Hyperbolic saturation kinetics

A drug that binds reversibly to a protein, as shown in Figure 1A, displays hyperbolic saturation kinetics. At equilibrium, the fraction bound is as described by Eq. (1), where Kh = kyi/ku, ES is the enzyme-substrate complex and E, is the total enzyme ... 

More complex enzymatic reactions usually display Michaelis-Menten kinetics and can be described by Eq. (2). However, the forms of constants Km and Vm can be very complicated, consisting of many individual rate constants. King and Altman (7) have provided a method to readily derive the steady-state equations for enzymatic reactions, including the forms that describe Km and Vm. The advent of symbolic mathematics programs makes the implementation of these methods routine, even for very complex reaction schemes. The P450 catalytic cycle (Fig. 2) is an example of a very complicated reaction scheme. However, most P450-mediated reactions display standard hyperbolic saturation kinetics. Therefore, although the rate constants that determine Km and Vm are... 

For a detailed review of simple to complex enzyme kinetics, a book by Segel (21) is recommended. Most P450 oxidations show hyperbolic saturation kinetics and competitive inhibition between substrates. Therefore, both Km values and drug interactions can be predicted from inhibition studies. Competitive inhibition suggests that the enzymes have a single binding site and only one substrate can bind at any one time. For the inhibition of substrate A by substrate B to be competitive, the following must be observed ... 

Substrate A has a hyperbolic saturation curve Enzymes that bind to only one substrate molecule will show hyperbolic saturation kinetics. However, the observation of hyperbolic saturation kinetics does not necessarily mean that only one substrate molecule is interacting with the enzyme (see discussion of non-Michaelis-Menten kinetics in sec. IV). 

As shown in Figure 8.4, the synthesis of NAD from tryptophan involves the nonenzymic cyclization of aminocarhoxymuconic semialdehyde to quinolinic acid. The alternative metahoUc fate of aminocarhoxymuconic semialdehyde is decarboxylation, catalyzed hy picolinate carboxylase, leading into the oxidative branch of the pathway, and catabolism via acetyl coenzyme A. There is thus competition between an enzyme-catalyzed reaction that has hyperbolic, saturable kinetics, and a nonenzymic reaction thathas linear, first-order kinetics. 

Vb,. (b) Verify that the enzyme obeys hyperbolic saturation kinetics, (c) Calculate the first-order rate constant for the enzyme concentration employed. 

In enzymatic processes, the rate equation typically displays hyperbolic saturation kinetics, better known as Michaelis-Menten kinetics (7.3) ... 

Key regulatory enzyme of the glycolysis in Claviceps is phosphofructokinase that has rather atypical regulatory properties—feedforward regulation and hyperbolic saturation kinetics—(Kfen and Rehacek, 1984). 

The synthesis of NAD from tryptophan involves the non-enzymic cyclization of aminocarboxymuconic semialdehyde to quinolinic acid. The alternative metabolic fate of aminocarboxymuconic semialdehyde is decarboxylation, catalysed by picolinate carboxylase, leading to acetyl CoA and total oxidation. There is thus competition between an enzyme-catalysed reaction, which has hyperbolic, saturable kinetics, and a non-enzymic reaction, which has linear kinetics. At low rates of flux through the pathway, most metabolism will be by way of the enzyme-catalysed pathway, leading to oxidation. As the rate of formation of aminocarboxymuconic semialdehyde increases, and picolinate carboxylase becomes more or less saturated, so an increasing proportion will be available to undergo cyclization to quinolinic acid and onward metabolism to NAD. There is thus not a simple stoichiometric relationship between tryptophan and niacin, and the equivalence of the two coenzyme precursors will vary as the amount of tryptophan to be metabolized and the rate of metabolism vary. 

Mathematically, the Michaelis-Menten equation is the equation of a rectangular hyperbola. Sometimes you ll here reference to hyperbolic kinetics, this means it follows the Michaelis-Menten equation. A number of other names also imply that a particular enzyme obeys the Michaelis-Menten equation Michaelis-Menten behavior, saturation kinetics, and hyperbolic kinetics. 

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. 

A further manifestation of an initial binding step is saturation kinetics. For example, in the presence of excess catalyst, (26) predicts a hyperbolic relation between fcobs and concentration of nucleophilic sites C0(=nP0). Such behavior is actually what is observed. 

The hyperbolic saturation curve that is commonly seen with enzymatic reactions led Leonor Michaelis and Maude Men-ten in 1913 to develop a general treatment for kinetic analysis of these reactions. Following earlier work by Victor Henri, Michaelis and Menten assumed that an enzyme-substrate complex (ES) is in equilibrium with free enzyme... 

