Antagonism and bistability in protein interaction networks

A protein interaction network (PIN) is a set of proteins that modulate one another's activities by regulated synthesis and degradation, by reversible binding to form complexes, and by catalytic reactions (e.g., phosphorylation and dephosphorylation). Most PINs are so complex that their dynamical characteristics cannot be deduced accurately by intuitive reasoning alone. To predict the properties of such networks, many research groups have turned to mathematical models (differential equations based on standard biochemical rate laws, e.g., mass-action, Michaelis-Menten, Hill). When using Michaelis-Menten rate expressions to model PINs, care must be exercised to avoid making inconsistent assumptions about enzyme-substrate complexes. We show that an appealingly simple model of a PIN that functions as a bistable switch is compromised by neglecting enzyme-substrate intermediates. When the neglected intermediates are put back into the model, bistability of the switch is lost. The theory of chemical reaction networks predicts that bistability can be recovered by adding specific reaction channels to the molecular mechanism. We explore two very different routes to recover bistability. In both cases, we show how to convert the original 'phenomenological' model into a consistent set of mass-action rate laws that retains the desired bistability properties. Once an equivalent model is formulated in terms of elementary chemical reactions, it can be simulated accurately either by deterministic differential equations or by Gillespie's stochastic simulation algorithm.     

Designing experiments that aid in the identification of regulatory networks.

Predictive mathematical models of the interactions of a genetic network can provide insight into the mechanisms of gene regulation, the role of various genes within a network and how multiple genes interact leading to complex traits. However, identification of the parameters and interactions is currently a limiting step in the development of such models. This work reviews the state of the art for design of experiments in biological systems and demonstrates the need for improved design of experiments through the use of a model system. Appropriate design of experiments has a profound impact on the ability to identify a model and on the quality of resulting identified model. Key issues include the selection of appropriate input sequences (e.g. random, independent multivariate inputs) and the selection of the sampling frequencies. This work demonstrates that these issues are especially important in the identification of biochemical networks and that the traditional biochemical approach is incapable of truly identifying the behavior present in such networks.     

Combinatorial complexity of pathway analysis in metabolic networks

Elementary flux mode analysis is a promising approach for a pathway-oriented perspective of metabolic networks. However, in larger networks it is hampered by the combinatorial explosion of possible routes. In this work we give some estimations on the combinatorial complexity including theoretical upper bounds for the number of elementary flux modes in a network of a given size. In a case study, we computed the elementary modes in the central metabolism of Escherichia coli while utilizing four different substrates. Interestingly, although the number of modes occurring in this complex network can exceed half a million, it is still far below the upper bound. Hence, to a certain extent, pathway analysis of central catabolism is feasible to assess network properties such as flexibility and functionality.     

From components to regulatory motifs in signalling networks.

The developments in biochemistry and molecular biology over the past 30 years have produced an impressive parts list of cellular components. It has become increasingly clear that we need to understand how components come together to form systems. One area where this approach has been growing is cell signalling research. Here, instead of focusing on individual or small groups of signalling proteins, researchers are now using a more holistic perspective. This approach attempts to view how many components are working together in concert to process information and to orchestrate cellular phenotypic changes. Additionally, the advancements in experimental techniques to measure and visualize many cellular components at once gradually grow in diversity and accuracy. The multivariate data, produced by experiments, introduce new and exciting challenges for computational biologists, who develop models of cellular systems made up of interacting cellular components. The integration of high-throughput experimental results and information from legacy literature is expected to produce computational models that would rapidly enhance our understanding of the detail workings of mammalian cells.     

Robustness analysis and behavior discrimination in enzymatic reaction networks

Characterizing the behavior and robustness of enzymatic networks with numerous variables and unknown parameter values is a major challenge in biology, especially when some enzymes have counter-intuitive properties or switch-like behavior between activation and inhibition. In this paper, we propose new methodological and tool-supported contributions, based on the intuitive formalism of temporal logic, to express in a rigorous manner arbitrarily complex dynamical properties. Our multi-step analysis allows efficient sampling of the parameter space in order to define feasible regions in which the model exhibits imposed or experimentally observed behaviors. In a first step, an algorithmic methodology involving sensitivity analysis is conducted to determine bifurcation thresholds for a limited number of model parameters or initial conditions. In a second step, this boundary detection is supplemented by a global robustness analysis, based on quasi-Monte Carlo approach that takes into account all model parameters. We apply this method to a well-documented enzymatic reaction network describing collagen proteolysis by matrix metalloproteinase MMP2 and membrane type 1 metalloproteinase (MT1-MMP) in the presence of tissue inhibitor of metalloproteinase TIMP2. For this model, our method provides an extended analysis and quantification of network robustness toward paradoxical TIMP2 switching activity between activation or inhibition of MMP2 production. Further implication of our approach is illustrated by demonstrating and analyzing the possible existence of oscillatory behaviors when considering an extended open configuration of the enzymatic network. Notably, we construct bifurcation diagrams that specify key parameters values controlling the co-existence of stable steady and non-steady oscillatory proteolytic dynamics.     

