Membrane transport models with fast and slow reactions: General analytical solution for a single relaxation

Summary Membrane transport models are usually expressed on the basis of chemical kinetics. The states of a transporter are related by rate constants, and the time-dependent changes of these states are given by linear differential equations of first order. To calculate the time-dependent transport equation, it is necessary to solve a system of differential equations which does not have a general analytical solution if there are more than five states. Since transport measurements in a complex system rarely provide all the time constants because some of them are too rapid, it is more appropriate to obtain approximate analytical solutions, assuming that there are fast and slow reaction steps. The states of the fast steps are related by equilibrium constants, thus permitting the elimination of their differential equations and leaving only those for the slow steps. With a system having only two slow steps, a single differential equation is obtained and the state equations have a single relaxation. Initial conditions for the slow reactions are determined after the perturbation which redistribute the states related by fast reactions. Current and zero-trans uptake equations are calculated. Curve fitting programs can be used to implement the general procedure and obtain the model parameters.     

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.     

An Efficient Legendre Wavelet-Based Approximation Method for a Few Newell–Whitehead and Allen–Cahn Equations

In this paper, a wavelet-based approximation method is introduced for solving the Newell–Whitehead (NW) and Allen–Cahn (AC) equations. To the best of our knowledge, until now there is no rigorous Legendre wavelets solution has been reported for the NW and AC equations. The highest derivative in the differential equation is expanded into Legendre series, this approximation is integrated while the boundary conditions are applied using integration constants. With the help of Legendre wavelets operational matrices, the aforesaid equations are converted into an algebraic system. Block pulse functions are used to investigate the Legendre wavelets coefficient vectors of nonlinear terms. The convergence of the proposed methods is proved. Finally, we have given some numerical examples to demonstrate the validity and applicability of the method.     

An Efficient Wavelet-Based Approximation Method to Gene Propagation Model Arising in Population Biology

In this paper, we have applied an efficient wavelet-based approximation method for solving the Fisher’s type and the fractional Fisher’s type equations arising in biological sciences. To the best of our knowledge, until now there is no rigorous wavelet solution has been addressed for the Fisher’s and fractional Fisher’s equations. The highest derivative in the differential equation is expanded into Legendre series; this approximation is integrated while the boundary conditions are applied using integration constants. With the help of Legendre wavelets operational matrices, the Fisher’s equation and the fractional Fisher’s equation are converted into a system of algebraic equations. Block-pulse functions are used to investigate the Legendre wavelets coefficient vectors of nonlinear terms. The convergence of the proposed methods is proved. Finally, we have given some numerical examples to demonstrate the validity and applicability of the method.     

Kinetic studies of enzymatic hydrolysis of insoluble cellulose: Derivation of a mechanistic kinetic model

A comprehensive mechanistic kinetic model for enzymatic hydrolysis of insoluble cellulose has been synthesized by combining models for several key aspects which have been derived independent of each other. The model takes into account the major contributing factors: the nature of the enzyme system, the structure of cellulose, and the mode of interaction between the enzyme and cellulose molecules. It consists of a set of simultaneously occurring ordinary differential equations with ten kinetic constants. All of the kinetic constants have been determined independently by carrying out critically designed experiments, and they appear in the comprehensive model without any arbitrary manipulations. The governing equations of the model have been numerically simulated by means of the computer subroutine CSMP III. The model predicts the progress of hydrolysis of cellulose over a wide range of experimental conditions and hydrolysis times reasonably well. The model can even be applied to predict the progress of hydrolysis for intensively pretreated cellulose with a minor adjustment. The applicability of the model for the actual process development is also discussed.     

Mathematical modeling and computer simulation of isoelectric focusing with electrochemically defined ampholytes

A mathematical model of isoelectric focusing at the steady state has been developed for an M-component system of electrochemically defined ampholytes. The model is formulated from fundamental principles describing the components' chemical equilibria, mass transfer resulting from diffusion and electromigration, and electroneutrality. The model consists of ordinary differential equations coupled with a system of algebraic equations. The model is implemented on a digital computer using FORTRAN-based simulation software. Computer simulation data are presented for several two-component systems showing the effects of varying the isoelectric points and dissociation constants of the constituents.     

THE THEORY OF DIFFUSION IN CELL MODELS

The differential equations which describe the simultaneous diffusion of water and a salt in a cell model have been formulated and solved. The equations have been derived from the general laws which describe diffusion processes, thereby furnishing a physical interpretation for the constants which enter into the theory. The theoretical time curves for the two diffusing substances are in good agreement with the experimentally determined curves and accurately reproduce all of the essential characteristics of the experiment.     

Corrected equations for calculation of constants in enzyme inhibition and activation

Equations for calculation of the constants of biparametrical types of enzyme inhibition and activation were obtained that take into account a ratio of the lengths of L vector projections representing such reactions in the three-dimensional K (m)V I coordinate system. This allows higher accuracy of calculation and is more correct for comparison of these constants. Examples of data analysis of enzyme inhibition and activation by using the traditional equations (they do not take into account the lengths of vector projections) and corrected ones (they take into account the lengths of vector projections) are given. The corrected and traditional equations are used for calculation of the constants of biparametrical types of enzyme inhibition and activation.     

