Hyperbolic kinetics

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. 

It has been demonstrated that lin-log equations capture hyperbolic kinetics slightly better than either a linear approximation or the power-law approach [318, 320 322]. One reason for the improved performance is that for lin-log kinetics the elasticities (and kinetic orders) are not constant, but change with changing metabolite concentrations. For a monosubstrate reaction v(Sj, and omitting the dependence on the enzyme concentration, we obtain... 

Hexokinase catalyzes the phosphorylation of glucose and fructose by ATP. However the Km for glucose is 0.13 mmol L-1, whereas that for fructose is 1.3 mmol L-1. Assume is the same for both glucose and fructose and the enzyme displays hyperbolic kinetics [Eq. (9.5)]. (a) Calculate the normalized initial velocity of the reaction (i.e., V(/Umax) for each substrate when [S]0 = 0.13, 1.3, and 13.0 mmol L-1. (b) For which substrate does hexokinase have the greater affinity ... 

These assumptions are really quite mild the / need only be increasing and sufficiently smooth. It is not even required that f, be bounded on IR. From a biological perspective, of course, the model loses relevance for really large values of S. In addition to the Monod functions, other functions that have been suggested include the exponential kinetics AM(1—exp(—S log 2/a)), hyperbolic kinetics m tanh(S log 3/2a), and piece-wise linear kinetics given by mS/2a for S<2a and by w for S > 2a. The piecewise linear kinetics fails to satisfy the strict monotonocity of (iii) and fails to satisfy (iv) at one point, but these assumptions could be weakened so as to include this case. 

The muscle and brain isoenzyme (M-type) shows hyperbolic kinetics with phosphoenolpyruvate and is inhibited by ATP, with an increase in for phosphoenolpyruvate and development of sigmoidal kinetics. Fructose-1,6-bisphosphate and alanine have no effect on this isoenzyme. 

Pyruvate carboxylase. Pyruvate carboxylase has hyperbolic kinetics in respect to its substrates pyruvate, C02, and ATP, but sigmoidal kinetics in respect to its essential allosteric activator acetyl CoA. The values of n for pyruvate carboxylase are plotted as a function of acetyl CoA concentration. The range of acetyl CoA concentrations in the mitochondria (subcellular localization of pyruvate carboxylase) is indicated by the shaded area. 

Atkins, W.M., W.D. Lu, and D.L. Cook (2002). Is there a toxicological advantage for non-hyperbolic kinetics in cytochrome P450 catalysis Functional allostery from distributive catalysis. J. Biol Chem. Ill, 33258-33266. 

In contrast with Michaelian enzymes, which have hyperbolic kinetics, allosteric enzymes, thanks to their sigmoidal kinetics, possess an enhanced sensitivity towards variations in the concentration of an effector or of the substrate. This is the reason why many enzymes that play an important role in the control of metabolism are of the allosteric type. 

Why do chymotrypsin and ATCase have different velocity curves Chymotrypsin and aspartate transcar-bamoylase exhibit different types of kinetics. Chymotrypsin is a nonallosteric enzyme and exhibits hyperbolic kinetics. ATCase is an allosteric enzyme. It has multiple subunits, and the binding of one molecule of substrate affects the binding of the next molecule of substrate. It exhibits sigmoidal kinetics. 

Allosteric enzymes display sigmoidal kinetics when rates are plotted versus substrate concentration. Michaelis-Menten enzymes exhibit hyperbolic kinetics. Allosteric enzymes usually have multiple subunits, and the binding of substrates or effector molecules to one subunit changes the binding behavior of the other subunits. 

Michaelis-Menten (M-M) hyperbolic kinetics (Eq. 13.7) is often assumed and directly applied in rate determination, particularly in early discovery when the enzyme kinetic behavior is usually unknown ... 

A large family of enzymes that deviate from hyperbolic kinetics (Michaelis) is the allosteric enzymes. These enzymes contain two or more topologically distinct binding sites that interact functionally with each other. Most commonly, sigmoidal or S-shaped curves are obtained, being indicative of positive substrate cooperativity. The reaction rate for these enzymes can be calculated by the Hill equation ... 

When the catalytic and regulatory subunits are dissociated, the catal3rtic activity, measured with saturating concentrations of carbamyl phosphate, displays conventional hyperbolic kinetics with respect to aspartate concentration. However, sigmoidal kinetics are shown by the native aspartate carbamyltransferase, in which form the subunits are associated. In the presence of CTP, the sigmoidicity of the rate-aspartate curve is further increased, with the effect that the apparent Michaelis constant for aspartate is increased. ATP competes with CTP for the regulatory site, with the result that the CTP inhibitory effect is antagonized. 

Since significant meaning is placed on these measured constants and parameters, it is important that they be determined accurately and unambiguously. It is also important that the reader or practitioner in the field of enzymology be able to assess if the measurement of these parameters is reliable. Furthermore, since enzyme behavior is often modeled as Michaelis-Menten (hyperbolic) kinetics, it seems reasonable that interpretations of observations should be made in the context of the Michaelis-Menten model. In some cases, alternative explanations for enzyme kinetic behavior may be appropriate and one may be inclined to select one interpretation over another (preferably based on a kinetic analysis, although too often this is done on intuition). 

The estimation and discussion of Km (the MichaeMs constant) may be irrelevant because Km is a constant defined by (and confined within) use of the Michaelis-Menten model (hyperbolic kinetics) in the first place. 

To make an appropriate assessment of the pattern of inhibition, one need only compare the pattern of reaction velocity versus [S] observed relative to the pattern predicted from an application of the hyperbolic kinetics model. This requires making an estimate of V ax and from the data available. Transforming the original data to a Lineweaver-Burke plot (despite the aforementioned limitations) indicates that only four data points (at low [S]) can be used to estimate Vmax and Km (as 3.58 units and 0.48 mM, respectively. Fig. 14.10). The predicted (uninhibited) behavior of the enzyme activity can now be calculated by applying the rectangular hyperbola [Eq. (14.5)] (yielding the upper curve in Fig. 14.11), and it becomes clear that inhibition was obvious at [S] <1 mM. The degree of inhibition is expressed appropriately as the difference between observed and predicted activity at any [S] value, if one makes interpretations within the context of the Michaelis-Menten model. 

In this situation, two different polypeptide chains interact to form a new specific complex, whose biologic activity can be modified by ligands that bind to the precursor subunits. This type of regulation has been extensively studied for the aspartate transcarbamylase of E. coli (Gerhart, 1970). This enzyme catalyzed the initial step in the synthesis of cytidine nucleotides it is allosterically inhibited by CTP and shows positive cooperativity for substrate. It may be dissociated by mercurials into catalytic subunits, which are insensitive to CTP and which exhibit hyperbolic kinetics for substrate, and into regulatory subunits which bind CTP. 

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