Improved non-parametric statistical methods for the estimation of Michaelis-Menten kinetic parameters by the direct linear plot.
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ABSTRACT:
SUBMITTER: Porter WR
PROVIDER: S-EPMC1164507 | biostudies-other | 1977 Feb
REPOSITORIES: biostudies-other
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Project description:Examining enzyme kinetics is critical for understanding cellular systems and for using enzymes in industry. The Michaelis-Menten equation has been widely used for over a century to estimate the enzyme kinetic parameters from reaction progress curves of substrates, which is known as the progress curve assay. However, this canonical approach works in limited conditions, such as when there is a large excess of substrate over enzyme. Even when this condition is satisfied, the identifiability of parameters is not always guaranteed, and often not verifiable in practice. To overcome such limitations of the canonical approach for the progress curve assay, here we propose a Bayesian approach based on an equation derived with the total quasi-steady-state approximation. In contrast to the canonical approach, estimates obtained with this proposed approach exhibit little bias for any combination of enzyme and substrate concentrations. Importantly, unlike the canonical approach, an optimal experiment to identify parameters with certainty can be easily designed without any prior information. Indeed, with this proposed design, the kinetic parameters of diverse enzymes with disparate catalytic efficiencies, such as chymotrypsin, fumarase, and urease, can be accurately and precisely estimated from a minimal amount of timecourse data. A publicly accessible computational package performing such accurate and efficient Bayesian inference for enzyme kinetics is provided.
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Project description:The statistical implications of the direct linear plot for enzyme kinetic data, described in the preceding paper (Eisenthal & Cornish-Bowden, 1974), are discussed for the case of the Michaelis-Menten equation. The plot is shown to lead directly to non-parametric confidence limits for the kinetic parameters, V and K(m), which depend on far less sweeping assumptions about the nature of experimental error than those implicit in the method of least squares. Median estimates of V and K(m) can also be defined, which are shown to be more robust than the least-squares estimates in a wide variety of experimental situations.
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Project description:[This corrects the article on p. 39 in vol. 26.].
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Project description:Biochemical transformation kinetics is based on the formation of enzyme-substrate complexes. We developed a robust scheme based on unit productions of enzymes and reactants in cyclic events to comply with mass action law to form enzyme-substrate complexes. The developed formalism supports a successful application of Michaelis-Menten kinetics in all biochemical transformations of single parameters. It is an essential tool to overcome some challenging healthcare and environmental issues. In developing the formalism, we defined the substrate [S]= [Product]3/4 and rate of reaction based on rate and time perspectives. It allowed us to develop two quadratic equations. The first, represents a body entity that gave a useful relationship of enzyme E?=?2S0.33, and the second nutrients/feed, each giving [Enzymes] and [Enzyme-substrate complexes], simulating rate of reaction, [substrate], and their differentials. By combining [Enzymes] and [Enzyme-substrate complexes] values, this quadratic equation derives a Michaelis-Menten hyperbolic function. Interestingly, we can derive the proportionate rate of reaction and [Enzymes] values of the quadratics resulting in another Michaelis-Menten hyperbolic. What is clear from these results is that between these two hyperbolic functions, in-competitive inhibitions exist, indicating metabolic activities and growth in terms of energy levels. We validated these biochemical transformations with examples applicable to day to day life.
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Project description:The aim was to elucidate how steps in drug translocation by a solute carrier transporter impact Michaelis-Menten parameters Km, Ki, and Vmax. The first objective was to derive a model for carrier-mediated substrate translocation and perform sensitivity analysis with regard to the impact of individual microrate constants on Km, Ki, and Vmax. The second objective was to compare underpinning microrate constants between compounds translocated by the same transporter. Equations for Km, Ki, and Vmax were derived from a six-state model involving unidirectional transporter flipping and reconfiguration. This unidirectional model is applicable to co-transporter type solute carriers, like the apical sodium-dependent bile acid transporter (ASBT) and the proton-coupled peptide cotransporter (PEPT1). Sensitivity analysis identified the microrate constants that impacted Km, Ki, and Vmax. Compound comparison using the six-state model employed regression to identify microrate constant values that can explain observed Km and Vmax values. Results yielded some expected findings, as well as some unanticipated effects of microrate constants on Km, Ki, and Vmax. Km and Ki were found to be equal for inhibitors that are also substrates. Additionally, microrate constant values for certain steps in transporter functioning influenced Km and Vmax to be low or high.
