Network Thermodynamics Research Articles

Network Derivation of Liquid Junction Potentials in Single-Membrane System.

Peusner's network thermodynamics (PNT) is one of the more important formalisms of nonequilibrium thermodynamics used to describe membrane transport and the conversion of the internal energy of the system into energy dissipated in the environment and free energy used for the work involved in the transport of solution components in membrane processes. A procedure of transformation the Kedem-Katchalsky (K-K) equations for the transport of binary electrolytic solutions through a membrane to the Kedem-Katchalsky-Peusner (K-K-P) equations based on the PNT formalism for liquid junction potentials was developed. The subject of the study was a membrane used for hemodialysis (Ultra Flo 145 Dialyser) and aqueous NaCl solutions. The research method was the L version of the K-K-P formalism for binary electrolyte solutions. The Peusner coefficients obtained from the transformations of the K-K formalism coefficients for the transport of electrolyte solutions through the artificial polymer membrane were used to calculate the coupling coefficients of the membrane processes and to calculate the dissipative energy flux. In addition, the dissipative energy flux, as a function of thermodynamic forces, made it possible to investigate the energy conversion of transport processes in the membrane system.

L version of the transformed Kedem-Katchalsky equations for membrane transport of electrolyte solutions and internal energy conversion.

One of the important formalisms of non-equilibrium thermodynamics is Peusner network thermodynamics. The description of the energy conversion in membrane processes, i.e., the conversion of the internal energy of the system into the dissipated energy and the free energy used for the work associated with the transport of solution components, allows us to describe the relationship between these energies and the thermodynamic forces acting in the membrane system. The aim of this study was to develop a procedure to transform the Kedem-Katchalsky equations for the transport of binary electrolytic solutions across a membrane into the Kedem-Katchalsky-Peusner equations based on Peusner network thermodynamics. The conversion of electrochemical energy to free energy in the membrane system was also determined. The nanobiocellulose biomembranes (Biofill) were the subject of the study with experimentally determined transport parameters for aqueous NaCl solutions. The research method is the Kedem-Katchalsky-Peusner formalism for binary electrolyte solutions with introduced Peusner coefficients. The coefficients of the L version of the membrane transport equations and the Peusner coupling coefficients were derived as functions of NaCl concentration in the membrane. Based on these coefficients, the fluxes of internal energy of the system, energy dissipated to the surroundings and free energy related to the transport of electrolyte across the membrane were calculated and presented as functions of the osmotic and electric forces on the membrane. The Peusner coefficients obtained from the transformations of the coefficients of the Kedem-Katchalsky formalism for the transport of electrolyte solutions through the Biofill membrane were used to calculate the coupling coefficients of the membrane processes and the dissipative energy flux. The dissipative energy flux takes the form of a quadratic form due to the thermodynamic forces on the membrane - second degree curves are obtained. Moreover, the dissipative energy flux as a function of thermodynamic forces allowed us to examine the energy conversion in transport processes in the membrane system.

A method for evaluating the transport and energy conversion properties of polymer biomembranes using the Kedem-Katchalsky-Peusner equations.

A basic parameter in non-equilibrium thermodynamics is the production of entropy (S-entropy), which is a consequence of the irreversible processes of mass, charge, energy, and momentum transport in various systems. The product of S-entropy production and absolute temperature (T) is called the dissipation function and is a measure of energy dissipation in non-equilibrium processes. This study aimed to estimate energy conversion in membrane transport processes of homogeneous non-electrolyte solutions. The stimulus version of the R, L, H, and P equations for the intensity of the entropy source achieved this purpose. The transport parameters for aqueous glucose solutions through Nephrophan® and Ultra-Flo 145 dialyser® synthetic polymer biomembranes were experimentally determined. Kedem-Katchalsky-Peusner (KKP) formalism was used for binary solutions of non-electrolytes, with Peusner coefficients introduced. The R, L, H, and P versions of the equations for the S-energy dissipation were derived for the membrane systems based on the linear non-equilibrium Onsager and Peusner network thermodynamics. Using the equations for the S-energy and the energy conversion efficiency factor, equations for F-energy and U-energy were derived. The S-energy, F-energy and U-energy were calculated as functions of osmotic pressure difference using the equations obtained and presented as suitable graphs. The R, L, H, and P versions of the equations describing the dissipation function had the form of second-degree equations. Meanwhile, the S-energy characteristics had the form of second-degree curves located in the 1st and 2nd quadrants of the coordinate system. These findings indicate that the R, L, H, and P versions of S-energy, F-energy and U-energy are not equivalent for the Nephrophan® and Ultra-Flo 145 dialyser® membranes.

