ENZO: a web tool for derivation and evaluation of kinetic models of enzyme catalyzed reactions

We describe a web tool ENZO (Enzyme Kinetics), a graphical interface for building kinetic models of enzyme catalyzed reactions. ENZO automatically generates the corresponding differential equations from a stipulated enzyme reaction scheme. These differential equations are processed by a numerical solver and a regression algorithm which fits the coefficients of differential equations to experimentally observed time course curves. ENZO allows rapid evaluation of rival reaction schemes and can be used for routine tests in enzyme kinetics. It is freely available as a web tool, at http://enzo.cmm.ki.si.     

Modeling the kinetics of acylation of insulin using a recursive method for solving the systems of coupled differential equations

This paper describes a theoretical method for solving systems of coupled differential equations that describe the kinetics of complicated reaction networks in which a molecule having multiple reaction sites reacts irreversibly with multiple equivalents of a ligand (reagent). The members of the network differ in the number of equivalents of reagent that have reacted, and in the patterns of sites of reaction. A recursive algorithm generates series, asymptotic, and average solutions describing this kinetic scheme. This method was validated by successfully simulating the experimental data for the kinetics of acylation of insulin.     

Global Kinetic Explorer: A new computer program for dynamic simulation and fitting of kinetic data

We describe a new dynamic kinetic simulation program that allows multiple data sets to be fit simultaneously to a single model based on numerical integration of the rate equations describing the reaction mechanism. Unlike other programs that allow fitting based on numerical integration of rate equations, in the dynamic simulation rate constants, output factors, and starting concentrations of reactants can be scrolled while observing the change in the shape of the simulated reaction curves. Fast dynamic simulation facilitates the exploration of initial parameters that serve as the starting point for nonlinear regression in fitting data and facilitates exploration of the relationships between individual constants and observable reactions. The exploration of parameter space by dynamic simulation provides a powerful tool for learning kinetics and for evaluating the extent to which parameters are constrained by the data. This feature is critical to avoid overly complex models that are not supported by the data.     

Kinetics of binding the mRNA cap analogues to the translation initiation factor eIF4E under second-order reaction conditions

The kinetics of binding of five analogues of the 5'-mRNA cap, differing in size and electric charge, to the eukaryotic initiation factor eIF4E, at 20 degrees C, pH 7.2, and ionic strength of 150 mM, were measured, after mixing solutions of comparable concentrations of the reagents, in a stopped-flow spectrofluorimeter. The registered stopped-flow signals were fitted using an efficient software package, called Dyna Fit, based on a numerical solution of the kinetic rate equations for assumed reaction mechanisms. One-, two-, and three-step binding models were considered. The quality of fits for these models were compared using two statistical criteria: Akaike's Information Criterion and Bayesian Information Criterion. Based on resulting probabilities of the models, it was concluded that for all investigated ligands a one-step binding model has essentially no support in the experimental observations. Our conclusions were also analysed from the perspective of kinetic transients obtained for cap-eIF4E systems under the so called pseudo-first order reaction condition, which result in the linear correlation of the observed association rate constant with ligand concentration. The existence of such a linear correlation is usually considered as proof of a one-step binding mechanism. The kinetic and optical parameters, derived from fitting a two-step cap-binding model with the DynaFit, were used to simulate kinetic transients under pseudo-first order reaction conditions. It appeared that the observed association rate constants derived from these simulated transients are also linearly correlated with the ligand concentration. This indicated that these linear dependencies are not sufficient to conclude a one-step binding.     

Global parameter optimization for cardiac potassium channel gating models.

