Models Of Biochemical Reaction Networks Research Articles

NO/peroxynitrite dynamics of high glucose-exposed HUVECs: Chemiluminescent measurement and computational model

Pathogenesis of many of diabetes-related vascular complications is associated with endothelial cell (EC) dysfunction, which is reduced bioavailability of EC-released nitric oxide (NO). Interaction dynamics of NO, superoxide (O 2 −) and peroxynitrite (ONOO −) are dependent on both their productions and consumptions through various pathways. Quantitative knowledge of these interaction dynamics in high glucose-induced EC dysfunction remains poorly understood. We developed an integrated experimental and computational approach to gain a quantitative understanding of the interactions of NO, O 2 − and ONOO − in high glucose-exposed ECs. End-products, nitrite and nitrate, were measured using a chemiluminescence analyzer. A computational biochemical reaction network model was developed to predict the effect of high glucose on ECs NO, O 2 − and ONOO −. ECs NO and O 2 − productions increased in high glucose as evidenced by increased total NOx concentration, primarily increasing nitrate concentration. The model predicted an increase in O 2 − and ONOO − concentrations and a decrease in NO concentration in high glucose conditions. Administration of superoxide dismutase (SOD) decreased O 2 − concentration and increased NO concentration, thus SOD improved high glucose-induced changes in these interactions. An important finding of this study was that the NO bioavailability decreased in high glucose conditions even though NO production of EC increased. The integrated approach provides a framework to predict NO, O 2 − and ONOO − concentrations and productions that are difficult to measure in one experiment and will be useful in further EC dysfunction studies.

An Observer for Mass-action Chemical Reaction Networks

In biological research, experimental data analysis plays an important role since it enables quantitative understanding of biochemical processes. On the other hand, today's measurement techniques, in continuous development, generally allow measuring a subset of the major system's variables. Such major issue can be tackled by relying on a system's and mathematical approach. For instance, first principles modelling of metabolic or signal transduction networks typically leads to a set of nonlinear differential equations. In this paper, we devise a nonlinear observer specifically suited for models of biochemical reaction networks. We show that the observer is locally convergent under certain observability conditions which can be inferred by elementary network analysis. The applicability and performance of the outlined observer are shown considering the state estimation problem for a benchmark biochemical reaction network.

Thermodynamic Constraints in Kinetic Modeling: Thermodynamic‐Kinetic Modeling in Comparison to Other Approaches

AbstractKinetic models of reaction networks may easily violate the laws of thermodynamics and the principle of detailed balance. In large network models, the constraints that are imposed by these laws are particularly difficult to address. This hinders modeling of biochemical reaction networks. Thermodynamic‐kinetic modeling is a method that provides a thermodynamically sound and formally appealing way for deriving dynamic model equations of reaction systems. State variables of this approach are thermokinetic potentials that describe the ability of compounds to drive a reaction. A compound has a parameter called capacity, which is the ratio of its concentration and thermokinetic potential. A reaction is described by its resistance which is the ratio of the thermokinetic driving force and flux. In these aspects, the formalism is similar to the modeling formalism for electrical networks and an analogous graphical representation is possible. The thermodynamic‐kinetic modeling formalism is equivalent to the traditional kinetic modeling formalism with the exception that it is not possible to build thermodynamically infeasible models. Here, the thermodynamic‐kinetic modeling formalism is reviewed, compared to other approaches, and some of its advantages are worked out. In contrast to other approaches, thermodynamic‐kinetic modeling does not rely on an explicit enumeration of stoichiometric cycles. It is capable of describing rate laws far from equilibrium. Further, the parameterization by capacities and resistances is particularly intuitive and powerful.

Homotopy methods for counting reaction network equilibria

Dynamical system models of complex biochemical reaction networks are usually high-dimensional, non-linear, and contain many unknown parameters. In some cases the reaction network structure dictates that positive equilibria must be unique for all values of the parameters in the model. In other cases multiple equilibria exist if and only if special relationships between these parameters are satisfied. We describe methods based on homotopy invariance of degree which allow us to determine the number of equilibria for complex biochemical reaction networks and how this number depends on parameters in the model.

