Regression Michaelis-Menten equation

Note the close analogy with the Lineweaver-Burk form of the simple Michaelis-Menten equation. In a diagram representing MV against MX one obtains a line which has the same intercept as in the simple case. The slope, however, is larger by a factor (1 + YIK-) as shown in Fig. 39.17b. Usually, one first determines and in the absence of a competitive inhibitor (F = 0), as described above. Subsequently, one obtains A" from a new set of experiments in which the initial rate V is determined for various levels of X in the presence of a fixed amount of inhibitor Y. The slope of the new line can be obtained by means of robust regression. 

Non-linear models, such as described by the Michaelis-Menten equation, can sometimes be linearized by a suitable transformation of the variables. In that case they are called intrinsically linear (Section 11.2.1) and are amenable to ordinary linear regression. This way, the use of non-linear regression can be obviated. As we have pointed out, the price for this convenience may have to be paid in the form of a serious violation of the requirement for homoscedasticity, in which case one must resort to non-parametric methods of regression (Section 12.1.5). 

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. 

However, the further increases in the substrate concentration to 15mM decreased the initial reaction rate. This behavior may be due to substrate or product inhibition. Since the Michaelis-Menten equation does not incorporate the inhibition effects, we can drop the last two data points and limit the model developed for the low substrate concentration range only (Cs < lOmM). Figure 2.8 shows the three plots prepared from the given data. The two data points which were not included for the linear regression were noted as closed circles. 

Table 2.4 shows the SAS NLIN specifications and the computer output. You can choose one of the four iterative methods modified Gauss-Newton, Marquardt, gradient or steepest-descent, and multivariate secant or false position method (SAS, 1985). The Gauss-Newton iterative methods regress the residuals onto the partial derivatives of the model with respect to the parameters until the iterations converge. You also have to specify the model and starting values of the parameters to be estimated. It is optional to provide the partial derivatives of the model with respect to each parameter, b. Figure 2.9 shows the reaction rate versus substrate concentration curves predicted from the Michaelis-Menten equation with parameter values obtained by four different... 

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). 

In the first approach, we examine the rate progress curves at various substrate concentrations, and use linear regression to evaluate initial rates. These initial rates are then fitted to the Michaelis-Menten equation (Eqn. 9.14) (Exercise 3 Michaelis-Menten kinetics I). This method has the advantage of being simple and robust. It has the disadvantage that the choice of data points used to obtain initial rates is often arbitrary, and also that the progress curves at low substrate concentrations show marked curvature because of substrate depletion. 

Cornish—Bowden—Eisenthal Method.9 This method is distinct from the previous three linear regression methods, in that each pair of (v, [S]) values is used to construct a separate line on a plot in which VW and Km form the y and x axes, respectively. Beginning with another version of the Michaelis-Menten equation, in which Vmax is the y value and Km is the x value as shown in Eq. 2.23,... 

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 ... 

Before the advent of computer technology and computational methods, the linear transformations of the Michaelis-Menten equation were extensively used for the calculation of kinetic parameters (Allison Puiich, 1979) with the aid of a linear transformation of rectangular hyperbola, one can calculate with precision the asymptotes (Kmax and Kjd by linear regression (Fig. 5). The merits of various transformations were estimated with respect to the statistical bias inherent in most linear transformations of the Michaelis-Menten equation (Wilkinson, 1961 Johanson Lumry, 1961 Johanson Faunt, 1992 Straume Johnson, 1992 Ritchie Prvan, 1996). The detailed statistical treatment of initial rate data, however, is presented in Chapter 18. 

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. 

The initial rate of BDNPP cleavage, determined at pH 10.4 as a function of substrate concentration, reveals typical saturation behavior (Fig. 5.21). The data were fitted by non-linear regression, using the Michaelis-Menten equation (Eq. 2.5, Chap. 2). Here = 9.4 2.1 mM, v ax = 3.76 x 10 M s with... 

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 ... 

Even though linearization methods are valuable tools for determining the mechanism of inhibition, once determined, kinetic parameters can better be evaluated by non-linear regression to the corresponding rate equations, as presented in section 3.2.2 for simple Michaelis-Menten kinetics. 

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