Parallel structures in multidimensional networks

The method of parallel representation of networks is used to determine the restrictions imposed by the space dimension on the variety of multidimensional regular networks. The method makes it possible to establish an analytic relation between the network dimension and the connectivity of vertices and the perimeter of elementary contours. It is proved that an infinite-dimensional network is equivalent to an infinite tree. In addition, the problem of closed regular polytopes inside networks is discussed.     

Petri net modelling of biological networks

Mathematical modelling is increasingly used to get insights into the functioning of complex biological networks. In this context, Petri nets (PNs) have recently emerged as a promising tool among the various methods employed for the modelling and analysis of molecular networks. PNs come with a series of extensions, which allow different abstraction levels, from purely qualitative to more complex quantitative models. Noteworthily, each of these models preserves the underlying graph, which depicts the interactions between the biological components. This article intends to present the basics of the approach and to foster the potential role PNs could play in the development of the computational systems biology.     

Redox reaction system conjugating electrochemical reduction of NADP and enzymatic reaction across the electron transfer membrane

The conjugated redox reaction was driven across the electron transfer membrane prepared from a urethane prepolymer to carry positive charge, where NADP⁺ as electron transfer carrier was adsorbed in the prepared polyurethane membrane. Glutathione reductase [NAD (P)H: oxidized-glutathione oxidoreductase (EC 1.6.4.1)] was used as the catalyst for production of the reduced form of glutathione (GSH) from the oxidized form (GSSG) in the objective reaction, and methyl viologen (MV) was used for the electrochemical regeneration of NADPH in the subreaction. The conjugated redox reaction in the separated reactions system, using the three-compartment cell with two membranes, was successful without MV contamination. Under the given conditions, the conversion ratio of GSH from GSSG reached 50% after 4 h of incubation at 30°C and the amount of GSH accumulated was 0.5 μmol ml⁻¹ of reaction mixture.     

Mutation in autocatalytic reaction networks

A class of kinetic equations describing catalysed and template induced replication, and mutation is introduced. This ODE in its most general form is split into two vector fields, a replication and a mutation field. The mutation field is considered as a perturbation of the replicator equation. The perturbation expansion is a Taylor series in a mutation parameter . First, second and higher order contributions are computed by means of the conventional Rayleigh-Schrödinger approach. Qualitative shifts in the positions of rest points and limit cycles on the boundary of the physically meaningful part of concentration space are predicted from flow topologies. The results of the topological analysis are summarized in two theorems which turned out to be useful in applications: the rest point migration theorem (RPM) and the limit cycle migration theorem (LCM). Quantitative expressions for the shifts of rest points are computed directly from the perturbation expansion. The concept is applied to a collection of selected examples from biophysical chemistry and biology.     

Modes and cuts in metabolic networks: complexity and algorithms

Constraint-based approaches recently brought new insight into our understanding of metabolism. By making very simple assumptions such as that the system is at steady-state and some reactions are irreversible, and without requiring kinetic parameters, general properties of the system can be derived. A central concept in this methodology is the notion of an elementary mode (EM for short) which represents a minimal functional subsystem. The computation of EMs still forms a limiting step in metabolic studies and several algorithms have been proposed to address this problem leading to increasingly faster methods. However, although a theoretical upper bound on the number of elementary modes that a network may possess has been established, surprisingly, the complexity of this problem has never been systematically studied. In this paper, we give a systematic overview of the complexity of optimisation problems related to modes. We first establish results regarding network consistency. Most consistency problems are easy, i.e., they can be solved in polynomial time. We then establish the complexity of finding and counting elementary modes. We show in particular that finding one elementary mode is easy but that this task becomes hard when a specific EM (i.e. an EM containing some specified reactions) is sought. We then show that counting the number of elementary modes is musical sharpP-complete. We emphasize that the easy problems can be solved using currently existing software packages. We then analyse the complexity of a closely related task which is the computation of so-called minimum reaction cut sets and we show that this problem is hard. We then present two positive results which both allow to avoid computing EMs as a prior to the computation of reaction cuts. The first one is a polynomial approximation algorithm for finding a minimum reaction cut set. The second one is a test for verifying whether a set of reactions constitutes a reaction cut; this test can be readily included in existing algorithms to improve their performance. Finally, we discuss the complexity of other cut-related problems.     

