Steady state King-Altman method

Comparison of Different Steady-State Methods. For relatively simple mechanisms, all the diagrammatic and systematic procedures illustrated in the foregoing sections are quite convenient. The King-Altman method is best suited for single-loop mechanisms, but becomes laborious for more complex cases with five or more enzyme forms because of the work involved in the calculation and drawing of valid patterns. With multiloop reaction schemes involving four to five enzyme species, the systematic approach requires the least effort, especially... 

The depicted mechanism is of course much more complex than the simple Michaelis-Menten scheme, but it can be simplified by employing the steady state approximation. This may be done in a relatively simple manner by applying either the King-Altman method or the less well-known Christiansen formalism. Applying the King-Altman method to the catalytic cycle given in Scheme 4.3 leads to a rate equation that is equal to the substi-tuted-enzyme mechanism, the detailed derivation of which was debated by Cornish-Bowden. ... 

In Chapter 3 (Section 3.4), we have derived the rate law for the reversible Michaelis-Menten mechanism with two central complexes (Reaction (3.31)) with the aid of the steady-state roximation. We shall proceed now with the derivation of the same rate law with the aid of the King-Altman method. In order to apply the King-Altman method efficiently to a specific mechanism, a rigorous procedure must be strictly followed (Fromm, 1975 Wong, 1975 Punch Allison, 2000). 

Rat equation in Enzyme kinetics (see), an equation expressing the rate of a reaction in terms of rate constants and the concentrations of enzyme spedes, substrate and product. When it is assumed that steady state conditions obtain, the Michaelis-Menten equation (see) is a suitable approximation. R.e. are represented graphically (see Enzyme graph) they may be derived by the King-Altman method (see). 

The King and Altman Method. King and Altman developed a systematic approach for deriving steady-state rate equations, which has contributed to the advance of enzyme kinetics. The first step of this method is to draw an enclosed geometric figure with each enzyme form as one of the corners. Equation (5), for instance, can be rewritten as ... 

A useful procedure for deriving steady-state rate expressions for enzyme-catalyzed reactions . Although not as commonly used as the King and Altman method, it is far more convenient (and less error-prone) when attempting to obtain expressions for complicated reaction schemes. One of its values is that the approach is very systematic and straightforward. The systematic nature of the procedure can be illustrated by the derivation of the steady-state ordered Bi Bi reaction. 

As has already been shown, graph theory methods were first used in chemical kinetics by King and Altman who applied them to linear enzyme mechanisms [1] to derive steady-state kinetic equations. Vol kenshtein and Gol dshtein in their studies during the 1960s [2 1] also elaborated a new formalism for the derivation of steady-state kinetic equations based on graph theory methods ("Mason s rule , etc.). 

The steady-state rate equation is obtained according to following rules (King and Altman method) ... 

More complex enzymatic reactions usually display Michaelis-Menten kinetics and can be described by Eq. (2). However, the forms of constants Km and Vm can be very complicated, consisting of many individual rate constants. King and Altman (7) have provided a method to readily derive the steady-state equations for enzymatic reactions, including the forms that describe Km and Vm. The advent of symbolic mathematics programs makes the implementation of these methods routine, even for very complex reaction schemes. The P450 catalytic cycle (Fig. 2) is an example of a very complicated reaction scheme. However, most P450-mediated reactions display standard hyperbolic saturation kinetics. Therefore, although the rate constants that determine Km and Vm are... 

In principle, the steady-state rate expression for any enzyme with any number of reactants can be derived using the methods of the previous section. In practice, the procedure is very laborious, so use is made of an algorithmic method, introduced by King and Altman in 1956 it is not applicable to (1) nonenzymatic reactions (each reactant concentration must be S>[E]0), (2) mixtures of enzymes, or (3) reactions with nonenzymatic steps. However, these are not severe restrictions. It is applied as follows ... 

From the four-state diagram of Figure 4.9, the expression for the steady state flux through the reaction can be obtained from the diagrammatic method of King and Altman [112]. The flux J may be expressed... 

At the time that the previous chapter in Volume 11 was written, the method of King and Altman (7) was the method of choice for deriving steady-state rate equations for enzymic reactions, and this is still true for any mechanism involving branched reaction pathways. The best description of this method may be found in Mahler and Cordes (8). A useful advance was made in 1975 with the introduction of the net rate constant method (9), and because it is the simplest method to use for any nonbranched mechanism, as well as for equations for isotopic exchange, positional isotopic exchange, isotope partitioning, etc., we shall present it here. 

The simple and elegant method of King and Altman allows the steady-state rate equations for mechanisms of considerable complexity to be written down in terms of the individual rate constants without going through complex algebraic expansions of large determinants. It was used to derive aU of the rate equations discussed in this and in the next chapters. 

Many methods and minor variants of methods, have appeared in the literature for the derivation of steady state equations (see for instance King Altman, 1956 Wong, 1975 Huang, 1979). They all depend on the solution of a set of linear equations. Matrix and other methods for deriving rate equations for the approach to the steady state are presented in section 5.1 and the principle used in equation (3.3.25) of writing rate equations in... 

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