A database-based approach for predicting coupled cascade reaction kinetics in polymers: application to oxidative degradation kinetics of high-performance polymers

Coupled cascade reactions forming complex reaction networks can be commonly found in polymerisation reactions and other reactions involving radical intermediates. Predicting the mechanism and kinetics of such reactions requires proper modelling of complex reaction networks. This becomes particularly difficult when coupled cascade reactions occur in polymeric systems containing different types of residues. Here, we propose a residue-based database approach to model such reactions in polymers, with the aid of a visual interface developed here. We demonstrate this approach by predicting the oxidative degradation kinetics of high-performance polymers (HPPs). First, we show that residue-based reaction database can be linked to construct the whole reaction network. For this purpose, we developed a database for oxidation reactions of commonly occurring residues in industrially important HPPs. Then we implement a visual interface which takes inputs from a user about residues in a polymer of interest and subsequently link appropriate databases to build reaction network. Finally, this program executes numerical integration of rate equations in the back-end. Application of this approach and the developed program is demonstrated by studying the oxidative degradation kinetics of three well-known HPPs- PMR-15, HFPE-30 and PMR-II, where the computations took less than a minute in a conventional desktop computer.     

Parameter estimation for ligand binding systems kinetics applied to 1,25-dihydroxycholecalciferol

A mathematical analysis of the kinetics of the hormone-receptor interaction was applied to the 1,25-dihydroxycholecalciferol-intestinal receptor system. The exact analytical solution and the numerical integration of the kinetic equation were installed in a Statistical Analysis System (SAS) computer program to estimate the rate constants of the reaction. Estimates of the parameters obtained by these two methods are similar, demonstrating that the numerical integration can be combined with the nonlinear regression procedure for least-squares parameter fitting using a simple SAS program. This enables estimation of kinetics rate constants when the kinetic equation cannot be solved analytically. The ratio of the rate constants (ka/kd) found by the nonlinear procedure is close to the independently determined equilibrium (Scatchard) constant in the nonlinear analysis.     

Free energy and the kinetics of biochemical diagrams, including active transport.

In earlier papers on muscle contraction it was found very useful to relate the actual (not standard) free energy levels of the different states in the biochemical diagram of the myosin cross-bridge to the first-order rate constants governing transitions between these states and to the details of the conversion of ATP free energy into mechanical work. This same approach is applied here to other macromolecular biochemical systems, for example, carriers in active transport, and simple enzyme reactions. With the definition of free energy changes between states of diagram used here (and in the muscle papers), the rate constants of the diagram are firat order, the macromolecular transitions are effectively isomeric, the equilibrium constants are dimensionless, the free energy changes are directly related to first-order rate constant ratios, and the ratio of products of forward and backward rate constants around any cycle of the diagram is related to operational free energy changes (e.g. the in vivo free energy of ADP HYDROLYSIS). These general points are illustrated by means of particular arbitrary models, especially transport models. In contrast to the muscle case, the free energy conversion question in other biochemical systems can be handled at the less detailed, complete-cycle level rather than at the elementary transition level. There is a corresponding complete-cycle kinetics, with composite first-order rate constants for the different possible cycles (in both directions). An introductory stochastic treatment of cycle kinetics is included.     

The computation of hyperbolic dependences in enzyme kinetics.

A computer program aimed at analysing results following Michaelis-Menten kinetics can be used unmodified in the treatment of other kinetic results provided that the kinetic equations in these cases can be written in the form of the Michaelis-Menten equation. A list is presented of the parameters to be set instead of substrate concentration and reaction rate, and of constants replacing Km and V, if such a program is applied in analysing enzyme inhibitions, activations and pH-dependences.     

