Expressions of the Michaelis-Menten equation when studying enzyme reactions in a variable-volume medium.
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ABSTRACT:
SUBMITTER: Gingold MP
PROVIDER: S-EPMC1168447 | biostudies-other | 1974 Dec
REPOSITORIES: biostudies-other
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Project description:Many chemical reactions in biological cells occur at very low concentrations of constituent molecules. Thus, transcriptional gene-regulation is often controlled by poorly expressed transcription-factors, such as E.coli lac repressor with few tens of copies. Here we study the effects of inherent concentration fluctuations of substrate-molecules on the seminal Michaelis-Menten scheme of biochemical reactions. We present a universal correction to the Michaelis-Menten equation for the reaction-rates. The relevance and validity of this correction for enzymatic reactions and intracellular gene-regulation is demonstrated. Our analytical theory and simulation results confirm that the proposed variance-corrected Michaelis-Menten equation predicts the rate of reactions with remarkable accuracy even in the presence of large non-equilibrium concentration fluctuations. The major advantage of our approach is that it involves only the mean and variance of the substrate-molecule concentration. Our theory is therefore accessible to experiments and not specific to the exact source of the concentration fluctuations.
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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 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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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)).
| S-EPMC3381512 | biostudies-literature
Project description:BackgroundStandard pediatric growth curves cannot be used to impute missing height or weight measurements in individual children. The Michaelis-Menten equation, used for characterizing substrate-enzyme saturation curves, has been shown to model growth in many organisms including nonhuman vertebrates. We investigated whether this equation could be used to interpolate missing growth data in children in the first three years of life and compared this interpolation to several common interpolation methods and pediatric growth models.MethodsWe developed a modified Michaelis-Menten equation and compared expected to actual growth, first in a local birth cohort (N = 97) then in a large, outpatient, pediatric sample (N = 14,695).ResultsThe modified Michaelis-Menten equation showed excellent fit for both infant weight (median RMSE: boys: 0.22 kg [IQR:0.19; 90% < 0.43]; girls: 0.20 kg [IQR:0.17; 90% < 0.39]) and height (median RMSE: boys: 0.93 cm [IQR:0.53; 90% < 1.0]; girls: 0.91 cm [IQR:0.50;90% < 1.0]). Growth data were modeled accurately with as few as four values from routine well-baby visits in year 1 and seven values in years 1-3; birth weight or length was essential for best fit. Interpolation with this equation had comparable (for weight) or lower (for height) mean RMSE compared to the best performing alternative models.ConclusionsA modified Michaelis-Menten equation accurately describes growth in healthy babies aged 0-36 months, allowing interpolation of missing weight and height values in individual longitudinal measurement series. The growth pattern in healthy babies in resource-rich environments mirrors an enzymatic saturation curve.
| S-EPMC10832211 | biostudies-literature
Project description:The Michaelis-Menten equation was fitted to simulated data containing different sorts of error by using the three linear transformations, and the methods of S. R. Cohen [Anal. Biochem. (1968) 22, 549-552], R. Eisenthal & A. Cornish-Bowden [Biochem. J. (1974) 139, 715-120], F. de M. Merino [Biochem. J. 143, 93-95] and G. N. Wilkinson [Biochem. J. (1961) 808 324-332). The best methods were those of Eisenthal & Cornish-Bowden (1974) and Wilkinson (1961).
| S-EPMC1165686 | biostudies-other
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: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%.
| S-EPMC6640786 | biostudies-literature
Project description:1. Experimental progress curves were simulated for a reaction obeying Michaelis-Menten kinetics. 2. K(m) and V were estimated (a) by fitting the integrated Michaelis-Menten equation to the progress curves, and (b) from the initial slopes of the curves (i.e. from initial velocities). 3. The integrated equation could not be fitted successfully by a non-linear method, so it was transformed and fitted by a linear method. 4. Provided that the initial substrate concentration was greater than K(m) and the data were precise enough, the integrated equation gave parameter estimates which were unbiased and as reliable as those derived from initial velocities although based on fewer experiments. 5. The integrated equation could be used for progress curves of unknown origin.
| S-EPMC1165894 | biostudies-other
Project description:The methods of Atkins & Nimmo (1973) and Fernley (1974) for fitting the integrated Michaelis-Menten equation were compared by using the same sets of simulated experimental data. The method of Fernley (1974) is to be preferred because it gives precise and unbiased estimates of the Michaelis-Menten parameters over a wide range of substrate concentrations. However, the estimates may not be symmetrically distributed, especially at low substrate concentrations.
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