Quantitative ion channel model evaluation requires the estimation of voltage dependent rate constants. We have tested whether a unique set of rate constants can be reliably extracted from nonstationary macroscopic voltage clamp potassium current data. For many models, the rate constants derived independently at different membrane potentials are not unique. Therefore, our approach has been to use the exponential voltage dependence predicted from reaction rate theory (Stevens, C. F. 1978. Biophys. J. 22:295-306; Eyring, H., S. H. Lin, and S. M. Lin. 1980. Basic Chemical Kinetics. Wiley and Sons, New York) to couple the rate constants derived at different membrane potentials. This constrained the solution set of rate constants to only those that also obeyed this additional set of equations, which was sufficient to obtain a unique solution. We have tested this approach with data obtained from macroscopic delayed rectifier potassium channel currents in voltage-clamped guinea pig ventricular myocyte membranes. This potassium channel has relatively simple kinetics without an inactivation process and provided a convenient system to determine a globally optimized set of voltage-dependent rate constants for a Markov kinetic model. The ability of the fitting algorithm to extract rate constants from the macroscopic current data was tested using "data" synthesized from known rate constants. The simulated data sets were analyzed with the global fitting procedure and the fitted rate constants were compared with the rate constants used to generate the data. Monte Carlo methods were used to examine the accuracy of the estimated kinetic parameters. This global fitting approach provided a useful and convenient method for reliably extracting Markov rate constants from macroscopic voltage clamp data over a broad range of membrane potentials. The limitations of the method and the dependence on initial guesses are described.     

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.     

Inversion of Markov processes to determine rate constants from single-channel data.

The determination of rate constants from single-channel data can be very difficult, in part because the single-channel lifetime distributions commonly analyzed by experimenters often have a complicated mathematical relation to the channel gating mechanism. The standard treatment of channel gating as a Markov process leads to the prediction that lifetime distributions are exponential functions. As the number of states of a channel gating scheme increases, the number of exponential terms in the lifetime distribution increases, and the weights and decay constants of the lifetime distributions become progressively more complicated functions of the underlying rate constants. In the present study a mathematical strategy for inverting these functions is introduced in order to determine rate constants from single-channel lifetime distributions. This inversion is easy for channel gating schemes with two or fewer states of a given conductance, so the present study focuses on schemes with more states. The procedure is to derive explicit equations relating the parameters of the lifetime distribution to the rate constants of the scheme. Such equations can be derived using the equality between symmetric functions of eigenvalues of a matrix and sums over principle minors, as well as expressions for the moments, derivatives, and weights of a lifetime distribution. The rate constants are then obtained as roots to this system of equations. For a gating scheme with three sequential closed states and a single gateway state, exact analytical expressions were found for each rate constant in terms of the parameters of the three-exponential closed-time distribution. For several other gating schemes, systems of equations were found that could be solved numerically to obtain the rate constants. Lifetime distributions were shown to specify a unique set of real rate constants in sequential gating schemes with up to five closed or five open states. For kinetic schemes with multiple gating pathways, the analysis of simulated data revealed multiple solutions. These multiple solutions could be distinguished by examining two-dimensional probability density functions. The utility of the methods introduced here are demonstrated by analyzing published data on nicotinic acetylcholine receptors, GABA(A) receptors, and NMDA receptors.     

Kinetic Analysis of Ligand-Receptor Association Data that Display An Overshoot Phenomenon

Abstract

A binding overshoot was frequently observed in the time course of association of diazepam with rat brain membrane receptors shortly after the start of the interaction. Such time profiles most likely reflect the “receptor switch” mechanism, assuming an equilibrium between two forms of a receptor (R and R*) that possess different affinities to the ligand (L) in question. Similar effects could be caused by the presence of a slowly dissociating competitor. The kinetics of these mechanisms were verified by simulation of theoretical time courses. A computer program for simulation of the time course, and estimation of rate constants of the individual reaction steps, was developed and is described in this communication. It employs the Euler-Cauchy integration for simulation of theoretical time courses. Optimised estimates of the rate constants were computed by simultaneous random variation of parameters within a pre-set interval. Stable solutions can be obtained for this system, thus enabling evaluation of equilibrium constants defined by the model. The source code is available in Turbo-Pascal. It can be used, after re-writing the rate equations, for fitting of similar kinetic models to suitable experimental data.     

