Alternative algebraic rate‐integration approach for progress‐curve analysis of enzyme kinetics

A recent article of Zavrel et al. in this journal (Eng. Life Sci. 2010, 10, 191–200) described a comparison of several computer programs for progress‐curve analysis with respect to different computational approaches for parameter estimation. The authors applied both algebraic and dynamic parameter estimations, although they omitted time‐course analysis through the integrated rate equation. Recently, it was demonstrated that progress‐curve analysis through the integrated rate equation can be considered a simple and useful alternative for enzymes that obey the generalized Michaelis–Menten reaction mechanism. To complete this gap, the time‐dependent solution of the generalized Michaelis–Menten equation is here fitted to the progress curves from the Zavrel et al. reference article. This alternative rate‐integration approach for determining the kinetics parameters of Michaelis–Menten‐type enzymes yields the values with the greatest accuracy, as compared with the results obtained by other (algebraic or dynamic) parameter estimations.     

Effects of periodic input on the quasi-steady state assumptions for enzyme-catalysed reactions

In this paper we investigate the validity of a quasi-steady state assumption in approximating Michaelis-Menten type kinetics for enzyme-catalysed biochemical reactions that are subject to periodic substrate input.Acknowledgement This work was funded by the EPSRC Grant GR/R53098. JLL is supported by the Scottish Executive Environment and Rural Affairs Department.     

Perspectives of data analysis of enzyme inhibition and activation,Part 3: Equations for calculation of the initial rates of enzymatic reactions

Equations for calculation of the initial rates of activated enzymatic reactions and the inhibited enzymatic reactions, unavailable in experimental enzymology, were obtained. Examples of practically using of these equations are given. © 2009 Wiley Periodicals, Inc. J Biochem Mol Toxicol 23:108–118, 2009; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jbt.20271     

Quasi-steady-state kinetics at enzyme and substrate concentrations in excess of the Michaelis-Menten constant

In vitro enzyme reactions are traditionally conducted under conditions of pronounced substrate excess since this guarantees that the bound enzyme is at quasi-steady-state (QSS) with respect to the free substrate, thereby justifying the Briggs-Haldane approximation (BHA). In contrast, intracellular reactions, amplification assays, allergen digestion assays and industrial applications span a range of enzyme-to-substrate ratios for which the BHA is invalid, including the extreme of enzyme excess. The quasi-equilibrium approximation (QEA) is valid for a subset of enzyme excess states. Previously, we showed that the total QSSA (tQSSA) overlaps and extends the validity of the BHA and the QEA, and that it is at least roughly valid for any total substrate and enzyme concentrations. The analysis of the tQSSA is hampered by square root nonlinearity. Previous simplifications of the tQSSA rate law are valid in a parameter domain that overlaps the validity domains of the BHA and the QEA and only slightly extends them. We now integrate the tQSSA rate equation in closed form, without resorting to further approximations. Moreover, we introduce a complimentary simplification of the tQSSA rate law that is valid in states of enzyme excess when the absolute difference between total enzyme and substrate concentrations greatly exceeds the Michaelis-Menten constant. This includes a wide range of enzyme and substrate concentrations where both the BHA and the QEA are invalid and allows us to define precisely the conditions for zero-order and first-order product formation. Remarkably, analytical approximations provided by the tQSSA closely match the expected stochastic kinetics for as few as 15 reactant molecules, suggesting that the conditions for the validity of the tQSSA and for its various simplifications are also of relevance at low molecule numbers.     

Analysis of progress curves of enzymatic reactions with unstable species

We present a general kinetic analysis of the Michaelis-Menten mechanism for the case in which the enzyme, the enzyme-substrate complex and the product are unstable. The kinetic data analysis which we suggest is based on the time progress curves of the product and/or on the time progress curve of the species into which the immediate product is transformed. This analysis, carried out under conditions of limiting enzyme concentration, allows the determination of the rate and equilibrium constants if adequate experimental results are available. We illustrate the method numerically by computer simulation of the reaction with added experimental errors.     

An algebraic model to determine substrate kinetic parameters by global nonlinear fit of progress curves

We propose that the time course of an enzyme reaction following the Michaelis-Menten reaction mechanism can be conveniently described by a newly derived algebraic equation, which includes the Lambert Omega function. Following Northrop's ideas [Anal. Biochem.321, 457–461, 1983], the integrated rate equation contains the Michaelis constant (K_M) and the specificity number (k_S≡k_cat/K_M$k_{\text{S}}\equiv k_{\text{cat}}/K_{\text{M}}$) as adjustable parameters, but not the turnover number k_cat. A modification of the usual global-fit approach involves a combinatorial treatment of nominal substrate concentrations being treated as fixed or alternately optimized model parameters. The newly proposed method is compared with the standard approach based on the “initial linear region” of the reaction progress curves, followed by nonlinear fit of initial rates to the hyperbolic Michaelis-Menten equation. A representative set of three chelation-enhanced fluorescence EGFR kinase substrates is used for experimental illustration. In one case, both data analysis methods (linear and nonlinear) produced identical results. However, in another test case, the standard method incorrectly reported a finite (50–70 μM) K_M value, whereas the more rigorous global nonlinear fit shows that the K_M is immeasurably high.     