Computing chemical organizations in biological networks

Novel techniques are required to analyze computational models of intracellular processes as they increase steadily in size and complexity. The theory of chemical organizations has recently been introduced as such a technique that links the topology of biochemical reaction network models to their dynamical repertoire. The network is decomposed into algebraically closed and self-maintaining subnetworks called organizations. They form a hierarchy representing all feasible system states including all steady states. We present three algorithms to compute the hierarchy of organizations for network models provided in SBML format. Two of them compute the complete organization hierarchy, while the third one uses heuristics to obtain a subset of all organizations for large models. While the constructive approach computes the hierarchy starting from the smallest organization in a bottom-up fashion, the flux-based approach employs self-maintaining flux distributions to determine organizations. A runtime comparison on 16 different network models of natural systems showed that none of the two exhaustive algorithms is superior in all cases. Studying a 'genome-scale' network model with 762 species and 1193 reactions, we demonstrate how the organization hierarchy helps to uncover the model structure and allows to evaluate the model's quality, for example by detecting components and subsystems of the model whose maintenance is not explained by the model. All data and a Java implementation that plugs into the Systems Biology Workbench is available from http://www.minet.uni-jena.de/csb/prj/ot/tools.

Multistationarity in the activation of a MAPK: Parametrizing the relevant region in parameter space

Mathematical models of biochemical reaction networks in the form of ordinary differential equations can exhibit all sorts of complex dynamical behaviour. It is for example known, that even a single layer of a MAPK cascade can exhibit bistability (i.e. there exist multiple (positive) steady state solutions). It is almost a common-place that bistability or some other form of multistationarity are observed in many biochemical reaction networks, especially if the focus is on signal transduction or cell cycle regulation. However, multistationarity is only exhibited if the parameter vector is located in an appropriate region of parameter space. To find these regions, for example by using numerical tools like bifurcation analysis, is a non-trivial task as it amounts to searching the whole parameter space. In this paper we show that for a model of a single layer of a MAPK cascade it is possible to derive analytical descriptions of these regions, if mass action kinetics are used. Moreover, our results give rise to a straightforward explanation for the ‘robust yet fragile’ behaviour in the activation of a MAPK.

SBMLmerge, a system for combining biochemical network models.

The Systems Biology Markup Language (SBML) is an XML-based format for representing mathematical models of biochemical reaction networks, and it is likely to become a main standard in the systems biology community. As published mathematical models in cell biology are growing in number and size, modular modelling approaches will gain additional importance. The main issue to be addressed in computer-assisted model combination is the specification and handling of model semantics. The software SBMLmerge assists the user in combining models of biological subsystems to larger biochemical networks. First, the program helps the user in annotating all model elements with unique identifiers pointing to databases such as KEGG or Gene Ontology. Second, during merging, SBMLmerge detects and resolves various syntactic and semantic problems. Typical problems are conflicting variable names, elements which appear in more than one input model, and mathematical problems arising from the combination of equations. If the input models make contradicting statements about a biochemical quantity, the user is asked to choose between them. In the end the merging process results in a new, valid SBML model.