利用酶催化反应过程曲线确定动力学参数的新方法

本文提出了一个利用过程曲线确定酶催化反应动力学参数的新方法.利用这一方法,仅仅根据两条实验曲线就可以确定单底物酶催化反应的全部动力学参数,并且所有的图形都是(?)     

Understanding glucose transport by the bacterial phosphoenolpyruvate:glycose phosphotransferase system on the basis of kinetic measurements in vitro

The kinetic parameters in vitro of the components of the phosphoenolpyruvate:glycose phosphotransferase system (PTS) in enteric bacteria were collected. To address the issue of whether the behavior in vivo of the PTS can be understood in terms of these enzyme kinetics, a detailed kinetic model was constructed. Each overall phosphotransfer reaction was separated into two elementary reactions, the first entailing association of the phosphoryl donor and acceptor into a complex and the second entailing dissociation of the complex into dephosphorylated donor and phosphorylated acceptor. Literature data on the K(m) values and association constants of PTS proteins for their substrates, as well as equilibrium and rate constants for the overall phosphotransfer reactions, were related to the rate constants of the elementary steps in a set of equations; the rate constants could be calculated by solving these equations simultaneously. No kinetic parameters were fitted. As calculated by the model, the kinetic parameter values in vitro could describe experimental results in vivo when varying each of the PTS protein concentrations individually while keeping the other protein concentrations constant. Using the same kinetic constants, but adjusting the protein concentrations in the model to those present in cell-free extracts, the model could reproduce experiments in vitro analyzing the dependence of the flux on the total PTS protein concentration. For modeling conditions in vivo it was crucial that the PTS protein concentrations be implemented at their high in vivo values. The model suggests a new interpretation of results hitherto not understood; in vivo, the major fraction of the PTS proteins may exist as complexes with other PTS proteins or boundary metabolites, whereas in vitro, the fraction of complexed proteins is much smaller.     

Apoflavodoxin folding mechanism: an alpha/beta protein with an essentially off-pathway intermediate.

The folding reaction of Anabaena apoflavodoxin has been studied by stopped-flow kinetics and site-directed mutagenesis. Although the urea unfolding equilibrium is two-state, a transient intermediate accumulates during the folding reaction. The intermediate is monomeric, and it is not related to proline isomerization. Unlike many cases where the presence of an intermediate has been detected either by a burst phase or by the curvature, at low urea concentration, of the otherwise only observable kinetic phase, two kinetic phases are observed in apoflavodoxin folding whose total amplitude equals the amplitude of unfolding. To determine the role of the intermediate in the folding reaction, the apoflavodoxin kinetic data have been fitted to all conceivable three-species kinetic models (either linear or triangular). Using a stepwise fitting procedure, we find that the off-pathway mechanism explains most of the kinetic data (not a slow unfolding phase), the on-pathway mechanism being rejected. By using global analysis, good overall agreement between data and fit is found when a triangular mechanism is considered. The fitted values of the microscopic constants indicate that most of the unfolded molecules refold from the denatured state. Apoflavodoxin thus folds via a triangular, but essentially off-pathway, mechanism. We calculate that the retardation of the folding caused by the off-pathway intermediate is not large. Some unusual properties of the intermediate are discussed.     

Integrated steady state rate equations and the determination of individual rate constants.

Integrated steady state rate equations have been used to determine the kinetic constants (Vs, Ks, Vp, and Kp) and rate constants (k1, k2, k3, and k4) of the reversible enzyme mechanism: (see article). The fumarase reaction has been used as a model to illustrate the procedures for determining these constants. In contrast to initial velocity studies, the values of the constants have been obtained by examining the enzyme reaction in only one direction rather than in both forward and reverse directions. To accomplish this, a new procedure is described for fitting data to integrated rate equations which eliminates problems encountered when data are analyzed graphically. The advantages of examining on enzyme reaction in one direction with these new procedures allow this method to be extended to the examination of enzymes with simple mechanisms where initial velocities are difficult to measure because either the substrate or product is not readily available, or because the reaction is not readily reversible.     

