Linear compartmental systems. III. Application to enzymatic reactions

In this paper we extend to enzyme systems the results previously obtained in paper I of this series for linear compartmental systems. We obtain the time course equations for both the enzyme and ligand species involved in the reaction mechanisms, which fit a general enzyme system model when the connections between the different enzyme species are of first or pseudofirst order. The kinetic equations obtained here for a given species, enzyme or ligand have the advantage over all previous equations described in the literature, in that they are in the most simplified form possible, since they only contain the kinetic parameters and initial concentrations of the enzymatic reaction which really have some influence on the time progress curves of the species under study. These kinetic equations are denominated optimized equation to distinguish them from the others, which shall call non-optimized equations. We discuss those cases when both types of equation coincide and we show how, when they do not coincide, the non-optimized equations can be simplified to the optimized ones. Therefore, we show that the optimized equations could be used in all cases to avoid the need of subsequent simplifications to eliminate the parameters that play no role in the corresponding time equations. To illustrate the use of this procedure we will apply it to two simple examples of enzymatic reactions.     

A symbolic algorithm for periodic tridiagonal systems of equations

In the current paper, we present a novel symbolic algorithm for solving periodic tridiagonal linear systems without imposing any restrictive conditions. The computational cost of the algorithm is less than or almost equal to those of three well-known algorithms given by Chawla and Khazal (Int. J. Comput. Math. 79(4):473–484, 2002) and by El-Mikkawy (Appl. Math. Comput. 161:691–696, 2005), respectively. In addition, the solution of periodic anti-tridiagonal linear systems is also discussed. Two numerical experiments are provided in order to illustrate the performance and effectiveness of our algorithm. All of the experiments were performed on a computer with aid of programs written in MATLAB.     

Statistical derivation of kinetic molecular weight development equations in nonlinear free-radical polymerization

We propose a statistical method to derive the differential equations that describe the weight-average molecular weight development during nonlinear free-radical polymerizations, by using the random sampling technique. We consider two types of nonlinear free-radical polymerization schemes, free-radical polymerization with chain transfer to polymer and free-radical crosslinking (co)polymerization. The obtained equations fully agree with those obtained through the kinetic approach using the method of moments. The physical meaning of each term in the differential equations as well as the implication of the functional form is discussed from the point of view of the statistical derivation.     

Mechanochemical synthesis of nanoparticles by a dilution method: derivation of kinetic equations

A kinetic equation was derived for the mechanochemical preparation of nanosystems by a dilution method; the equation was used to calculate mechanical activation times necessary for the completion of exchange reactions and mass-transfer coefficients in ball milling.     

Linear compartmental systems. IV. A software,under MS-Windows,for obtaining the instantaneous species concentrations in enzyme systems

Software application is implemented in this work to take full advantage of the characteristics of current operating systems and to provide the optimized symbolic kinetic equations for both enzyme and ligand species involved in enzyme reactions. This software, called SKEE-w2013, is implemented using C# language and runs under all operating systems from Windows XP up to Windows 8. It is applicable to any enzyme reaction mechanism that fits the general reaction scheme proposed previously by our group. It can be downloaded, free of charge, from http://oretano.iele-ab.uclm.es/~BioChem-mg/software.php. Besides the optimized equations, the software can provide non-optimized equations, so that the user can compare the advantage of using optimized equations rather than the non-optimized ones, whenever they do not coincide. Moreover, the software circumvents the limitations of other existing previous software tools implemented with what are nowadays obsolete programming languages and that, moreover, are limited to non-optimized kinetic equations.     

Global sensitivity analysis of nonlinear chemical kinetic equations using lie groups: I. Determination of one-parameter groups

We introduce one-parameter groups of transformations that effect wide-ranging changes in the rate constants and input/output fluxes of homogeneous chemical reactions involving an arbitrary number of species in reactions of zero, first and second order. Each one-parameter group is required to convert every solution of such elementary rate equations into corresponding solutions of a one-parameter family of altered elementary rate equations. The generators of all allowed one-parameter groups are obtained for systems withN species using an algorithm which exactly determines their action on the rate constants, and either exactly determines or systematically approximates their action on the concentrations. Compounding the one-parameter groups yields all many-parameter groups of smooth time-independent transformations that interconvert elementary rate equations and their solutions.     