Continuous measurement and analysis of staining kinetics by flow cytometry

The measurement of color development with time in cells following the start of a staining reaction is of interest in a number of biological systems. These include the subsets of peripheral white blood cells after acridine orange staining, the uptake by cells and nuclei of fluorescent agents, especially antitumor drugs, and measurement of intracellular enzyme kinetics using fluorogenic or absorbing substrates. The present work describes a simple computer program for analyzing flow cytometric (FCM) data versus time, including both the population kinetics of color development and the variability of staining speed within one population of cells. A single-channel absorption measurement in flow (Technicon Hemalog D) was used to record peroxidase kinetics in peripheral blood cells. Every 5 s, a 64-channel absorption histogram was recorded, up to a maximum of 64 histograms. The data were then analyzed by a computer program which searched for the peak channel of each histogram. A least-squares fit was computed for these maxima. The asymmetries of the 64 absorption histograms were compared to see if there was more than one population present with different time constants. Although developed for enzyme kinetic measurements, this program may have wider usefulness in any measurements of time-dependent phenomena by FCM.     

Determination of thermodynamics and kinetics of RNA reactions by force

Single-molecule methods have made it possible to apply force to an individual RNA molecule. Two beads are attached to the RNA; one is on a micropipette, the other is in a laser trap. The force on the RNA and the distance between the beads are measured. Force can change the equilibrium and the rate of any reaction in which the product has a different extension from the reactant. This review describes use of laser tweezers to measure thermodynamics and kinetics of unfolding/refolding RNA. For a reversible reaction the work directly provides the free energy; for irreversible reactions the free energy is obtained from the distribution of work values. The rate constants for the folding and unfolding reactions can be measured by several methods. The effect of pulling rate on the distribution of force-unfolding values leads to rate constants for unfolding. Hopping of the RNA between folded and unfolded states at constant force provides both unfolding and folding rates. Force-jumps and force-drops, similar to the temperature jump method, provide direct measurement of reaction rates over a wide range of forces. The advantages of applying force and using single-molecule methods are discussed. These methods, for example, allow reactions to be studied in non-denaturing solvents at physiological temperatures; they also simplify analysis of kinetic mechanisms because only one intermediate at a time is present. Unfolding of RNA in biological cells by helicases, or ribosomes, has similarities to unfolding by force.     

Presteady-state kinetics and carrier-mediated transport: A theoretical analysis

Summary Kinetic studies of cotransport mechanisms have so far been limited to the conventional steady-state approach which does not allow in general to resolve either isomerization or ratelimiting steps and to determine the values of the individual rate constants for the elementary reactions involved along a given transport pathway. Such questions can only be answered using presteady-state or relaxation experiments which, for technical reasons, have not yet been introduced into the field of cotransport kinetics. However, since two recent reports seem compatible with the observation of such transient kinetics, it would appear that theoretical studies are needed to evaluate the validity of such claims and to critically evaluate the expectations from a presteady-state approach. We thus report such a study which was performed on a simple four-state mechanism of carrier-mediated transport. The time-dependent equation for zero-trans substrate uptake was thus derived and then extended to models withp intermediary steps. It is concluded that (p-1) exponential terms will describe the approach to the steady state but that such equations have low analytical value since the parameters of the flux equation cannot be expressed in terms of the individual rate constants of the elementary reactions for models withp>5. We thus propose realistic simplifications based on the time-scale separation hypothesis which allows replacement of the rate constants of the rapid steps by their equilibrium constants, thereby reducing the complexity of the kinetic system. Assuming that only one relaxation can be observed, this treatment generates approximate models for which analytical expressions can easily be derived and simulated through computer modeling. When performed on the four-state mechanism of carrier-mediated transport, the simulations demonstrate the validity of the approximate solutions derived according to this hypothesis. Moreover, our approach clearly shows that presteady-state kinetics, should they become applicable to (co)transport kinetics, could be invaluable in determining more precise transport mechanisms.     

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.     

A microscopic model of enzyme kinetics.