Determination of accurate KI values for tight-binding enzyme inhibitors: an in silico study of experimental error and assay design

Determination of accurate K(I) values for tight-binding enzyme inhibitors is important from both a basic biochemistry point of view (understanding the differences in affinity of related molecules) and a medicinal chemistry vantage (developing structure-activity relationships (SAR)). It is advantageous to directly fit the quadratic equation describing tight-binding behavior, known commonly as the Morrison equation, to obtain these K(I) values. The results of simulated experiments that examine the effect of assay design and experimental error on the ability to accurately determine K(I) values at several [E]0/K(I-app) ratios are described. Input ("true") values of the uninhibited velocity, inhibition constant, and total enzyme concentration were used to calculate the velocity at various inhibitor concentrations. Gaussian error was introduced into the velocities and the simulated reactions were fit to estimate upsilon0, K(I), and [E]0. Recommendations for optimizing the inhibitor dilutions within the context of a 96-well-plate format and simple serial dilution steps are made. These include using three points to determine the enzyme concentration ([I]=0, 0.5[E]0, and [E]0), using a narrow dilution series with only two or three points to determine the asymptote at high inhibitor concentration, and avoiding fixing [E]0 to a constant value in the fitting if at all possible. The risks and rewards of fixing [E]0 to a constant value, especially the effect on SAR, are also examined.     

The comparison of the estimation of enzyme kinetic parameters by fitting reaction curve to the integrated Michaelis-Menten rate equations of different predictor variables

The estimation of enzyme kinetic parameters by nonlinear fitting reaction curve to the integrated Michaelis-Menten rate equation ln(S(0)/S)+(S(0)-S)/K(m)=(V(m)/K(m))xt was investigated and compared to that by fitting to (S(0)-S)/t=V(m)-K(m)x[ln(S(0)/S)/t] (Atkins GL, Nimmo IA. The reliability of Michaelis-Menten constants and maximum velocities estimated by using the integrated Michaelis-Menten equation. Biochem J 1973;135:779-84) with uricase as the model. Uricase reaction curve was simulated with random absorbance error of 0.001 at 0.075 mmol/l uric acid. Experimental reaction curve was monitored by absorbance at 293 nm. For both CV and deviation <20% by simulation, K(m) from 5 to 100 micromol/l was estimated with Eq. (1) while K(m) from 5 to 50 micromol/l was estimated with Eq. (2). The background absorbance and the error in the lag time of steady-state reaction resulted in negative K(m) with Eq. (2), but did not affect K(m) estimated with Eq. (1). Both equations gave better estimation of V(m). The computation time and the goodness of fit with Eq. (1) were 40-fold greater than those with Eq. (2). By experimentation, Eq. (1) yielded K(m) consistent with the Lineweaver-Burk plot analysis, but Eq. (2) gave many negative parameters. Apparent K(m) by Eq. (1) linearly increased, while V(m) were constant, vs. xanthine concentrations, and the inhibition constant was consistent with the Lineweaver-Burk plot analysis. These results suggested that the integrated rate equation that uses the predictor variable of reaction time was reliable for the estimation of enzyme kinetic parameters and applicable for the characterization of enzyme inhibitors.     

A method to describe enzyme-catalyzed reactions by combining steady state and time course enzyme kinetic parameters

Background

Complete analysis of single substrate enzyme-catalyzed reactions has required a separate use of two distinct approaches. Steady state approximations are employed to obtain substrate affinity and initial velocity information. Alternatively, first order exponential decay models permit simulation of the time course data for the reactions. Attempts to use integrals of steady state equations to describe reaction time courses have so far met with little success.

Methods

Here we use equations based on steady state approximations to directly model time course plots.

Results

Testing these expressions with the enzyme β-galactosidase, which adheres to classical Michaelis–Menten kinetics, produced a good fit between observed and calculated values.

General significance

This study indicates that, in addition to providing information on initial kinetic parameters, steady state approximations can be employed to directly model time course kinetics.Integrated forms of the Michaelis–Menten equation have previously been reported in the literature. Here we describe a method to directly apply steady state approximations to time course analysis for predicting product formation and simultaneously obtain multiple kinetic parameters.     

A semi-integrated method for the determination of enzyme kinetic parameters and graphical representation of the Michaelis-Menten equation

A semi-integrated method for the determination of the enzyme kinetics parameters (Km and V) and graphical representation of the Michaelis-Menten equation is proposed as a variation of determination of initial reaction rate (v) as a function of initial substrate concentration ([S]0). The method is based on the determination of the time required to exhaust half of the initial substrate concentration as a function of the initial substrate concentration. The advantages and limitations of this method are discussed.     

