Sigmoidal rate plot

For an enzyme that follows MichaeHs—Menten kinetics, R = SI. For a regulatory enzyme that gives a sigmoidal rate plot, Rj < 81 if the enzyme is exhibiting positive cooperativity, a term that means that the substrate and enzyme bind in such a way that the rate increases to a greater extent with increasing [S] than the MichaeHs—Menten model predicts. Cases with R-s > 81 indicate negative cooperativity so that the catalytic effect becomes less than that found in MichaeHs—Menten kinetics. In these cases, kinetic analysis is usually carried out by means of the HiU equation. 

It should be pointed out that sigmoidal rate plots are sometimes observed for reactions of solids. One of the rate laws used to model such reactions is the Prout-Tompkins equation, the left-hand side of which contains the function ln(a/ (1 — a)) where a is the fraction of the sample reacted (see Section 7.4). The left-hand sides of Eqs. (6.72) and (7.68) have the same form, and both result in sigmoidal rate plots. These cases illustrate once again how gready different types of chemical processes can give rise to similar rate expressions. 

Having a set of a,t) data available, a graph was prepared to illustrate the type of plot that can be expected when a reaction follows an Avrami-Erofeev rate law. The result is shown in Figure 7.2, and the sigmoidal curve is characteristic of a reaction that follows a nucleation rate law. In Chapter 2, it was shown that a sigmoidal rate plot results from autocatalysis, but for reactions in the solid state such plots are more likely to indicate that the reaction is controlled by some type of nucleation process. 

When the data are presented in this way, the unmistakable sigmoidal nature of the rate plots for dequation-anation of [Co(NH3)5H20]Cl3 suggests that the process obeys an Avrami—Erofeev type of rate law. In... 

In isotope exchange studies, if we are plotting v x versus a substrate-product pair, we can encounter a number of types of exchange rate profiles, including h5 erbolic (H), or sigmoidal (S) plots with no, partial or complete depression of... 

It states that the rate is proportional to the fraction x that has decomposed (which is dominant early in the reaction) and to the fraction not decomposed (which is dominant in latter stages of reaction). The decomposition of potassium permanganate and some other solids is in accordance with this equation. The shape of the plot of x against t is sigmoid in many cases, with slow reactions at the oeginning and end, but no theory has been proposed that explains everything. 

If k is much larger than k", Eq. (6-64) takes the form of Eq. (6-61) for the fraction Fhs thus we may expect the experimental rate constant to be a sigmoid function of pH. If k" is larger than k, the / -pH plot should resemble the Fs-pH plot. Equation (6-64) is a very important relationship for the description of pH effects on reaction rates. Most sigmoid pH-rate profiles can be quantitatively accounted for with its use. Relatively minor modifications [such as the addition of rate terms first-order in H or OH to Eq. (6-63)] can often extend the description over the entire pH range. 

The above equation then represents the balanced conditions for steady-state reactor operation. The rate of heat loss, Hl, and the rate of heat gain, Hq, terms may be calculated as functions of the reactor temperature. The rate of heat loss, Hl, plots as a linear function of temperature and the rate of heat gain, Hq, owing to the exponential dependence of the rate coefficient on temperature, plots as a sigmoidal curve, as shown in Fig. 3.14. The points of intersection of the rate of heat lost and the rate of heat gain curves thus represent potential steady-state operating conditions that satisfy the above steady-state heat balance criterion. 

In autocatalytic reaction the graph plotted between the rate of reaction and time shows a sigmoid curve as shown in Fig. 6.1. As the product (catalyst) concentration increases the rate increases, and reaches to a maximum when the reaction is complete. 

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... 

When the sigmoidal shape of the rate constant/pH profile associated with (1.207) or the simpler derivatives (1.208) or (1.209) give way to a bell-shape or inverted bell-shape plot, the reactions of at least three acid-base-related species (two equilibria) have to be considered. This may involve acid-base forms of (a) one reactant or (b) two different reactants. 

