APOPTO-CELL--a simulation tool and interactive database for analyzing cellular susceptibility to apoptosis

We have developed a web service that provides a comprehensive analysis of the susceptibility of cells to undergo apoptosis in response to an activation of the mitochondrial apoptotic pathway. Based on ordinary differential equations, (pre-determined) protein concentrations and release kinetics of mitochondrial pro-apoptotic factors, a network of 52 reactions and 19 reaction partners can be employed as a tool to display temporal protein profiles, to identify key regulatory proteins and to determine critical threshold concentrations required for the execution of apoptosis in HeLa cancer cells or other cell types. The web service also provides an interactive database function for the deposition of cell-type-specific quantitative data. In addition, the web service provides an output that can be compared directly to experimental results obtained from real-time single-cell experiments, making this a widely applicable systems biology tool for apoptosis and cancer researchers. AVAILABILITY: http://systemsbiology.rcsi.ie/apopto-cell.html. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.     

An Effective Automatic Procedure for Testing Parameter Identifiability of HIV/AIDS Models

Realistic HIV models tend to be rather complex and many recent models proposed in the literature could not yet be analyzed by traditional identifiability testing techniques. In this paper, we check a priori global identifiability of some of these nonlinear HIV models taken from the recent literature, by using a differential algebra algorithm based on previous work of the author. The algorithm is implemented in a software tool, called DAISY (Differential Algebra for Identifiability of SYstems), which has been recently released (DAISY is freely available on the web site ). The software can be used to automatically check global identifiability of (linear and) nonlinear models described by polynomial or rational differential equations, thus providing a general and reliable tool to test global identifiability of several HIV models proposed in the literature. It can be used by researchers with a minimum of mathematical background.     

Graphical rules for non-steady state enzyme kinetic

The exiting graphical methods in enzyme kinetics can be used only within the scope of steady state reactions. In this paper, two graphical rules are presented to deal with the non-steady state enzyme catalysed reaction systems. According to Rule 1 we can immediately write out the phase concentration of enzyme species. The calculation work such as setting up differential equations, making Laplace transformation, expanding determinants, which are both tedious and liable to error, are completely saved. By means of Rule 2 the secular equations for the consecutive first-order reactions can be written out directly without need of setting up differential equations, expanding determinants, etc., that would otherwise be laborious and prone to errors. In addition, two check formulae are also presented for these two graphical methods, respectively. They are useful in order for avoiding the omission of terms during calculations, especially, for complicated mechanisms.     

Enzyme kinetics for a two-step enzymic reaction with comparable initial enzyme-substrate ratios

We extend the validity of the quasi-steady state assumption for a model double intermediate enzyme-substrate reaction to include the case where the ratio of initial enzyme to substrate concentration is not necessarily small. Simple analytical solutions are obtained when the reaction rates and the initial substrate concentration satisfy a certain condition. These analytical solutions compare favourably with numerical solutions of the full system of differential equations describing the reaction. Experimental methods are suggested which might permit the application of the quasi-steady state assumption to reactions where it may not have been obviously applicable before.     

A mathematical model for the branched chain amino acid biosynthetic pathways of Escherichia coli K12

As a first step toward the elucidation of the systems biology of the model organism Escherichia coli, it was our goal to mathematically model a metabolic system of intermediate complexity, namely the well studied end product-regulated pathways for the biosynthesis of the branched chain amino acids L-isoleucine, L-valine, and L-leucine. This has been accomplished with the use of kMech (Yang, C.-R., Shapiro, B. E., Mjolsness, E. D., and Hatfield, G. W. (2005) Bioinformatics 21, in press), a Cellerator (Shapiro, B. E., Levchenko, A., Meyerowitz, E. M., Wold, B. J., and Mjolsness, E. D. (2003) Bioinformatics 19, 677-678) language extension that describes a suite of enzyme reaction mechanisms. Each enzyme mechanism is parsed by kMech into a set of fundamental association-dissociation reactions that are translated by Cellerator into ordinary differential equations. These ordinary differential equations are numerically solved by Mathematica. Any metabolic pathway can be simulated by stringing together appropriate kMech models and providing the physical and kinetic parameters for each enzyme in the pathway. Writing differential equations is not required. The mathematical model of branched chain amino acid biosynthesis in E. coli K12 presented here incorporates all of the forward and reverse enzyme reactions and regulatory circuits of the branched chain amino acid biosynthetic pathways, including single and multiple substrate (Ping Pong and Bi Bi) enzyme kinetic reactions, feedback inhibition (allosteric, competitive, and non-competitive) mechanisms, the channeling of metabolic flow through isozymes, the channeling of metabolic flow via transamination reactions, and active transport mechanisms. This model simulates the results of experimental measurements.     