Use of symbolic and numeric computation techniques in analysis of biochemical reaction networks

AbstractMathematical models of biochemical reaction networks involve nonlinear differential equations with a number of parameters whose values are generally not known a priori. Analysis of such networks includes both analytical and numerical work. The analytical part of such analysis often requires manipulation with massive algebraic expressions that may become challenging due to the inherent nonlinearities and unknown parameters in the model. Computer algebra systems (CASs), which are also capable of performing calculations with symbols and to generate ready‐to‐use computer codes for numerical computations, are excellent tools in such analysis. The present work gives a brief description of how we can benefit from computer algebra to overcome such difficulties and combine symbolic and numeric computation techniques in such analysis with several illustrative biochemical reaction networks. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

K-Cone Analysis: Determining All Candidate Values for Kinetic Parameters on a Network Scale

The absence of comprehensive measured kinetic values and the observed inconsistency in the available in vitro kinetic data has hindered the formulation of network-scale kinetic models of biochemical reaction networks. To meet this challenge we present an approach to construct a convex space, termed the k-cone, which contains all the allowable numerical values of the kinetic constants in large-scale biochemical networks. The definition of the k-cone relies on the incorporation of in vivo concentration data and a simplified approach to represent enzyme kinetics within an established constraint-based modeling approach. The k-cone approach was implemented to define the allowable combination of numerical values for a full kinetic model of human red blood cell metabolism and to study its correlated kinetic parameters. The k-cone approach can be used to determine consistency between in vitro measured kinetic values and in vivo concentration and flux measurements when used in a network-scale kinetic model. k-Cone analysis was successful in determining whether in vitro measured kinetic values used in the reconstruction of a kinetic-based model of Saccharomyces cerevisiae central metabolism could reproduce in vivo measurements. Further, the k-cone can be used to determine which numerical values of in vitro measured parameters are required to be changed in a kinetic model if in vivo measured values are not reproduced. k-Cone analysis could identify what minimum number of in vitro determined kinetic parameters needed to be adjusted in the S. cerevisiae model to be consistent with the in vivo data. Applying the k-cone analysis a priori to kinetic model development may reduce the time and effort involved in model building and parameter adjustment. With the recent developments in high-throughput profiling of metabolite concentrations at a whole-cell scale and advances in metabolomics technologies, the k-cone approach presented here may hold the promise for kinetic characterization of metabolic networks as well as other biological functions at a whole-cell level.

Modeling and simulation of intracellular dynamics: choosing an appropriate framework.

Systems biology is a reemerging paradigm which, among other things, focuses on mathematical modeling and simulation of biochemical reaction networks in intracellular processes. For most simulation tools and publications, they are usually characterized by either preferring stochastic simulation or rate equation models. The use of stochastic simulation is occasionally accompanied with arguments against rate equations. Motivated by these arguments, we discuss in this paper the relationship between these two forms of representation. Toward this end, we provide a novel compact derivation for the stochastic rate constant that forms the basis of the popular Gillespie algorithm. Comparing the mathematical basis of the two popular conceptual frameworks of generalized mass action models and the chemical master equation, we argue that some of the arguments that have been put forward are ignoring subtle differences and similarities that are important for answering the question in which conceptual framework one should investigate intracellular dynamics.

The JigCell model builder and run manager.

We describe the JigCell Model Builder (JCMB), a tool for creating biochemical reaction network models. JCMB is designed for ease of use and its interface uses the standard spreadsheet metaphor. The JigCell Run Manager (JCRM) is a tool for organizing the large collections of simulation runs typically required by reaction network modeling activities. JCMB and JCRM are part of the JigCell suite available at http://jigcell.biol.vt.edu.

In silico model-driven assessment of the effects of single nucleotide polymorphisms (SNPs) on human red blood cell metabolism.

The completion of the human genome project and the construction of single nucleotide polymorphism (SNP) maps have lead to significant efforts to find SNPs that can be linked to pathophysiology. In silico models of complete biochemical reaction networks relate a cell's individual reactions to the function of the entire network. Sequence variations can in turn be related to kinetic properties of individual enzymes, thus allowing an in silico model-driven assessment of the effects of defined SNPs on overall cellular functions. This process is applied to defined SNPs in two key enzymes of human red blood cell metabolism: glucose-6-phosphate dehydrogenase and pyruvate kinase. The results demonstrate the utility of in silico models in providing insight into differences between red cell function in patients with chronic and nonchronic anemia. In silico models of complex cellular processes are thus likely to aid in defining and understanding key SNPs in human pathophysiology.

