Compare Figure 3 for synergy effects in E. coli.

**Figure 6.**Static changes and dynamic fluctuations of metabolite concentrations. (

**a**) Variability of metabolic states caused by perturbed reaction rates (schematic drawing). An enzyme activation (left) increases a single reaction rate (solid arrow), which changes the steady-state metabolite concentrations and fluxes (dotted arrows). Similarly, variability in enzyme activities (quasi-static random changes) causes slow correlated metabolite variability (curved lines on the right, blue: positive; red: negative); (

**b**) variation of metabolite concentrations in E. coli central metabolism (Figure 3) caused by uncorrelated variable enzyme concentrations (geometric standard deviation of enzyme levels: 2). Standard deviations of log concentrations are shown by colours; (

**c**) correlated metabolite variations (covariances of log concentrations) are shown as coloured lines (values below 10% of the maximal value were removed for clarity). Metabolite variances (in (

**b**)) and covariances (in (

**c**)) computed from first-order response coefficients. Local enzyme fluctuations lead to network-wide metabolite fluctuations: the frequency spectra are related by spectral response coefficients. (

**d**) Thermodynamic forces (same data as in Figure 3d). Effects of chemical noise in a cell volume of 2 $\mathsf{\mu}$m${}^{3}$ and with a glycolytic flux of 1 mM/min; (

**e**) reactions close to equilibrium (small thermodynamic forces) produce strong chemical noise because of their large forward and backward fluxes. Spectral power density of the original noise; (

**f**) resulting metabolic fluctuations: fast fluctuations at a frequency of 1 s. Formulae are given in Supplementary Section S5.2.

**Figure 7.**Metabolite fluctuations due to chemical noise. Chemical noise causes random fluctuations of metabolite concentrations and fluxes. In a chemical Langevin equation, chemical noise is modelled by adding white-noise terms to the reaction rates. Fast and slow fluctuations are damped differently in the network. (

**a**) High-frequency metabolite fluctuations (spectral densities at oscillation freuency f = 1 s

^{−1}) exist only close to the noise source. Circle colors show the spectral densities at 1 s

^{−1}, line colors show covariances (blue: positive; red: negative). (

**b**) Metabolite fluctuations decrease with the frequency (see Figure 6). (

**Top**): spectral densities, each curve corresponds to the variance of one metabolite over a range of frequencies. (

**Bottom**): each curve corresponds to the standard deviation of a metabolite concentration (square root of spectral density, shown on the y-axis), computed for different time resolutions of observation (x-axis). For details, see Supplementary Figure S5a. (

**c**) Low-frequency metabolite fluctuations (oscillation period of 17 min) are correlated along the entire pathway. Results for flux fluctuations are shown in Supplementary Figure S5b. Smoothing at different time resolutions changes the variance of metabolite fluctuations. At low time resolutions, high-frequency fluctuations are filtered out (see Supplementary Figure S5a).

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Liebermeister, W. Structural Thermokinetic Modelling. *Metabolites* **2022**, *12*, 434.
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Liebermeister W. Structural Thermokinetic Modelling. *Metabolites*. 2022; 12(5):434.
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Liebermeister, Wolfram. 2022. "Structural Thermokinetic Modelling" *Metabolites* 12, no. 5: 434.
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