Model Balancing: A Search for In-Vivo Kinetic Constants and Consistent Metabolic States
Abstract
: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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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
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 StyleLiebermeister, 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