Nonlinear regression Michaelis-Menten equation

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. 

Hyperbolic curve fits to control enzymatic data and to data obtained in the presence of a competitive inhibitor. Curve fitting to the Michaelis-Menten equation results in two different values for Km- However, Km does not, in actuality, change, and the value in the presence of inhibitor (15 uiM) is an apparent value. Fitting with the correct equation, that for turnover in the presence of a competitive inhibitor ( Eq. 5), results in plots identical in appearance to those obtained with the Michaelis-Menten equation. However, nonlinear regression now reveals that Km remains constant at 5 ulM and that [l]/Ki = 2.5 with knowledge of [/], calculation of K is straightforward... 

It is apparent that O Eq. 5 is a variation of the MichaeUs-Menten equation. The inhibitor data shown in O Figure 4-7 can instead be fitted to O Eq. 5, holding Km (and Vjnax) constant to their control values (5 pM and 20 nmol/min/mg, respectively). The curve obtained is identical to that fitted with the Michaelis-Menten equation (O Figure 4-7), but nonlinear regression now yields the information that K, of the inhibitor equals 40% of the concentration at which it was included in the assay to obtain the best-fit curve. In other words, if the concentration of inhibitor present in the experiment shown in O Figure 4-7 was 25 pM, the Ki for the inhibitor is 10 pM. 

Another approach for the determination of the kinetic parameters is to use the SAS NLIN (NonLINear regression) procedure (SAS, 1985) which produces weighted least-squares estimates of the parameters of nonlinear models. The advantages of this technique are that (1) it does not require linearization of the Michaelis-Menten equation, (2) it can be used for complicated multiparameter models, and (3) the estimated parameter values are reliable because it produces weighted least-squares estimates. 

These hyperbolic equations are analogous to the Michaelis-Menten equation. Nonlinear regression is preferable to the method proposed in the 1960s by Kitz and Wilson, which necessitates a double-reciprocal linear transformation of the data (analogous to a Lineweaver-Burk plot) that can bias the estimates of /clnact and A). 

Spreadsheet Summary The second exercise in Chapter 13 of Applications of Microsoft Excel in Analytical Chemistry involves enzyme catalysis. A linear transformation is made so that the Michaelis constant, K, and the maximum velocity, can be determined from a least-squares procedure. The nonlinear regression method is used with Excel s Solver to find these parameters by fitting them into the nonlinear Michaelis-Menten equation. 

Characteristic Quantities In principle, the quantities and ro,max that characterize an enzyme can be determined by directly fitting the Michaelis-Menten equation to the measured data using computer supported methods of nonlinear regression. We can simplify the analysis in a manner suggested by Hans Lineweaver and Dean Burke in 1934 by linearizing the relation. In order to do this, we must find the reciprocal of the Michaelis-Menten equation. After transforming, we have ... 

The kinetic parameters in Equations 4.21 and 4.22 can be determined from experimental data using nonlinear regression techniques. Nevertheless, these equations can be simplified by considering the excess concentration of one of the substrates. For example, at high values of [52], the reaction rate can be simplified to a Michaelis-Menten equation form. 

Although hnearized plot are useful for viewing the data, the best way of analyzing such kinetic data is to fit them directly to the Michaelis-Menten equation by using nonlinear regression, because today there are computers to do this. 

Fig. 12 Kinetics of deltamethrin hydrolysis by hCE-1 and hydrolase A. Velocity was measured by the amount of PBald (phenoxy benzaldehyde) released during the reaction. Data symbols) were fit to the Michaelis-Menten equation and the nonlinear regression lines plotted. Each point represents the mean S.D. n = 3). This figure is published with permission (Godin et al. 2006)...

In former days, before electronic caelulations came into existance all calibrations and evaluations had to be carried out manually by graphical methods Linearized solutions had been used instead of nonlinear regression. Lineweaver and Burk 73) derived the linearized equation (19). It was introduced for calibration purposes to TLC by Kufner and Schlegel74). Kaiserf,3), Hulpke and Stegh 64) also used calibration techniques with reciprocal transformations of R and m without reference to the Michaelis-Menten transformation. 

Kinetic parameters are calculated with Prism 4.00 (Graph Pad Software, Inc., San Diego, CA), using nonlinear regression of the Michaebs-Menten equation. Representative Michaelis-Menten kinetics of SN-38 glucuronidation by UGTlAls are shown in Fig. 2. 

A nonlinear regression analysis can be applied to the relation between 7sjim and [Mred] to evaluate kcat and K- separately. However, the parameter / may be assumed to be zero at the first approximation even with this assumption, the maximum error of 7s um is only about 5% around [Mred] /- M 4—5 [9]. Equation 5 can then be rewritten as a Michaelis-Menten type expression ... 

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