Solving the SAT problem using a DNA computing algorithm based on ligase chain reaction

A new DNA computing algorithm based on a ligase chain reaction is demonstrated to solve an SAT problem. The proposed DNA algorithm can solve an n-variable m-clause SAT problem in m steps and the computation time required is O (3m+n). Instead of generating the full-solution DNA library, we start with an empty test tube and then generate solutions that partially satisfy the SAT formula. These partial solutions are then extended step by step by the ligation of new variables using Taq DNA ligase. Correct strands are amplified and false strands are pruned by a ligase chain reaction (LCR) as soon as they fail to satisfy the conditions. If we score and sort the clauses, we can use this algorithm to markedly reduce the number of DNA strands required throughout the computing process. In a computer simulation, the maximum number of DNA strands required was 2(0.48n) when n=50, and the exponent ratio varied inversely with the number of variables n and the clause/variable ratio m/n. This algorithm is highly space-efficient and error-tolerant compared to conventional brute-force searching, and thus can be scaled-up to solve large and hard SAT problems.     

Visual exploration of isotope labeling networks in 3D

Isotope labeling networks (ILNs) are graphs explaining the flow of isotope labeled molecules in a metabolic network. Moreover, they are the structural backbone of metabolic flux analysis (MFA) by isotopic tracers which has been established as a standard experimental tool in fluxomics. To configure an isotope labeling experiment (ILE) for MFA, the structure of the corresponding ILN must be understood to a certain extent even by a practitioner. Graph algorithms help to analyze the network structure but produce rather abstract results. Here, the major obstruction is the high dimension of these networks and the large number of network components which, consequently, are hard to figure out manually. At the interface between theory and experiment, the three-dimensional interactive visualization tool CumoVis has been developed for exploring the network structure in a step by step manner. Navigation and orientation within ILNs are supported by exploiting the natural 3D structure of an underlying metabolite network with stacked labeled particles on top of each metabolite node. Network exploration is facilitated by rotating, zooming, forward and backward path tracing and, most important, network component reduction. All features of CumoVis are explained with an educational example and a realistic network describing carbon flow in the citric acid cycle.     

A study on enzymatic reaction using a liquid emulsion membrane technique

An enzymatic reaction using a liquid emulsion membrane technique was studied to investigate the effects of some experimental variables on the stability of liquid membrane, enzyme deactivation, and transport of substrates and products. The hydrolysis of L-phenylalanine methyl ester by alpha-chymotrypsin was selected as a model reaction system. First, a transport mechanism for the substrates and products across the membrane was qualitatively identified. Second, it was found that the pH of the internal phase was one of the most important variables to determine the enzyme activity in a liquid membrane. Third, the effect of membrane phase which consists of surfactant, carrier, and organic solvent on the emulsion stability was investigated. It was found that the properties of the organic solvents greatly affect the emulsion stability. For an optimum condition, it was possible to reuse the emulsion which consists of membrane phase and internal phase without further separation. It was finally concluded that the enzyme in a liquid membrane retained 60% of its native activity in spite of vigorous mixing during the emulsification step.     

Role of conformational fluctuations in the enzymatic reaction of HIV-1 protease

The emergence of compensatory drug-resistant mutations in HIV-1 protease challenges the common view of the reaction mechanism of this enzyme. Here, we address this issue by performing classical and ab initio molecular dynamics simulations (MD) on a complex between the enzyme and a peptide substrate. The classical MD calculation reveals large-scale protein motions involving the flaps and the cantilever. These motions modulate the conformational properties of the substrate at the cleavage site. The ab initio calculations show in turn that substrate motion modulates the activation free energy barrier of the enzymatic reaction dramatically. Thus, the catalytic power of the enzyme does not arise from the presence of a pre-organized active site but from the protein mechanical fluctuations. The implications of this finding for the emergence of drug-resistance are discussed.     

