Planck’s radiation law provides an equation for the intensity of the electromagnetic radiation from a physical body as a function of frequency and temperature. The frequency that corresponds to the maximum intensity ...Planck’s radiation law provides an equation for the intensity of the electromagnetic radiation from a physical body as a function of frequency and temperature. The frequency that corresponds to the maximum intensity is a function of temperature. At a specific temperature, for the frequencies correspond to much less than the maximum intensity, an equation was derived in the form of the Lambert <em>W</em> function. Numerical calculations validate the equation. A new form of solution for the Euler’s transcendental equation was derived in the form of the Lambert <em>W</em> function with logarithmic argument. Numerical solutions to the Euler’s equation were determined iteratively and iterative convergences were investigated. Numerical coincidences with physical constants were explored.展开更多
Series of exponential equations in the form of were solved graphically, numerically and analytically. The analytical solution was derived in terms of Lambert-W function. A general numerical solution for any y is found...Series of exponential equations in the form of were solved graphically, numerically and analytically. The analytical solution was derived in terms of Lambert-W function. A general numerical solution for any y is found in terms of n or in base y. A solution is close to the fine structure constant. The equation which provided the solution as the fine structure constant was derived in terms of the fundamental constants.展开更多
Conventional analysis of enzyme-catalyzed reactions uses a set of initial rates of product formation or substrate decay at a variety of substrate concentrations. Alternatively to the conventional methods, attempts hav...Conventional analysis of enzyme-catalyzed reactions uses a set of initial rates of product formation or substrate decay at a variety of substrate concentrations. Alternatively to the conventional methods, attempts have been made to use an integrated Michaelis-Menten equation to assess the values of the Michaelis-Menten KM and turnover kcat constants directly from a single time course of an enzymatic reaction. However, because of weak convergence, previous fits of the integrated Michaelis-Menten equation to a single trace of the reaction have no proven records of success. Here we propose a reliable method with fast convergence based on an explicit solution of the Michaelis-Menten equation in terms of the Lambert-W function with transformed variables. Tests of the method with stopped-flow measurements of the catalytic reaction of cytochrome c oxidase, as well as with simulated data, demonstrate applicability of the approach to de termine KM and kcat constants free of any systematic errors. This study indicates that the approach could be an alternative solution for the characterization of enzymatic reactions, saving time, sample and efforts. The single trace method can greatly assist the real time monitoring of enzymatic activity, in particular when a fast control is mandatory. It may be the only alternative when conventional analysis does not apply, e.g. because of limited amount of sample.展开更多
文摘Planck’s radiation law provides an equation for the intensity of the electromagnetic radiation from a physical body as a function of frequency and temperature. The frequency that corresponds to the maximum intensity is a function of temperature. At a specific temperature, for the frequencies correspond to much less than the maximum intensity, an equation was derived in the form of the Lambert <em>W</em> function. Numerical calculations validate the equation. A new form of solution for the Euler’s transcendental equation was derived in the form of the Lambert <em>W</em> function with logarithmic argument. Numerical solutions to the Euler’s equation were determined iteratively and iterative convergences were investigated. Numerical coincidences with physical constants were explored.
文摘Series of exponential equations in the form of were solved graphically, numerically and analytically. The analytical solution was derived in terms of Lambert-W function. A general numerical solution for any y is found in terms of n or in base y. A solution is close to the fine structure constant. The equation which provided the solution as the fine structure constant was derived in terms of the fundamental constants.
文摘Conventional analysis of enzyme-catalyzed reactions uses a set of initial rates of product formation or substrate decay at a variety of substrate concentrations. Alternatively to the conventional methods, attempts have been made to use an integrated Michaelis-Menten equation to assess the values of the Michaelis-Menten KM and turnover kcat constants directly from a single time course of an enzymatic reaction. However, because of weak convergence, previous fits of the integrated Michaelis-Menten equation to a single trace of the reaction have no proven records of success. Here we propose a reliable method with fast convergence based on an explicit solution of the Michaelis-Menten equation in terms of the Lambert-W function with transformed variables. Tests of the method with stopped-flow measurements of the catalytic reaction of cytochrome c oxidase, as well as with simulated data, demonstrate applicability of the approach to de termine KM and kcat constants free of any systematic errors. This study indicates that the approach could be an alternative solution for the characterization of enzymatic reactions, saving time, sample and efforts. The single trace method can greatly assist the real time monitoring of enzymatic activity, in particular when a fast control is mandatory. It may be the only alternative when conventional analysis does not apply, e.g. because of limited amount of sample.