Modeling and characterization of glucose-sensitive hydrogel: Effect of Young's modulus

A multiphysics model concerning the diffusion and enzyme reaction simultaneously is developed in this paper to characterize the equilibrium behavior of the glucose-sensitive hydrogel, which is called the multi-effect-coupling glucose-stimulus (MECglu) model. The responsive behavior of the hydrogel in the chemo-electro-mechanical coupled energy domains is modeled by the nonlinear coupled partial differential equations. They include the Nernst–Planck equations for the diffusion of mobile species and the enzyme reaction catalyzed by the glucose oxidase and the catalase, the Poisson equation for electric potential, and the mechanical equilibrium equation for finite deformation of the glucose-oxidase-loaded pH-sensitive hydrogel. Numerical simulations demonstrate that the MECglu model can consist well with the published experiment for the practical physiological glucose concentration ranging from 0 to 16.5 mM (300 mg/ml). The effect of Young's modulus of the hydrogel is investigated on the distributive concentrations of reacting and diffusive species and the deformation of the glucose-sensitive hydrogels.     

The kinetics of invertase action

A method of obtaining better fits to experimental curves for the hydrolysis of sucrose in the presence of invertase is found by assuming that the reaction is first-order, but that there is an initial time delay before this form is assumed. More rigorously, a system of two differential equations describing the reaction is established, assuming that the formation of enzyme-substrate complex is irreversible, and a solution is obtained relating, at any time, the concentration of the products, the total concentration of complex which has been formed in that time, and the time. This solution is compared with other equations for the reaction and is used to calculate reaction constants. Curves showing the concentration of complex existing at timet are given and are employed in calculating an “equivalent” enzyme concentration. This expression is discussed. Supported in part by a contract from the Atomic Energy Commission.     

Kinetic model of laccase-catalyzed oxidation of aqueous phenol

Laccase from Trametes versicolor (EC 1.10.3.2) catalyzes the oxidation of aqueous phenol by oxygen and has demonstrated good potential for applications in various industrial and environmental processes. A kinetic model of this system has been developed to facilitate a better understanding of the mechanisms and rate-limiting steps of enzyme-catalyzed transformation and to eventually assist in the choice and design of suitable reactor systems. A kinetic model was derived based on the differential and mass balance equations that describe the interactions of various forms of the enzyme with the aromatic substrate and oxygen. This model also incorporated an expression accounting for enzyme inactivation over time due to the reaction environment. The model was validated by comparing model predictions with experimental observations of phenol transformation and oxygen consumption over time at a variety of enzyme concentrations. Excellent agreement was found between experimental data and predictions of the kinetic model. Sensitivity analyses demonstrated that the reaction between oxidized-laccase and phenol was the rate-limiting step.     

Entwicklung von Berechnugsmodellen für Enzymreaktoren (Teil 2)

Based on the partial differential equations for the transport of enthalpy and material a calculation model for the examination of the change of temperature and concentration in axial and radial direction was developed for the continuously working enzyme reactors. The pecularity of the kinetics of processes which are catalyzed by enzymes was considered due to the reaction velocity in modelling. In order to induce the numeric solution, an explicit method of differences was selected, the applicability of which was proved under consideration of appropriate initial and marginal conditions.     

Pre-steady state kinetics of the ferroxidase reaction catalyzed by ceruloplasmin

Analysis of the presteady-state step of the ferroxidase reaction catalyzed by ceruloplasmin revealed that the kinetic curve for the reaction product accumulation versus time is characterized by an induction period, tau, whose value does not change within the pH range of 5.0-7.5. The value of tau decreases with a rise in the enzyme concentration and increases with a rise in the substrate concentration at constant Fe(II) and E0 concentrations. A reaction scheme is proposed which points to the existence of an induction period in the ceruloplasmin-catalyzed ferroxidase reaction. It was shown that the calculated values of tau = f([Fe(II)]0) and tau = f([E]0) obtained through the use of differential equations qualitatively coincide with the corresponding experimental values.     

