Hill equation enzyme kinetics

A linear form of the Hill equation is used to evaluate the cooperative substrate-binding kinetics exhibited by some multimeric enzymes. The slope n, the Hill coefficient, reflects the number, nature, and strength of the interactions of the substrate-binding sites. A... 

Two other general ways of treating micellar kinetic data should be noted. Piszkiewicz (1977) used equations similar to the Hill equation of enzyme kinetics to fit variations of rate constants and surfactant concentration. This treatment differs from that of Menger and Portnoy (1967) in that it emphasizes cooperative effects due to substrate-micelle interactions. These interactions are probably very important at surfactant concentrations close to the cmc because solutes may promote micellization or bind to submicellar aggregates. Thus, eqn (1) and others like it do not fit the data for dilute surfactant, especially when reactants are hydrophobic and can promote micellization. 

In order to understand the mechanisms of the enzymatic process and also predict the reaction characteristics, one needs to understand the kinetics of the reaction. The important factor that effects the enzyme reaction is the availability and concentration of the substrates. An important model that gives a mathematical relationship is the Michaelis-Menten and Hill equation. The equation is denoted as... 

It was noted earlier that Michaelis-Menten kinetics and its linear transformations are not valid for allosteric enzymes. Instead, the Hill equation, an equation originally empirically developed to describe the cooperative binding of Oz to hemoglobin (Chapter 7), is used. The expression describing such a straight-line plot is... 

The Hill equation is used to estimate Km for allosteric enzymes. Equations based on classic Michaelis-Menten kinetics are not applicable. 

These conclusions are supported by a series of related experiments on the same recombinant pea cytosolic enzyme in which the steady state oxidation of / -cresol has also been found to exhibit sigmoidal kinetics (Celik et al., 1999). In this case, the data were satisfactorily fitted to the Hill equation, Eq. (2),... 

Piskiewicz [119] has developed a kinetic model of micellar catalysis, based on the Hill equation of enzyme kinetics, which assumes a cooperative interaction between reactants and surfactant to form reactive substrate-micelle complexes. This model is probably not applicable to systems in which the surfactant is in large excess over substrate, as in most micellar mediated reactions, but it gives a very reasonable explanation of the rate effects of very dilute surfactants. 

The two-variable model for birhythmicity is built on the basis of eqns (2.7) by incorporating into them a term related to the transformation of product into substrate, in a reaction catalysed by an enzyme whose cooperative kinetics is described by a Hill equation, characterized by a degree of cooperativity n. The kinetic equations of the model thus take the form of eqns (3.1) where the various parameters remain defined as for eqns (2.7) and (2.11) ... 

One can probably guess that in relation to reality, the reaction examples of the illustration or of equation (3-73) are much simplified. Many enzymes of known function catalyze reactions involving more than one substrate. The mechanisms can be quite complex, however, the rate laws do generally follow the form of equation (3-73) if the composition of only one substrate is varied at one time. A good discussion of such multisubstrate enzyme-catalyzed reactions is given by Plowman [K.M. Plowman, Enzyme Kinetics, McGraw-Hill Book Co., New York, NY, (1972)]. There is a strong family resemblance between these enzymatic sequences and those encountered in the detailed collision theory of Benson and Axworthy in Chapter 1. 

Prove that for > 1 the Hill equation describes positive cooperativity in binding, or kinetics, for an enzyme. SOLUTION... 

In this case, the homotropic cooperativity occurs with a monomeric enzyme through substrate binding to both a catalytic and an effector site, and these sites may not be equivalent (Shou et al., 1999). This results in two parameter estimates and two y ax parameter estimates, as discussed below. One of the most commonly used equations describing sigmoidal kinetic profiles is the Hill equation (Eq. 4.5) ... 

Should one use the Hill plot in practice to examine the initial velocity behavior of enzymes Because infinite cooperativity is assumed to be the basis of the Hill treatment, only rapidly equilibrating systems are suitable for the Hill analysis. However, enzyme systems displaying steady-state kinetic behavior will not satisfy this requirement for this reason, one must avoid the use of kinetic data in any application of the Hill equation to steady-state enzyme systems. 

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 ... 

The Hill coefficient is an index of the cooperativity in the substrate binding process—the greater the value of n, the higher the cooperativity. For the case where n = 1 (no cooperativity), the Hill equation reduces to the Michaehs-Menten model. If the cooperativity of the sites is low, n will not correspond to the number of substrate-binding sites, but the minimum number of effective substrate-binding sites. Regardless of this limitation, the Hill equation can still be used to characterize the kinetic behavior of a cooperative enzyme. In this case, n becomes merely an index of cooperativity, which can have noninteger values. 

The Hill equation is a three-parameter function k, n, Vniax), and constitutes the simplest equation that describes the kinetic behavior of cooperative enzymes. From a practical point of view, the next most useful model is the syimnetiy model. Even though it only accounts for positive cooperativity and is based on somewhat arbitrary assumptions, this model can account for aUosteric effects. 

Fig. 6.35 Kinetic modei for an enzyme with two reactant sites and two modifier sites. (From J.-H.S. Hofmeyr, A. Cornish-Bowden, The reversible Hill equation how to incorporate cooperative enzymes into metabolic models, Comput. Appl. Biosci. 13 (1997) 377-385. Copyright 1997 Oxford University Press).

An attempt was made recently to find out how sensitively the regulatory system that controls the cell cycle and cell proliferation responds to signal input. Ferrell et found that, in intact oocytes, the response is ultrasensitive , a kinetic characterization introduced by Daniel E. Koshland Jr.26 (Ultrasensitivity has been defined as the response of an enzyme that is more sensitive to changes in the concentration of the substrate than an enzyme with a normal hyperbolic response, according to the Michaelis-Menten equation. One can also use the Hill coefficient (wh) to indicate hyperbolic (Michaelis-Menten) sensitivity ( h = 1-0), ultrasensitivity ( h > l)j and subsensitivity ( h <... 

---