Thermodynamics of stoichiometric biochemical networks in living systems far from equilibrium

The principles of thermodynamics apply to both equilibrium and nonequilibrium biochemical systems. The mathematical machinery of the classic thermodynamics, however, mainly applies to systems in equilibrium. We introduce a thermodynamic formalism for the study of metabolic biochemical reaction (open, nonlinear) networks in both time-dependent and time-independent nonequilibrium states. Classical concepts in equilibrium thermodynamics-enthalpy, entropy, and Gibbs free energy of biochemical reaction systems-are generalized to nonequilibrium settings. Chemical motive force, heat dissipation rate, and entropy production (creation) rate, key concepts in nonequilibrium systems, are introduced. Dynamic equations for the thermodynamic quantities are presented in terms of the key observables of a biochemical network: stoichiometric matrix Q, reaction fluxes J, and chemical potentials of species mu without evoking empirical rate laws. Energy conservation and the Second Law are established for steady-state and dynamic biochemical networks. The theory provides the physiochemical basis for analyzing large-scale metabolic networks in living organisms.     

Some problems in the usage of Gibbs free energy in biochemistry

The usage of Gibbs free energy (G) in biochemistry is examined critically. The textbook formulation of the Second Law of Thermodynamics as applied to chemically-reacting systems is reviewed. Cognizance of the established theory and terminology of chemical thermodynamics leads to the conclusion that the symbol "delta G", as used in most biochemical calculations of free-energy change (e.g. in freeze-clamp study of steady-state metabolic processes), is erroneous. The instantaneous change, symbolized by the expression (delta G/delta xi) (with xi the degree of advancement of the reaction), is seen to be the correct form for describing the thermodynamic quality of the reactions of cell metabolism. Mathematical and graphical analysis of a sample reaction demonstrates the fundamental difference between delta G and (delta G/delta xi). Some problems in the application and interpretation of free-energy change in biochemical systems are reviewed: (1) Advances in protein dynamics have revealed the free-energy linkage properties of the enzyme molecule in binding/catalytic events of catalysis, demanding that we view the thermodynamics of elementary enzyme reactions with a finer eye. (2) The reality of metabolic microenvironments in vivo leads to equivocation in the significance of free-energy changes measured under macroscopic conditions in vitro. (3) The physicochemical character of reaction dynamics in the living cell may in some cases exceed the domain of validity of such thermodynamic state functions as Gibbs free energy.     

Thermodynamics of microbial growth and metabolism: an analysis of the current situation

This paper attempts to review in how far thermodynamic analysis can be used to understand and predict the performance of microorganisms with respect to growth and bio-product synthesis. In the first part, a simple thermodynamic model of microbial growth is developed which explains the relationship between the driving force for growth in terms of Gibbs energy dissipation and biomass yield. From the currently available literature, it appears that the Gibbs energy dissipation per C-mol of biomass grown, which represents the driving force for chemotrophic growth, may have been adapted by evolutionary processes to strike a reasonable compromise between metabolic rate and growth efficiency. Based on empirical correlations of the C-molar Gibbs energy dissipation, the wide variety of biomass yields observed in nature can be explained and roughly predicted. This type of analysis may be highly useful in environmental applications, where such wide variations occur. It is however not able to predict biomass yields in very complex systems such as mammalian cells nor is it able to predict or to assess bio-product or recombinant protein yields. For this purpose, a much more sophisticated treatment that accounts for individual metabolic pathways separately is required. Based on glycolysis as a test example, it is shown in the last part that simple thermodynamic analysis leads to erroneous conclusions even in well-known, simple cases. Potential sources for errors have been analyzed and can be used to identify the most important needs for future research.     

