Hanes plots

Fig. 8.7. Kinetic model for batch fermentation, Langmuir-Hanes plot. Reprinted from Najafpour et al. (2004).18 Copyright with permission from Elsevier.

A Langmuir-Hanes plot based on the Monod rate equation is presented in Figure 8.7. The Monod kinetic model can be used for microbial cell biocatalyst and is described as follows ... 

Equation 7.3.31 gives rise to what is known as an Eadie or Hofstee plot, while equation 7.3.32 gives rise to a Hanes plot. The Eadie plot has the advantage of spreading the points out more evenly and of determining K and J ax separately. The three types of plots are shown schematically in Figure 7.4. The Lineweaver-Burk and Eadie plots are the ones used most frequently in data analysis. 

Methods of plotting data obtained from enzyme catalyzed reactions, (a) Lineweaver-Burk plot, (b) Eadie or Hofstee plot, (c) Hanes plot. 

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 ". ... 

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, ... 

However, note that is replaced with Xia (the dissociation constant of A for the free enzyme) in the rapid-equilibrium equation. A standard double-reciprocal plot (1/v v. 1/[A]) at different concentrations of inhibitor will yield a series of parallel lines. A vertical intercept v. [I] secondary replot will provide a value for X on the horizontal axis. If questions arise as to whether the lines are truly parallel, one possibility is to replot the data via a Hanes plot ([A]/v v. [A]). In such a plot, the lines of an uncompetitive inhibitor intersect on the vertical axis. 

ACIDITY FUNCTION BUNNETT-OLSEN EQUATIONS DECREE OF DISSOCIATION HAMMETT EQUATION HAMMOND PRINCIPLE/POSTULATE LEFFLER S ASSUMPTION REACTING BOND RULES HANES PLOT... 

Figure 8.4 The Lineweaver-Burk plot (A) and the Hanes plot (B) of typical enzyme kinetics in presence of a competitve (a) noncompetive (b), mixed type (c) and uncompetitive (d) inhibitor.

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 ). 

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 ... 

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. 

Hanes Method.7 A different rearrangement of the Michaelis-Menten equation yields the form that is used in the Hanes plot ... 

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). 

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... 

Similarly, the Hanes analysis yields slope 1 / vm and intercept Kl vm, from which you again compute Kand vm, and the corresponding standard deviations. Note that the roles of slope and intercept are interchanged in the Iineweaver-Burk and Hanes plots. Compare the results of the two approaches. 

The unweighted Lineweaver-Burk and Hanes plots yield different answers even though they analyze the same data set. On the other hand, when we use appropriate global weighting factors, both methods yield identical results, as they should. Need we say more about the importance of proper weight-... 

Fig. 3.5 Spreadsheet for the analysis of the data on the kinetics of nicotinamide mononucleotide adenyl transferase by Atkinson etal, Biochem.J. 80 (1961) 318. The results should be read as, e.g., on line 30 K = 0.4406 0.0906 for the unweighted Lineweaver-Burk method, K = 0.5821 0.0522 for the unweighted Hanes plot, or, on line31, vm = 0.5853 0.1039 for Lineweaver-Burk, vm= 0.6848 0.0383 forHanes.

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. 

What type of inhibition is taking place (2) Sketch the curves for no inhibition. competitive, uncompetitive, noncompetitive (mixed) inhibition, and substrate inhibition on a Woolf-Hanes plot and on an Eadie-Hofstee plot. 

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