Linearization Hanes

Values for Km and kcat were derived from the linearized Hanes equation. 

In general terms, the assumption of a linear relationship implies that a dependent variable T, is a linear function of an independent variable X. In the case of the linear Lineweaver-Burkplot,i/uo is the dependent variable while /A is the independent variable. In the case of the linear Hanes plot, A/i>o is the dependent and A is the independent variable. The linear regression, as presented here, is based on four assumptions ... 

This is referred to as a Hanes, Hildebrand-Benesi, or Scott plot. From this linear plot, Kd = intercept/slope and 1/slope =Bmax. 

On the other hand, the macrolides showed unusual enzymatic reactivity. Lipase PF-catalyzed polymerization of the macrolides proceeded much faster than that of 8-CL. The lipase-catalyzed polymerizability of lactones was quantitatively evaluated by Michaelis-Menten kinetics. For all monomers, linearity was observed in the Hanes-Woolf plot, indicating that the polymerization followed Michaehs-Menten kinetics. The V, (iaotone) and K,ax(iaotone)/ m(iaotone) values increased with the ring size of lactone, whereas the A (iactone) values scarcely changed. These data imply that the enzymatic polymerizability increased as a function of the ring size, and the large enzymatic polymerizability is governed mainly by the reachon rate hut not to the binding abilities, i.e., the reaction process of... 

The suggestion of a helical host molecule was originally put forward by Hanes 136) and then developed by Freudenberg and his colleagues 137). Chemical 138 140) and X-ray diffraction studies by Rundle et al.141 143) and by Bear 144,145) demonstrated that these ideas were correct, and revealed that the helical structure had an outer diameter of 13.0 A, an inner diameter of 5 A, and a pitch of 8.0 A with six glucose units per turn. The iodine atoms were arranged in a linear fashion with an average I-I separation of approximately 3.1 A. These early results have been reviewed 146, 147). They represent the first confirmed example of helical structure for a biopolymer. 

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,... 

See Double-Reciprocal Plot Hanes Plot Direct Linear Plot Dixon Plot Dixon-Webb Plot Eadie-Hofstee Plot Substrate Concentration Range Frieden Protocol Fromm Protocol Point-of-Convergence Method Dal-ziel Phi Relationships Scatchard Plots Hill Plots... 

A linear graphical method for analyzing the initial rate kinetics of enzyme-catalyzed reactions. In the Hanes plot, [A]/v is plotted as a function of [A], where v is the initial rate and [A] is the substrate concentration ". ... 

DALZIEL PHI RELATIONSHIPS DIRECT LINEAR RLOT DIXON RLOT DIXON-WEBB RLOT DOUBLE-RECIRROCAL RLOT EADIE-HOFSTEE RLOT FRIEDEN RROTOCOL FROMM RROTOCOL HANES RLOT HILL RLOT... 

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 ). 

For two-carbon alkanes, log P increases as the number of chlorines increases. The dataset consists of 1,1,1,2-tetrachloroethane 1,1,1-trichloroet-hane 1,1,2,2-tetrachloroethane 1,1,2-trichloroethane 1,1-dichloroethane 1,2-dichloroethane, chloroethane hexachloroethane and pentachloroethane. Log P represents 93.52% of the variance in the linear regression equation. The probability of getting a correlation of 0.9671 for a sample size of nine is less than 1%. 

There are well-established methods for obtaining the type of inhibition and the value of the inhibition constants from initial-rate kinetics, often from linearized plots such as lineweaver-Burk, Eadie-Hofstee, or Hanes. As these procedures are covered very well by a range of basic textbooks on biochemistry and kinetics (see the list of Suggested Further Reading ) we will not repeat these procedures here. Instead, we will discuss the situation in which an enzyme reaction is followed over more than just the initial range of conversion. Towards this end, the rate equation,... 

In addition to being easier to fit than the hyperbolic Michaelis-Menten equation, Lineweaver-Burk graphs clearly show differences between types of enzyme inhibitors. This will be discussed in Section 4.5. However, Lineweaver-Burk equations have their own distinct issues. Nonlinear data, possibly indicating cooperative multiunit enzymes or allosteric effects, often seem nearly linear when graphed according to a Lineweaver-Burk equation. Said another way, the Lineweaver-Burk equation forces nonlinear data into a linear relationship. Variations of the Lineweaver-Burk equation that are not double reciprocal relationships include the Eadie-Hofstee equation7 (V vs. V7[S]) (Equation 4.14) and the Hanes-Woolf equation8 ([S]/V vs. [S]) (Equation 4.15). Both are... 

The Hanes-Woolf equation is another transformation of the Michaelis-Menten equation that yields a linear graph of the appropriate transformed variables. The equation is ... 

Another linear representation of the Michaelis-Menten equation is the Hanes-Woolf plot (Equation 17.14). 

Competitive inhibitors do not change the value of Vmax> which is reached when sufficiently high concentrations of the substrate are present so as to completely displace the inhibitor. However, the affinity of the substrate for the enzyme appears to be decreased in the presence of a competitive inhibitor. This happens because the free enzyme E is not only in equilibrium with the enzyme-substrate complex E. S, but also with the enzyme-inhibitor complex E. L Competitive inhibitors increase the apparent of the substrate by a factor of (1 + The evaluation of the kinetics is again greatly facilitated by the conversion of Equation 17.15 into a linear form using Line-weaver-Burk, Eadie-Hofstee, or Hanes-Woolf plots, as shown in Fig. 17.7. 

