Michaelis-Menten equation transformations

Because of the hyperbolic shape of versus [S] plots, Vmax only be determined from an extrapolation of the asymptotic approach of v to some limiting value as [S] increases indefinitely (Figure 14.7) and is derived from that value of [S] giving v= V(nax/2. However, several rearrangements of the Michaelis-Menten equation transform it into a straight-line equation. The best known of these is the Lineweaver-Burk double-reciprocal plot ... 

The above equation can be transformed into the Michaelis-Menten equation by multiplying the numerator and denominator by Km ... 

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). 

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, ... 

GRAPHICAL REPRESENTATION. The above expression represents the equation of a hyperbola (i.e., f(x) = axl(b + x) where a and b are constants) and a plot of the initial velocity as a function of [S] will result in a rectangular hyperbola. Another way for representing the Michaelis-Menten equation is by using the doublereciprocal (or, Lineweaver-Burk ) transformation ... 

Two characteristics, the Michaelis constant KM and the maximal velocity V are the most important numeric data. The well-known Michaelis-Menten equation describes the relationship between the initial reaction rate and the substrate concentration with these two constants. The actual form of the rate equation of an enzymic process depends on the chemical mechanism of the enzymic transformation of the substrate to product (Table 8.1). 

The Michaelis-Menten equation (Eqn 6-9) can be algebraically transformed into versions that are useful in the practical determination of Km and Vmax (Box 6-1) and, as we describe later, in the analysis of inhibitor action (see Box 6-2 on page 210). TABLE... 

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 ... 

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... 

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 ... 

The Hanes plot also starts with the Lineweaver-Burk transformation (equation 5.15) of the Michaelis-Menten equation which in this instance is multiplied by S throughout on simplification this yields ... 

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). 

Understand the Michaelis-Menten equation, its derivation, and its transformations. 

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]). 

The most commonly used transformation of the Michaelis-Menten equation is the Lineweaver-Burk double reciprocal equation. 

For the Michaelis-Menten equation there are algebraic transformations, in addition to the Lineweaver-Burk equation, that yield straight line plots from enzyme kinetic data. One such plot is due to Eadie and Hofstee their equation takes the following form ... 

The Michaelis-Menten equation is often employed in soil-water systems to describe kinetics of ion uptake by plant roots and microbial cells, as well as microbial degradation-transformation of organics (e.g., pesticides, industrial organics, nitrogen, sulfur, and natural organics) and oxidation or reduction of metals or metalloids. Derivation of the Michaelis-Menten equation(s) is demonstrated below. 

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. ... 

Michaelis-Menten equation is transformed into one that gives a straight-line plot. Taking the reciprocal of both sides of equation 23 gives... 

The Michaelis-Menten equation can be algebraically transformed into more useful way to plot the experimental data. Lineweaver and Burk have taken the reciprocal of both [S] and v of the Michaelis-Menten equation to give Double Reciprocal or Lineweaver-Burke Plot Need in form y = ax + b, so take reciprocals of both sides (Fig. 6.4) and have -... 

Because of the difficulty in determining Vm from a hyperbolic curve, the Michaelis-Menten equation was transformed by Lineweaver and Burk into an equation for a straight line... 

Although it is quite simple to set up an experiment to determine the variation of v with [5], the exact value of V,n,y is not easily determined from hyperbolic curves. Furthermore, many enzymes deviate from ideal behavior at high substrate concentrations and indeed may be inhibited by excess substrate, so the calculated value of cannot be achieved in practice. In the past it was common practice to transform die Michaelis-Menten equation (9) into one of several reciprocal forms (equations [10] and [11]), and either 1/v was plotted against 1/[S], or [S]/v was plotted against [S]. 

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. 

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. 

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). 

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