Development of thermodynamic optimum searching (TOS) to improve the prediction accuracy of flux balance analysis

Flux balance analysis (FBA) has been widely used in calculating steady‐state flux distributions that provide important information for metabolic engineering. Several thermodynamics‐based methods, for example, quantitative assignment of reaction directionality and energy balance analysis have been developed to improve the prediction accuracy of FBA. However, these methods can only generate a thermodynamically feasible range, rather than the most thermodynamically favorable solution. We therefore developed a novel optimization method termed as thermodynamic optimum searching (TOS) to calculate the thermodynamically optimal solution, based on the second law of thermodynamics, the minimum magnitude of the Gibbs free energy change and the maximum entropy production principle (MEPP). Then, TOS was applied to five physiological conditions of Escherichia coli to evaluate its effectiveness. The resulting prediction accuracy was found significantly improved (10.7–48.5%) by comparing with the ¹³C‐fluxome data, indicating that TOS can be considered an advanced calculation and prediction tool in metabolic engineering. Biotechnol. Bioeng. 2013; 110: 914–923. © 2012 Wiley Periodicals, Inc.     

Physico-chemical properties of organic compounds and the energetics of metabolism

The existing reference system of thermodynamic potentials of substance formation does not outline some important properties inherent to energy transformation and utilization in the cell metabolism. To elicit these properties, a new reference system is suggested. On its basis, a generalized unit of chemical substance reductance, called redoxon, has been developed. The molecules of more reduced substances contain a higher number of redoxons. Energy value of one redoxon is nearly constant in organic compounds but strongly varying in inorganic ones. The stoichiometric and thermodynamic balances of biochemical reactions in the new reference system have been obtained. The suggested approach has been shown to be an adequate tool for the analysis of the mass-energy balance of metabolic processes including heterotrophic and autotrophic growth of intact cells and organisms.     

A scalable algorithm to explore the Gibbs energy landscape of genome-scale metabolic networks

The integration of various types of genomic data into predictive models of biological networks is one of the main challenges currently faced by computational biology. Constraint-based models in particular play a key role in the attempt to obtain a quantitative understanding of cellular metabolism at genome scale. In essence, their goal is to frame the metabolic capabilities of an organism based on minimal assumptions that describe the steady states of the underlying reaction network via suitable stoichiometric constraints, specifically mass balance and energy balance (i.e. thermodynamic feasibility). The implementation of these requirements to generate viable configurations of reaction fluxes and/or to test given flux profiles for thermodynamic feasibility can however prove to be computationally intensive. We propose here a fast and scalable stoichiometry-based method to explore the Gibbs energy landscape of a biochemical network at steady state. The method is applied to the problem of reconstructing the Gibbs energy landscape underlying metabolic activity in the human red blood cell, and to that of identifying and removing thermodynamically infeasible reaction cycles in the Escherichia coli metabolic network (iAF1260). In the former case, we produce consistent predictions for chemical potentials (or log-concentrations) of intracellular metabolites; in the latter, we identify a restricted set of loops (23 in total) in the periplasmic and cytoplasmic core as the origin of thermodynamic infeasibility in a large sample (10(6)) of flux configurations generated randomly and compatibly with the prior information available on reaction reversibility.     

The thermodynamic meaning of metabolic exchange fluxes

Metabolic flux analysis (MFA) deals with the experimental determination of steady-state fluxes in metabolic networks. An important feature of the ¹³C MFA method is its capability to generate information on both directions of bidirectional reaction steps given by exchange fluxes. The biological interpretation of these exchange fluxes and their relation to thermodynamic properties of the respective reaction steps has never been systematically investigated. As a central result, it is shown here that for a general class of enzyme reaction mechanisms the quotients of net and exchange fluxes measured by ¹³C MFA are coupled to Gibbs energies of the reaction steps. To establish this relation the concept of apparent flux ratios of enzymatic isotope-labeling networks is introduced and some computing rules for these flux ratios are given. Application of these rules reveals a conceptional pitfall of ¹³C MFA, which is the inherent dependency of measured exchange fluxes on the chosen tracer atom. However, it is shown that this effect can be neglected for typical biochemical reaction steps under physiological conditions. In this situation, the central result can be formulated as a two-sided inequality relating fluxes, pool sizes, and standard Gibbs energies. This relation has far-reaching consequences for metabolic flux analysis, quantitative metabolomics, and network thermodynamics.     