Kinetic study of a substrate cycle involving a chemical step: highly amplified determination of phenolic compounds

A kinetic analysis of a substrate cycle in which one of the two steps was substituted by a chemical reaction has been made. The model is illustrated by the amplified determination, in a continuous assay, of phenolic compounds at low concentrations using the enzyme tyrosinase and β-NADH to reduce the o-quinone product of catalytic activity. Progress curves corresponding to β-NADH disappearance were not linear and followed first-order kinetics. Knowledge of the kinetics of the reaction has allowed us to achieve detection limits as low as 50 nM in a simple 10-min assay. There is no analytical solution to the non-linear differential equation system that describes the kinetics of the reaction, therefore, computer simulations of its dynamic behaviour are also presented, good agreement with the experimental results being obtained. The method is applicable to the measurement of any other metabolite, and its amplification capacity as well as the simplicity of determining kinetic parameters enable it to be implemented in a bioreactor for automation purposes.     

Re-interpretation of the logistic equation for batch microbial growth in relation to Monod kinetics

Aims:  To determine the underlying substrate utilization mechanism in the logistic equation for batch microbial growth by revealing the relationship between the logistic and Monod kinetics. Also, to determine the logistic rate constant in terms of Monod kinetic constants.
Methods and Results:  The logistic equation used to describe batch microbial growth was related to the Monod kinetics and found to be first-order in terms of the substrate and biomass concentrations. The logistic equation constant was also related to the Monod kinetic constants. Similarly, the substrate utilization kinetic equations were derived by using the logistic growth equation and related to the Monod kinetics.
Conclusion:  It is revaled that the logistic growth equation is a special form of the Monod growth kinetics when substrate limitation is first-order with respect to the substrate concentration. The logistic rate constant ( k ) is directly proportional to the maximum specific growth rate constant ( μ _m) and initial substrate concentration ( S ₀) and also inversely related to the saturation constant ( K _s).
Significance and Impact of the Study:  The semi-empirical logistic equation can be used instead of Monod kinetics at low substrate concentrations to describe batch microbial growth using the relationship between the logistic rate constant and the Monod kinetic constants.     

A population pharmacokinetic model with time-dependent covariates measured with errors

We propose a population pharmacokinetic (PK) model with time-dependent covariates measured with errors. This model is used to model S-oxybutynin's kinetics following an oral administration of Ditropan, and allows the distribution rate to depend on time-dependent covariates blood pressure and heart rate, which are measured with errors. We propose two two-step estimation methods: the second-order two-step method with numerical solutions of differential equations (2orderND), and the second-order two-step method with closed form approximate solutions of differential equations (2orderAD). The proposed methods are computationally easy and require fitting a linear mixed model at the first step and a nonlinear mixed model at the second step. We apply the proposed methods to the analysis of the Ditropan data, and evaluate their performance using a simulation study. Our results show that the 2orderND method performs well, while the 2orderAD method can yield PK parameter estimators that are subject to considerable biases.     

Metal reduction kinetics in Shewanella

MOTIVATION: Metal reduction kinetics have been studied in cultures of dissimilatory metal reducing bacteria which include the Shewanella oneidensis strain MR-1. Estimation of system parameters from time-series data faces obstructions in the implementation depending on the choice of the mathematical model that captures the observed dynamics. The modeling of metal reduction is often based on Michaelis-Menten equations. These models are often developed using initial in vitro reaction rates and seldom match with in vivo reduction profiles. RESULTS: For metal reduction studies, we propose a model that is based on the power law representation that is effectively applied to the kinetics of metal reduction. The method yields reasonable parameter estimates and is illustrated with the analysis of time-series data that describes the dynamics of metal reduction in S.oneidensis strain MR-1. In addition, mixed metal studies involving the reduction of Uranyl (U(VI)) to the relatively insoluble tetravalent form (U(IV)) by S. alga strain (BR-Y) were studied in the presence of environmentally relevant iron hydrous oxides. For mixed metals, parameter estimation and curve fitting are accomplished with a generalized least squares formulation that handles systems of ordinary differential equations and is implemented in Matlab. It consists of an optimization algorithm (Levenberg-Marquardt, LSQCURVEFIT) and a numerical ODE solver. Simulation with the estimated parameters indicates that the model captures the experimental data quite well. The model uses the estimated parameters to predict the reduction rates of metals and mixed metals at varying concentrations. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.     

