King and Altman method

This appendix illustrates the steps involved in deriving the reaction rate equation (Equation 17.11) from the reaction scheme given in Section 17.3.2 using the King and Altman method.41 This... 

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. 

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

The Systematic Approach. The systematic approach for deriving rate equations was first devised by Fromm based on certain concepts advanced by Volkenstein and Goldstein. Its underlying principles, however, are more akin to the graphic method of King and Altman. The procedure to be described here is a modified method that includes the contributions from the aforementioned workers and from Wong and Hanes. ... 

King and Altman [5] and Temkin developed a method to represent a reaction mechanism as a graph. Its nodes are intermediates and its edges are steps. Reaction directions are marked by arrows on the edges. 

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.). 

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... 

The flux expression for this mechanism follows from application of the method of King and Altman [112] ... 

Many enzyme reactions have more than one intermediate for which King and Altman (1956) devised a method, based on matrix algebra, by establishing the rate equation of a given enzymic reaction simply by inspecting all complexes and the reactions between them. 

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. 

Mechanism I is referred to as a compulsory-pathway mechanism since the order of addition of substrates to the enzyme is fixed mechanism II is often called a shuttle or ping pong mechanism because part of a substrate is shuttled back and forth between substrates and enzyme and mechanism III involves a random addition of substrates to the enzyme. Obviously a large number of additional mechanisms could be written by permuting the substrates and by combining two of the mechanisms. If (C) = (D) = 0, the initial velocities for the first two mechanisms can be easily obtained using the method of King and Altman ... 

The method of King and Altman rendered an invaluable service to enzymology because, with its help, the rate laws for many major reaction mechanisms in enzyme kinetics were developed. It is not necessary to understand the theory of the King-Altman method in order to apply it in practice, and indeed the theory is considerably more difficult than the practice. Therefore, we shall describe in the following sections the derivation of rate laws for several simple mechanisms... 

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. 

The equations for any other exchange reaction, and indeed for any other mechanism, can be derived similarly. However, in most cases, the rate equations for isotope exchange are derived by the method of King and Altman and various extensions of the same also, the rate equations may be derived efficiently by the net rate constant method (Chapter 4). In most cases, the rate equations for isotope exchange are far too complicated to permit the determination of the usual kinetic constants. Nevertheless, ffiere are a number of simplifying assumptions which will permit the derivation of manageable rate equations in specific cases (Boyer, 1959 Fromm eta/., 1964 Darvey, 1973). 

The graph approach to solving the concentrations is the powerful method of King and Altman [15, 19-21]. We will not pursue this method, but we will build from this a new perspective based on the -representation. The TOF of the... 

King, E.I. and Altman, C., A schematic method of deriving rate laws for enzyme-catalyzed reactions,... 

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... 

For mechanisms involving random addition of substrates, the King-Altman method gives squared terms in numerator and denominator of the rate equations, which are messy and difficult to work with. The method of Cha (10) treats each random segment as if it were in rapid equilibrium, and this simplifies the rate equation. The fact that data fit such a simplified equation does not prove that the mechanism is a rapid equilibrium one (see the rules in Section V,A,2 below) but does facilitate initial velocity analysis. 

Please note, these approximations are all dealing with the simple reaction sequence from Scheme 4.2. The application of e.g. the King-Altman method for multistep reactions will lead to the Michaelis-Menten equation in its typical form but with much more complex values for and Vkf The typical curve resulting from applying eqn (4.1) with = 4at = 1 is shown in Figure 4.1 adapted from ref. 26. 

Let us proceed with the derivation of a rate law for this mechanism with the aid of the King-Altman method. First, let us draw a master pattern as a closed loop, showing all enzyme forms and the reaction between them (Fig. 1). 

The last example in Figure 2 clearly shows why, in complex cases, the finding of partial patterns from the master pattern is the most difficult part of the King-Altman method. Often, especially in complex mechanisms, it is not easy to write down all possible King-Altman patterns, and errors are a common occurrence. 

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