Hills Equation

This may be based on Eq. XVI-2 [232] or on related equations with film thickness given by some version of the Frenkel-Halsey-Hill equation (Eq. XVII-79) [233,234],... 

For an applied stress of 1 MN/m and letting X be the multiplier on this stress, we can determine the value of X to make the Tsai-Hill equation become equal to 1. 

FIGURE 11.17 Symmetrical and asymmetrical dose-response curves, (a) Symmetrical Hill equation with n = 1 and EC5o = 1.0. Filled circle indicates the EC50 (where the abscissa yields a half maximal value for the ordinate). Below this curve is the second derivative of the function (slope). The zero ordinate of this curve indicates the point at which the slope is zero (inflection point of the curve). It can be seen that the true EC50 and the inflection match for a symmetrical curve, (b) Asymmetrical curve (Gompertz function with m = 0.55 and EC50= 1.9). The true EC50 is 1.9, while the point of inflection is 0.36. 

In this model, the factor s introduces the asymmetry. Alternatively, a modified Hill equation can be used [1] ... 

TED50= Tl[2 (IM/H) ln(2 + (Cpeak/CE50) AH) TED90 = 7T/2 (IM/H) ln(10+9 (Cpeak/CE50)A//) For single dose monoexponential kinetics and direct effect conditions, the area under the effect time curve (AUEC) can be derived by integration of the Hill equation. 

THE MICHAELIS-MENTEN HILL EQUATIONS MODEL THE EFFECTS OF SUBSTRATE CONCENTRATION... 

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... 

This is usually described as the Hill equation (see also Appendix 1.2C [Section 1.2.4.3]). Here, H is again the Hill coefficient, and y and vm l, are, respectively, the observed response and the maximum response to a large concentration of the agonist, A. [A]50 is the concentration of A at which y is half maximal. Because it is a constant for a given concentration-response relationship, it is sometimes denoted by K. While this is algebraically neater (and was the symbol used by Hill), it should be remembered that K in this context does not necessarily correspond to an equilibrium constant. Employing [A]50 rather than K in Eq. (1.6) helps to remind us that the relationship between... 

It is worth noting the distinction between the Hill equation and the logistic equation, which was first formulated in the 19th century as a means of describing the time-course of population increase. It is defined by the expression ... 

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. 

Weiss N (1997) The Hill equation revisited Uses and misuses. FASEB J 11 835—841. 

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... 

The linear form of this equation is denoted hy the Lineweaver-Burk or double reciprocal plot, which is derived from the Michaelis-Menten and Hill equation and is denoted as ... 

For simplicity, a linear relationship between concentration and effect is often assumed, reducing the problem of PK/PD to the pharmacokinetics. However, the concentration-effect relationship of any drug tends towards a plateau, and a sigmoidal model (sigmoid E ax model or Hill equation) is more appropriate [21-24] ... 

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