Analysis of solvent-free ethyl oleate enzymatic synthesis at equilibrium conditions

This work reports experimental equilibrium data for the esterification of pure oleic acid and a fatty acid mixture with ethanol, using an immobilized Candida antarctica B lipase as catalyst. Reactions are performed in a solvent-free system, containing a mixture of substrates and different amounts of distilled water. According to the initial amount of water and the extent of the reaction, one or two liquid phases are present. Therefore, when the equilibrium is achieved, the liquid–liquid and chemical reaction equilibria have to be simultaneously satisfied.

Several reports dealing with enzymatic reactions performed in two-phase systems have found that the value of the reaction equilibrium constant calculated from overall experimental concentrations varies not only with temperature but also with substrate ratio and water content. Although this approach is a valuable way to explore equilibrium shifts in biphasic systems, it is limited to ideal systems with constant partition coefficients. The aim of this work is to consider the biphasic nature of the reactive mixture through a computational procedure that simultaneously takes into account liquid–liquid and reaction equilibria. This approach enables the determination of a classical temperature-dependent thermodynamic equilibrium constant, which accurately fits experimental equilibrium conversions over a wide range of operating conditions.     

A Petri net approach to the study of persistence in chemical reaction networks

Persistence is the property, for differential equations in R(n), that solutions starting in the positive orthant do not approach the boundary of the orthant. For chemical reactions and population models, this translates into the non-extinction property: provided that every species is present at the start of the reaction, no species will tend to be eliminated in the course of the reaction. This paper provides checkable conditions for persistence of chemical species in reaction networks, using concepts and tools from Petri net theory, and verifies these conditions on various systems which arise in the modeling of cell signaling pathways.     

Sensitivity and control analysis of periodically forced reaction networks using the Green's function method

A general sensitivity and control analysis of periodically forced reaction networks with respect to small perturbations in arbitrary network parameters is presented. A well-known property of sensitivity coefficients for periodic processes in dynamical systems is that the coefficients generally become unbounded as time tends to infinity. To circumvent this conceptual obstacle, a relative time or phase variable is introduced so that the periodic sensitivity coefficients can be calculated. By employing the Green's function method, the sensitivity coefficients can be defined using integral control operators that relate small perturbations in the network's parameters and forcing frequency to variations in the metabolite concentrations and reaction fluxes. The properties of such operators do not depend on a particular parameter perturbation and are described by the summation and connectivity relationships within a control-matrix operator equation. The aim of this paper is to derive such a general control-matrix operator equation for periodically forced reaction networks, including metabolic pathways. To illustrate the general method, the two limiting cases of high and low forcing frequency are considered. We also discuss a practically important case where enzyme activities and forcing frequency are modulated simultaneously. We demonstrate the developed framework by calculating the sensitivity and control coefficients for a simple two reaction pathway where enzyme activities enter reaction rates linearly and specifically.     

Integrated analysis of multiple data sources reveals modular structure of biological networks

It has been a challenging task to integrate high-throughput data into investigations of the systematic and dynamic organization of biological networks. Here, we presented a simple hierarchical clustering algorithm that goes a long way to achieve this aim. Our method effectively reveals the modular structure of the yeast protein-protein interaction network and distinguishes protein complexes from functional modules by integrating high-throughput protein-protein interaction data with the added subcellular localization and expression profile data. Furthermore, we take advantage of the detected modules to provide a reliably functional context for the uncharacterized components within modules. On the other hand, the integration of various protein-protein association information makes our method robust to false-positives, especially for derived protein complexes. More importantly, this simple method can be extended naturally to other types of data fusion and provides a framework for the study of more comprehensive properties of the biological network and other forms of complex networks.     

Reaction heat measurements in the enzymatic hydrolysis reaction of penicillin GK to obtain 6-APA

The purpose of this study was to measure heats involved in the hydrolysis process for the industrial production of 6-aminopenicillanic acid (6-APA) using dynamic calorimetry techniques. The experimental design was planned using Hess' law. The information derived from the calorimeter was correlated mathematically to determine the heat released during enzymatic hydrolysis. This is important for temperature control systems and reactor design. The results obtained with the calorimetric measurements at 308 K and pH 7.5 are the penicillin hydrolysis, ΔH_hydrol, at 35.9 ± 5.7 kJ mol⁻¹ and phenyl acetic acid neutralization, ΔH_neut, at −47.1 ± 3.8 kJ mol⁻¹.