Evaluation of Transport Properties and Energy Conversion of Bacterial Cellulose Membrane Using Peusner Network Thermodynamics.

We evaluated the transport properties of a bacterial cellulose (BC) membrane for aqueous ethanol solutions. Using the Rr version of the Kedem-Katchalsky-Peusner formalism (KKP) for the concentration polarization (CP) conditions of solutions, the osmotic and diffusion fluxes as well as the membrane transport parameters were determined, such as the hydraulic permeability (Lp), reflection (σ), and solute permeability (ω). We used these parameters and the Peusner (Rijr) coefficients resulting from the KKP equations to assess the transport properties of the membrane based on the calculated dependence of the concentration coefficients: the resistance, coupling, and energy conversion efficiency for aqueous ethanol solutions. The transport properties of the membrane depended on the hydrodynamic conditions of the osmotic diffusion transport. The resistance coefficients R11r, R22r, and Rdetr were positive and higher, and the R12r coefficient was negative and lower under CP conditions (higher in convective than nonconvective states). The energy conversion was evaluated and fluxes were calculated for the U-, F-, and S-energy. It was found that the energy conversion was greater and the S-energy and F-energy were lower under CP conditions. The convection effect was negative, which means that convection movements were directed vertically upwards. Understanding the membrane transport properties and mechanisms could help to develop and improve the membrane technologies and techniques used in medicine and in water and wastewater treatment processes.

Energy conversion in Textus Bioactiv Ag membrane dressings using Peusner's network thermodynamic descriptions.

The Textus Bioactiv Ag membrane is an active dressing for the treatment of chronic wounds such as venous stasis ulcers and burns. Determination of the transport and internal energy conversion properties of the Textus Bioactiv Ag membrane using the Kedem-Katchalsky-Peusner model. This model introduces the coefficients Lij necessary to calculate the degree of coupling (lij, QL), energy conversion efficiency (eij), dissipated energy (S-energy), free energy (F-energy), and internal energy (U-energy). The research material was the Textus Bioactiv Ag membrane that is used as an active dressing in the treatment of difficult-to-heal wounds, and KCl aqueous solutions. The research methods employed Peusner's formalism of network thermodynamics and Kedem and Katchalsky's thermodynamics of membrane processes. To calculate the Lij coefficients, we used hydraulic conductivity (Lp), diffusion conductivity (ů) and reflection (ó) coefficients to perform experimental measurements in different conditions. The Lp coefficient for the Textus Bioactiv Ag membrane is nonlinearly dependent on the average concentrations of the solutions. In turn, the ů and ó coefficients are nonlinearly dependent on the differences in osmotic pressures (Äđ). An increase in the Äđ causes the Textus Bioactiv Ag membrane to become more permeable and less selective for KCl solutions. The coefficients of Peusner (Lij), couplings (lij, QL), energy conversion efficiency (eij), S-energy, F-energy, and U-energy also depend nonlinearly on Äđ. Our results showed that for higher concentrations of KCl solutions transported through the Textus Bioactiv Ag membrane, the coupling and energy conversion coefficients were greater for larger Äđ up to their maximum values for large Äđ. Coupling of the membrane structure with the electrolyte flux through the membrane is observed for Äđ greater than 10 kPa. Textus Bioactiv Ag membrane dressings possess the properties of a solution component separator as well as an internal energy converter.