The combination of transformed and constrained Gibbs energies

Gibbs free energy is the thermodynamic potential representing the fundamental equation at constant temperature, pressure, and molar amounts. Transformed Gibbs energies are important for biochemical systems because the local concentrations within cell compartments cannot yet be determined accurately. The method of Constrained Gibbs Energies adds kinetic reaction extent limitations to the internal constraints of the system thus extending the range of applicability of equilibrium thermodynamics from predefined constraints to dynamic constraints, e.g., adding time-dependent constraints of irreversible chemical change. In this article, the implementation and use of Transformed Gibbs Energies in the Gibbs energy minimization framework is demonstrated with educational examples. The combined method has the advantage of being able to calculate transient thermodynamic properties during dynamic simulation.     

Does microbial life always feed on negative entropy? Thermodynamic analysis of microbial growth.

Schr?dinger stated in his landmark book, What is Life?, that life feeds on negative entropy. In this contribution, the validity of this statement is discussed through a careful thermodynamic analysis of microbial growth processes. In principle, both feeding on negative entropy, i.e. yielding products of higher entropy than the substrates, and generating heat can be used by microorganisms to rid themselves of internal entropy production resulting from maintenance and growth processes. Literature data are reviewed in order to compare these two mechanisms. It is shown that entropy-neutral, entropy-driven, and entropy-retarded growth exist. The analysis of some particularly interesting microorganisms shows that enthalpy-retarded microbial growth may also exist, which would signify a net uptake of heat during growth. However, the existence of endothermic life has never been demonstrated in a calorimeter. The internal entropy production in live cells also reflects itself in the Gibbs energy dissipation accompanying growth, which is related quantitatively to the biomass yield. An empirical correlation of the Gibbs energy dissipation in terms of the physico-chemical nature of the growth substrate has been proposed in the literature and can be used to predict the biomass yield approximately. The ratio of enthalpy change and Gibbs energy change can also be predicted since it is shown to be approximately equal to the same ratio of the relevant catabolic process alone.     

Modeling and analysis of biopathways dynamics

Cellular processes are governed and coordinated by a multitude of biopathways. A pathway can be viewed as a complex network of biochemical reactions. The dynamics of this network largely determines the functioning of the pathway. Hence the modeling and analysis of biochemical networks dynamics is an important problem and is an active area of research. Here we review quantitative models of biochemical networks based on ordinary differential equations (ODEs). We mainly focus on the parameter estimation and sensitivity analysis problems and survey the current methods for tackling them. In this context we also highlight a recently developed probabilistic approximation technique using which these two problems can be considerably simplified.     

Metabolic systems cost-benefit analysis for interpreting network structure and regulation

MOTIVATION: Interpretation of bioinformatics data in terms of cellular function is a major challenge facing systems biology. This question is complicated by robust metabolic networks filled with structural features like parallel pathways and isozymes. Under conditions of nutrient sufficiency, metabolic networks are well known to be regulated for thermodynamic efficiency however; efficient biochemical pathways are anabolically expensive to construct. While parameters like thermodynamic efficiency have been extensively studied, a systems-based analysis of anabolic proteome synthesis 'costs' and the cellular function implications of these costs has not been reported. RESULTS: A cost-benefit analysis of an in silico Escherichia coli network revealed the relationship between metabolic pathway proteome synthesis requirements, DNA-coding sequence length, thermodynamic efficiency and substrate affinity. The results highlight basic metabolic network design principles. Pathway proteome synthesis requirements appear to have shaped biochemical network structure and regulation. Under conditions of nutrient scarcity and other general stresses, E. coli expresses pathways with relatively inexpensive proteome synthesis requirements instead of more efficient but also anabolically more expensive pathways. This evolutionary strategy provides a cellular function-based explanation for common network motifs like isozymes and parallel pathways and possibly explains 'overflow' metabolisms observed during nutrient scarcity. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.     

