Sigmoidal rate curves

Many sigmoid rate curves have been reported. A typical example is provided by the hydrolysis of phthalamic acid. ... 

Sigmoid rate curve. The purple line shows how the reaction rate, for a reaction catalyzed by a cooperative enzyme, varies with the substrate concentration. For comparison, the black line shows the shape of the hyperbolic curve seen with reactions catalyzed by enzymes that obey the Michaelis-Menten equation. 

Figure 8.31 Classical sigmoidal rate curves indicative of bio-catalysis with strong positive cooperativity involved in the catalytic mechanism. Curves move to the right as constant K increases.

A frequently encountered pH-rate profile exhibits a bell-like shape or hump, with two inflection points. This graphical feature is essentially two sigmoid curves back-to-back. By analogy with the earlier analysis of the sigmoid pH-rate curve, where the shape was ascribed to an acid-base equilibrium of the substrate, we find that the bell-shaped curve can usually be accounted for in terms of two acid-base dissociations of the substrate. The substrate can be regarded, for this analysis, as a dibasic acid H2S, where the charge type is irrelevant we take the neutral molecule as an example. The acid dissociation constants are... 

Provided that the time-temperature curve obtained from the calorimetric experiments is wholly of first-order, or comprises a first-order section, usually after the inflection point of sigmoid reaction curves, a conventional analysis yields a first-order rate constant ku which is related to the concentration of monomer, m, and the initial concentration of initiator, c0, by the equations... 

The rate constants for micelle-catalyzed reactions, when plotted against surfactant concentration, yield approximately sigmoid-shaped curves. The kinetic model commonly used quantitatively to describe the relationship of rate constant to surfactant, D, concentration assumes that micelles, D , form a noncovalent complex (4a) with substrate, S, before catalysis may take place (Menger and Portnoy, 1967 Cordes and Dunlap, 1969). An alternative model... 

Table 12.2 lists a number of rate equations that are commonly applied to dissolution processes.Two equations that are used quite frequently are the cube rate law (Hixon Crowell, 1931) which takes the geometry of the dissolving particle into account, and the Avrami-Erofejev law which applies to sigmoidal dissolution curves. 

These features led to a net increase in the surface area of the sample during the early stages of dissolution and this could explain the sigmoidal dissolution curve observed for some of the samples where the maximum in surface area coincided approximately with the maximum in dissolution rate, i. e. with the inflection point of the... 

The Richards model reduces the unexplained statistical variation in the accumulation of PCBs by phytoplankton, but it does not provide any information about the mechanisms responsible for the observed pattern. Numerous causes are possible for deviation from the classical pattern of accumulation. However, violations of assumptions associated with the classical model (i.e., constant uptake rate, instantaneous mixing within a single compartment, and a time-independent probability of depuration) are most likely the cause. With phytoplankton, several physiological mechanisms can potentially contribute to a sigmoidal accumulation curve. 

Although the Richards model does not provide mechanistic information, it does point out a need for further study and understanding of the uptake mechanisms. Brisbin et al. (30) reported that one consequence of sigmoidal accumulation is that there must be a period of accelerated or enhanced accumulation after the lag period in order to attain equilibrium levels similar to the classical model. In contrast to reports that cellular processes play no role in the accumulation of contaminants by phytoplankton (23, 25, 34-36), a sigmoidal accumulation curve may indicate that cellular processes such as the cycling of materials within the cell may enhance the rate of accumulation or depuration to a level above that which is attainable by diffusion alone. 

A plot of VQ against [S] for an allosteric enzyme gives a sigmoidal-shaped curve. Allosteric enzymes often have more than one active site which co-operatively bind substrate molecules, such that the binding of substrate at one active site induces a conformational change in the enzyme that alters the affinity of the other active sites for substrate. Allosteric enzymes are often multi-subunit proteins, with an active site on each subunit. In addition, allosteric enzymes may be controlled by effector molecules (activators or inhibitors) that bind to a site other than the active site and alter the rate of enzyme activity. Aspartate transcarbamoylase is an allosteric enzyme that catalyzes the committed step in pyrimidine biosynthesis. This enzyme consists of six catalytic subunits each with an active site and six regulatory subunits to which the allosteric effectors cytosine triphosphate (CTP) and ATP bind. Aspartate transcarbamoylase is feedback-inhibited by the end-product of the pathway, CTP, which acts as an allosteric inhibitor. In contrast, ATP an intermediate earlier in the pathway, acts as an allosteric activator. 

These observations can be represented as a special case of the general rate equation derived by the application of order-disorder theory to diffusionless transitions in solids.3 According to this equation, the shape of the rate curve is determined by the relative numerical values of zkp/kn and of c. The larger the factor is relative to c, the more sigmoidal the curves become. This is understandable since the propagation effect which is responsible for the autocatalytic character of the transformation becomes more noticeable when kPlkn is large and c small. Under these conditions some time elapses before a sufficient number of nucleation sites are formed then the... 

Representative for systems exhibiting sigmoidal conversion curves Fig. 1 shows experimental results for the rate constant of the reaction of TS, evaluated from thermal and y-polymerization data according to K = (1 — X) dX(t)/dt, and normalized to the rate constant in the low conversion limit. It is obvious, that at low conversion K depends on X, contrary to what is to be expected for a simple first order reaction. The functional form of KPC) is different for the two modes of polymerization. The overall increase of K with increasing X reveals an autocatalytic reaction enhancement. A measure for its efficiency is the ratio K(X = 0.5)/K(X = 0) which tirnis out to be about 200 for TS under thermal polymerization conditions. This effect is often observed with disubstituted diacetylenes albeit with different kinetic... 

The change in CD with time was analyzed by fitting the early linear portion of the sigmoidal decay curve. These rates scaled as WT concentration. Relative activities were calculated by dividing the rate of the mutant by the rate of WT at the same protein concentration. 

A second approach is to calculate the change in potential-per-unit change in volume in reagent (AE/AV). By inspection, the endpoint can be located from the inflection point of the titration curve. This is the point that corresponds to the maximum rate of change of cell emf per unit volume of titrant added (usually 0.05 or 0.1 mL). The first-derivative method is based on the sigmoid shaped curve. 

Molecular oxygen is the major cause of irreversible deterioration of hydrocarbon substrates, leading to the loss of useful properties and to the ultimate failure of the substrate. The oxidation process of hydrocarbons is autocatalytic oxidation starts slowly, sometimes with a short induction period, followed by a gradual increase in the rate, concomitant with the build up of hydroperoxides, which eventually subside, giving rise to a sigmoidal oxidation curve. When initiators such as peroxides are present, the length of the induction period is absent, or very short, but it can be prolonged by antioxidants, as shown in Fig. 1. The basic autoxidation theory of hydrocarbons involves a complex set of elementary reaction steps in a free radical-initiated chain reaction mechanism the basic tenets of this theory apply equally to polymer oxidation. 

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