Many in vivo enzymatic processes, such as those of the tissue factor pathway of blood coagulation, occur in environments with facilitated substrate delivery or enzymes bound to cellular or lipid surfaces, which are quite different from the ideal fluid environment for which the Michaelis-Menten equation was derived. To describe the kinetics of such reactions, we propose a microscopic model that focuses on the kinetics of a single-enzyme molecule. This model provides the foundation for macroscopic models of the system kinetics of reactions occurring in both ideal and nonideal environments. For ideal reaction systems, the corresponding macroscopic models thus derived are consistent with the Michaelis-Menten equation. It is shown that the apparent Km is in fact a function of the mechanism of substrate delivery and should be interpreted as the substrate level at which the enzyme vacancy time equals the residence time of ES-complexes; it is suggested that our microscopic model parameters characterize more accurately an enzyme and its catalytic efficiency than does the classical Km. This model can also be incorporated into computer simulations of more complex reactions as an alternative to explicit analytical formulation of a macroscopic model.     

Quantitative measurements of NO reaction kinetics with a Clark-type electrode.

Nitric oxide (NO) plays important physiological roles in the body. Knowledge regarding the kinetics of NO catabolism is important for understanding the biological functions of NO. Clark-type NO electrodes have been frequently employed in measuring the kinetics of NO reactions; however, the slow response time of these electrodes can cause measurement errors and limit the application of the electrode in measurements of fast NO reactions. In this study, a simplified diffusion model is given for describing the response process of the NO electrode to the change of NO concentration. The least-square method is used in fitting the currents calculated from the diffusion equation to the experimental curves for determining the diffusion parameters and rate constants. The calculated currents are in excellent accordance with the experimental curves for different NO reaction kinetics. It has been demonstrated that when using an NO electrode with a response time of approximately 6 s to measure fast NO reactions with a half-life of approximately 1s, the response currents of the electrode have large differences compared to the curve of actual NO concentration in the solution; however, the rate constant of NO decay can still be accurately determined by computer simulations with the simplified diffusion model. Theoretical analysis shows that an NO electrode with a response time of 6 s (D/L2=0.06 s-1) and the lowest detection limit of 1 nM NO can be used in measuring kinetics of extremely rapid NO reactions with a half-life below 10 ms.     

Maximum likelihood estimation of molecular motor kinetics from staircase dwell-time sequences

Molecular motors, such as kinesin, myosin, or dynein, convert chemical energy into mechanical energy by hydrolyzing ATP. The mechanical energy is used for moving in discrete steps along the cytoskeleton and carrying a molecular load. High resolution single molecule recordings of motor steps appear as a stochastic sequence of dwells, resembling a staircase. Staircase data can also be obtained from other molecular machines such as F1 -ATPase, RNA polymerase, or topoisomerase. We developed a maximum likelihood algorithm that estimates the rate constants between different conformational states of the protein, including motor steps. We model the motor with a periodic Markov model that reflects the repetitive chemistry of the motor step. We estimated the kinetics from the idealized dwell-sequence by numerical maximization of the likelihood function for discrete-time Markov models. This approach eliminates the need for missed event correction. The algorithm can fit kinetic models of arbitrary complexity, such as uniform or alternating step chemistry, reversible or irreversible kinetics, ATP concentration and mechanical force-dependent rates, etc. The method allows global fitting across stationary and nonstationary experimental conditions, and user-defined a priori constraints on rate constants. The algorithm was tested with simulated data, and implemented in the free QuB software.     