A theoretical model of the competition between hydrolase and carboxylesterase in protection against organophosphorus poisoning

A system of coupled non-linear differential equations describing interactions between organophosphorus compounds (OPs), OP hydrolase, acetylcholinesterase (AChE), and carboxylesterase (CaE) in a single compartment was derived incorporating irreversible combination of OP with AChE, hydrolytic breakdown of OP, and irreversible combination of OP with CaE. The equations were then uncoupled, providing non-linear differential equations on AChE, CaE and OP concentrations. One steady state solution of the AChE equation provided theoretical expressions for the amounts of OP hydrolyzed, bound with CaE, and bound with AChE. Assuming that the LD50 of an OP reflects the dose that depletes AChE to a 'minimal essential' level and that a single compartment model is applicable in vivo, the steady state solution becomes an equation predicting the LD50 from rate constants, initial enzyme levels, and the allowable AChE depletion. Normalization by initial AChE concentration produced a dimensionless relationship describing an 'OP toxicity surface' that clearly demonstrates regions where hydrolysis and CaE offer protection against OP poisoning. The surface can be used to theoretically predict an LD50 given only kinetic rate constants and effective whole-body AChE and CaE levels. Predictions of LD50s of seven OPs in rats were compared with published data. The relationship was found to adequately predict published LD50s spanning 5 orders of magnitude. The OP toxicity surface relationship provides a conceptual tool for use in OP toxicity research but should be particularly useful in predicting the relative protective effects of catalytic and stoichiometric scavenger mechanisms for an OP.     

NMR spin exchange kinetics at equilibrium in membrane transport and enzyme systems

A set of differential equations is formulated to describe the rapid exchange (time scale, approximately 0.01 to approximately 10 s) of a labelled solute across the membranes of cells in suspension. The labelling is achieved with nuclear magnetic resonance by exposure of the system to a high intensity radio-frequency pulse, and the excited nuclei relax to the equilibrium state with a short half life. An analytical expression for the decay of the magnetic resonance signal is presented; the solution involves the determination of eigenvalues, of an array of Laplace-Carson transformed differential equations, by use of the general solution of a quartic polynomial. Simulations of the behaviour of the exchange system using various conditions of cell number, rate constants and nuclear magnetic relaxation times are presented. The marked concentration dependence of the extent of reaction at a given time has not previously been reported for nuclear magnetic resonance exchange systems and is a feature anticipated from the known saturability of several membrane transport systems including glucose transport into human erythrocytes. The theory is readily generalized to other model systems by appropriate reinterpretation of the physical meaning of various parameters; the general form of the solution holds in many biological contexts other than membrane transport and includes equilibrium enzyme kinetics.     

Computational multiple steady states for enzymatic esterification of ethanol and oleic acid in an isothermal CSTR

The capacity of complex biochemical reaction networks (consisting of 11 coupled non-linear ordinary differential equations) to show multiple steady states, was investigated. The system involved esterification of ethanol and oleic acid by lipase in an isothermal continuous stirred tank reactor (CSTR). The Deficiency One Algorithm and the Subnetwork Analysis were applied to determine the steady state multiplicity. A set of rate constants and two corresponding steady states are computed. The phenomena of bistability, hysteresis and bifurcation are discussed. Moreover, the capacity of steady state multiplicity is extended to the family of the studied reaction networks.     

Assessment of formal and low structured kinetic modeling of polyhydroxyalkanoate synthesis from complex substrates

A formal kinetic mathematical model for poly-(3-hydroxybutyrate-co-3-hydroxyvalerate-co-4-hydroxybutyrate) [P(3HB-co-3HV-co-4HB)] terpolyester synthesis from glucose and galactose derived from whey permeate supplemented with gamma-butyrolactone by the archaeon Haloferax mediterranei was created. Further, a low structured mathematical model for poly-3-hydroxybutyrate synthesis from whey permeate by Pseudomonas hydrogenovora was developed. In both cases, biosyntheses for obtaining the experimental data used for compiling the models were performed via fed-batch cultivations. The model developed for H. mediterranei consists of 10 differential and 11 algebraic equations, including 27 kinetic constants. The model compiled for P. hydrogenovora encompasses 10 differential and 3 algebraic equations, including 36 kinetic constants. Both models were solved by Runge-Kuta variable step numerical integration with Monte Carlo parameter optimization procedure. Difficulties arising from the modeling of redirection of metabolic fluxes from biomass growth toward polyhydroxyalkanoate synthesis and byproducts are discussed.     

Kinetics of unfolding and refolding of proteins. I. Mathematical analysis

A mathematical method for the phenomenological analysis of unimolecular reaction kinetics is presented. Starting from the solution to the differential equations of the system, equations have been developed to relate the unknown parameters inherent to any reaction mechanism, such as the microscopic rate constants for individual reaction steps, to the experimental quantities obtained from the rate of change of average physical properties of the system. Provided that kinetic measurements can be made in both forward and reverse directions, all unknown parameters can be evaluated for many simple mechanisms, and for some of them characteristic relations between observable quantities can be established, which can serve as criteria for immediate rejection. The results will be applied in the following papers to experimental studies of denaturation and renaturation of several proteins.     