Appraisal the effective diffusion coefficients of Li-ions by PITT and network thermodynamic methods in thin-layer Fe-sulfidic electrodes of Li-accumulator

In this work, we explored the possibilities of network thermodynamics to determine the most important kinetic parameter of the diffusion process, the diffusion coefficient of lithium ions (DLi) in iron sulfide. Iron sulfides have been electrolytically synthesized in thin aluminum-based layers for implementation in miniature lithium batteries. A comparison is made of the results obtained by the method of network thermodynamics and the method of potentiostatic pulse titration PITT. The theoretical aspects of both methods are presented. The task was reduced to obtaining curves I (current) - time (t), their analysis and calculation of coefficient values (DLi), using the theoretical foundations of diffusion and the methods used. At the early stages of applying the method of network thermodynamics to the study of diffusion in lithium current sources, it was not clear why the method is suitable for determining DLi not in the entire working range of potentials. A wide range of studies using the PITT method helps to answer questions related to the application of the network thermodynamic method. When using both methods, it is important to establish the potential range with diffusion kinetics and to reveal the accompanying electrode processes. Thus, it was found that both methods used are unable to provide reliable results in the FexSy electrode potential range of 2.8–1.8 V, since the diffusion nature of the electrode process is not a priority in this potential range.

Network Thermodynamics-Based Scalable Compartmental Model for Multi-Strain Epidemics

SARS-CoV-2 continues to upend human life by posing novel threats related to disease spread and mutations. Current models for the disease burden of SARS-CoV-2 consider the aggregate nature of the virus without differentiating between the potency of its multiple strains. Hence, there is a need to create a fundamental modeling framework for multi-strain viruses that considers the competing viral pathogenic pathways. Alongside the consideration that other viral pathogens may coexist, there is also a need for a generalizable modeling framework to account for multiple epidemics (i.e., multi-demics) scenarios, such as influenza and COVID-19 occurring simultaneously. We present a fundamental network thermodynamics approach for assessing, determining, and predicting viral outbreak severity, which extends well-known standard epidemiological models. In particular, we use historical data from New York City’s 2011–2019 influenza seasons and SARS-CoV-2 spread to identify the model parameters. In our model-based analysis, we employ a standard susceptible–infected–recovered (SIR) model with pertinent generalizations to account for multi-strain and multi-demics scenarios. We show that the reaction affinities underpinning the formation processes of our model can be used to categorize the severity of infectious or deceased populations. The spontaneity of occurrence captured by the change in Gibbs free energy of reaction (∆G) in the system suggests the stability of forward occurring population transfers. The magnitude of ∆G is used to examine past influenza outbreaks and infer epidemiological factors, such as mortality and case burden. This method can be extrapolated for wide-ranging utility in computational epidemiology. The risk of overlapping multi-demics seasons between influenza and SARS-CoV-2 will persist as a significant threat in forthcoming years. Further, the possibility of mutating strains requires novel ways of analyzing the network of competing infection pathways. The approach outlined in this study allows for the identification of new stable strains and the potential increase in disease burden from a complex systems perspective, thereby allowing for a potential response to the significant question: are the effects of a multi-demic greater than the sum of its individual viral epidemics?

Network thermodynamics of biological systems: A bond graph approach

Edmund Crampin (1973–2021) was at the forefront of Systems Biology research and his work will influence the field for years to come. This paper brings together and summarises the seminal work of his group in applying energy-based bond graph methods to biological systems. In particular, this paper: (a) motivates the need to consider energy in modelling biology; (b) introduces bond graphs as a methodology for achieving this; (c) describes extensions to modelling electrochemical transduction; (d) outlines how bond graph models can be constructed in a modular manner and (e) describes stoichiometric approaches to deriving fundamental properties of reaction networks. These concepts are illustrated using a new bond graph model of photosynthesis in chloroplasts.