The ALIS-based off-rate measurement method was applied to a proprietary series of Zap-70 Kinase inhibitors. First, an ACE50 experiment was conducted to demonstrate that the compounds bind the same site as the quench reagent staurospor-ine. As shown in Fig. 3.15, sigmoidal plots indicate that, with the exception of one compound, the ACE50 values were all very similar to one-another. Linear ratio plots of the same ACE50 data confirm that the compounds all bind isosteri-cally with respect to the quench reagent, a necessary prerequisite for effective competition. 

For example, Bachelard used [Mgtotai]/[ATPtotai ] = 1 in his rate studies, and he obtained a slightly sigmoidal plot of initial velocity versus substrate ATP concentration. This culminated in the erroneous proposal that brain hexokinase was allosterically activated by magnesium ions and by magnesium ion-adenosine triphosphate complex. Purich and Fromm demonstrated that failure to achieve adequate experimental control over the free magnesium ion concentration can wreak havoc on the examination of enzyme kinetic behavior. Indeed, these investigators were able to account fully for the effects obtained in the previous hexokinase study. ... 

FIGURE 16.6 Dependence of reaction showing sigmoidal (bottom) and hyperbolic (top) behavior. The top plot also shows the initial rate of reaction as a function of reactant concentration when the concentration of enzyme remains constant. 

M aqueous NaOH (done quickly before the subsequent hydrolysis could occur to any extent) showed the monodeprotonation with pKa value of 9.1, which was assigned to the 25a = 25b equilibrium. The pK value was higher than that of 7.3 for 24a under the same conditions, which is ascribable to the proximate phosphate anion interaction with zinc(II) (like 25c). The pendent phosphodiester in 25b underwent spontaneous hydrolysis in alkaline buffer to yield a phosphomonoester-pendent zinc(II) complex 26. Plots of the first-order rate constants vs pH (=7.5 -10.5) gave a sigmoidal curve with an inflection point at pH... 

Allosteric enzymes do not follow the Michaelis-Menten kinetic relationships between substrate concentration Fmax and Km because their kinetic behaviour is greatly altered by variations in the concentration of the allosteric modulator. Generally, homotrophic enzymes show sigmoidal behaviour with reference to the substrate concentration, rather than the rectangular hyperbolae shown in classical Michaelis-Menten kinetics. Thus, to increase the rate of reaction from 10 per cent to 90 per cent of maximum requires an 81-fold increase in substrate concentration, as shown in Fig. 5.34a. Positive cooperativity is the term used to describe the substrate concentration-activity curve which is sigmoidal an increase in the rate from 10 to 90 per cent requires only a nine-fold increase in substrate concentration (Fig. 5.346). Negative cooperativity is used to describe the flattening of the plot (Fig. 5.34c) and requires requires over 6000-fold increase to increase the rate from 10 to 90 per cent of maximum rate. 

This behavior can be clearly seen in Fig. 6.22b, in which experimental chargetime curves of quinizarin monolayers on mercury obtained for different values of t in order to obtain several values of the dimensionless rate constant k°i for this system have been plotted [46]. It can be seen that the charge-time curves are dramatically affected by the time length, so the curve corresponding to t = 10 ms can be considered as practically reversible (stepped sigmoid), whereas that corresponding tor = 1 ms presents an almost continuous feature, which is typical of irreversible processes. 

The second-order dependence of the TNP hydrolysis rate catalyzed by 11 fits the kinetic equation 7. A plot of the observed rate constants kobsi as a function of the pH using 0.1 mM 11 and 10 pAf TNP gave a similar sigmoidal curve to that found in the earlier NA hydrolysis with an inflection point at pH 8.3 (see Figure 10). Therefore, the same species lib is concluded to react with NA and TNP. The second-order rate constant TNP (see Eq. [8]) was extremely large 1.1 X 103 AT1 sec"1 at 25°C and pH 10.2 with I = 0.10 (NaN03). 

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. 

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