Determination of rate constants of antigen-antibody reaction. A theory

A new method of determination of rate constants for antigen-antibody interactions is proposed. This method is based on a solid phase immunoenzymatic analysis of the dynamics of elution of immobilized antigen-bound antibodies in the presence of a free antigen. The kinetics of this process is described by a system of differential equations, whose solution results in expression defining the dynamics of antibody interaction with immobilized and free antigens. Simple formulas were derived for the calculation of the rate and equilibrium constants for the antibody-antigen reaction on the basis of experimental kinetic curves. The use of theoretical kinetic curves for antibody elution showed that these formulas reflect with a high degree of accuracy the kinetic properties of the reaction under study.     

Kinetic modeling and fitting software for interconnected reaction schemes: VisKin

Reaction kinetics for complex, highly interconnected kinetic schemes are modeled using analytical solutions to a system of ordinary differential equations. The algorithm employs standard linear algebra methods that are implemented using MatLab functions in a Visual Basic interface. A graphical user interface for simple entry of reaction schemes facilitates comparison of a variety of reaction schemes. To ensure microscopic balance, graph theory algorithms are used to determine violations of thermodynamic cycle constraints. Analytical solutions based on linear differential equations result in fast comparisons of first order kinetic rates and amplitudes as a function of changing ligand concentrations. For analysis of higher order kinetics, we also implemented a solution using numerical integration. To determine rate constants from experimental data, fitting algorithms that adjust rate constants to fit the model to imported data were implemented using the Levenberg-Marquardt algorithm or using Broyden-Fletcher-Goldfarb-Shanno methods. We have included the ability to carry out global fitting of data sets obtained at varying ligand concentrations. These tools are combined in a single package, which we have dubbed VisKin, to guide and analyze kinetic experiments. The software is available online for use on PCs.     

A dynamic model for functional mapping of biological rhythms

Functional mapping is a statistical method for mapping quantitative trait loci (QTLs) that regulate the dynamic pattern of a biological trait. This method integrates mathematical aspects of biological complexity into a mixture model for genetic mapping and tests the genetic effects of QTLs by comparing genotype-specific curve parameters. As a way of quantitatively specifying the dynamic behaviour of a system, differential equations have proved to be powerful for modelling and unravelling the biochemical, molecular, and cellular mechanisms of a biological process, such as biological rhythms. The equipment of functional mapping with biologically meaningful differential equations provides new insights into the genetic control of any dynamic processes. We formulate a new functional mapping framework for a dynamic biological rhythm by incorporating a group of ordinary differential equations (ODE). The Runge–Kutta fourth-order algorithm was implemented to estimate the parameters that define the system of ODE. The new model will find its implications for understanding the interplay between gene interactions and developmental pathways in complex biological rhythms.     

A dynamic model for functional mapping of biological rhythms

Functional mapping is a statistical method for mapping quantitative trait loci (QTLs) that regulate the dynamic pattern of a biological trait. This method integrates mathematical aspects of biological complexity into a mixture model for genetic mapping and tests the genetic effects of QTLs by comparing genotype-specific curve parameters. As a way of quantitatively specifying the dynamic behavior of a system, differential equations have proven to be powerful for modeling and unraveling the biochemical, molecular, and cellular mechanisms of a biological process, such as biological rhythms. The equipment of functional mapping with biologically meaningful differential equations provides new insights into the genetic control of any dynamic processes. We formulate a new functional mapping framework for a dynamic biological rhythm by incorporating a group of ordinary differential equations (ODE). The Runge-Kutta fourth order algorithm was implemented to estimate the parameters that define the system of ODE. The new model will find its implications for understanding the interplay between gene interactions and developmental pathways in complex biological rhythms.     

Metabolism of apolipoproteins A-I and A-II in human high-density lipoprotein: a mathematical approach for analysis of their specific activity decay curves

The differential rate equations describing the compartmental model of human high-density lipoprotein (HDL) were integrated by means of Laplace transforms and an exponential equation was obtained for each of the three compartments. These equations were used to fit the observed plasma decay data and give estimates for the rate constants of the system by means of a written computer program. Furthermore, these estimates were used to calculate the exponential constants of the integrated equations. Consequently, the amount of label in any of the intravascular, extravascular, and urine compartments can be calculated as a fraction of the original dose of label at any time point. This method was tested using data for the (AI)HDL subclass because it contains only apolipoprotein A-I as the major apolipoprotein and does not contain apolipoprotein A-II. The calculated plasma and urine radioactivity data were compared with the experimentally obtained data from two normolipoproteinemic subjects and found to be in good agreement. The significance of this method is its application to the analysis of the decay data of the individual apolipoproteins of (AI + AII) HDL subclass where the urinary radioactivity data resulting from the individual apolipoprotein breakdown on the native particle cannot be measured experimentally at present. Such data are essential for the detailed calculation of the kinetic parameters of these apolipoproteins.