An Algebraic-combinatorial Model for the Identification and Mapping of Biochemical Pathways

We develop the mathematical machinery for the construction of an algebraic-combinatorial model using Petri nets to construct an oriented matroid representation of biochemical pathways. For demonstration purposes, we use a model metabolic pathway example from the literature to derive a general biochemical reaction network model. The biomolecular networks define a connectivity matrix that identifies a linear representation of a Petri net. The sub-circuits that span a reaction network are subject to flux conservation laws. The conservation laws correspond to algebraic-combinatorial dual invariants, that are called S- (state) and T- (transition) invariants. Each invariant has an associated minimum support. We show that every minimum support of a Petri net invariant defines a unique signed sub-circuit representation. We prove that the family of signed sub-circuits has an implicit order that defines an oriented matroid. The oriented matroid is then used to identify the feasible sub-circuit pathways that span the biochemical network as the positive cycles in a hyper-digraph.

A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation

A generalization of Wegschcider's condition concerning equilibrium constants in chemically reacting systems is formulated, which is then proved to be a necessary and sufficient condition for detailed balancing. In order to include a large multitude of rate laws, a generalized mass action kinetics is considered which comprises usual mass action kinetics and all reversible enzyme kinetics and which is consistent with basic postulates of irreversible thermodynamics for ideal mixtures. Reaction systems of arbitrary stoichiometry are considered. They may contain reactants with fixed concentrations, as is characteristic for models of biochemical reaction networks. Existence, uniqueness, and global asymptotic stability of equilibrium states for reaction systems endowed with generalized man action kinetics are proved. Using these results, he generalized Wegscheider condition is shown to be a sufficient criterion for the applicability of the quasi-steady-state approximation.

Quasi-steady-state approximation in the mathematical modeling of biochemical reaction networks

In the mathematical modeling of biochemical reaction networks the application of the quasi-steady-state approximation permits a reduction of the number of dynamic variables as well as of the number of parameters. It is shown that the quasi-steady-state approximation represents the zeroth approximate solution of the perturbation problem dX dt = RV(X)+ 1 μ S W ̃ (X) with μ ⪡ 1. The perturbation equation develops by subdivision of the flux rates of the model into the rates w i(X) = (1/μ) w ̃ i(X) of fast reactions and the rates v j ( X) of slow reactions. The matrix C=( R⋮ S) denotes the stochiometric matrix of the reaction network. The analysis of this perturbation problem provides conditions for the applicability of the quasi-steady-state approximation in a given network. The paper presents a practical guide for the construction of the approximate solution.

On the statistical foundation of the thermodynamic evolution-criterion of Glansdorff and Prigogine: Friedrich Schogl. Institut fur Theoretische Physik, Lehrestuhl B, Reinisch-Westalische Technische Hoshschule, 51 Aachen, Temlergraben 55, Germany

Local stability analysis of steady states in mathematical models of biochemical reaction networks is an important tool for systems biology. The second variation of the Gibbs energy around a steady state is a positive definite function and a candidate for a Lyapunov function. A sufficient condition for the local stability is the local negative definiteness of the time derivative of this function. This is expressed by the Glansdorff–Prigogine stability criterion. Previously, the criterion was criticized to be overly conservative and difficult to check. Here, we derive an easily testable form of the criterion for models of biochemical networks. The criterion can be evaluated with incomplete knowledge of the parameters. For ideal mass-action kinetics, it depends only on the steady state fluxes. For reaction systems in ideal solutions, the Glansdorff–Prigogine criterion is overly conservative and we give a tighter criterion that depends on the same subset of the parameters as the Glansdorff–Prigogine criterion. Whenever these criteria are indefinite, there exist parameter values such that the steady state is unstable. By means of simple example systems we explore these aspects and discuss the possible uses of the criteria for systems biology.