The Hill model

To take into account a system s divariance. Hill deems it necessaiy to take surface effects into consideration. With this aim in mind, he supposes that the adsorption film, that is to say the adsoibent + adsotbed block, is easily assimilated to a solution where the adsoibent is formed by the free sites on the solid s surface, and the adsorbed species are the gas molecules that have settled on those sites. 

In this case, the possible variables are pressme, temperature and the quantities of matter for the adsorbent (iis) and the adsorbed (ria) species. 

There are 2 independent components (adsorbent + adsorbed + gas - an equilibrium relation between these three components). 

There are 2 exterior parameters (P and T) and there are 2 phases (solid and gaseous). 

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A model analogous to the Hill model (for enzymatic reactions), which describes a more accurate dependence of the observed rate constants on surfactant concentration, was developed by Piszkiewicz. This model is applicable especially at low surfactant concentration and the data may be treated without reference to CMC. According to this model, a substrate (S) and n number of detergent molecules (D), aggregate to form critical micelle (D S), which may react to yield the product... 

Potency comprises both achon and inhibition of achon and is predicted by the Hill model though a 50% level is chosen, it is an arbitrary percentage and other values such as 60 or 40% action can also be calculated and used. Potency is not a relevant factor xmless it is so low that the dose requirement is very high (to a level where nonlinear binding with albumin can be observed, resulting in nonlinear kinetics) or where the serious side effects are dose-dependent and make an effechve dose unacceptably toxic. The potency, EC50, is expressed in a mechanishc equilibrium model where the achon is direct ... 

A more searching analysis of the Poole-Frenkel mechanism performed for polysiloxane on the basis of the Hill model ( ) showed that charge carrier emission should proceed from the isolated Coulomb centre and should take place in the hemisphere related to that centre. The depth of the centres, determined form the activation dependence of the temperature, was =... 

FIGURE 48.3 The Hill model of muscle separates the artive properties of muscle into a contractile element, in series with a purely elastic element. The properties of the passive muscle are represented by the parallel elastic element. 

The influence of experimental conditions on Hill parameter estimation is described by Glende and Reich (1972). According to Reich et al. (1972), a nonlinear regression with a minimum of 15 measuring points should be used for an unbiased determination of parameters. Also, the Hill model has been extended by Adair (1925) to allow for the existence of stable, partly liganded intermediates. 

A novel kinetic model for micellar catalysis has been developed based on the assumption that the Stem layer is always saturated with respect to counterions. This means that the ground state for ions is the ion bound to the micellar surface and not the free-ion in the bulk phase. An analogy between micellar reactions and reactions catalysed by regulatory enzymes has led to the application of the Hill model to the dependence rate constants of micellar catalysed reactions upon the detergent concentration. The decrease in rate at high concentrations of detergent is interpreted in terms of substrate inhibition. ... 

If a P molecule binds to exactly n ligand molecules at a time, then the Hill model is useful ... 

Figure 28.8 Cooperativity described by the Hill model, (a) When the fraction v of sites filled is plotted versus ligand concentration. v. the steepness of the curve increases with n, the Hill coefficient, (b) The Hill plot linearizes the data. The slope gives the Hill coefficient i and the intercept gives the binding constant K.

Figure 28.9 For the maturation of frog eggs, the production of phosphorylated mitogen-activated protein kinase (MAPK-P) is governed by small changes in the concentration of the MOS protein. This relationship can be described by the Hill model. Source JE Ferrel and EM Machleder, Science 280, 895-898 (1998). 

If an enzyme requires n identical substrate molecules to bind at the same time before it can catalyze a reaction, and if each molecule binds with affinity K, the Hill model can be used to predict the rate v of product formation ... 

Figure 8.8. Analysis of the initial velocity versus substrate concentration data for a cooperative enz3me using (a) the Hill model and (b) the MWC model.

The Hill model is composed of three elements two of which are arranged in series which, in turn, are in parallel with the third element. The contractile element is freely extendable when at rest, but capable of shortening when activated by an electrical stimnli. The contractile element is cormected to an elastic serial element. 

The Hill model was adopted for the compression creep using the 0°, 45 , 90 results at 1473 K, 30 MPa (Figure 6) and a stress exponent of 1.9 which corresponds to the exponent of the 90 orientation. Ideally, the Hill parameters are constant for a material. However, in this case parameters decreased with an increasing creep strain (Figure 6, left). For further calculations, mean Hill parameters were used (Table II). The determination of the average was conducted for strain from 2.4% to 3.2% because of the higher reliability of the measured values. 

The description of anisotropic creep via an equivalent stress according to Hill was tested for compression creep of WHIPOX . It allows a fast numerical estimation of the CMCs deformation behavior. The Hill model consists of few parameters which can easily be determined with a couple of measurements. One further advantage is the built-in implementation in commercial finite element software, e.g. ANSYS. However, the reliability of the results is limited. For WHIPOX , the most important limitations are the assumed isochoric behavior of the material, tension and compression symmetry as well as the fact that only one stress exponent and creep equation can be specified. 

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