Kinetics of nitrification and denitrification reactions

Nitrification and denitrification are important microbiological reactions of nitrogen. In this work, the kinetics of these reactions have been investigated based on a Monod-type expression involving two growth limiting substrates: ammonium nitrogen and dissolved oxygen for nitrification and nitrate nitrogen and dissolved organic carbon for denitrification. The kinetic constants and yield coefficients were evaluated for both these reactions. Past experimental work was used to determine the constants for the nitrification reaction. For the denitrification reaction, experiments were performed in a stirred tank reactor under conditions such that only one substrate was growth limiting. Steady-state values of the substrate concentrations in the reactor were determined at various dilution rates. These data were analyzed to obtain the kinetic and stoichiometric constants. From these constants it was concluded that in the range of nitrate nitrogen concentrations encountered in waste water, the denitrification reaction can be considered a first-order reaction. It was also found that three times as much organic carbon is required as nitrate nitrogen for complete nitrogen removal.     

Steady-state kinetics of the anaerobic reaction of soybean lipoxygenase-1 with linoleic acid and 13-L-hydroperoxylinoleic acid.

The steady-state kinetics of the anaerobic reaction of soybean lipoxygenase-1 with linoleic acid and 13-L-hydroperoxylinoleic acid were studied. Initial rates of the formation of oxodienoic acids**, absorbing at 285 nm, were measured at pH 10. About 50% of the consumed 13-L-hydroperoxylinoleic acid was converted into oxodienoic acids regardless of the initial ratio of the two substrates. A linear inhibition by both linoleic acid and 13-L-hydroperoxylinoleic acid was observed in the concentration range studied, which is on the upper side limited by the concentrations at which micelle- or acid-soap formation starts. A kinetic scheme is proposed based on one active site in lipoxygenase-1 which alternately binds the two substrates. Values for the kinetic constants were calculated by fitting simultaneously the complete set of data to the appropriate rate equation.     

Kinetics of the creatine kinase reaction in neonatal rabbit heart: an empirical analysis of the rate equation

Here we define the kinetics of the creatine kinase (CK) reaction in an intact mammalian heart containing the full range of CK isoenzymes. Previously derived kinetic constants [Schimerlik, M. I., & Cleland, W. W. (1973) J. Biol. Chem. 248, 8418-8423] were refit for the reaction occurring at 37 degrees C. Steady-state metabolite concentrations from 31P NMR and standard biochemical techniques were determined. 31P magnetization transfer data were obtained to determine unidirectional creatine kinase fluxes in hearts with differing total creatine contents and differing mitochondrial CK activities during KCl arrest and isovolumic work for both the forward reaction (MgATP synthesis) and reverse reaction (phosphocreatine synthesis). The NMR kinetic data and substrate concentration data were used in conjunction with a kinetic model based on MM-CK in solution to determine the applicability of the solution-based kinetic models to the CK kinetics of the intact heart. Our results indicated that no single set of rate equation constants could describe both the KCl-arrested and working hearts. We used our experimental data to constrain the solution-derived kinetic model and derived a second set of rate equation constants, which describe the isovolumic work state. Analysis of our results indicates that the CK reaction is rate limited in the direction of ATP synthesis, the size of the guanidino substrate pool drives the measured CK flux in the intact heart, and during isovolumic work the CK reaction operates under saturating conditions; that is, the substrate concentrations are at least 2-fold greater than the Km or Kim for each substrate.(ABSTRACT TRUNCATED AT 250 WORDS)