Perspective: a stirring role for metabolism in cells

Based on recent findings indicating that metabolism might be governed by a limit on the rate at which cells can dissipate Gibbs energy, in this Perspective, we propose a new mechanism of how metabolic activity could globally regulate biomolecular processes in a cell. Specifically, we postulate that Gibbs energy released in metabolic reactions is used to perform work, allowing enzymes to self‐propel or to break free from supramolecular structures. This catalysis‐induced enzyme movement will result in increased intracellular motion, which in turn can compromise biomolecular functions. Once the increased intracellular motion has a detrimental effect on regulatory mechanisms, this will establish a feedback mechanism on metabolic activity, and result in the observed thermodynamic limit. While this proposed explanation for the identified upper rate limit on cellular Gibbs energy dissipation rate awaits experimental validation, it offers an intriguing perspective of how metabolic activity can globally affect biomolecular functions and will hopefully spark new research.     

dGPredictor: Automated fragmentation method for metabolic reaction free energy prediction and de novo pathway design

Group contribution (GC) methods are conventionally used in thermodynamics analysis of metabolic pathways to estimate the standard Gibbs energy change (Δ_rG′^o) of enzymatic reactions from limited experimental measurements. However, these methods are limited by their dependence on manually curated groups and inability to capture stereochemical information, leading to low reaction coverage. Herein, we introduce an automated molecular fingerprint-based thermodynamic analysis tool called dGPredictor that enables the consideration of stereochemistry within metabolite structures and thus increases reaction coverage. dGPredictor has comparable prediction accuracy compared to existing GC methods and can capture Gibbs energy changes for isomerase and transferase reactions, which exhibit no overall group changes. We also demonstrate dGPredictor’s ability to predict the Gibbs energy change for novel reactions and seamless integration within de novo metabolic pathway design tools such as novoStoic for safeguarding against the inclusion of reaction steps with infeasible directionalities. To facilitate easy access to dGPredictor, we developed a graphical user interface to predict the standard Gibbs energy change for reactions at various pH and ionic strengths. The tool allows customized user input of known metabolites as KEGG IDs and novel metabolites as InChI strings (https://github.com/maranasgroup/dGPredictor).     

Fundamental equation of thermodynamics for protein-ligand binding

For the internal energy and every thermodynamic potential that can be defined by a Legendre transform, there is a fundamental equation that contains all the thermodynamic information about a system. For a system involving the binding of molecular oxygen and hydrogen ions by a protein, fundamental equations are given for the Gibbs energy G, the transformed Gibbs energy G' at specified pH, and the further transformed Gibbs energy G" at specified pH and specified concentration of molecular oxygen. The Maxwell equations for these various Gibbs energies are important because they provide the connection with experimentally determined properties and increase our understanding of these properties. Measurements of the average number of oxygen molecules bound as a function of T, pH and concentration of molecular oxygen make it possible to calculate Delta(f)G"(o) of the reactant. Maxwell equations make it possible to calculate the average number of hydrogen ions bound, Delta(f)S"(o), Delta(f)H"(o) and their partial derivatives. These relations are illustrated with numerical calculations on a simple reaction system.     

Thermodynamic Analysis of Energy Transfer in Acidogenic Cultures

A global thermodynamic analysis, normally used for pure cultures, has been performed for steady‐state data sets from acidogenic mixed cultures. This analysis is a combination of two different thermodynamic approaches, based on tabulated standard Gibbs energy of formation, global stoichiometry and medium compositions. It takes into account the energy transfer efficiency, ?, together with the Gibbs free energy dissipation, ΔG_o, analysis of the different data. The objective is to describe these systems thermodynamically without any heat measurement. The results show that ? is influenced by environmental conditions, where increasing hydraulic retention time increases its value all cases. The pH effect on ? is related to metabolic shifts and osmoregulation. Within the environmental conditions analyzed, ? ranges from 0.23 for a hydraulic retention time of 20 h and pH 4, to 0.42 for a hydraulic retention time of 8 h and a pH ranging from 7–8.5. The estimated values of ΔG_o are comparable to standard Gibbs energy of dissipation reported in the literature. For the data sets analyzed, ΔG_o ranges from –1210 kJ/mol_x, corresponding to a stirring velocity of 300 rpm, pH 6 and a hydraulic retention time of 6 h, to –20744 kJ/mol_x for pH 4 and a hydraulic retention time of 20 h. For average conclusions, the combined approach based on standard Gibbs energy of formation and global stoichiometry, used in this thermodynamic analysis, allows for the estimation of Gibbs energy dissipation values from the extracellular medium compositions in acidogenic mixed cultures. Such estimated values are comparable to the standard Gibbs energy dissipation values reported in the literature. It is demonstrated that ? is affected by the environmental conditions, i.e., stirring velocity, hydraulic retention time and pH. However, a relationship that relates this parameter to environmental conditions was not found and will be the focus of further research.     

