Kinetic investigation of a solvent-free, chemoenzymatic reaction sequence towards enantioselective synthesis of a β-amino acid ester

A solvent-free, chemoenzymatic reaction sequence for the enantioselective synthesis of β-amino acid esters has been kinetically and thermodynamically characterized. The coupled sequence comprises a thermal aza-Michael addition of cheap starting materials and a lipase catalyzed aminolysis for the kinetic resolution of the racemic ester. Excellent ee values of >99% were obtained for the β-amino acid ester at 60% conversion. Kinetic constants for the aza-Michael addition were obtained by straightforward numerical integration of second-order rate equations and nonlinear fitting of the progress curves. A different strategy had to be devised for the biocatalytic reaction. Initially, a simplified Michaelis-Menten model including product inhibition was developed for the reaction running in THF as an organic solvent. Activity based parameters were used instead of concentrations in order to facilitate the transfer of the kinetic model to the solvent-free system. Observed solvent effects not accounted for by the use of thermodynamic activities were incorporated into the kinetic model. Enzyme deactivation was observed to depend on the ratio of the applied substrates and also included in the kinetic model. The developed simple model is in very good agreement with the experimental data and allows the simulation and optimization of the solvent-free process.     

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.     

Parameter identification of the calvin photosynthesis cycle

Summary This article is concerned with the determination of kinetic parameters of the Calvin photosynthesis cycle which is described by seventeen nonlinear ordinary differential equations. It is shown that the task requires dynamic data for several sets of initial conditions. The numerical technique is based upon an algorithm for non-linear optimization and Gear's numerical integration scheme for stiff systems of differential equations. The sensitivity of the parameters to noise in the data is tested with a method adapted from Rosenbrook and Storey. A preliminary set of parameters has been obtained from a preliminary set of experimental data. The numerical methods are then tested with synthetic data derived from these parameters. The mathematical model and the results obtained in the simulation are used as an aid in designing new experiments.     

A century of enzyme kinetic analysis, 1913 to 2013

This review traces the history and logical progression of methods for quantitative analysis of enzyme kinetics from the 1913 Michaelis and Menten paper to the application of modern computational methods today. Following a brief review of methods for fitting steady state kinetic data, modern methods are highlighted for fitting full progress curve kinetics based upon numerical integration of rate equations, including a re-analysis of the original Michaelis–Menten full time course kinetic data. Finally, several illustrations of modern transient state kinetic methods of analysis are shown which enable the elucidation of reactions occurring at the active sites of enzymes in order to relate structure and function.     

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.     

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.     

Analysis of progress curves by simulations generated by numerical integration.

A highly flexible computer program written in FORTRAN is presented which fits computer-generated simulations to experimental progress-curve data by an iterative non-linear weighted least-squares procedure. This fitting procedure allows kinetic rate constants to be determined from the experimental progress curves. Although the numerical integration of the rate equations by a previously described method [Barshop, Wrenn & Frieden (1983) Anal. Biochem. 130, 134-145] is used here to generate predicted curves, any routine capable of the integration of a set of differential equations can be used. The fitting program described is designed to be widely applicable, easy to learn and convenient to use. The use, behaviour and power of the program is explored by using simulated test data.     

A mathematical model of actin filament turnover for fitting FRAP data

A novel mathematical model of the actin dynamics in living cells under steady-state conditions has been developed for fluorescence recovery after photobleaching (FRAP) experiments. As opposed to other FRAP fitting models, which use the average lifetime of actins in filaments and the actin turnover rate as fitting parameters, our model operates with unbiased actin association/dissociation rate constants and accounts for the filament length. The mathematical formalism is based on a system of stochastic differential equations. The derived equations were validated on synthetic theoretical data generated by a stochastic simulation algorithm adapted for the simulation of FRAP experiments. Consistent with experimental findings, the results of this work showed that (1) fluorescence recovery is a function of the average filament length, (2) the F-actin turnover and the FRAP are accelerated in the presence of actin nucleating proteins, (3) the FRAP curves may exhibit both a linear and non-linear behaviour depending on the parameters of actin polymerisation, and (4) our model resulted in more accurate parameter estimations of actin dynamics as compared with other FRAP fitting models. Additionally, we provide a computational tool that integrates the model and that can be used for interpretation of FRAP data on actin cytoskeleton.     