Sensitivity analysis of complex kinetic systems. Tools and applications

Sensitivity analysis investigates the effect of parameter change on the solution of mathematical models. In chemical kinetics, models are usually based on differential equations and the results are concentration-time curves, reaction rates, and various kinetic features of the reaction. This review discusses in detail the concentration sensitivity, rate sensitivity, and feature sensitivity analysis of spatially homogeneous constant-parameter reaction systems. Sensitivity analyses of distributed parameter systems and of stochastic systems are also briefly described. Special attention is paid to the interpretation of sensitivity coefficients which can provide information about the importance and interconnection of parameters and variables. Applications of sensitivity analysis to uncertainty analysis, parameric scaling, parameter estimation, experimental design, stability analysis, repro-modeling, and investigation and reduction of complex reaction mechanisms are discussed profoundly.     

Parametric analysis of kinetic models. I. The simplest autocatalytic oscillator

A series of mechanisms allowing oscillations and differring in the form of their buffer steps (step 3)

Reaction rate analysis of complex kinetic systems

Using the elementary sensitivity densities, a reaction rate sensitivity gradient is obtained which is the derivative of the rate of species concentration change with respect to the rate coefficient. The dimensionless (log-normalized) form of the reaction rate sensitivity gradient is the ratio of the rate of concentration change of species i due to elementary reaction j and the net rate of concentration change of species i. This result provides a mathematical basis for the use of various forms of reaction rate analyses in the study of complex reaction mechanisms. The kinetic information inherent in the relative reaction rate matrix is extracted by principal component analysis. The method is used to analyze the mechanism of high-temperature formaldehyde oxidation and high-temperature propane pyrolysis. Ranking of the elementary reactions allowed us to reduce significantly the original mechanisms and a detailed study of the results revealed the reaction structures and the major reaction paths of the species.     

Harmonic analysis of large systems. I. Methodology

Methods have been developed for the determination of vibrational frequencies and normal modes of large systems in the full conformational space (including all degrees of freedom) and in a reduced conformational space (reducing the number of degrees of freedom). The computational method, which includes Hessian generation and storage, full and iterative diagonalization techniques, and the refinement of the results, is presented. A method is given for the quasiharmonic analysis and the reduced basis quasiharmonic analysis. The underlying principle is that from the atomic fluctuations, an effective harmonic force field can be determined relative to the dynamic average structure. Normal mode analysis tools can be used to characterize quasiharmonic modes of vibration. These correspond to conventional normal modes except that anharmonic effects are included. Numerous techniques for the analyses of vibrational frequencies and normal modes are described. Criteria for the analysis of the similarity of low-frequency normal modes is presented. The approach to determining the natural frequencies and normal modes of vibration described here is general and applicable to any large system. © 1995 John Wiley & Sons, Inc.

1 This article is a U.S. Government work and, as such, is in the public domain in the United States of America.

     

A computer program system for kinetic analysis of non-isothermal thermogravimetric data: I. Data acquisition and preparation for kinetic analysis

FORTRAN software is described which enables the generation of rate of weight change data (DTG) from percentage weight change measurements (TG), obtained under non-isothermal conditions. The program also transposes this information into the dimensionless extent and rate of reaction at unit temperature intervals by means of a cubic spline interpolation. A simple search routine identifies all DTG spikes in the thermogravimetric record, and the temperature and extent of reaction at which the rate attains its maximum value. This total information serves as input data for the kinetic analysis software to be discussed in part II of this communication. An example of the application of this program to the pyrolysis of bituminous coal is presented.     

Ab initio calculation of anisotropic magnetic properties of complexes. I. Unique definition of pseudospin Hamiltonians and their derivation

A methodology for the rigorous nonperturbative derivation of magnetic pseudospin Hamiltonians of mononuclear complexes and fragments based on ab initio calculations of their electronic structure is described. It is supposed that the spin-orbit coupling and other relativistic effects are already taken fully into account at the stage of quantum chemistry calculations of complexes. The methodology is based on the establishment of the correspondence between the ab initio wave functions of the chosen manifold of multielectronic states and the pseudospin eigenfunctions, which allows to define the pseudospin Hamiltonians in the unique way. Working expressions are derived for the pseudospin Zeeman and zero-field splitting Hamiltonian corresponding to arbitrary pseudospins. The proposed calculation methodology, already implemented in the SINGLE_ANISO module of the MOLCAS-7.6 quantum chemistry package, is applied for a first-principles evaluation of pseudospin Hamiltonians of several complexes exhibiting weak, moderate, and very strong spin-orbit coupling effects.     

