Transformations of the Michaelis-Menten equation

Graphical transformation of the representation of enzyme kinetics is useful as the value of V max is impossible to obtain directly from practical measurements. A series of graphical transformations/linearisations may be used to overcome this problem. Lineweaver and Burk (see reference(,7)) simply inverted the Michaelis-Menten equation (equation 5.10). Thus  

A plot — against - gives a straight line graph with an intercept of —— and a slope y8o S V max 

This transformation suffers from a number of disadvantages. The data are reciprocals of measurements, and small experimental errors can lead to large errors in the graphically determined values of K, , especially at low substrate concentrations. Departures from linearity are also less obvious than on other kinetic plots such as the Eadie-Hofstee and Hanes plots (see reference 7 ). 

If the Lineweaver-Burk double-reciprocal equation (equation 5.17) is multiplied by Vmax throughout then, after rearrangement, the following equation is obtained  

The Eadie-Hofstee plot of Mo against Mo/S has an intercept of Vmax on the Mo axis, while on the other axis the intercept is Vmax/Km (Fig. 5.13). 

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Confirming that curved lines are the nemesis of the biochemist, at least three or more different transformations of the Michaelis-Menten equation have been invented (actually four)—each one of which took two people to accomplish (Fig. 8-5). The purpose of these plots is to allow you to determine the values of Km and VmaK with nothing but a ruler and a piece of paper and to allow professors to take a straightforward question about the Michaelis-Menten expression and turn it upside down (and/or backward). You might think that turning a backward quantity like Km upside down would make everything simpler—somehow it doesn t work that way. 

Initial values for a non-linear fit of Eq. (1) can be achieved by linearizations. Most conventional linearizations result from the transformation of the Michaelis-Menten equation, and are plotted according to ... 

This transformation of the Michaelis-Menten equation for a one-substrate enzyme can be written as ... 

Also referred to as the Hanes-Hultin plot and the Hanes-Woolf (or, Woolf-Hanes) plot, the method is based on a transformation of the Michaelis-Menten equation i.e., the expression for the Uni Uni mechanism) [A]/v = (i a/ max) + ([A]/Umax) whcrc U ax IS the maximum forward velocity and is the Michaelis constant for A. In the Hanes plot, the slope of the line is numerically equal to Umax, the vertical intercept is equivalent to, ... 

Transformations of the Michaelis-Menten Equation The Double-Reciprocal Plot... 

Other transformations of the Michaelis-Menten equation have been derived, each with some particular advantage in analyzing enzyme kinetic data. (See Problem 11 at the end of this chapter.)... 

The Eadie-Hofstee Equation One transformation of the Michaelis-Menten equation is the Lineweaver-Burk, or double-reciprocal, equation. Multiplying both sides of the Lineweaver-Burk equation by Umax and rearranging gives the Eadie-Hofstee equation ... 

One common transformation of the Michaelis-Menten equation is the double-reciprocal plot of Lineweaver and Burk, which is obtained by taking the reciprocal of both sides of Equation (5.24) to yield... 

Enzyme kinetics and the mode of inhibition are well described by transformation of the Michaelis-Menten equation. The binding affinity of the inhibitor to the enzyme is defined as the inhibition constant Ki, whereas the affinity, with which the substrate binds, is referred to the Michaelis-Menten coefficient Km. Michaelis-Menten kinetics base on three assumptions ... 

Based on the result from the IC50 determination, determination of additional kinetic parameters such as Ki and the inhibition mode are useful (variation of the substrate concentration e.g. Km/4 1 Km with time). Transformation of the Michaelis-Menten equation are used both for calculation the Ki value as well as for graphical depiction of the type of inhibition (e.g. direct plot ([rate]/[substrate], Dixon plot [l/rate]/[inhibitor], Linewaver-Burk plot [l/rate]/[l/substrate] or Eadie-Hofstee plot [rate]/[rate/substrate]). 

Lineweaver-Burk equation. An algebraic transformation of the Michaelis-Menten equation (plot of 1/V vs 1/[S]), allowing determination of Vmax and Km by extrapolation of [S] to infinity. 

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 most commonly used transformation of the Michaelis-Menten equation is that due to Hans Line-weaver and Dean Burk. Their equation was introduced in 1934 by taking reciprocals of both sides of Eq. (5.9) we obtain... 

Figure 5, Linear transformations of the Michaelis-Menten equation (Eqs. C3.59)-(3,6i)),...

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 oxidation rate was also dependent on the concentration of hypotaurine in the incubation medium. The reaction appeared to obey simple Michaelis-Menten kinetics (Fig. 2). The kinetic constants were estimated from a linear transformation of the Michaelis-Menten equation in a t against v/s plot. The apparent Michaelis constant ( ) was about 0.2 ramol/1 and the maximal velocity (F) about 0.1 ymol/s X kg. In our crude liver homogenate the apparent for hypotaurine oxidation was of the same order of magnitude as for the partially purified L-cysteine sulphinate decarboxylase (Jacobsen et al., 1964), the preceding enz3nne in the biosynthesis pathway. 

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