# Model Balancing: A Search for In-Vivo Kinetic Constants and Consistent Metabolic States

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Metabolic Model and Statistical Estimation Model

#### 2.2. Model Balancing

#### 2.3. A Convex Version of the Score Functions

#### 2.4. Details and Variants of Model Balancing

## 3. Results

#### 3.1. Model Balancing

#### 3.2. Tests with Artificial Data

#### 3.3. Model Fitting with Experimentally Measured Data

## 4. Discussion

#### 4.1. Model Balancing in Relation to Other Methods

- Parameter balancing. Parameter balancing determines consistent kinetic constants from kinetic and thermodynamic data. Unlike model balancing, it does not use rate laws or flux data. All multiplicative constants (such as Michaelis–Menten constants or catalytic constants) are described by log-values, which leads to a linear regression problem. The equilibrium constants are parameterised directly by standard chemical potentials rather than independent variables that can be adjusted if needed [24]. With Gaussian priors and measurement errors (in log-scale), likelihood and posterior terms are quadratic and convex. Parameter balancing can handle either kinetic and thermodynamic constants (“kinetic parameter balancing”), metabolite concentrations and thermodynamic forces (“state balancing”), or kinetic constants and metabolic states (“state/parameter balancing”). With known signs of thermodynamic forces, defined by the flux directions, parameter balancing can predict thermodynamically feasible kinetic constants and metabolite concentrations. While its optimisation takes place on the same set as in model balancing, it does not consider rate laws and cannot be used to fit kinetic constants to flux data. As a post-processing step, balanced kinetic constants can be adjusted to rate laws and flux data, but this works only for a single metabolic state so, unlike in model balancing, data from multiple states cannot be combined.
- Enzyme cost minimisation. Enzyme cost minimisation (ECM) [43] predicts optimal enzyme and metabolite concentrations in kinetic models with given parameter values. ECM determines metabolite and enzyme concentrations that realise predefined fluxes at a minimal cost, for instance, at a minimal total enzyme and metabolite concentration. The optimisation is carried out in (log-)metabolite space. In contrast to parameter balancing, ECM assumes given kinetic constants and optimises a biological cost rather than a goodness of fit. With given rate laws, the cost function (a weighted sum of enzyme and metabolite concentrations) is convex in log-metabolite space.

#### 4.2. Model Balancing in Practice

- Inferring missing data types If fluxes and two of the data types are given, the third type can be estimated. For example, we may estimate in-vivo kinetic constants from metabolite concentrations and enzyme concentrations; we may estimate metabolite concentrations from enzyme concentrations and enzyme kinetics; or we may estimate enzyme concentrations from metabolite concentrations and enzyme kinetics. If the data were complete and precise, the third type of variables could be directly computed, and model balancing would not be necessary. But when data are uncertain and incomplete, model balancing allows us to infer the missing data while completing and adjusting the others.
- Adjusting omics data to obtain complete, consistent metabolic states Given a model with known kinetic constants, we can translate metabolite and enzyme data into complete, consistent metabolic states. Again, fluxes must be given and thermodynamically realisable with the assumed equilibrium constants and metabolite bounds. We can even estimate metabolic states without any enzyme or metabolite data: in this case, model balancing predicts plausible states with the given fluxes, relying on priors for enzyme or metabolite concentrations.
- Imposing thermodynamic constraints and bounds on data To build consistent metabolic models, we may collect data for kinetic and state variables and apply model balancing. The resulting kinetic constants and state variables satisfy the rate laws, agree with physical and physiological constraints, and resemble data and prior values. Above we used this to construct a physically and biologically plausible model of E. coli central metabolism. Posterior sampling (as in [16]) might be used to assess uncertainties in model parameters.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ECM | Enzyme Cost Minimisation |

FBA | Flux Balance Analysis |

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**Figure 1.**Model balancing relies on a dependency schema for kinetic constants and state variables. All kinetic constants are described in logarithmic scale. A subset of kinetic constants determines all other kinetic constants through linear relationships. If kinetic constants, metabolite concentrations, and fluxes are known, the enzyme concentrations can be computed from rate laws and fluxes: each enzyme concentration is a convex function of the (logarithmic) kinetic constants and metabolite concentrations. The signs of thermodynamic forces are constrained by the flux directions.

