Cooperativity Hill equation

SCATCHARD PLOT KLOTZ PLOT ALLOSTERISM COOPERATIVITY HILL EQUATION AND PLOT WOMACK-COLOWICK DIALYSIS METHOD... 

The Hill Equation Describes the Behavior of Enzymes That Exhibit Cooperative Binding of Substrate... 

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... 

In Chapter 1 (Section 1.2.4.3), the Hill equation and the Hill coefficient, nH, are described. Hill coefficients greater than or less than unity are often interpreted as indicating positive or negative cooperativity, respectively, in the relationship between receptor occupancy and response. For example, positive cooperativity could arise due to amplification in a transduction mechanism mediated by G-proteins and changes in cell calcium concentration. 

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. 

Linked-function mechanisms for cooperative binding interaction of metabolites and/or drugs, based on the presence of two or more different conformational states of the protein or receptor. See Adair Equation Cooperative Ligand Binding Hemoglobin Hill Equation Plot Koshland-Nemethy-Filmer Model Monod-Wyman-Changeux Model Negative Cooperativity Positive Cooperativity... 

ADAIR EQUATION COOPERATIVE LIGAND BINDING HILL EQUATION PLOT KOSHLAND-NEMETHY-EILMER MODEL MONOD-WYMAN-CHANCEUX MODEL NEGATIVE COOPERATIVITY POSITIVE COOPERATIVITY HEMOGLOBIN ALLOTOPIC EFFECT Allowed electronic transitions,... 

Figure 8. Two routes to ultrasensitivity. A, Cooperativity generates an ultrasensitive dose-response curve. Plots were generated for the Hill equation, y = x / ( C + x"), for C = 20 for n = 1, 2 and 3. B, An exponential function also generates an ultrasensitive dose-response curve. Plot was generated for the equation y = e / (e + e ) for C = 20.

Equation (8) is an approximation because it ignores intermediate species that have some, but not all, of the binding sites occupied. Even so, the Hill coefficient provides a useful measure of cooperativity. The binding of 02 to hemoglobin is described well by the Hill equation with n 2.8. In the case of phosphofructokinase, which has four subunits, the dependence of the rate on the fructose-6-phos-phate concentration at a fixed, relatively high concentration of ATP is described well with n 3.8. 

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... 

Prove that for n > 1, the Hill equation conforms to a positively cooperative binding or enzyme system. 

Figure 13.3a shows a plot of the rate of packaging as a function of the ATP concentration under a constant force of 5pN. This data is well described by the characteristic Michaelis-Menten behavior with a Pmax 100 bp s and a Km 30 gM. Interestingly, the fit, done to a Michaelis-Menten-Hill equation reveals a Hill coefficient n I, indicating that the binding of the ATP to the motor is not cooperative. These same studies revealed that ADP is a competitive inhibitor of the motor and that phosphate release should be a nearly irreversible step [55], as its concentration in solution can be varied three orders of magnitude without affecting the rate of the motor. 

The plot of log[Y/(l - Y)] versus log[L] is linear (Fig. 9.3) with a slope of n. The all-or-none binding assumes extreme cooperativity in ligand binding. Under such conditions n should equal the number of sites within the protein. Few proteins exhibit extreme cooperativity. Therefore, the value for n is usually less than the number of sites but, when positive cooperativity occurs, n is greater than 1. In the case of negative cooperativity n is less than 1. An n value of 1 reduces the fractional saturation for the Hill equation (Equation 9.7) to that of simple ligand binding (Equation 9.4). 

A second way to characterize cooperativity involves fitting the oxygen-binding data at intermediate saturation (0.2 < 0 < 0.8)—that is, about the inflection point in a Hill plot—to the Hill equation... 

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. 

Dependent multiple-site cooperative binding and Hill equation... 

This is the Hill equation for cooperative binding of ligands by a receptor (see Chapter 8 for the Hill equation equivalent for cooperative biocatalysis). The term n represents the maximum value of B at saturation as well as the effective number ofbinding sites involved in cooperative binding activity (Figure 7.19). For the sake of completeness, when there is a tendency for ligand... 

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) ... 

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