Mathematical description of modifying synapses of hippocampal pyramidal neurons

The system of differential equations describing the plasticity of the hippocampal pyramidal neuron CA3, developed before, was analyzed. The system was divided into two groups according to magnitudes of the biochemical reaction constants. The first group with large values of the constants was transformed into quasi stationary algebraic equations. This allowed one to transform the system of 32 differential equations to a system containing only 4 differential equations, which can be used for modeling of learning processes in various parts of the brain.     

Enzyme Kinetics for Complex System Enables Accurate Determination of Specificity Constants of Numerous Substrates in a Mixture by Proteomics Platform

Many important experiments in proteomics including protein digestion, enzyme substrate screening, enzymatic labeling, etc., involve the enzymatic reactions in a complex system where numerous substrates coexists with an enzyme. However, the enzyme kinetics in such a system remains unexplored and poorly understood. Herein, we derived and validated the kinetics equations for the enzymatic reactions in complex system. We developed an iteration approach to depict the enzymatic reactions in complex system. It was validated by 630 time-course points from 24 enzymatic reaction experiments and was demonstrated to be a powerful tool to simulate the reactions in the complex system. By applying this approach, we found that the ratio of substrate depletion is independent of other coexisted substrates under specific condition. This observation was then validated by experiments. Based on this striking observation, a simplified model was developed to determine the catalytic efficiencies of numerous competing substrates presented in the complex enzyme reaction system. When coupled with high-throughput quantitative proteomics technique, this simplified model enabled the accurate determination of catalytic efficiencies for 2369 peptide substrates of a protease by using only one enzymatic reaction experiment. Thus, this study provided, in the first time, a validated model for the large scale determination of specificity constants which could enable the enzyme substrate screening approach turned from a qualitative method of identifying substrates to a quantitative method of identifying and prioritizing substrates. Data are available via ProteomeXchange with identifier PXD004665.     

Simplifying principles for chemical and enzyme reaction kinetics

Tihonov's Theorems for systems of first-order ordinary differential equations containing small parameters in the derivatives, which form the mathematical foundation of the steady-state approximation, are restated. A general procedure for simplifying chemical and enzyme reaction kinetics, based on the difference of characteristic time scales, is presented. Korzuhin's Theorem. which makes it possible to approximate any kinetic system by a closed chemical system, is also reported. The notions and theorems are illustrated with examples of Michaelis-Menten enzyme kinetics and of a simple autocatalytic system. Another example illustrates how the differences in the rate constants of different elementary reactions may be exploited to simplify reaction kinetics by using Tihonov's Theorem. All necessary mathematical notions are explained in the appendices. The most simple formulation of Tihonov's 1st Theorem ‘for beginners’ is also given.     

An enzyme mechanism language for the mathematical modeling of metabolic pathways

MOTIVATION: As a first step toward the elucidation of the systems biology of complex biological systems, it was our goal to mathematically model common enzyme catalytic and regulatory mechanisms that repeatedly appear in biological processes such as signal transduction and metabolic pathways. RESULTS: We describe kMech, a Cellerator language extension that describes a suite of enzyme mechanisms. Each enzyme mechanism is parsed by kMech into a set of fundamental association-dissociation reactions that are translated by Cellerator into ordinary differential equations that are numerically solved by Mathematica. In addition, we present methods that use commonly available kinetic measurements to estimate rate constants required to solve these differential equations.     

Kinetics of the hydrolysis of lean meat protein by alcalase: Derivation of two alternative rate equations and their fit to experimental data

Two rate equations have been developed to model the hydrolysis of ground lean meat protein by Alcalase. The first equation was based on classical Michaelis-Menten kinetics and the second on the adsorption of enzyme to the protein prior to reaction. It was assumed that this adsorption could be modelled by a Langmuir-type adsorption isotherm. Each equation considered the enzyme to be competitively inhibited by reaction product, and considered enzyme inactivation to be first order. Both rate equations have been fitted to experimental data obtained from the hydrolysis of meat protein by Alcalase. Initial rate data indicated that the adsorption model was more appropriate. However, both equations gave satisfactory fits to 11 reaction progress curves determined over a wide range of enzyme and substrate concentrations.     

