Eadie transformation

When the incorporation of [ H]NAD into TCA insoluble products was measured as a function of NAD concentration in permeabilised LI 210 cells, a pronouncedly biphasic plot was obtained (Fig. 1). This data was analysed using a Hofstee-Eadie transformation (v vs v/s) and was shown to divide quite clearly into two data sets, corresponding to NAD concentrations above and below 10 juM. The kinetic constants calculated from this analysis are shown in Table 1. Below 10 ijM the reaction has a of 9.45 (jM whereas above 10 juM the is 29.9 ijM. 

Xanthan Gum. As a result of a project to transform agriculturally derived products into industrially usefiil products by microbial action, the Northern Regional Research Laboratories of the USDA showed that the bacterium TCanthomonas campestris - noduces a polysaccharide with industrially usefiil properties (77). Extensive research was carried out on this interesting polysaccharide in several industrial laboratories during the eady 1960s, culminating in commercial production in 1964. 

If instead we use the Eadie-Hofstee transformation, Equation 17.2 becomes... 

Historically, data have been transformed to facilitate plotting on linear plots such as Lineweaver-Burk (1/y versus 1/[S]), Hanes-Woolf ([S]/y versus [S]), or Eadie-Hofstee (v/[S] versus y). However, with the present availability of affordable nonlinear regression and graphing software packages such as GraphPad Prism,... 

Figure 3.1 Ol Example of a Fourier smoother with boxcar apodization (zeroing 70 points on eadi end), (a) Spectrum in Figure 3.9 in apodization domain with first and last 70 points set to zero (b) after transformation back to original units.

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 ... 

Reduction of fr (Eq. 3.1) shows that it is composed of ee, and bin. The character table reveals that no orbitals transform as but that p. belongs to while df. and dn belong to eg. That these three orbitals on platinum are allowed by symmetry to participate in out-of-plane it bonding is reasonable since they are all oriented perpendicular to the plane of the ion (the xy plane). Selection of orbitals on platinum suitable for in-plane it bonds is left as an exercise. (Hint In choosing vectors to represent the suitable atomic orbitals, remember that the in-plane and out-of-p[ane ir bonds will be perpendicular to each other and that the regions of overlap lor the former will be on eadi side of a hording axis. Thus the in-plane vectors should be positioned perpendicular to the bonding axes.)30... 

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 ). 

Figure 22 Examples of enzyme kinetic plots used for determination of Km and Vmax for a normal and an allosteric enzyme Direct plot [(substrate) vs. initial rate of product formation] and various transformations of the direct plot (i.e., Eadie-Hofstee, Lineweaver-Burk, and/or Hill plots) are depicted for an enzyme exhibiting traditional Michaelis-Menten kinetics (coumarin 7-hydroxylation by CYP2A6) and one exhibiting allosteric substrate activation (testosterone 6(3-hydroxylation by CYP3A4/5). The latter exhibits an S-shaped direct plot and a hook -shaped Eadie-Hofstee plot such plots are frequently observed with CYP3A4 substrates. Km and Vmax are Michaelis-Menten kinetic constants for enzymes. K is a constant that incorporates the interaction with the two (or more) binding sites but that is not equal to the substrate concentration that results in half-maximal velocity, and the symbol n (the Hill coefficient) theoretically refers to the number of binding sites. See the sec. III.C.3 for additional details.

The Eadie-Hofstee Equation One transformation of the Michaelis-Menten equation is the... 

Fig. 4 Nitrate-specific growth rates (/iN) of conlonial Phaeocystis in the incubation bottles between days 25 and 31, versus initial dissolved iron concentrations. The Monod hyperbola (dashed line) was fitted using an Eadie-Hofstee linear transformation (r2 = 0.85), and excludes the +1.8 nM Fe datum (in parentheses). The half-saturation constant for growth (K ) and maximum nitrate-specific growth rate (/ Nmax) are indicated...

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]). 

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 ... 

A spectrum in a specified ranalogue signals from eadi photodiode are digitised and transferred to a computer, where they e corrected for dark current response and transformed to absorbance. A number of digital techniques are available to increase sensitivity and to extend the use of rapid-scanning detectors to multicomponent analysis, reaction kinetics, tablet dissolution tests, process control, and detection in HPLC (A. F. Fell et al, Chrom-atographia, 1982, 16, 69-78). 

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). 

In this section we wish to show that an alternative form for the ponderomotive force, proposed by Helmholtz, can also be justified by statistical-mechanical methods and that its relation to the ponderomotive force and pressure derived in the previous section is the same as foimd m purely thermod mamical arguments. Helmholtz arrives at an expression for the ponderomotive force on the basis of macroscopic energy considerations for a dielectric subjected to reversible transformations. We are thus led to develop this part of the theory by considering a s tem in equilibrium. Since the S3rstein need not be uniform (due to the presence of a nonuniform external field), we shall divide it into a number of cells. Eadi cell contains a large number of atoms, but is sufficiently small to be considered macroscopically uniform. This means that... 

Biochemical Plots Several methods are readily applied to the determination of kinetic parameters and Tmax)- Traditionally, these terms are determined using the classic biochemical plots, particularly those transformed from the well-known Michaelis-Menten plot, for example, Lineweaver-Burk and Eadie-Hofstee plots (Li et ah, 1995 Nakajima et ah, 2002 Nnane et ah, 2003 Yamamoto et ah, 2003). 

Fig. C.3. Reducible representation, block form, and irreducible representation. In the first row, the matrices F(Ri) are displayed that form a reilucible representation (eadi matrix corresponds to the symmetry operation Rj) the matrix elements are in general nonzero. The central row shows arepresentation F equivalent to the first one i.e., related by a similarity transformation (with matrix P). The new representation exhibits a block form i.e., in this particular case each matrix has two blocks of zeros that are identical in all matrices. The last row shows an equivalent representation F that corresponds to the smallest square blocks (of nonzeros) i.e., the maximum number of the blocks, of the form identical in all the matrices. Not only F, F, and F" are representations of the group, but also any sequence of individual blocks (as that shadowed) is a representation. Thus, F is decomposed into the four irreducible representations.

A more satisfactory distribution of error is found for S/v versus S, known as the Hanes plot, or the Hanes-Wilkinson plot (Table Fig.). A further linear transformation is represented by v versus v/S, known as the Eadie-Hofstee plot. The error increases with v/S, but since v is a component of both coordinates, the errors vary with respect to the origin rather than the axis, i.e. all error bars converge on the origin (Table Fig.). 

Linear transformations of the equation v=V S/Km + S. Error bars are also shown. 1. Lineweaver-Burk, or double reciprocal plot 2. Eadie-Hofstee plot 3. Hanes-Wilkinson plot 4. Eisenthal-Cornish Bowden, or direct linear plot 5. The Scatchard plot which is used for determination of ligand binding constants. 

One striking difference between the pn orbitals of the cyclic polyene and those of the H molecules is that the symmetry labels are different. For example, the pir levels of cyclopropenyl transform as aS + e" (double primes since they are anti-83nmnetric with respect to reflection in the plane perpendicular, to the threefold rotation axis) but the 5 orbitals of H3 transform as a[ e (slnj primes since these orbitals are symmetric with respect to this symmetry operation). The breakdown into nondegenerate and degenerate orbitals (one of eadi) is the same in both cases. 

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