Gibbs Free Energy, a Thermodynamic Measure of Protein–Protein Interactions, Correlates with Neurologic Disability

Modern network science has been used to reveal new and often fundamental aspects of brain network organization in physiological as well as pathological conditions. As a consequence, these discoveries, which relate to network hierarchy, hubs and network interactions, have begun to change the paradigms of neurodegenerative disorders. In this paper, we explore the use of thermodynamics for protein–protein network interactions in Alzheimer’s disease (AD), Parkinson’s disease (PD), multiple sclerosis (MS), traumatic brain injury and epilepsy. To assess the validity of using network interactions in neurological diseases, we investigated the relationship between network thermodynamics and molecular systems biology for these neurological disorders. In order to uncover whether there was a correlation between network organization and biological outcomes, we used publicly available RNA transcription data from individual patients with these neurological conditions, and correlated these molecular profiles with their respective individual disability scores. We found a linear correlation (Pearson correlation of −0.828) between disease disability (a clinically validated measurement of a person’s functional status) and Gibbs free energy (a thermodynamic measure of protein–protein interactions). In other words, we found an inverse relationship between disease disability and thermodynamic energy. Because a larger degree of disability correlated with a larger negative drop in Gibbs free energy in a linear disability-dependent fashion, it could be presumed that the progression of neuropathology such as is seen in Alzheimer’s disease could potentially be prevented by therapeutically correcting the changes in Gibbs free energy.

Energy-Based Modeling of the Feedback Control of Biomolecular Systems With Cyclic Flow Modulation

Energy-based modelling brings engineering insight to the understanding of biomolecular systems. It is shown how well-established control engineering concepts, such as loop-gain, arise from energy feedback loops and are therefore amenable to control engineering insight. In particular, a novel method is introduced to allow the transfer function based approach of classical linear control to be utilised in the analysis of feedback systems modelled by network thermodynamics and thus amalgamate energy-based modelling with control systems analysis. The approach is illustrated using a class of metabolic cycles with activation and inhibition leading to the concept of Cyclic Flow Modulation.

Evaluation of transport properties of biomembranes by means of Peusner network thermodynamics

The R version of the Kedem-Katchalsky-Peusner (KKP) network equations is one of the basic research tools for membrane transport. For binary solutions of non-electrolytes containing a solvent and one solute, these equations include the Peusner resistance coefficients. The aim of the study was to assess the transport properties of biomembranes on the basis of the concentration characteristics of the coefficients: resistance, coupling, energy conversion efficiency and degraded and free energy fluxes. The subject of the study were polymer biomembranes used as a membrane dressing (Bioprocess) and used in hemodialysis (Nephrophan, Ultra-flo) with the coefficients of hydraulic permeability (Lp), reflection () and diffusion permeability () for aqueous glucose solutions. The research method was the R version of the KKP network equations for binary solutions of non-electrolytes. We developed a procedure for evaluation the transport properties of membranes. This procedure requires the calculation of the dependence of the following coefficients: Peusner resistance, Kedem-Caplan-Peusner coupling, Caplan-Peusner energy conversion efficiency, Peusner coupling, and the dissipated energy and free energy fluxes on the mean glucose concentration. Results show that the values of the Peusner resistance coefficients, the Kedem-Caplan-Peusner coupling, the Caplan-Peusner energy conversion efficiency, and the Peusner coupling depend on the mutual relationship between the coefficients Lp, ,  and C. In turn, the value of the dissipated energy and free energy fluxes it is also determined by the values of the volume and diffusion fluxes. The presented procedure for evaluation transport properties of membranes can be helpful in explaining the mechanisms of membrane transport and conducting energy analyzes of membrane processes. Therefore, this procedure can be used for selection of a suitable membrane for practical (eg., industrial or medical) applications.

Network Thermodynamical Modeling of Bioelectrical Systems: A Bond Graph Approach.

Interactions among biomolecules, electrons, and protons are essential to many fundamental processes sustaining life. It is therefore of interest to build mathematical models of these bioelectrical processes not only to enhance understanding but also to enable computer models to complement in vitro and in vivo experiments. Such models can never be entirely accurate; it is nevertheless important that the models are compatible with physical principles. Network Thermodynamics, as implemented with bond graphs, provide one approach to creating physically compatible mathematical models of bioelectrical systems. This is illustrated using simple models of ion channels, redox reactions, proton pumps, and electrogenic membrane transporters thus demonstrating that the approach can be used to build mathematical and computer models of a wide range of bioelectrical systems.

