Enzyme reactions general rate equation

A reaction which follows power-law kinetics generally leads to a single, unique steady state, provided that there are no temperature effects upon the system. However, for certain reactions, such as gas-phase reactions involving competition for surface active sites on a catalyst, or for some enzyme reactions, the design equations may indicate several potential steady-state operating conditions. A reaction for which the rate law includes concentrations in both the numerator and denominator may lead to multiple steady states. The following example (Lynch, 1986) illustrates the multiple steady states... 

The kinetics of the general enzyme-catalyzed reaction (equation 10.1-1) may be simple or complex, depending upon the enzyme and substrate concentrations, the presence/absence of inhibitors and/or cofactors, and upon temperature, shear, ionic strength, and pH. The simplest form of the rate law for enzyme reactions was proposed by Henri (1902), and a mechanism was proposed by Michaelis and Menten (1913), which was later extended by Briggs and Haldane (1925). The mechanism is usually referred to as the Michaelis-Menten mechanism or model. It is a two-step mechanism, the first step being a rapid, reversible formation of an enzyme-substrate complex, ES, followed by a slow, rate-determining decomposition step to form the product and reproduce the enzyme ... 

In this case, most of the enzyme is in the EP form at saturation, and Fmax = /c3[Et]. It is useful to define a more general rate constant, kcat, to describe the limiting rate of any enzyme-catalyzed reaction at saturation. If the reaction has several steps and one is clearly rate-limiting, fccat is equivalent to the rate constant for that limiting step. For the simple reaction of Equation 6-10, kCat = k2. For the reaction of Equation 6-25, kcat = k3. When several steps are partially rate-limiting, kcat can become a complex function of several of the rate constants that define each individual reaction step. In the Michaelis-Menten equation, kcat = Fmax/[Et], and Equation 6-9 becomes... 

Two possible explanations can be readily put forward as to why this form of equation should be suitable for describing the dependence of microbial growth rate on feed concentration. The first of these is that the equation has the same form as the theoretically based Michaelis-Menten equation used to describe enzyme kinetics. The chemical reactions occurring inside a microbial cell are generally mediated by enzymes, and it would be reasonable to suppose that one of these reactions is for some reason slower than the others. As a result the growth kinetics of the micro-organism would be expected to reflect the kinetics of this enzyme reaction, probably modified in some way, but in essence having the form of the Michaelis-Menten equation. 

The general rule for writing the rate equation according to the quasi-equilibrium treatment of enzyme kinetics can be exemplified for the random bisubstrate reaction with substrates A and B forming products P and Q (Figure 7.1), where KaKab = KbKba and KpKpq = KqKqp. 

Enzyme kinetics are normally determined under steady-state, initial-rate conditions, which place several constraints on the incubation conditions. First, the amount of substrate should greatly exceed the enzyme concentration, and the consumption of substrate should be held to a minimum. Generally, the amount of substrate consumed should be held to less than 10%. This constraint ensures that accurate substrate concentration data are available for the kinetic analyses and minimizes the probability that product inhibition of the reaction will occur. This constraint can be problematic when the Km of the reaction is low, since the amount of product (10% of a low substrate concentration) may be below that needed for accurate product quantitation. One method to increase the substrate amount available is to use larger incubation volumes. For example, a 10-mL incubation has 10 times more substrate available than a 1-mL incubation. Another method is to increase the sensitivity of the assay, e.g., using mass spectral or radioisotope assays. When more than 10% of the substrate is consumed, the substrate concentration can be corrected via the integrated form of the rate equation (Dr. James Gillette, personal communication) ... 

Homogeneous catalyzed reactions (other than enzyme-catalyzed reactions) have been used extensively in trace analysis. The general approach is to employ a reaction in which the species of analytical interest acts as a catalyst. Since the mechanisms of such reactions are varied and complex and often are not completely known, it is impractical or impossible to give exact rate equations for such mechanisms here. However, the rate expressions in terms of the rate of formation of product can usually be written in the general form... 

The molecular details of enzymatic reactions are currently of great interest and particularly so to kineticists because of the special kinetic problems involved. First, the complex mechanisms of enzymatic reactions require rather special phenomenological rate equations second, since enzymes are catalysts, the nature of catalysis in general must be considered and finally, the complex nature of even the simplest enzymatic mechanism makes its clarification a challenge. 

In the general case, the Rapid Equilibrium Random Bi Bi system with both dead-end complexes ElAP and EBQ (Reaction (8.35)), the denominator of the rate equation has nine terms, each for one form of enzyme the unity represents the free enzyme (Eq. (8.37)). From the general rate Eq. (8.37), one can write down directly the rate equations for all other possible combinations of the Rapid Equilibrium Random Bi Bi system. 

Following Widdas °, we have assumed that the carrier equilibrates relatively fast with the substrate compared with the rate of translocation, as in most enzyme reactions. When this is not so, a new rate-limiting term enters into the equations. This has been discussed in some detail by Rosenberg and Wilbrandt , Wilbrandt and Rosenberg , and Dawson and Widdas . It does not seem to have been generally appreciated that translocator-catalysed reactions in which the carrier site does not equilibrate completely... 

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