Rate equations linearization, 49 Michaelis parameters

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. 

Figure 7. The Michaelis Menten rate equation as a function of substrate concentration S (in arbitrary units). Parameters are Km 1 and Vm 1. A A linear plot. B A semilogarithmic plot. At a concentration S Km, the rate attains half its maximal value Vm.

Figure 27. Interpretation of the saturation parameter. Shown is a Michaelis Menten rate equation (solid line) and the corresponding saturation parameter d (dashed line). For small substrate concentration S Km the reaction acts in the linear regime. For increasing concentrations the saturation parameter d ... 

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. 

This equation is known as the Michaelis-Menton equation. A plot of rate r versus substrate concentration [S] (Figure 2.20) shows that the rate equation follows first-order kinetics fp = (k/kuAS] at low substrate concentrations and zero-order kinetics r = k ed. high substrate concentrations. The tangent to versus [S] plot drawn at [S] = 0 intersects r = k at a point corresponding to [S] = k - This method of tangent can be used to determine the kinetic parameters k and k - Alternatively, we can write the rate equation in linear form as (l/fp) = kM/k) l/[S]) + (1/k) by inverting Equation 2.169. Thus, by making a linear plot of... 

However, the above equations will only give reasonable values for the parameters in the unlikely case when the reaction is both essentially irreversible and not subject to product inhibition. Much more complex equations with many more parameters have to be used fora fit of the whole time course of the reaction to a realistic mechanism. There are hazards, too, in the determination of true initial rates. It is necessary to check that the change in substrate concentration during the period of initial rate measurement is sufficiently small so that the observed rate is the correct one for each specified value of Cs(0). After a preliminary estimate for and one can readily calculate whether these parameters would cause a significant deviation from true linearity of initial rates, that is adherence to the steady state condition because of the difference in substrate concentration at the beginning and end of the measurement. This must not be left to the appearance of the initial slope. It is also important, for the correct evaluation of the Michaelis parameters, to heed the advice of Dowd Riggs (1965) that the range of Cg(0) must straddle both sides of by a factor of 10. 

A plot of 1 /Vp versus 1/S produces a straight line with slope Km/Vp and y intercept 1 /Vp (Fig. 7.22), Thus, the linear form of the Michaelis-Menten equation allows estimation of the so-called adjustable parameters (Segel, 1976). The adjustable parameters include Km (in units of concentration) and Vp (in units of product quantity per unit surface per unit time, or quantity of product per unit time), which denotes maximum rate of product formation. 

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. 

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