The role of fluctuations in determining cellular network thermodynamics.

The steady state distributions of phenotypic responses within an isogenic population of cells result from both deterministic and stochastic characteristics of biochemical networks. A biochemical network can be characterized by a multidimensional potential landscape based on the distribution of responses and a diffusion matrix of the correlated dynamic fluctuations between N-numbers of intracellular network variables. In this work, we develop a thermodynamic description of biological networks at the level of microscopic interactions between network variables. The Boltzmann H-function defines the rate of free energy dissipation of a network system and provides a framework for determining the heat associated with the nonequilibrium steady state and its network components. The magnitudes of the landscape gradients and the dynamic correlated fluctuations of network variables are experimentally accessible. We describe the use of Fokker-Planck dynamics to calculate housekeeping heat from the experimental data by a method that we refer to as Thermo-FP. The method provides insight into the composition of the network and the relative thermodynamic contributions from network components. We surmise that these thermodynamic quantities allow determination of the relative importance of network components to overall network control. We conjecture that there is an upper limit to the rate of dissipative heat produced by a biological system that is associated with system size or modularity, and we show that the dissipative heat has a lower bound.

Energy-based analysis of biochemical cycles using bond graphs.

Thermodynamic aspects of chemical reactions have a long history in the physical chemistry literature. In particular, biochemical cycles require a source of energy to function. However, although fundamental, the role of chemical potential and Gibb's free energy in the analysis of biochemical systems is often overlooked leading to models which are physically impossible. The bond graph approach was developed for modelling engineering systems, where energy generation, storage and transmission are fundamental. The method focuses on how power flows between components and how energy is stored, transmitted or dissipated within components. Based on the early ideas of network thermodynamics, we have applied this approach to biochemical systems to generate models which automatically obey the laws of thermodynamics. We illustrate the method with examples of biochemical cycles. We have found that thermodynamically compliant models of simple biochemical cycles can easily be developed using this approach. In particular, both stoichiometric information and simulation models can be developed directly from the bond graph. Furthermore, model reduction and approximation while retaining structural and thermodynamic properties is facilitated. Because the bond graph approach is also modular and scaleable, we believe that it provides a secure foundation for building thermodynamically compliant models of large biochemical networks.

Irreversible thermodynamics of open chemical networks. I. Emergent cycles and broken conservation laws.

In this paper and Paper II, we outline a general framework for the thermodynamic description of open chemical reaction networks, with special regard to metabolic networks regulating cellular physiology and biochemical functions. We first introduce closed networks "in a box", whose thermodynamics is subjected to strict physical constraints: the mass-action law, elementarity of processes, and detailed balance. We further digress on the role of solvents and on the seemingly unacknowledged property of network independence of free energy landscapes. We then open the system by assuming that the concentrations of certain substrate species (the chemostats) are fixed, whether because promptly regulated by the environment via contact with reservoirs, or because nearly constant in a time window. As a result, the system is driven out of equilibrium. A rich algebraic and topological structure ensues in the network of internal species: Emergent irreversible cycles are associated with nonvanishing affinities, whose symmetries are dictated by the breakage of conservation laws. These central results are resumed in the relation a + b = s(Y) between the number of fundamental affinities a, that of broken conservation laws b and the number of chemostats s(Y). We decompose the steady state entropy production rate in terms of fundamental fluxes and affinities in the spirit of Schnakenberg's theory of network thermodynamics, paving the way for the forthcoming treatment of the linear regime, of efficiency and tight coupling, of free energy transduction, and of thermodynamic constraints for network reconstruction.