Two different ways that hydrogen ions are involved in the thermodynamics and rapid-equilibrium kinetics of the enzymatic catalysis of S=P and S+H2O=P

Hydrogen ions are involved in two different ways in the thermodynamics and rapid-equilibrium kinetics of enzyme-catalyzed reactions. The two ways are through pKs and through the production or consumption of hydrogen ions in the mechanism. These ways are examined for the catalyzed reactions S=P and S+H2O=P. Since the apparent equilibrium constant K' can be calculated from the kinetic parameters by use of the Haldane equation, the treatment of the effects of pH must be consistent in thermodynamics and kinetics. This leads to a new kind of Haldane equation that involves 10(pH) or 10(-pH) in addition to the kinetic parameters when hydrogen ions are produced or consumed. These concepts are applicable to more complicated reactions and rate equations. Derivations of equations for calculating these two types of pH effects are discussed in thermodynamics and rapid-equilibrium kinetics. A computer program is used to make four plots of apparent equilibrium constants and changes in the binding of hydrogen ions in the catalyzed reaction.     

Single turnover kinetics of the reaction between oxycytochrome P-450cam and reduced putidaredoxin

A study of the single turnover kinetics of the reaction between oxycytochrome P-450cam and reduced putidaredoxin was performed using the inhibitor metyrapone to trap the cytochrome immediately after release of the product, 5-exo-hydroxycamphor. EPR determinations of the concentrations of reduced putidaredoxin and ferric metyrapone-bound cytochrome at the same time points showed that there is no time lag between the oxidation of reduced putidaredoxin and the appearance of metyrapone-bound cytochrome. This implies that the rate constant for electron transfer is smaller than the rate constant for the later processes involved in product formation and release, lumped into a single step. Taking this restriction into account and doing computer simulation of absorbance versus time curves, previously obtained at various putidaredoxin concentrations using stopped-flow spectrophotometry, allowed bounds to be determined for rate constants of the processes within the reaction. At 4 degrees C in buffer at pH 7.4 with 0.50 M KCl, the rate constant for the bimolecular association of the two enzymes is between 3 and 20/microM.s; the rate constant for dissociation is between 12 and 600/s; the rate constant for electron transfer is between 60 and 100/s; and the rate constant for the later processes is at least 200/s.     

Thermodynamics and kinetics of the glyoxylate cycle

Because the standard Gibbs energies of formation of all the species of reactants in the glyoxylate cycle are known at 298.15 K, it is possible to calculate the apparent equilibrium constants of the five reactions in the cycle in the pH range 5-9 and ionic strengths from 0 to approximately 0.35 M. In making calculations on such a system, it is convenient to specify concentrations of coenzymes like NADox and NADred because they are involved in many reactions and may be in steady states. Calculations are given for [NADox] = 1000[NADred] and [NADox] = 10[NADred]. Equilibrium compositions are calculated using computer programs when all the reactants are present initially and when only glyoxylate and CoA are present initially. The kinetics of the reactions in the glyoxylate cycle at specified concentrations of NADox and NADred are calculated by numerical solution of the steady-state rate equations for the case where the reactant concentrations are below their Michaelis constants and only glyoxylate and CoA are present initially.     

Steady magnetic fields effect on lipid peroxidation kinetics

The effect of steady magnetic fields (ranging from 0 to 280 mT) has been investigated on the kinetics of non-enzymatic lipid peroxidation occurring in a model system consisting of liposomes obtained from 1, 2-dioleoylphosphatidylcholine by oxygen consumption. The process was found to be accelerated by weak steady magnetic fields. A computer simulation method was employed to detect the reactions that govern the process kinetics, to elucidate magneto-sensitive stages (initiation and reduction of iron(III), as well as lipid peroxide radical recombination) and to determine their rate constants at various external magnetic fields. The kinetics of peroxidation of lipid cell membranes have been modeled mathematically at oxygen and ‘free’ iron concentrations close to those in the cells and also at increased free iron concentrations at different external magnetic field values.     

pH modulation of transient state kinetics of enzymes. I. Simple theoretical models

The effect of proton concentration on pre-steady-state kinetics has been investigated theoretically for enzyme reactions involving the breaking of one substrate into two products. Even for the simple double-intermediate mechanism the approach to the steady state may exhibit a rather complex kinetics, which is pH-dependent. This process may even exhibit damped oscillations. A change of pH may completely change this transient kinetics and even suppresses the oscillatory regime. A simple method is presented which allows estimation of the values of the rate and ionization constants. This procedure allows one to distinguish the simple double-intermediate mechanism from a more complex process where the 'fast' binding of the substrate induces a 'slow' conformation change of the enzyme.     