The Lineweaver-Burk reciprocal plot is not the only linear transformation of the basic velocity (or ligand binding) equation. Indeed, under some circumstances one of the other linear plots described below may be more suitable or may yield more reliable estimates of the kinetic constants. For example, the Hanes-Woolf plot of [S]/u versus [S] may be more convenient... 

The Lineweaver-Burk equation may be rearranged to yield the linear equation for the Hanes-Woolf plot ... 

It appears that phosphorus is related to the formation of starch in the plant. Thus Hanes has synthesized a linear polysaccharide from a-D-glucopyranosc 1-phosphate (Cori ester) through the action of potato phosphorylase. Dunlap and Beckmann and likewise Cori have found that the B-fraction activates this enzymic synthesis, while the A-frac-tion is inactive. It has not been established whether this effect is due to the branched character of the B-fraction or to the presence of phosphate in its structure. 

A plot of v (as y) against v/[S] (as x) will yield, after linear regression, a y intercept of Emax and a slope of A nl (Fig. 2.7). This plot is the preferred linear regression method for determining Km and V/ln ix, since precision and accuracy are somewhat better than those obtained using the Hanes plot, and much better than those found using the Lineweaver-Burk method. 

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). 

Fig. 9.3. Determination of the parameters of the Michaelis-Menten equation and by the Lineweaver-Burk (A), Eadie-Hofstee (B), Hanes (C), and direct linear (D) plots. The error bars in A, B and C represent a variation of 5 /, of Vmi, and show the large effect small errors at low [S] may have on the estimates. Outlying lines obtained in the direct linear plot (D) are easily recognized, at least if a large fraction of the lines do converge in the same intersection.

Despite their appealing simplicity, these methods have serious limitations. The Lineweaver-Burk and Hanes plots are unreliable, e.g., the variation of the variance almost certainly results in an incorrect weighting, whereas in the Eadie-Hofstee plot Vo is present in both variables. The direct linear plot of Eisenthal and Cornish-Bowden (1974), for which the Michaelis-Menten equation is rearranged to relate to A , i.e., = Vo -f- Vo A ,/[S] is very simple but... 

In the context of the analysis of enzyme kinetics it is sometimes stated that one should always use a non-linear least-squares method for such data, because the usual, unweighted least-squares fits depend on the particular analysis method (Lineweaver-Burk, Hanes, etc.) used. We have seen in section 3.5 that the latter part of this statement is correct. But how about the former ... 

Hanes plot, and 1/ fin the Lineweaver-Burk plot, and they did not give the same results. The weighting converted both to use v as the dependent variable, and then they agreed, of course, and we obtain the same result with the non-linear least-squares fit to (3.5-1). But when we use S as our dependent variable, we get a different result with the non-linear least squares, because we again compare apples and pears. And, of course, when we use non-linear least squares on (3.5-2) we get the same result as with the linear least-squares analysis of the Lineweaver-Burk plot, and non-linear least-squares of (3.5-2) would yield the same answers as linear least squares based on a Hanes plot. 

Figure 8.25 Linear, single-reciprocal Hanes plot.

A Lineweaver-Burk plot ( ) indicates that with D-glucose as the substrate, the enzyme obeys Michaelis-Menten kinetics with a Km value of 3.2 + 0.08 mM and a Vmax of 126.0 + 0.02 micromol/mg protein/min (Figure 11). Similar results were obtained by the direct linear plot (88), Hanes and Woolf ( ) or Eadie-Hofstee plots (90). All the kinetic data reported here and subsequently, were based on the initial rates of hydrogen peroxide formation... 

Other methods of linearization of Eq. 3.11 are presented in Table 3.1. The method of Hanes (Hanes 1932) and Eaddie-Hofstee (Hofstee 1959) have been the most used alternatives. 

Other methods of linearization (i.e. Eaddie-Hofstee Hanes) can be used, as well as integrated methods, as shown in section 3.2.2 for simple Michaelis-Menten kinetics. 

Most common Michaelis-Menten analytics are associated with the typical linearization plots namely Lineweaver-Burk (or double-reciprocal plot), Eadie-Hofstee " and Hanes plots. All three have in common that they are using the experimental differential values of either product formation or substrate uptake. Very often these values cannot be determined directly and must be differentiated from the corresponding concentration-time values numerically. This can naturally cause large errors, in particular when the measured data are very noisy and/or just a very low number of data points exist. Nevertheless, the Hanes plot is the most favoured one due to its much more uniform error distribution (see eqn (4.6) with v = d[P]/dt). 

Figure 1 shows the graphical presentation of Eq. (5.3), in three different linear forms the Lineaweaver-Burk plot, the Hanes plot, and the Dixon plot. 

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 ... 

Figure 4 shows a plot of initial rate data in the linear form, in the Lineweaver-Burk and the Hanes fashion, replotted from the data in Table 1. 

In the graphical analysis of initial rate data, it is pradent to use all three plots shown in Figs. 3 and 4. The direct plot of versus [A] wiU show directly the influence of substrate concentration on initial rate of reaction. The two linear plots should be used together, because the Lineweaver-Burk plot serves to visualize the influence of low concentrations whereas the Hanes plot serves to visualize the influence of high concentrations of substrates. The third plot, the Eadie-Hofstee plot, is useful in detecting exceptionally bad measurements (Section 3.11). 

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