On the network thermodynamics of mass action chemical reaction networks

In this paper we elaborate on the mathematical formulation of mass action chemical reaction networks as recently given in van der Schaft, Rao, Jayawardhana (2012). We show how the reference chemical potentials define a specific thermodynamical equilibrium, and we discuss the port-Hamiltonian representation of the formulation. This leads to the network interpretation as a number of energy-storing elements (corresponding to the chemical complexes), together with a nonlinear energy-dissipating relation corresponding to the reactions with associated conductances for the edges of the graph of complexes. The resulting system cannot be put into standard Brayton-Moser gradient system form, but it does allow a generalized gradient system representation with respect to the Gibbs' free energy.

Determination of flux directions by thermodynamic network analysis: Computing informative metabolite pools

Abstract Network thermodynamics focuses on the energetic analysis of complex metabolic networks. The method connects free Gibbs energies, metabolite concentrations and flux directions by fundamental thermodynamic laws. Here, a new application of network thermodynamics is presented that identifies those metabolite pools that have to be measured in order to determine as many flux directions as possible. For a medium-scaled reaction network such informative metabolite pools are computed with an approach based on Monte Carlo sampling. It turns out that some reactions can be directed with only a few measurements whereas other reactions cannot be directed even with a complete data set. High connectivity in metabolic reaction networks in alliance with concentration ranges make it impossible to intuitively foresee such results. In particular, the impact of measurements of a special type of metabolites being involved in many reactions, so called energetic currency metabolites is investigated.

Kinetic Models of (M-R)-Systems

Kinetic models using enzyme kinetics are developed for the three ways that Louie proved that Rosen’s minimal (M-R)-System can be closed to efficient cause; i.e., how the “replication” component can itself be entailed from within the system. The kinetic models are developed using the techniques of network thermodynamics. As a demonstration, each model is simulated using a SPICE circuit simulator using arbitrarily chosen rate constants. The models are built from SPICE sub-circuits representing the key terms in the chemical rate equations. The models include the addition of an ad hoc semi-permeable membrane so the system can achieve steady state fluxes and also to illustrate the need for all the efficient cause agents to be continually replaced. Comments are made about exactly what is being simulated.

A gentle introduction to the thermodynamics of biochemical stoichiometric networks in steady state

Reaction networks in thermodynamic equilibrium under isothermal and isobaric conditions minimize the Gibbs free energy, but chemical reactions in living organisms operate typically far from equilibrium. Currently, there is no general optimization principle for nonequilibrium systems which can be used in the analysis of biochemical networks. Motivated by the avalailabity of whole genome reconstructions of metabolic reactions, the thermodynamics of biochemical stoichiometric networks has made significant progress in the last decade. These include the consistent formulation of conservation conditions resembling Kirchhoff’s law for electrical networks. In addition, Beard and Qian suggested that the flow force relationship Δμ = RT log(J + /J − ) between the forward and backward fluxes J + and J − and the chemical potential difference of a chemical reaction can be extended from mass action kinetics to more general reactions schemes. In this tutorial review we summarize the recent literature on reaction network thermodynamics and discuss its implications to the analysis of large biochemical systems. In addition, we discuss some recent work on flow-force relationships and global variational principles characterizing nonequilibrium steady states of reaction networks.

Chemometric analysis of triglycerides hydrolysis in an enzyme membrane reactor

The work concerns hydrolysis of oil in a reactor with an ultrafiltration enzyme membrane. A phenomenological model of the reactor performance and its mathematical representation have earlier been elaborated according to a network thermodynamics approach by Schnakenberg.We try to explain when and why the process in the membrane reactor exhibits a typical consecutive kinetics, i.e., mono- and diglycerides are produced together with fatty acids and glycerol or only the latter two products are formed.The detailed discussion on the reactor performance at different levels of selected parameters is based on chemometric simulation procedure. The parameters were: the load density of lipase immobilised in the membrane, the rate constant of uncompetitive inhibition of the lipase catalyst, and the rate of convection streams of the circulating oil and aqueous phases.Planning of the simulation experiments has been done by the full factorial design technique while the cluster analysis has been used to discuss the resulted features of the reactor performances. This was possible after selection of the linkage distances between the particular clusters of simulated experiments, represented by “cumulative” points defined as the functions of all the product time courses in the simulated processes versus the normalised time.