Hyperbolic concentrator

Several workers have concluded that under conditions used in their study ion-pairing in the mobile phase between amphiphilic hetaeron ions and oppositely charged sample components governed retention. Horvath et al. (34) examined the effect of alkyl sulfates and other alkyl anions on the retention of catecholamines in which both the concentration and the length of the alkyl chains of the hetaerons were varied. The hyperbolic concentration dependence of the retention factor shown in Fig. 48, was found to be similar to that reported by others. 

FIGURE 18 (a) Practical thermal secondary concentrator referred to as the trumpet, which has been built and tested at the University of Chicago, (b) Profiles for a DCPC and two dielectric compound hyperbolic concentrators (DCHCs) used for photovoltaic secondaries. 

Plot the relative amount of bound fluorescent ligand (FL) as a function of ligand concentration in solution for each protein in the array (7) and fit to a simple hyperbolic concentration-response curve according to ... 

Black and Leff [11] presented a model, termed the operational model, that avoids the inclusion of ad hoc terms for efficacy. This model is based on the experimental observation that the relationship between agonist concentration and tissue response is most often hyperbolic. This allows for response to be expressed in terms of... 

FIGURE 3.6 Classical model of agonism. Ordinates response as a fraction of the system maximal response. Abscissae logarithms of molar concentrations of agonist, (a) Effect of changing efficacy as defined by Stephenson [24], Stimulus-response coupling defined by hyperbolic function Response = stimulus/(stimulus-F 0.1). (b) Dose-response curves for agonist of e = 1 and various values for Ka. 

It can be seen that if KA< v then negative and/or infinite values for response are allowed. No physiological counterpart to such behavior exists. This leaves a linear relationship between agonist concentration and response (where Ka = v) or a hyperbolic one (KA>v). There are few if any cases of truly linear relationships between agonist concentration and tissue response. Therefore, the default for the relationship is a hyperbolic one. 

General Procedure Dose-response curves are obtained for an agonist in the absence and presence of a range of concentrations of the antagonist. The dextral displacement of these curves (ECSo values) are fit to a hyperbolic equation to yield the potency of the antagonist and the maximal value for the cooperativity constant (a) for the antagonist. 

The preferred kinetic model for the metathesis of acyclic alkenes is a Langmuir type model, with a rate-determining reaction between two adsorbed (complexed) molecules. For the metathesis of cycloalkenes, the kinetic model of Calderon as depicted in Fig. 4 agrees well with the experimental results. A scheme involving carbene complexes (Fig. 5) is less likely, which is consistent with the conclusion drawn from mechanistic considerations (Section III). However, Calderon s model might also fit the experimental data in the case of acyclic alkenes. If, for instance, the concentration of the dialkene complex is independent of the concentration of free alkene, the reaction will be first order with respect to the alkene. This has in fact been observed (Section IV.C.2) but, within certain limits, a first-order relationship can also be obtained from many hyperbolic models. Moreover, it seems unreasonable to assume that one single kinetic model could represent the experimental results of all systems under consideration. Clearly, further experimental work is needed to arrive at more definite conclusions. Especially, it is necessary to investigate whether conclusions derived for a particular system are valid for all catalyst systems. 

It was found out that reaction of the hydrolysis of highlymetoxilated beet pectin (catalyzed by P. fellutanum pectinesterase) obeyed Michaelis—Menten equation only under low substrate concentrations (up to 1.2%), when graph of the dependence of reaction speed was hyperbolic in form. 

These relationships are identical to Haldane relationships, but unlike the latter, their validity does not derive from a proposed reaction scheme, but merely from the observed hyperbolic dependence of transport rates upon substrate concentration. Krupka showed that these relationships were not obeyed by the set of data previously used by Lieb [64] to reject the simple asymmetric carrier model for glucose transport. Such data therefore cannot be used either to confirm or refute the model. 

The inhibition modality for a slow binding inhibitor is easily determined from the effects of substrate concentration on the value of k0bs at any fixed inhibitor concentration (Tian and Tsou, 1982 Copeland, 2000). For a competitive inhibitor the value of fcobs will diminish hyperbolically with increasing substrate concentration according to Equation (6.15) ... 

These results show that if the relationship between the concentration of an agonist and the proportion of receptors that it occupies is measured directly (e.g., using a radioligand binding method), the outcome should be a simple hyperbolic curve. Although the curve is describable by the Hill-Langmuir equation, the dissociation equilibrium constant for the binding will be not KA but Ke, which is determined by both E and KA. 

Though this looks complicated, it still predicts a simple hyperbolic relationship (as with the Hill-Langmuir equation see Figure 1.1 and the accompanying text) between agonist concentration and the proportion of receptors in the state (AR G ) that leads to a response. If a very large concentration of A is applied, so that all the receptors are occupied, the value of pAR.G. asymptotes to ... 

Now consider the other extreme condition where diffusion is rapid relative to chemical reaction [i.e., hT( 1 — a) is small]. In this situation the effectiveness factor will approach unity for both the poisoned and unpoisoned reactions, and we must retain the hyperbolic tangent terms in equation 12.3.124 to properly evaluate Curve C in Figure 12.11 is calculated for a value of hT = 5. It is apparent that in this instance the activity decline is not nearly as sharp at low values of a as it was at the other extreme, but it is obviously more than a linear effect. The reason for this result is that the regions of the catalyst pore exposed to the highest reactant concentrations do not contribute proportionately to the overall reaction rate because they have suffered a disproportionate loss of activity when pore-mouth poisoning takes place. 

A plot of the initial reaction rate, v, as a function of the substrate concentration [S], shows a hyperbolic relationship (Figure 4). As the [S] becomes very large and the enzyme is saturated with the substrate, the reaction rate will not increase indefinitely but, for a fixed amount of [E], it reaches a plateau at a limiting value named the maximal velocity (vmax). This behavior can be explained using the equilibrium model of Michaelis-Menten (1913) or the steady-state model of Briggs and Haldane (1926). The first one is based on the assumption that the rate of breakdown of the ES complex to yield the product is much slower that the dissociation of ES. This means that k2 tj. 

COOPERATIVE ENZYMES do not show a hyperbolic dependence of the velocity on substrate concentration. If the binding of one substrate increases the affinity of an oligomeric enzyme for binding of the next substrate, the enzyme shows positive cooperativity. If the first substrate makes it harder to bind the second substrate, the enzyme is negatively cooperative. 

Probably the most important variable to consider in defining optimal conditions or standard conditions is the initial substrate concentration. Most enzymes show a hyperbolic curve as relation between initial reaction velocity and substrate concentration, well known now as the Michaelis-Menten curve. With increasing substrate concentration (S) the velocity (o) rises asymptotically to a maximum value (V) (Fig. 3), according to the expression ... 

The Thiele modulus is related to the concentration dependence in a catalyst body by the following equations representing the ratios of the hyperbolic cosines ... 

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