Sedimentation equilibrium in nonideal heterogeneous systems. I. Fundamental equations for heterogeneous solute systems and some preliminary results

The sedimentation-equilibrium method is extended to treat nonideal solutions of heterogeneous macromolecules. The solute is assumed to be heterogeneous not only in molecular weight but also in other quantities such as partial specific volume, second virial coefficient and specific refractive increment. General expressions for various observable molecular weights, especially for weight-average, z-average, and number-average molecular weights, are derived. Their dependences on sedimentation parameter and solute concentration are discussed in detail. For the extrapolation of observable molecular weights, giving a type of weight-average, and z-average, to infinite dilution to estimate the molecular weight and the second virial coefficient, average concentration is superior as a concentration variable to original concentration. The plots of observable molecular weight versus average concentration are usually less influenced by the choice of the sedimentation parameter, especially of rotor speed. The general expressions are applied to a few special cases; monodisperse polymer, polydisperse homologous polymer, and polymer blend. The results are compared with experiments on a monodisperse, polystyrene, a polydisperse poly(methyl methacrylate), and a mixture of the two polymers, all in 2-butanone at 25°C. The agreement between the theory and experiments is satisfactory.     

Reaction kinetics in thermal analysis. Part 1. The sensitivity of kinetic equations to experimental errors. A mathematical analysis

The frequent publication of contradicting or meaningless kinetic parameters and the resulting criticism of the “ill-conditioned nature” of non-isothermal reaction kinetics led the authors to an examination of the sensitivity of kinetic parameters to experimental errors. Using simple mathematical deductions, conditions were given at which about 10% precision of the kinetic parameters can easily be achieved. To obtain a graphic picture about the information content of a thermoanalytical curve and the effect of the systematic measurement errors, mathematical relationships were deduced to show the dependence of the kinetic parameters on the formal (geometric) characteristics of the thermoanalytical curves.     

The analysis of the derivation principles of kinetic equations based on exactly solvable models of the bulk reaction A + B → Product

We have considered two many-particle models of the irreversible reaction A + B → Product for which closed kinetic equations for the mean concentration N_A(t) of A species can be exactly obtained. These equations are identically recast into a unified form of integro-differential equation of general kinetic theory. It is shown that the memory functions for both models under consideration can be represented as a sum of the Markovian and non-Markovian parts. It is essential that the Markovian part of the Laplace transform of any kernel can be obtained using the Laplace transform of the kernel itself, and is the root of the non-Markovian part of the Laplace transform of the kernel.The properties established allowed us to perform correct approximation of the memory functions at small concentrations [B] of B species and derive the binary non-Markovian integro-differential equation. Within the binary theory accuracy this equation has been rewritten in a regular frame of a familiar rate equation satisfying general principles of binary kinetic equations.Thus using particular exactly solvable many-particle models, we have reproduced the most essential steps of the known general way for the derivation of the binary kinetic equation avoiding the sophisticated many-particle technique and the corresponding approximations. Besides, the results obtained can serve as an additional evidence of the approximations made in a general many-particle approach to the derivation of the binary kinetic equation.     

Multicomponent spectrometric analysis of riboflavin and photoproducts and their kinetic applications

Riboflavin (RF) is a light sensitive compound and is known to form a number of photoproducts. These photoproducts possess the same nucleus and may interfere in the analysis of RF by UV and visible spectrometry. Therefore, it is necessary to apply the methods of multicomponent spectrometric analysis to quantify the vitamin and its photoproducts accurately. Such methods are useful in the study of the kinetics of photodegradation reactions of RF to obtain accurate and reliable results. Any interference in these methods due to linear or nonlinear irrelevant absorption of the minor unknown products can be accounted for by the application of appropriate correction procedures prior to kinetic treatment. Various factors affecting the accuracy, precision and selectivity of these analytical procedures are also discussed. This review highlights the principles and applications of multicomponent spectrometric methods and their application to the simultaneous determination of RF and its major photoproducts in degraded solutions to evaluate the kinetics of degradation.      

A collision theory-based derivation of semiempirical equations for modeling dispersive kinetics and their application to a mixed-phase crystal decomposition

In recent works, the author has shown the utility of new, semiempirical kinetic model equations for treating dispersive chemical processes ranging from slow (minute/hour time scale) solid-state phase transformations to ultrafast (femtosecond) reactions in the gas phase. These two fundamental models (one for homogeneous/deceleratory sigmoidal conversion kinetics and the other for heterogeneous/acceleratory sigmoidal kinetics; isothermal conditions), based on the assumption of a "Maxwell-Boltzmann-like" distribution of molecular activation energies, provide a novel, quantum-based interpretation of the kinetics. As an extension to previous work, it is shown here that the derivation of these dispersive kinetic equations is supported by classical collision theory (i.e., for gas-phase applications). Furthermore, the successful application of the approach to the kinetic modeling of the solid-state decomposition of a binary system, CO2.C2H2, is demonstrated. Finally, the models derived appear to explain some of the (solid-state) kinetic data collected using isoconversional techniques such as those often reported in the thermal analysis literature.