Competitive and uncompetitive inhibitors simultaneously acting on an autocatalytic zymogen activation reaction

The time course of the residual enzyme activity of a general model consisting of an autocatalytic zymogen activation process inhibited by an irreversible competitive inhibitor and an irreversible uncompetitive inhibitor has been studied. Approached analytical expressions which furnish the time course of the residual enzyme activity from the onset of the reaction depending on the rate constants and initial concentration have been obtained. The goodness and limitations of the analytical equations were checked by comparing with the results obtained from the numerical integration, i.e. with the simulated progress curves. A dimensionless parameter giving the relative contributions of both the activation and the inhibitions routes is suggested, so that the value of this parameter determines whether the activation or the inhibitions routes prevail or if both processes are balanced during the time for which the analytical expressions are valid. The effects of the initial zymogen, free enzyme and inhibitors concentrations are analysed. Finally an experimental design and kinetic data analysis is proposed to evaluate simultaneously the kinetic parameters involved and to discriminate between different zymogen activation processes which can be considered particular cases of the general model.     

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.     

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.     

The kinetic pathway of folding of barnase

To search for folding intermediates, we have examined the folding and unfolding kinetics of wild-type barnase and four representative mutants under a wide range of conditions that span two-state and multi-state kinetics. The choice of mutants and conditions provided in-built controls for artifacts that might distort the interpretation of kinetics, such as the non-linearity of kinetic and equilibrium data with concentration of denaturant. We measured unfolding rate constants over a complete range of denaturant concentration by using by 1H/2H-exchange kinetics under conditions that favour folding, conventional stopped-flow methods at higher denaturant concentrations and continuous flow. Under conditions that favour multi-state kinetics, plots of the rate constants for unfolding against denaturant concentration fitted quantitatively to the equation for three-state kinetics, with a sigmoid component for a change of rate determining step, as did the refolding kinetics. The position of the transition state on the reaction pathway, as measured by solvent exposure (the Tanford beta value) also moved with denaturant concentration, fitting quantitatively to the same equations with a change of rate determining step. The sigmoid behaviour disappeared under conditions that favoured two-state kinetics. Those data combined with direct structural observations and simulation support a minimal reaction pathway for the folding of barnase that involves two detectable folding intermediates. The first intermediate, I(1), is the denatured state under physiological conditions, D(Phys), which has native-like topology, is lower in energy than the random-flight denatured state U and is suggested by molecular dynamics simulation of unfolding to be on-pathway. The second intermediate, I(2), is high energy, and is proven by the change in rate determining step in the unfolding kinetics to be on-pathway. The change in rate determining step in unfolding with structure or environment reflects the change in partitioning of this intermediate to products or starting materials.     

FitSpace Explorer: An algorithm to evaluate multidimensional parameter space in fitting kinetic data

Fitting several sets of kinetic data directly to a model based on numerical integration provides the best method to extract kinetic parameters without relying on the simplifying assumptions required to achieve analytical solutions of rate equations. However, modern computer programs make it too easy to enter an overly complex model, and standard error analysis grossly underestimates errors when a system is underconstrained and fails to reveal the full degree to which multiple parameters are linked through the complex relationships common in kinetic data. Here we describe the application of confidence contour analysis obtained by measuring the dependence of the sum square error on each pair of parameters while allowing all remaining parameters to be adjusted in seeking the best fit. The confidence contours reveal complex relationships between parameters and clearly outline the space over which parameters can vary (the “FitSpace”). The utility of the method is illustrated by examples of well-constrained fits to published data on tryptophan synthase and the kinetics of oligonucleotide binding to a ribozyme. In contrast, analysis of alanine racemase clearly refutes claims that global analysis of progress curves can be used to extract the free energy profiles of enzyme-catalyzed reactions.     

Moment arms and musculotendon lengths estimation for a three-dimensional lower-limb model

This paper presents a set of polynomial expressions that can be used as regression equations to estimate length and three-dimensional moment arms of 43 lower-limb musculotendon actuators. These equations allow one to find, at a low computational cost, the musculotendon geometric parameters required for numerical simulation of large musculoskeletal models. Nominal values for these biomechanical parameters were established using a public-domain musculoskeletal model of the lower limb (IEEE Trans. Biomed. Eng. 37 (1990) 757). To fit these nominal values, regression equations with different levels of complexity were generated, based on the number of generalized coordinates of the joints spanned by each musculotendon actuator. Least squares fitting was used to identify regression equation coefficients. The goodness of the fit and confidence intervals were assessed, and the best fitting equations selected.     

