Hyperbolic rate plot

The Michaelis-Menten equation has the same form as the equation for a rectangular hyperbola graphical analysis of reaction rate (v) versus substrate concentration [S] produces a hyperbolic rate plot (Figure 9.3). 

Although computer software is now readily available to fit enzyme kinetic data to the Michaelis-Menten and related equations, it can be instructive to use simple graphical methods in some cases. The most convenient of these (though not necessarily the most accurate) are based on doublereciprocal methods that convert the hyperbolic rate equations into much simpler linear forms for plotting. 

Usually, one plots the initial rate V against the initial amount X, which produces a hyperbolic curve, such as shown in Fig. 39.17a. The rate and amount at time 0 are larger than those at any later time. Hence, the effect of experimental error and of possible deviation from the proposed model are minimal when the initial values are used. The Michaelis-Menten equation can be linearized by taking reciprocals on both sides of eq. (39.114) (Section 8.2.13), which leads to the so-called Lineweaver-Burk form ... 

A plot of the initial reaction rate, v, as a function of the substrate concentration [S], shows a hyperbolic relationship (Figure 4). As the [S] becomes very large and the enzyme is saturated with the substrate, the reaction rate will not increase indefinitely but, for a fixed amount of [E], it reaches a plateau at a limiting value named the maximal velocity (vmax). This behavior can be explained using the equilibrium model of Michaelis-Menten (1913) or the steady-state model of Briggs and Haldane (1926). The first one is based on the assumption that the rate of breakdown of the ES complex to yield the product is much slower that the dissociation of ES. This means that k2 tj. 

Experimental values of H, obtained from equation (4.46) and plotted against the rate of flow of mobile phase for a given solute and set of conditions, produce a hyperbolic curve showing an optimum flow rate for maximum efficiency (Figure 4.15). The position of the maximum varies with the solute, and a family of curves can be derived for the components of a mixture. The most efficient flow rate for a particular sample is, therefore, a matter of compromise. The equation also indicates that the highest... 

Full and partial noncompetitive inhibitory mechanisms, (a) Reaction scheme for full noncompetitive inhibition indicates binding of substrate and inhibitor to two mutually exclusive sites. The presence of inhibitor prevents release of product, (b) Lineweaver-Burk plot for full noncompetitive inhibition reveals a common intercept with the 1/[S] axis and an increase in slope to infinity at infinitely high inhibitor concentrations. In this example, K =3 IulM. (c) Replot of Lineweaver-Burk slopes from (b) is linear, confirming a full inhibitory mechanism, (d) Reaction scheme for partial noncompetitive inhibition indicates binding of substrate and inhibitor to two mutually exclusive sites. The presence of inhibitor alters (reduces) the rate of release of product by a factor p. (e) Lineweaver-Burk plot for partial noncompetitive inhibition reveals a common intercept with the 1/[5] axis and an increase in slope to a finite value at infinitely high inhibitor concentrations. In this example, /Cj= 3 iulM and P = 0.5. (f) Replot of Lineweaver-Burk slopes from (e) is hyperbolic, confirming a partial inhibitory mechanism... 

Fig. 1.1 The characteristics of (a) zero- (b) first- and (c) second-order reactions. In (a) the concentration of A decreases linearly with time until it is all consumed at time T. The value of the zero-order rate constant is given by Aq/T. In (b) the loss of A is exponential with time. The plot of In [A], vs time is linear, the slope of which is k, the first-order rate constant. It obviously does not matter at which point on curve (b) the first reading is taken. In (c) the loss of A is hyperbolic with time. The plot of [A], vs time is linear with a slope equal to k, the second-order rate constant.

Figure 5.6 Plot of the rate of glucose uptake (i.e. activity of the carrier) against glucose concentration by a skeletal muscle cell. The curve obeys a hyperbolic equation. The is about 5 mM, which is similar for the enzyme glucokinase for glucose when its activity is plotted against the glucose concentration (see Chapter 3).

This method is widely used because it provides hnear transformation of the hyperbolic function describing the rate saturation process. Double-reciprocal plots can be reasonably accurate if rate data can be obtained over a reasonable range of saturation, say from 0.3 E ax to 0.8 E ax. 

An enzyme is said to obey Michaelis-Menten kinetics, if a plot of the initial reaction rate (in which the substrate concentration is in great excess over the total enzyme concentration) versus substrate concentration(s) produces a hyperbolic curve. There should be no cooperativity apparent in the rate-saturation process, and the initial rate behavior should comply with the Michaelis-Menten equation, v = Emax[A]/(7 a + [A]), where v is the initial velocity, [A] is the initial substrate concentration, Umax is the maximum velocity, and is the dissociation constant for the substrate. A, binding to the free enzyme. The original formulation of the Michaelis-Menten treatment assumed a rapid pre-equilibrium of E and S with the central complex EX. However, the steady-state or Briggs-Haldane derivation yields an equation that is iso-... 

Referring to an enzyme whose kinetic properties do not yield hyperbolic saturation curves in plots of the initial rate as a function of the substrate concentration. 

FIGURE 16.6 Dependence of reaction showing sigmoidal (bottom) and hyperbolic (top) behavior. The top plot also shows the initial rate of reaction as a function of reactant concentration when the concentration of enzyme remains constant. 

In the case of most enzymic transformations the reaction rate can be described as a hyperbolic function of the concentration of substrate the characteristic parameters of these hyperboles are the and the KM values, which can be determined easily by different linearized plots. Different factors such as temperature, pH, chemical modification of the functional groups in the side chains of the protein, reversible inhibitors, activators, allosteric effectors, influence the catalytic activity of the enzymes. 

Figure 17.16 Relationships of biodegradation rate, v, to substrate concentration, [/], when Michaelis-Menten enzyme kinetics is appropriate (a) when plotted as hyperbolic relationship (Eq. 17-79 in text), or (b) when plotted as inverse equation, Vv =...

The rate (v) of many enzyme-catalyzed reactions can be described by the Michaelis-Menten equation. For enzymes that exhibit Michaelis-Menten kinetics, plots of velocity versus substrate concentration are hyperbolic. 

Comparison of hyperbolic and sigmoidal enzyme kinetics. The plot of velocity, rate of reaction, versus concentration of substrate or activator shows the response of a hyperbolic (Michaelis-Menten) reaction in black. The response of a sigmoidal reaction is shown by the purple curve. The shaded area indicates the range of physiological concentrations of substrate or activator. 

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