**Figure 2.**Model balancing and its convex variants. Top right: Example model. Given the fluxes and external metabolite concentrations, the log concentration x determines the enzyme log concentrations ${y}_{1}$ and ${y}_{2}$. (

**a**) Posterior score for the enzyme level in reaction 1. Top: the log enzyme level ${y}_{1}$ depends on the metabolite log concentration x (solid curve). Horizontal lines show the preposterior mean (solid) and standard deviation (dashed). Black line: the posterior score term is zero where ${y}_{1}$ matches the preposterior mean and increases for smaller and larger values. Since the left part of the curve is negatively curved, the function is non-convex. In the plots below, this left part is decreased by a factor $\alpha =0.2$ (dark blue) or removed completely by setting $\alpha =0$ (cyan). The resulting last curve is convex; (

**b**) same as (

**a**), for reaction 2. The posterior score has its minimum at a lower x value than for reaction 1; (

**c**) Posterior score for x. Top: The enzyme posterior scores for reaction 1 and 2 (solid and dotted black curves from (

**a**,

**b**)) are added (thick solid line). By further adding the metabolite term (which is strictly convex), we obtain the posterior score (dashed line), which is non-convex. In the case of $\alpha =0.2$ (dark blue), the enzyme term is still non-convex, but the total posterior score becomes convex. In the case $\alpha =0$, the enzyme term is convex, and the total posterior score is convex as well.

**Figure 3.**Model balancing results for E. coli model with artificial data (stringency parameter $\alpha =0.5$). The model structure is shown in [48]. Each subfigure shows “true” artificial values (x-axis) versus reconstructed values (y-axis). The quality of fits or predictions is quantified by geometric standard deviations and by Pearson correlations on logarithmic scale. (

**a**) Metabolite concentrations; (

**b**) enzyme concentrations; (

**c**–

**f**) equilibrium constants, ${k}_{\mathrm{cat}}$ values, and Michaelis–Menten constants. The three rows show different estimation scenarios (see Supplementary Materials Figure S2). Upper row: simple scenario S1 (noise-free artificial data, data for all kinetic constants). Centre row: scenario S1K (noise-free artificial data, kinetic data given only for equilibrium constants). Lower row: scenario S2 (noise-free artificial data, no data for kinetic constants). Depending on the scenario, kinetic constants are either fitted (dots) or predicted (crosses).

**Figure 4.**Model balancing results for E. coli model with artificial data (stringency parameter $\alpha =0.5$). Same as Figure 3, but with noisy kinetic data and noisy state data.

**Figure 5.**Results for E. coli central metabolism with experimental data (aerobic growth on glucose). The kinetic data stem from previous parameter balancing based on in-vitro data.

**Top**: estimation using kinetic data.

**Centre**: model fit using equilibrium constants as data.

**Bottom**: estimation without usage of kinetic data. Metabolite, enzyme, and kinetic data were taken from [43]. A stringency parameter $\alpha =0.5$ was used. The fact that our kinetic data were obtained from parameter balancing, based on the same network model and the same priors, leads to a bias. However, a test with original in-vitro kinetic data (and thus fewer data points, not shown) yielded similar results. Predicted kinetic constants may be validated by cross-validation; the results can be expected to lie in between the results from our “fitting” (dots) and “prediction” scenarios (crosses).

**Figure 6.**Estimation methods for kinetic constants and metabolic states. (

**a**) Kinetic model and metabolic states. A model is parameterised by kinetic constants (e.g., equilibrium constants, catalytic constants, and Michaelis–Menten constants) and is used to generate a number of metabolic states (characterised by enzyme concentrations, metabolite concentrations, and fluxes). States may be stationary (with steady-state fluxes) or not (e.g., states during dynamic time courses); (

**b**) dependencies between kinetic constants and state variables (see Figure 1); (

**c**) model balancing. Kinetic constants and metabolite concentrations (for several states) are the free variables of a statistical model. Dependent kinetic constants, thermodynamic driving forces, and enzyme concentrations (bottom) are the dependent variables, and fluxes (top right) are predefined. Priors and data may be used in the estimation. The other graphics show (

**d**) parameter balancing [21,24] for kinetic data and metabolite concentrations; (

**e**) model balancing with given kinetic constants; (

**f**) enzyme cost minimisation [43], in which enzyme and metabolite concentrations are optimised for a low enzyme and metabolite cost.

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**MDPI and ACS Style**

Liebermeister, W.; Noor, E.
Model Balancing: A Search for In-Vivo Kinetic Constants and Consistent Metabolic States. *Metabolites* **2021**, *11*, 749.
https://doi.org/10.3390/metabo11110749

**AMA Style**

Liebermeister W, Noor E.
Model Balancing: A Search for In-Vivo Kinetic Constants and Consistent Metabolic States. *Metabolites*. 2021; 11(11):749.
https://doi.org/10.3390/metabo11110749

**Chicago/Turabian Style**

Liebermeister, Wolfram, and Elad Noor.
2021. "Model Balancing: A Search for In-Vivo Kinetic Constants and Consistent Metabolic States" *Metabolites* 11, no. 11: 749.
https://doi.org/10.3390/metabo11110749