Substrates Hill model

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... 

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 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 operational model allows simulation of cellular response from receptor activation. In some cases, there may be cooperative effects in the stimulus-response cascades translating activation of receptor to tissue response. This can cause the resulting concentration-response curve to have a Hill coefficient different from unity. In general, there is a standard method for doing this namely, reexpressing the receptor occupancy and/or activation expression (defined by the particular molecular model of receptor function) in terms of the operational model with Hill coefficient not equal to unity. The operational model utilizes the concentration of response-producing receptor as the substrate for a Michaelis-Menten type of reaction, given as... 

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

Similar to Eq. (67), the first reaction (incorporating the enzyme phosphofructo-kinase) exhibits a Hill-type inhibition by its substrate ATP [126]. The overall ATP utilization v3 (ATP) is modeled by a saturable Michaelis Menten function. The system is specified by five kinetic parameters (with Gx lumped into Vm ), the Hill coefficient n, and the total concentration, 4 / = [ATP] + [ADP]. Note that the model is not intended to capture biological realism, rather it serves as a paradigmatic example to identify dynamic behavior in metabolic pathways. 

For the irreversible reactions, we assume Michaelis Menten kinetics, giving rise to 15 saturation parameters O1. C [0, 1] for substrates and products, respectively. In addition, the triosephospate translocator is modeled with four saturation parameters, corresponding to the model of Petterson and Ryde-Petterson [113]. Furthermore, allosteric regulation gives rise to 10 additional parameters 7 parameters 9" e [0, — n for inhibitory interactions and 3 parameters 0" [0, n] for the activation of starch synthesis by the metabolites PGA, F6P, and FBP. We assume n = 4 as an upper bound for the Hill coefficient. 

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... 

An entirely distinct series of model complexes has been carried out in order to show that metal porphyrins will actually bind to the type of substrate with which P-450 interacts. Hill, Macfarlane, Mann, and Williams (51) have studied molecular complex formation between such molecules as quinones and sterols and several metal porphyrins. The complexes between some of the porphyrins and sterols are remarkable strong. At the same time they have devised NMR methods for the elucidation of the structures of these complexes. 

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. 

Several models have been proposed to account for the sigmoidal kinetics observed as a function of the concentration in substrate or effector. The first, developed by Hill (1910) for the binding of oxygen to... 

For the allosteric model based on a concerted transition between the two conformational states, as considered here in the model for glycolytic oscillations, application of definition (2.27) in the case where the two states of the enzyme differ only by the affinity toward the substrate (0 = 1) leads to expression (2.28) for the Hill coefficient related to saturation of the enzyme by the substrate (Goldbeter, 1976,1977) ... 

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) ... 

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. 

An advantage of the CT model, however, is the fact that it is possible to estimate the magnitude of the enzyme-substrate dissociation constant of the enzyme. This is not possible with the Hill equation. As described before, the Hill constant is a complex term that is related but is not equivalent to, the enzyme-substrate dissociation constant. By using the CT model, it is also possible to obtain estimates of the allosteric constant, L. This may prove useful in the study of allosteric modulators of enzyme activity. 

Addition of substrate, which here is synonymous to the allosteric effector, shifts the equilibrium from the low affinity T-form to the substantially more catalytically active R-form. Since one substrate molecule activates four catalytically active sites, the steep rise in enzyme activity after only a slight increase in substrate concentration is not unexpected. In this model it is important that the RT conformation is not permitted. All subunits must be in the same conformational state at one time to conserve the symmetry of the protomers. The equation given by Hill in 1913, derived from the sigmoidal absorption of oxygen by hemoglobin, is also suitable for a quantitative description of allosteric enzymes with sigmoidal behavior ... 

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