| S-EPMC4204736 | biostudies-literature
Project description:Nearly 100 years ago Michaelis and Menten published their now classic paper [Michaelis, L., and Menten, M. L. (1913) Die Kinetik der Invertinwirkung. Biochem. Z. 49, 333-369] in which they showed that the rate of an enzyme-catalyzed reaction is proportional to the concentration of the enzyme-substrate complex predicted by the Michaelis-Menten equation. Because the original text was written in German yet is often quoted by English-speaking authors, we undertook a complete translation of the 1913 publication, which we provide as Supporting Information . Here we introduce the translation, describe the historical context of the work, and show a new analysis of the original data. In doing so, we uncovered several surprises that reveal an interesting glimpse into the early history of enzymology. In particular, our reanalysis of Michaelis and Menten's data using modern computational methods revealed an unanticipated rigor and precision in the original publication and uncovered a sophisticated, comprehensive analysis that has been overlooked in the century since their work was published. Michaelis and Menten not only analyzed initial velocity measurements but also fit their full time course data to the integrated form of the rate equations, including product inhibition, and derived a single global constant to represent all of their data. That constant was not the Michaelis constant, but rather V(max)/K(m), the specificity constant times the enzyme concentration (k(cat)/K(m) × E(0)).
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Project description:A new plot is described for analysing the results of kinetic experiments in which the Michaelis-Menten equation is obeyed. Observations are plotted as lines in parameter space, instead of points in observation space. With appropriate modifications the plot is applicable to most problems of interest to the enzyme kineticist. It has the following advantages over traditional methods of plotting kinetic results: it is very simple to construct, because it is composed entirely of straight lines and requires no calculation or mathematical tables; the kinetic constants are read off the plot directly, again without calculation; it may be used during the course of an experiment to judge the success of the experiment, and to modify the experimental design; it provides clear and accurate information about the quality of the observations, and identifies aberrant observations; it provides a clear indication of the precision of the kinetic constants; constructed with care, it provides unbiased estimates of the kinetic constants, the same as those provided by a computer program; it may be used to simulate results for illustrative purposes very rapidly and simply.
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Project description:Nonlinear ordinary differential equations (ODEs) are common in pharmacokinetic-pharmacodynamic systems. Although their exact solutions cannot generally be determined via algebraic methods, their rapid and accurate solutions are desirable. Thus, numerical methods have a critical role. Inductive Linearization was proposed as a method to solve systems of nonlinear ODEs. It is an iterative approach that converts a nonlinear ODE into a linear time-varying (LTV) ODE, for which a range of standard integration techniques can then be used to solve (e.g., eigenvalue decomposition [EVD]). This study explores the properties of Inductive Linearization when coupled with EVD for integration of the LTV ODE and illustrates how the efficiency of the method can be improved. Improvements were based on three approaches, (1) incorporation of a convergence criterion for the iterative linearization process (for simulation and estimation), (2) creating more efficient step sizes for EVD (for simulation and estimation), and (3) updating the initial conditions of the Inductive Linearization (for estimation). The performance of these improvements were evaluated using single subject stochastic simulation-estimation with an application to a simple pharmacokinetic model with Michaelis-Menten elimination. The reference comparison was a standard non-stiff Runge-Kutta method with variable step size (ode45, MATLAB). Each of the approaches improved the speed of the Inductive Linearization technique without diminishing accuracy which, in this simple case, was faster than ode45 with comparable accuracy in the parameter estimates. The methods described here can easily be implemented in standard software programme such as R or MATLAB. Further work is needed to explore this technique for estimation in a population approach setting.
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Project description:An analytic solution to enzyme kinetics expressed by the Michaelis-Menten-Monod mathematical framework is presented. The analytic solution describes the implicit problem with the independent variable, normally time, substituted with the concentration of the reaction product. The analytic solution provides the substrate, enzyme, and microbial biomass concentration instantaneously and over all time domains without the use of numerical integration schemes or iterative solvers required to overcome transcendental functions. Experiments of NO2 - nitrification by Candidatus Nitrospira defluvii at temperatures ranging between 10 and 32 °C were used for validation tests with the numerical solution by finite differences and the implicit analytic solution presented here. Results showed that both finite differences and analytic solutions matched the experiments particularly well, with a correlation coefficient greater than 0.99 and residuals smaller than 2.75%.
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Project description:The Michaelis-Menten equation provides a hundred-year-old prediction by which any increase in the rate of substrate unbinding will decrease the rate of enzymatic turnover. Surprisingly, this prediction was never tested experimentally nor was it scrutinized using modern theoretical tools. Here we show that unbinding may also speed up enzymatic turnover--turning a spotlight to the fact that its actual role in enzymatic catalysis remains to be determined experimentally. Analytically constructing the unbinding phase space, we identify four distinct categories of unbinding: inhibitory, excitatory, superexcitatory, and restorative. A transition in which the effect of unbinding changes from inhibitory to excitatory as substrate concentrations increase, and an overlooked tradeoff between the speed and efficiency of enzymatic reactions, are naturally unveiled as a result. The theory presented herein motivates, and allows the interpretation of, groundbreaking experiments in which existing single-molecule manipulation techniques will be adapted for the purpose of measuring enzymatic turnover under a controlled variation of unbinding rates. As we hereby show, these experiments will not only shed first light on the role of unbinding but will also allow one to determine the time distribution required for the completion of the catalytic step in isolation from the rest of the enzymatic turnover cycle.
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