Full-scale model of glycolysis in Saccharomyces cerevisiae.

We present a powerful, general method of fitting a model of a biochemical pathway to experimental substrate concentrations and dynamical properties measured at a stationary state, when the mechanism is largely known but kinetic parameters are lacking. Rate constants and maximum velocities are calculated from the experimental data by simple algebra without integration of kinetic equations. Using this direct approach, we fit a comprehensive model of glycolysis and glycolytic oscillations in intact yeast cells to data measured on a suspension of living cells of Saccharomyces cerevisiae near a Hopf bifurcation, and to a large set of stationary concentrations and other data estimated from comparable batch experiments. The resulting model agrees with almost all experimentally known stationary concentrations and metabolic fluxes, with the frequency of oscillation and with the majority of other experimentally known kinetic and dynamical variables. The functional forms of the rate equations have not been optimized.     

Computer simulation of purine metabolism

A computer model of purine metabolism, including catabolism, salvage pathways and interconversion among nucleotides, is given. Steady-state rate equations corresponding to metabolic enzymes are written based on information from the literature about their kinetic behaviour. Numerical integration of this set of equations is performed employing selected parameters taken from the literature. After stabilization of purine compound concentrations is reached, simulation of enzyme deficit and enzyme overproduction is carried out. The latter is calculated by varying specified maximum velocities in the numerical integration. A pattern of intermediate metabolite concentrations is found. These results form a basis for the comparison of normal patterns or patterns reflecting the effects of inborn errors of metabolism. The aim of this paper is to demonstrate the usefulness of this computer simulation method in complex metabolism pathways.     

Analysis of three- and four-compartment models forin vivo radioligand-neuroreceptor interaction

Currently applied three-copartment models for analyzing kinetic data derived fromin vivo positron emission tomographic (PET) studies of radioligand-neuroreceptor interactions require assumptions which may not be strictly valid. Such assumptions include very rapid kinetics for nonspecific binding and the absence of multiple specific receptors or subtypes. Computer simulations, based on an exact analytical solution of the relevant differential equations, indicate the numerical errors that can arise when the assumptions are invalid. We propose a fourcompartment model which requires fewer assumptions. A simple relationship is derived for expressing the microscopic rate constants of either the three- or four-compartment model as explicit functions of the experimentally-observed macroscopic rate constants. This could eliminate the need for time-consuming, iterative, non-linear, curve-fitting approaches and numerical integration. The usefulness of the four-compartment model is limited, however, by the sensitivity and temporal resolution of current PET imaging devices.     

Production of biodiesel by lipase-catalyzed transesterification of vegetable oils: a kinetics study

Kinetics of production of biodiesel by enzymatic methanolysis of vegetable oils using lipase has been investigated. A mathematical model taking into account the mechanism of the methanolysis reaction starting from the vegetable oil as substrate, rather than the free fatty acids, has been developed. The kinetic parameters were estimated by fitting the experimental data of the enzymatic reaction of sunflower oil by two types of lipases, namely, Rhizomucor miehei lipase (RM) immobilized on ion-exchange resins and Thermomyces lanuginosa lipase (TL) immobilized on silica gel. There was a good agreement between the experimental results of the initial rate of reaction and those predicted by the proposed model equations, for both enzymes. From the proposed model equations, the regions where the effect of alcohol inhibition fades, at different substrate concentrations, were identified. The proposed model equation can be used to predict the rate of methanolysis of vegetable oils in a batch or a continuous reactor and to determine the optimal conditions for biodiesel production.     

Kinetic analysis of enzyme reactions with slow-binding inhibition.

In this paper we present a general kinetic study of slow-binding inhibition processes, i.e. enzyme reactions that do not respond instantly to the presence of a competitive inhibitor. The analysis that we present is based on the equation that describes the formation of products with time in each case on the experimental progress curve. It is carried out under the condition of limiting enzyme concentration and allows the discrimination between the different cases of slow-binding inhibition. The mechanism in which the formation of complex enzyme-inhibitor is a single or two slow steps or follow a rapid equilibrium, has been considered. The corresponding explicit equations of each case have been obtained and checked by numerical integration. A kinetic data analysis to evaluate the corresponding kinetic parameters is suggested. We illustrate the method, numerically by computer simulation, of the reaction and present some numerical examples that demonstrate the applicability of our procedure.