Linearized Form of the Michaelis-Menten Equation

A Linear Form of the Michaelis-Menten Equation Is Used to Determine... 

The direct measurement of the numeric value of and therefore the calculation of often requires im-practically high concentrations of substrate to achieve saturating conditions. A linear form of the Michaelis-Menten equation circumvents this difficulty and permits and to be extrapolated from initial velocity data obtained at less than saturating concentrations of substrate. Starting with equation (29),... 

A linear form of the Michaelis-Menten equation simplifies determination of and V. ... 

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. 

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. 

Although Km may be determined using data similar to that presented in Figure II-8, it is more accurate and convenient to use one of the linear forms of the Michaelis-Menten equation to determine Km- Lineweaver and Burk first pointed out that equation II-8 can be obtained by inversion of the Michaelis-Menten equation ... 

Bioorganic Synthesis Engineering 655 Table 20.4 Linearized forms of the Michaelis-Menten equation... 

Because any linear equation can be expressed in terms of its slope (m) and vertical intercept (Z ), such that y = mx + b, the double-reciprocal form of the Michaelis-Menten equation yields a slope of a vertical-... 

This is in the form of the Michaelis-Menten equation with being divided by the Thus, for a simple linear non-competitive inhibitor, Km remains unchanged while that factor (1 + (//, )). mu s altered so... 

The linear response range of the glucose sensors can be estimated from a Michaelis-Menten analysis of the glucose calibration curves. The apparent Michaelis-Menten constant KMapp can be determined from the electrochemical Eadie-Hofstee form of the Michaelis-Menten equation, i = i - KMapp(i/C), where i is the steady-state current, i is the maximum current, and C is the glucose concentration. A plot of i versus i/C (an electrochemical Eadie-Hofstee plot) produces a straight line, and provides both KMapp (-slope) and i (y-intercept). The apparent Michaelis-Menten constant characterizes the enzyme electrode, not the enzyme itself. It provides a measure of the substrate concentration range over which the electrode response is approximately linear. A summary of the KMapp values obtained from this analysis is shown in Table I. 

Another method to obtain estimates for Km and is the rearrangement of the Michaelis-Menten equation to a linear form. The estimation for the initial velocities, Vo, from progress curves is not a particularly reliable method. A better way to estimate Vn is by the integrated Michaelis-Menten equation (Cornish-Bowden, 1975). Nevertheless, the graphical methods are popular among enzymolo-gists. The three most common linear transformations of the Michaelis-Menten equation are the Lineweaver-Burk plot of 1/Vo vs. 1/[S] (sometimes called the double-reciprocal plot), the Eadie-Hofstee plot, i.e. v vs. vo/[S], and the Hanes plot, i.e., [SJ/vo vs. [S] (Fig. 9.3). 

The linear response range of sensors was estimated from a Adichaelis-Menten analysis of the glucose calibration curves in Figure 4. The apparent Nfichaelis-Menten constant Kj pp can be determined from the electrochemical Eadie-Hofstee form of the Michaelis-Menten equation. ... 

For a constant-volume BR, integration of the Michaelis-Menten equation leads to a form that can also be linearized. Thus, from equation 10.2-9,... 

To deduce the enzyme parameters, fit the laboratory data using a variation of the Michaelis-Menten equation (Eq. 17-81) in which that hyperbolic equation is inverted to yield a linear form ... 

Even this scheme represents a complex situation, for ES can be arrived at by alternative routes, making it impossible for an expression of the same form as the Michaelis-Menten equation to be derived using the general steady-state assumption. However, types of non-competitive inhibition consistent with the Michaelis-Menten type equation and a linear Linweaver-Burk plot can occur if the rapid-equilibrium assumption is valid (Appendix S.A3). In the simplest possible model, involving simple linear non-competitive inhibition, the substrate does not affect the inhibitor binding. Under these conditions, the reactions... 

In Equation 11.13, A = k2k3[E]0[S]0/(k2 + k3) / [S]0 + k3Ks/(k2 + k3), which has the form of a Michaelis-Menten equation, B = [E]0[S]0 /(fe + 3) 2/([S]0+ Km(apparent)), and b is a composite rate constant describing the build-up of the acyl enzme intermediate (or, in the general case, the covalently bound enzyme intermediate). The non-linear plot of [Lg ] against time is shown in Fig. 11.10A for a typical substrate of a-chymotrypsin extrapolation of the linear portion gives the intercept shown which allows evaluation of B. 