Ab initio prediction of thermodynamically feasible reaction directions from biochemical network stoichiometry

Analysis of the stoichiometric structure of metabolic networks provides insights into the relationships between structure, function, and regulation of metabolic systems. Based on knowledge of only reaction stoichiometry, certain aspects of network functionality and robustness can be predicted. Current theories focus on breaking a metabolic network down into non-decomposable pathways able to operate in steady state. The physics underlying these theories is based on mass balance and the laws of thermodynamics. However, due to the inherent nonlinearity of the thermodynamic constraints on metabolic fluxes, computational analysis of large-scale biochemical systems can be expensive. In this study, it is shown how the feasible reaction directions may be determined by either computing the allowable ranges under the mass-balance and thermodynamic constraints or by analyzing the stoichiometric structure of the network. The computed reaction directions translate into a set of linear constraints necessary for thermodynamic feasibility. This set of necessary linear constraints is shown to be sufficient to guarantee feasibility in certain cases, thus translating the nonlinear thermodynamic constraints to linear. We show that for a reaction network of 44 internal reactions representing energy metabolism, the computed linear inequality constraints represent necessary and sufficient conditions for thermodynamic feasibility.     

WinBEST-KIT: Windows-based biochemical reaction simulator for metabolic pathways

We have implemented an efficient, user-friendly biochemical reaction simulator called Web-based BEST-KIT (Biochemical Engineering System analyzing Tool-KIT) for analyzing large-scale nonlinear networks such as metabolic pathways. Users can easily design and analyze an arbitrary reaction scheme through the Internet and an efficient graphical user interface without considering the mathematical equations. The reaction scheme can include several reaction types, which are represented by both the mass action law (mass balance) and approximated velocity functions of enzyme kinetics at steady state, such as Michaelis-Menten, Hill cooperative, Competitive inhibition. However, since all modules in Web-based BEST-KIT have been developed in Java applet style, users cannot optionally make use of original mathematical equations in addition to the prepared equations. In the present study, we have developed a new version of BEST-KIT (for Microsoft Windows called WinBEST-KIT) to allow users to define original mathematical equations and to customize these equations very easily as user-defined reaction symbols. The following powerful system-analytical methods are prepared for system analysis: time-course calculation, parameter scanning, estimation of the values of unknown kinetic parameters based on experimentally observed time-course data of reactants, dynamic response of reactants against virtual external perturbations, and real-time simulation (Virtual Dry Lab).     

Thermodynamic analysis of trinitrotoluene biodegradation and mineralization pathways

Biodegradation of 2,4,6-trinitrotoluene (TNT) proceeds through several different metabolic pathways. However, the reaction steps which are considered rate-controlling have not been fully determined. Glycolysis and other biological pathways contain biochemical reactions which are acutely rate-limiting due to enzyme control. These rate-limiting steps also have large negative Gibbs free energy changes. Because xenobiotic compounds such as TNT can be used by biological systems as nitrogen, carbon, and energy sources, it is likely that their degradation pathways also contain acutely rate-limiting steps. Identification of these rate-controlling reactions will enhance and better direct genetic engineering techniques to increase specific enzyme levels.This article identifies likely rate-controlling steps (or sets of steps) in reported TNT biodegradation pathways by estimating the Gibbs free energy change for each step and for the overall pathways. The biological standard Gibbs free energy change of reaction was calculated for each pathway step using a group contribution method specifically tailored for biomolecules. The method was also applied to hypothetical "pathways" constructed to mineralize TNT using several different microorganisms. Pathways steps that have large negative Gibbs free energy changes are postulated to be potentially rate-controlling. The microorganisms which utilize degradation pathways with the largest overall (from TNT to citrate) negatiave Gibbs free energy changes were also determined. Such microorganisms can extract more energy from the starting substrate and are thus assumed to have a competitive advantage over other microorganisms. Results from this modeling-based research are consistent with much experimental work available in the literature. (c) 1996 John Wiley & Sons, Inc.