Sigmoid rate equations

To account for positive cooperativity and sigmoidal rate equations, a number of theoretical models for allosteric regulation have been developed. Common to most models is the assumption (and requirement) that enzymes act as multimers and exhibit interactions between the units. We briefly mention the most... 

Figure 10. A sigmoidal rate equation as a function of the substrate concentration S. Shown is the rate for n 1 (Michaelis Menten, dotted line), n 2 (dashed line), and n 4 (solid line). For increasing n, the rate equation is increasingly switch like. Parameters are Vm 1 and Ks 1.

Reference to the decomposition of KMn04 has already been made in the discussion of chain branching reactions (Chap. 3, Sect. 3.2) in which the participation of a highly reactive intermediate was postulated. This work provided a theoretical explanation of the Prout—Tompkins rate equation [eqn. (9)]. Isothermal decomposition in vacuum of freshly prepared crystals at 473—498 K gives symmetrical sigmoid a time curves which are described by the expression... 

The limiting cases are limvo 0 a = 1 and limy. x a = 0. To evaluate the saturation matrix we restrict each element to a well-defined interval, specified in the following way As for most biochemical rate laws na nt 1, the saturation parameter of substrates usually takes a value between zero and unity that determines the degree of saturation of the respective reaction. In the case of cooperative behavior with a Hill coefficient = = ,> 1, the saturation parameter is restricted to the interval [0, n] and, analogously, to the interval [0, n] for inhibitory interaction with na = 0 and n = , > 1. Note that the sigmoidality of the rate equation is not specifically taken into account, rather the intervals for hyperbolic and sigmoidal functions overlap. 

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 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... 

The rate equations which have found most widespread application to solid state reactions are summarized in Table 3.3. Other functions can be found in the literature. The expressions are grouped according to the shape of the isothermal a-time curves as acceleratory, sigmoid or deceleratory. The deceleratory group is further subdivided according to the controlling factor assumed in the derivation, as geometrical, diffusion or reaction order. 

Some of the isothermal relative rate [= (dfli d/V(da/d/LJ - time plots corresponding to the rate equations in Table 3.3. (a) sigmoid models (b) geometrical and reaction order (RO) models. (NOTE (i) the relative rate - time plots for the third-order rate equation and the difhision models are too deceleratory for usehil comparison and (ii) calculation on an arbitrary basis, e.g. that a = 0.98 at / = 100 min., results in plots of relative rates against a, in place of time, having similar shapes). 

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. 

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. 

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. 

Enzymatic reactions frequently undergo a phenomenon referred to as substrate inhibition. Here, the reaction rate reaches a maximum and subsequently falls as shown in Eigure 11-lb. Enzymatic reactions can also exhibit substrate activation as depicted by the sigmoidal type rate dependence in Eigure 11-lc. Biochemical reactions are limited by mass transfer where a substrate has to cross cell walls. Enzymatic reactions that depend on temperature are modeled with the Arrhenius equation. Most enzymes deactivate rapidly at temperatures of 50°C-100°C, and deactivation is an irreversible process. 

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 kinetic analysis of the sigmoid pH-rate profile will yield numerical estimates of the pH-independent parameters K, k, and k". With these estimates the apparent constant k is calculated using the theoretical equation over the pH range that was explored experimentally. Quantitative agreement between the calculated line and the experimental points indicates that the model is a good one. A further easy, and very pertinent, test is a comparison of the kinetically determined value with the value obtained by conventional methods under the same conditions. 

Bircumshaw and Edwards [1029] showed that the rate of nickel formate decomposition was sensitive to reactant disposition, being relatively greater for the spread reactant, a—Time curves were sigmoid and obeyed the Prout—Tompkins equation [eqn. (9)] with values of E for spread and aggregated powder samples of 95 and 110 kJ mole-1, respectively. These values are somewhat smaller than those subsequently found [375]. The decreased rate observed for packed reactant was ascribed to an inhibiting effect of water vapour which was most pronounced during the early stages. 

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. 

The answer to question (1) is illustrated graphically in Figure 14.6 (not specifically for Example 14-8). Again, the sigmoidal curve is constructed from the material balance, equation 14.3-5 (equation (A) in Example 14-8 or its equivalent for other rate laws). The two straight lines corresponding to feed temperatures T 0 and T are constructed... 

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... 

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