Complex Nonlinear Regression

Complex Rate Equations Complex rate equations may require individual treatment, although the examples in Fig. 7-2 are aU hn-earizable. A perfectly general procedure is nonlinear regression. For instance, when r =f(C, a, b,. . . ) where a,h,. . , ) are the constants to be found, the condition is... 

All these methods give similar results but their sensitivities and resolutions are different. For example, UV-Vis spectrophotometry gives good results if a single colorant or mixture of colorants (with different absorption spectra) were previously separated by SPE, ion pair formation, and a good previous extraction. Due to their added-value capability, HPLC and CE became the ideal techniques for the analysis of multicomponent mixtures of natural and synthetic colorants found in drinks. To make correct evaluations in complex dye mixtures, a chemometric multicomponent analysis (PLS, nonlinear regression) is necessary to discriminate colorant contributions from other food constituents (sugars, phenolics, etc.). 

A major limitation of the linearized forms of the Michaelis-Menten equation is that none provides accurate estimates of both Km and Vmax. Furthermore, it is impossible to obtain meaningful error estimates for the parameters, since linear regression is not strictly appropriate. With the advent of more sophisticated computer tools, there is an increasing trend toward using the integrated rate equation and nonlinear regression analysis to estimate Km and While this type of analysis is more complex than the linear approaches, it has several benefits. First, accurate nonbiased estimates of Km and Vmax can be obtained. Second, nonlinear regression may allow the errors (or confidence intervals) of the parameter estimates to be determined. 

Martin H) has written a perceptive analysis of the possible ways in which an ionized species may behave in various models and contribute to or be responsible for a given activity. QSAR studies that have dealt with ion-pair partitioning include a study of fibrinolytics ( ) and the effect of benzoic acids on the K ion flux in mollusk neurons ( ). Schaper (10) recently reanalyzed a large number of absorption studies to include terms for the absorption of ionized species. Because specific values were not available for log Pj, he let the relation between log Pi and log P be a parameter in a nonlinear regression analysis. In most cases he used the approximation that the difference between the two values is a constant in a given series. This same assumption was made in the earlier studies (, ) Our work suggests that the pKa of an acid can influence this differential (see below). The influence of structure on the log P of protonated bases or quaternary ammonium compounds is much more complex (11,12) and points out the desirability of being able to easily measure these values. 

Nonlinear regression analysis of a plot of /x or Y against [L] provides the association constant KA, and the electrophoretic mobility of the pure complex /jbSL from the experimental data. /x5 corresponds to the substrate mobility at [L] = 0. [L] can be approximated as the added ligand concentration only when the ligand concentration is much greater than the solute concentration or when the binding constant is small. 

Dansyl-amino acids HP-/3 (DS 6.3) Nonlinear regression 59 pH dependence, pKa shift during complexation... 

Similar equations can be written for both enantiomers of chiral analyte. Based on Eq. (14), nonlinear regression techniques allow one to determine the enantioselective binding constants (KR and Ks) and the mobilities of related transient diastereomeric complexes (/4°mplex and /x ""plex). 

Fio. 2. Proposed kinetic model for yeast inorganic pyrophosphatase. Here M represents Mg + but may also apply to any divalent cation with which the enzyme is active. In the rate equation A represents all mono-magnesium PPi complexes, B represents the di-magnesium complex, and I represents free PPi. Hydrogen ion equilibria are not considered. Kinetic runs were done at pH 7.4, 30° (9). Best values for kinetic constants were obtained from a computer program for nonlinear regression (9S). 

Complex nonlinear least-squares regression (CNLS)... 

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