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,... 

In the preceding sections, we have shown that all the rate equations, in the presence of a competitive, noncompetitive, or an uncompetitive inhibitor, have a form of a Michaelis-Menten equation, and can be linearized in the Lineweaver-Burk manner, in the fashion of Hanes, or in the form of Dixon plots ... 

LINEARIZED FORM OF THE INTEGRATED MICHAELIS-MENTEN (MM) EQUATION... 

Usually, one plots the initial rate V against the initial amount X, which produces a hyperbolic curve, such as shown in Fig. 39.17a. The rate and amount at time 0 are larger than those at any later time. Hence, the effect of experimental error and of possible deviation from the proposed model are minimal when the initial values are used. The Michaelis-Menten equation can be linearized by taking reciprocals on both sides of eq. (39.114) (Section 8.2.13), which leads to the so-called Lineweaver-Burk form ... 

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). 

Linearized Form of the Integrated Michaelis-Menten Equation... 

After rearranging Eq. (2), the values of ixDCD and KB can be estimated by nonlinear least squares curve-fitting methods (similar to the Michaelis-Menten equation) or the expression can be rearranged under different linear forms (y = rnx I n), where y = (jueff yiD) and x = [CD], Well known are... 

There are still other causes of nonlinearities than (apparent or real) higher-order transformation kinetics. In Section 12.3 we discussed catalyzed reactions, especially the enzyme kinetics of the Michaelis-Menten type (see Box 12.2). We may also be interested in the modeling of chemicals which are produced by a nonlinear autocatalytic reaction, that is, by a production rate function, p(Q, which depends on the product concentration, C,. Such a production rate can be combined with an elimination rate function, r(C,), which may be linear or nonlinear and include different processes such as flushing and chemical transformations. Then the model equation has the general form ... 

It is very useful to transform the Michaelis-Menten equation into a linear form for analyzing data graphically and detecting deviations from the ideal behavior. One of the best known methods is the double-reciprocal or Lineweaver-Burk plot. Inverting both sides of equation 3.1 and substituting equation 3.2 gives the Lineweaver-Burk plot 4... 

The most straightforward way is to plot r against Cs as shown in Figure 2.2. The asymptote for r will be rmax and KM is equal to Cs when r = 0.5 rmax. However, this is an unsatisfactory plot in estimating rmax and KM because it is difficult to estimate asymptotes accurately and also difficult to test the validity of the kinetic model. Therefore, the Michaelis-Menten equation is usually rearranged so that the results can be plotted as a straight line. Some of the better known methods are presented here. The Michaelis-Menten equation, Eq. (2.11), can be rearranged to be expressed in linear form. This can be achieved in three ways ... 

The linearized form of the uncompetitive Michaelis-Menten equation is given by taking its inverse with respect to reaction velocity (V ) and concentration of S, giving... 

When the substrate concentration is large, the reaction rate is dependent on the substrate concentration. This represents zero-order kinetic behavior. When the concentration is very low, then the kinetics may be represented by first-order kinetic behavior. At Jr = Jrmax/2, the value of the Michaelis constant KM is obtained as S. The Michaelis-Menten equation can be linearized, and the Lineweaver-Burk plot (Figure 8.3) is obtained from the following form... 

There are many ways of estimating the parameters KM and Vmax. Most biochemists have used plots derived by transforming the Michaelis-Menten equation into linear forms, e.g. ... 

Which of these plots should be used To generally understand the behavior of enzymes, use the simple graph of initial velocity against substrate concentration. The linearized forms are useful for calculation of ATM and Fmax. The Lineweaver-Burke plot is useful for distinguishing between types of inhibition (Chapter 8). The Eadie-Hofstee plot is better than the Lineweaver-Burke plot at picking up deviations from the Michaelis-Menten equation. 

It is sometimes useful to transform the Michaelis-Menten equation into a linear form. One of the best known is the Lineweaver-Burk plot. 

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