Rapid equilibrium hypothesis

It should also be noted that kf2 is equivalent to kat in the Michaelis-Menten rapid-equilibrium hypothesis when the decomposition rate of the enzyme-substrate complex is fast, as described above in Scheme 1. However, the value of kcax may be attributable to more complex situations involving several decomposition terms. 

A few years later, Briggs and Haldane (1925) argued against the validity of the rapid equilibrium hypothesis and proposed a steady-state hypothesis according to which, after a very short transient phase, the ES complex remains constant throughout the whole reaction period, as shown in Fig. 3.2. 

Kinetic models can be derived from the above mechanisms according to the rapid equilibrium hypothesis. 

Kekule explained the second objection away by maintaining that 11 and 12 were in rapid equilibrium through concerted bond shifts, in something like the same manner as the free-rotation hypothesis mentioned previously ... 

Inherently, the orf/zo-phenylene spacer induces only a weak energy difference between the closed and open forms, but the ground-state structure of all MPB 7b-d was found to be well-defined, with or without P-B interaction. A different situation was observed for the related DPB 8a and TPB 9.24 The corresponding nB NMR signals (<5 = 43.1 ppm for 8a and 50.1 ppm for 9) are in between those of triarylboranes ( 70 ppm) and tetracoordinate derivatives thereof ( 0 ppm), suggesting a rapid equilibrium in solution between the open and closed forms (Figure 12). This hypothesis was further corroborated by low-temperature NMR experiments and DFT calculations. The minima associated with the open and closed forms were found to be almost isoenergetic (AG < 3 kcal/mol at 25 °C). 

A simple model of the chemical processes governing the rate of heat release during methane oxidation will be presented below. There are simple models for the induction period of methane oxidation (1,2.>.3) and the partial equilibrium hypothesis (4) is applicable as the reaction approaches thermodynamic equilibrium. However, there are apparently no previous successful models for the portion of the reaction where fuel is consumed rapidly and heat is released. There are empirical rate constants which, due to experimental limitations, are generally determined in a range of pressures or concentrations which are far removed from those of practical combustion devices. To calculate a practical device these must be recalibrated to experiments at the appropriate conditions, so they have little predictive value and give little insight into the controlling physical and chemical processes. 

This equilibrium hypothesis is, however, not necessarily valid for rapid chemical reactions. This brings us to the second way in which solvents can influence reaction rates, namely through dynamic or frictional effects. For broad-barrier reactions in strongly dipolar, slowly relaxing solvents, non-equilibrium solvation of the activated complex can occur and the solvent reorientation may also influence the reaction rate. In the case of slow solvent relaxation, significant dynamic contributions to the experimentally determined activation parameters, which are completely absent in conventional transition-state theory, can exist. In the extreme case, solvent reorientation becomes rate-limiting and the transition-state theory breaks down. In this situation, rate con-... 

In the derivation of the rate equation, it is assumed that the surface reaction is irreversible and rate determining, whereas the adsorption steps of hydrogen and aldol are rapid enough for the quasi-equilibrium hypothesis to be applied. The desorption step of triol is assumed to be irreversible and very rapid (cr —>0). 

For simple kinetic mechanisms, like irreversible one-substrate reactions, both rapid equilibrium and steady-state hypothesis lead to rate equations that are formally equal in parametric terms, so when those parameters are experimentally determined, results are the same no matter what hypothesis is considered. Kinetic parameters are to be experimentally determined to obtain validated rate expressions to be used for the design or performance evaluation of enzyme reactors. 

The above analysis was based on the hypothesis of rapid equilibrium. Analysis according to steady-state hypothesis can be quite cumbersome and mathematical algorithms are required to solve the equations. These can be done rather easily by using the algorithm of King and Altmann (Cornish-Bowden 1995) even though compact parametric expressions may be hard to obtain. 

The concentration of the intermediate A is obtained after applying the quasi-equilibrium hypothesis on the rapid step(6) ... 

For rapidly reacting intermediates, the quasi-steady-state hypothesis can be applied to eliminate the concentrations of the intermediates from the rate equations. For rapid reaction steps, the quasi-equilibrium hypothesis is used to eliminate the concentrations of the intermediates. 

From the general solution obtained with the quasi-steady-state hypothesis, the solutions corresponding to the quasi-equilibrium hypothesis can be obtained as special cases. If step I is much more rapid than step II, fc i fc+2CB in Equation 2.27, the reaction rate becomes... 

The quasi-equilibrium assumption is frequently used in the heterogeneous catalysis, since the surface reaction steps are often rate-Hmiting, while the adsorption steps are rapid. This is not necessarily true for large molecules. Here we consider the application of the quasi-equilibrium hypothesis on two kinds of reaction mechanisms, an Eley-Rideal mechanism and a Langmuir-Hinshelwood mechanism. The rate expressions obtained with this approach are referred to as Langmuir-Hinshelwood-Hougen-Watson (LHHW) equations in the literature, in honor of the pioneering researchers. 

Suppose that B is highly reactive. When formed, it rapidly reverts back to A or transforms into C. This implies kr > kf and ks kf. The quasi-steady hypothesis assumes that B is consumed as fast as it is formed so that its time rate of change is zero. More specifically, we assume that the concentration of B rises quickly and achieves a dynamic equilibrium with A, which is consumed at a much slower rate. To apply the quasi-steady h)rpothesis to component B, we set dbldt = 0. The ODE for B then gives... 

We cannot answer the question posed by Anfin-sen s hypothesis. Does the native state have a minimum value of the Gibbs energy Nevertheless, it is observed that proteins usually behave as if folded, unfolded forms are in a true thermodynamic equilibrium, and that this equilibrium is attained rapidly. The difference AG between a folded and a denatured protein is only 21-63 kj mol-1, which shows that folded proteins are only marginally more stable than are unfolded polypeptide chains.645 The value of AG of unfolding as a function of temperature T is given by Eq. 29-13, where AH(T) and ACp are the changes in enthalpy and heat capacity upon unfolding.645 646... 

The fundamental equation in enzyme kinetics is the Michaelis-Menten equation. These workers worked on the hypothesis that the reaction proceeds through an enzyme-substrate complex (ES) that forms rapidly from the free enzyme (E) and the substrate (S) and may be described by an equilibrium (or Michaelis) constant AT/. Upon reaction, the ES complex then decomposes and is converted to product by a rate-determining step with a rate constant Acat. The scheme is shown below ... 

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