The kinetics of binding of o-methyl red to the outer surface of unilamellar spherical phospholipid vesicles

The kinetics of adsorption of the proton carrier o-methyl red to the surface of unilamellar spherical phospholipid vesicles have been investigated by means of the temperature-jump relaxation technique with absorbance detection. Single-exponential relaxation curves were observed with time constants in the range 30-130 microseconds. o-Methyl red binds in both its anionic form A- and protonated form AH. Adsorption-desorption of the two species is coupled by two fast protolytic reactions, occurring in the aqueous bulk phase and in the surface region of the membrane. The rate constants for adsorption and desorption of the two species were obtained from the dependences of the relaxation time on lipid concentration at different pH values. The analysis yielded apparent adsorption rate constants of kasAH = 9.8 X 10(6) M-1 s-1 and kasA = 1.3 X 10(6) M-1 s-1 (expressed in terms of monomeric lipid), and kasAH = 1.2 X 10(11) M-1 s-1 and kasA = 1.6 X 10(10) M-1 s-1 (expressed in terms of vesicle concentration). From the order of these rate constants it is concluded that adsorption of both species is actually diffusion-controlled. The peculiar pH dependence of the relaxation time is a consequence of the protolytic reaction in the surface region of the membrane. Its implication for the kinetics of adsorption-desorption processes are discussed.     

Feedback stabilization of fed-batch bioreactors: non-monotonic growth kinetics

This paper deals with the design of a feedback controller for fed-batch microbial conversion processes that forces the substrate concentration C(S) to a desired setpoint, starting from an arbitrary (initial) substrate concentration when non-monotonic growth kinetics apply. This problem is representative for a lot of industrial fermentation processes, with the baker's yeast fermentation as a well-known example. It is assumed that the specific growth rate mu is function of the substrate concentration only. A first approach exploits the availability of on-line measurements of both the substrate and biomass concentration. A second approach is merely based on on-line measurements of the biomass concentration, which provide an estimate for the specific growth rate. After a reformulation of the substrate concentration setpoint into a specific growth rate setpoint, it is demonstrated that the fed-batch process can still be stabilized around any desired operating point along the non-monotonic kinetics.     

Enzyme-catalyzed and binding reaction kinetics determined by titration calorimetry

BackgroundIsothermal calorimetry allows monitoring of reaction rates via direct measurement of the rate of heat produced by the reaction. Calorimetry is one of very few techniques that can be used to measure rates without taking a derivative of the primary data. Because heat is a universal indicator of chemical reactions, calorimetry can be used to measure kinetics in opaque solutions, suspensions, and multiple phase systems and does not require chemical labeling. The only significant limitation of calorimetry for kinetic measurements is that the time constant of the reaction must be greater than the time constant of the calorimeter which can range from a few seconds to a few minutes. Calorimetry has the unique ability to provide both kinetic and thermodynamic data.Scope of reviewThis article describes the calorimetric methodology for determining reaction kinetics and reviews examples from recent literature that demonstrate applications of titration calorimetry to determine kinetics of enzyme-catalyzed and ligand binding reactions.Major conclusionsA complete model for the temperature dependence of enzyme activity is presented. A previous method commonly used for blank corrections in determinations of equilibrium constants and enthalpy changes for binding reactions is shown to be subject to significant systematic error.General significanceMethods for determination of the kinetics of enzyme-catalyzed reactions and for simultaneous determination of thermodynamics and kinetics of ligand binding reactions are reviewed. This article is part of a Special Issue entitled Microcalorimetry in the BioSciences — Principles and Applications, edited by Fadi Bou-Abdallah.