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.     

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.     

Inadequacy of Rubisco initial and total activities to account for observed rates of photosynthetic carbon dioxide assimilation by Scenedesmus ecornis

Both initial and total activity of ribulose-1,5, bisphosphate carboxylase/oxygenase (Rubisco) measured for the green alga Scenedesmus ecornis are affected by the experimental procedure and they are not sufficiently high to account for the rates of ¹⁴C fixation by photosynthesis. The very low β-carboxylase activities detected (less than 3% of the Rubisco total activity) cannot explain the difference in CO₂ fixation. Attempts to obtain possible optimal conditions (pH, duration of activation with Mg²⁺ and HCO^- ₃, absence of proteases, linearity of ¹⁴C fixation with time) did not lead to increased activity yields. The substrate ribulose-1,5-bisphosphate was found to decrease the initial activity at concentrations higher than 25 μM for algae harvested by centrifugation and having thus experienced several minutes of darkness. Deactivation seems to be primarily responsible for this loss of activity. Furthermore, initial and total activities decrease when the delay before freezing increases, suggesting accumulation of an inhibitor from the light-dark transition metabolism during the first minutes of harvesting.     

A generalized numerical approach to steady-state enzyme kinetics: Applications to protein kinase inhibition

A generalized numerical treatment of steady-state enzyme kinetics is presented. This new approach relies on automatic computer derivation of the underlying mathematical model (a system of simultaneous nonlinear algebraic equations) from a symbolic representation of the reaction mechanism (a system of biochemical equations) provided by the researcher. The method allows experimental biochemists to analyze initial-rate enzyme kinetic data, under the steady-state approximation, without having to use any mathematical equations. An illustrative example is based on the inhibition kinetics of p56^lck kinase by an ATP competitive inhibitor. A computer implementation of the new method, in the modified software package DYNAFIT [Kuzmi?, P. (1996) Anal. Biochem. 237, 260–273], is freely available to all academic researchers.     

A developmental approach to the problem of variation in evolutionary rates

Certain general features are widely recognized in evolution, one of which is the variability in the rate at which morphological characters evolve and taxa are replaced by others. Although some rate-variability in evolution no doubt arises because of different rates of ecological change, it is proposed that some of the variability also arises from developmental, rather than ecological, sources. A theory is outlined whereby early-acting genes influencing the course of development evolve more slowly, but have individually larger effects, than genes affecting development at a later stage in the life-cycle. The erratic course of morphological evolution that results is illustrated by computer simulation. It is suggested that the applicability of the theory is restricted to long-term evolution and that variability in the rate of evolution over shorter periods may be of an entirely different nature.     

A simple hierarchical approach to modeling distributions of substitution rates

Genetic sequence data typically exhibit variability in substitution rates across sites. In practice, there is often too little variation to fit a different rate for each site in the alignment, but the distribution of rates across sites may not be well modeled using simple parametric families. Mixtures of different distributions can capture more complex patterns of rate variation, but are often parameter-rich and difficult to fit. We present a simple hierarchical model in which a baseline rate distribution, such as a gamma distribution, is discretized into several categories, the quantiles of which are estimated using a discretized beta distribution. Although this approach involves adding only two extra parameters to a standard distribution, a wide range of rate distributions can be captured. Using simulated data, we demonstrate that a "beta-" model can reproduce the moments of the rate distribution more accurately than the distribution used to simulate the data, even when the baseline rate distribution is misspecified. Using hepatitis C virus and mammalian mitochondrial sequences, we show that a beta- model can fit as well or better than a model with multiple discrete rate categories, and compares favorably with a model which fits a separate rate category to each site. We also demonstrate this discretization scheme in the context of codon models specifically aimed at identifying individual sites undergoing adaptive or purifying evolution.     

A comparison of methods for fitting allometric equations to field metabolic rates of animals

We re-examined data for field metabolic rates of varanid lizards and marsupial mammals to illustrate how different procedures for fitting the allometric equation can lead to very different estimates for the allometric coefficient and exponent. A two-parameter power function was obtained in each case by the traditional method of back-transformation from a straight line fitted to logarithms of the data. Another two-parameter power function was then generated for each data-set by non-linear regression on values in the original arithmetic scale. Allometric equations obtained by non-linear regression described the metabolic rates of all animals in the samples. Equations estimated by back-transformation from logarithms, on the other hand, described the metabolic rates of small species but not large ones. Thus, allometric equations estimated in the traditional way for field metabolic rates of varanids and marsupials do not have general importance because they do not characterize rates for species spanning the full range in body size. Logarithmic transformation of predictor and response variables creates new distributions that may enable investigators to perform statistical analyses in compliance with assumptions underlying the tests. However, statistical models fitted to transformations should not be used to estimate parameters of equations in the arithmetic domain because such equations may be seriously biased and misleading. Allometric analyses should be performed on values expressed in the original scale, if possible, because this is the scale of interest.