Transition state of the glycolytic pathway under FDP saturating conditions: experimental studies and a theoretical model

1. The transition state of the glycolytic pathway, under FDP saturating conditions, from no ADP to ADP-saturating levels, is studied in a metabolic model in vitro obtained from rat skeletal muscle. 2. When ADP is absent from the reaction mixture a steady state for NADH concentration is observed. After ADP addition, a new steady state is reached. The transition state from the first steady state to the second one shows a pulse of NADH. Both the profile and the size of this pulse depend on the enzyme concentration. 3. A kinetic model of the lower part of glycolysis (after PFK reaction) is proposed, and this is described by a set of first order coupled nonlinear differential equations. The results obtained through stability analysis and numerical integration of these equations agree with the experimental ones. 4. The possible role of the above mentioned transition state on the transmitter mechanism of glycolytic oscillations from PFK to the lower part of the glycolysis is discussed.     

The adenylate kinase reaction acts as a frequency filter towards fluctuations of ATP utilization in the cell

The buffering ability of the adenylate kinase reaction with respect to the phosphate potential and efficiency of oxidative phosphorylation in the presence of a fluctuating load conductance were studied by computer simulations. Fluctuations of the load conductance, i.e., of the irreversible ATP-utilizing reactions in the cell, were generated by integrating an Ornstein-Uhlenbeck diffusion process. This real or colored noise was then injected into the set of differential equations describing the rate laws for the changes of the adenine nucleotide concentrations based on a simple nonequilibrium thermodynamic model of oxidative phosphorylation. Numerical integration of this system of stochastic differential equations allowed us to investigate the influence of different parameters on the performance of this energy converter. Probability density estimates revealed that the variance of the efficiency about its optimal value was significantly reduced by the adenylate kinase reaction. It was found that the buffering ability of this enzyme is restricted to a specific frequency domain of the fluctuations of the load conductance. This frequency filtering was confirmed by substituting the random fluctuations of the load conductance by simple sinusoidal perturbations. All these studies revealed that for each domain of frequencies of the load perturbations there exists an optimal activity of the adenylate kinase which minimizes deviations from optimal efficiency of oxidative phosphorylation.     

Systems approach to the study of drug transport across membranes using suspension cultures of mammalian cells V. Simultaneous passive transport and biosynthesis

A physical model is described for the simultaneous enzymatic bioconversion of a nonelectrolyte solute and the passive transport of both the solute and product of the enzymatic reaction out of cells in culture suspension. The plasma membrane is assumed to be the rate-determining transport barrier. This model provides the basis for the experimental design and analysis of the Michaelis-Menten kinetic parameters of simple enzymatic reactions in situ, the phenomenological transport parameters and other factors. The primary set of differential equations describing the quasisteady state rate of change in the concentration of the solute and product within the cell due to enzyme reaction and transport are given. These are nonlinear and must be solved by numerical methods. However, analytical mathematical expressions have been derived for various cases in the limit when the rate of enzymatic reaction is first or zero order.     

The kinetic effect of product instability in a Michaelis-Menten mechanism with competitive inhibition

In most kinetic studies it is assumed that both the reactant and the products are stable. However, under certain conditions spontaneous decomposition or deterioration caused by one of the participating species occurs. Studies, in which a species (the free enzyme, the enzyme-substrate complex, an inhibitor or the product of the reaction) is unstable, have appeared in the literature. However, to our knowledge, the enzymatic systems, in which a competitive inhibition and a decomposition or transformation of the products take place simultaneously, have not been studied so far. In this paper, we present a kinetic analysis of an enzyme reaction that follows a Michaelis-Menten mechanism, in which the free enzyme suffers a competitive inhibition simultaneously with the decomposition of the immediate product. In this study, we have linearised the differential equations that describe the kinetics of the process. Under the assumption of limiting concentration of enzyme, we have obtained and tested the explicit equation describing the time dependence of the product concentration using numerical calculus. With this equation and the experimental progress curve of the product, we constructed an easy procedure for the evaluation of the principal kinetic parameters of the process.