In the kinetic considerations discussed above, a plot of 1 /V0 vs 1/[S0] yields a straight line, and the enzyme exhibits Michaelis-Menten (hyperbolic or saturation) kinetics. It is implicit in this result that all the enzyme-binding sites have the same affinity for the substrate and operate independently of each other. However, many enzymes exist as oligomers containing subunits or domains that function in the regulation of the catalytic site. Such enzymes do not exhibit classic Michaelis-Menten saturation kinetics. 

Figure 5. Saturation kinetics the dependence of enzyme catalysis on the concentration of substrate. Reaction velocity represents the rate at which product is formed. (A) shows a hyperbolic saturation curve for two hypothetical enzymes. One binds its substrate more tightly than the other and reaches saturation at lower substrate concentration. This enzyme has a lower value, the substrate concentration where the reaction is half of maximum. The other binds the substrate more loosely and reaches the same velocity but requires higher substrate concentrations. (B) shows hypothetical velocities for cooperative enzymes. Although more complex, these enzymes also show the phenomenon of saturation.

FIGURE 13.2 Biochemical plots for the enz5me kinetic characterizations of biotransformation, (a) Direct concentration-rate or Michaelis-Menten plot (b), Eadie-Hofstee plot (c), double-reciprocal or Lineweaver-Burk plot. The Michaelis-Menten plot (a), typically exhibiting hyperbolic saturation, is fundamental to the demonstration of the effects of substrate concentration on the rates of metabolism, or metabolite formation. Here, the rates at 1 mM were excluded for the parameter estimation because of the potential for substrate inhibition. Eadie-Hofstee (b) and Lineweaver-Burk (c) plots are frequently used to analyze kinetic data. Eadie-Hofstee plots are preferred for determining the apparent values of and Umax- The data points in Lineweaver-Burk plots tend to be unevenly distributed and thus potentially lead to unreliable reciprocals of lower metabolic rates (1 /V) these lower rates, however, dictate the linear regression curves. In contrast, the data points in Eadie-Hofstee plot are usually homogeneously distributed, and thus tend to be more accurate. 

Enzymes (see Chapter 23 on the accompanying website) show saturation kinetics. At high substrate concentrations the enzyme is said to be saturated with respect to substrate. At low concentrations of substrate the enzyme activity is first order with respect to substrate but becomes almost zero order with respect to substrate concentration at higher concentrations. Plot the following enzyme kinetic data and use a logarithmic function to plot a hyperbolic curve of substrate concentration on the x-axis and enzyme rate or activity on they-axis. Extrapolate the curve to estimate the maximum rate of enzyme activity. 

In the semilogarithmic plot (Figure 112.1a), the curves have the same sigmoidal shape and differ only by lateral displacement. The symmetrical sigmoidal shape is a property of the hyperbolic saturation function, which is frequently used to fit fluence-response curves and other types of stimulus-response relationships in sensory physiology, - in photochemical kinetics, and in other areas of biophysics and biology, including the well-known MichaeHs-Menten enzyme kinetics (see below). 

Different from conventional chemical kinetics, the rates in biochemical reactions networks are usually saturable hyperbolic functions. For an increasing substrate concentration, the rate increases only up to a maximal rate Vm, determined by the turnover number fccat = k2 and the total amount of enzyme Ej. The turnover number ca( measures the number of catalytic events per seconds per enzyme, which can be more than 1000 substrate molecules per second for a large number of enzymes. The constant Km is a measure of the affinity of the enzyme for the substrate, and corresponds to the concentration of S at which the reaction rate equals half the maximal rate. For S most active sites are not occupied. For S >> Km, there is an excess of substrate, that is, the active sites of the enzymes are saturated with substrate. The ratio kc.AJ Km is a measure for the efficiency of an enzyme. In the extreme case, almost every collision between substrate and enzyme leads to product formation (low Km, high fccat). In this case the enzyme is limited by diffusion only, with an upper limit of cat /Km 108 — 109M. v 1. The ratio kc.MJKm can be used to test the rapid... 

Equation (4) corresponds to saturation-type (Michaelis-Menten) kinetics and rate constants obtained over a suitable range of [CD], sufficient to reflect the hyperbolic curvature, can be analysed to provide the limiting rate constant, kc, and the dissociation constant, Ks (VanEtten et al., 1967a Bender and Komiyama, 1978 Szejtli, 1982 Sirlin, 1984 Tee and Takasaki, 1985). The rate constant ku is normally determined directly (at zero [CD]), and sometimes Ks can be corroborated by other means (Connors, 1987). 

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