The method of King and Altman

To effectively express concentrations and fluxes for this example, we introduce the shorthand notations LJ,tJ,n,n,I],Z],ll ,C,LJ,-U,rt, n,I, U, fl, and C, where each of these symbols represents a product of three first-order or pseudo-first-order rate constants. These symbols (called directional diagrams) represent the product of rate constants along the path defined by the diagram. The symbol U represents the product +2 +3 +4. Where substrate or product concentrations participate in one of the state transitions for a diagram, the pseudo-first-order rate constant (which incorporates the steady state reactant concentration) is used. Therefore the diagram U represents the product A 4/C 3A 21I, 

The steady state concentrations of any of the enzyme states Ei, EA, EB, or E2 can be expressed as the sum of all the directional diagrams that feed into a given state divided by the sum of all diagrams. For example, the directional diagrams that feed into state EA are C. U. D. and I f. The concentration [EA] is given by 

To illustrate this point, we consider another four-state mechanism for A = B  

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

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 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 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. ... 

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 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 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... 

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.). 

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 ... 

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. 

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. 

In this respect, very efficient modifications of the King-Altman method have been described by Wong and Hanes (1962), Volkenstein and Goldstein (1966), Gulbinsky and Cleland (1968), Fromm (1975) and others. A special attention deserves the method of Cha (1968) which is a very useful and a very widespread simplification of the King-Altman method (Topham Brockelhurst, 1992). 

Cha (1968) has described a method for analyzing mechanisms that contain steps in equilibrium that is much simpler than the complete King-Altman andysis because each group of enzyme forms at equilibrium can be treated as a single species. Thus, the method of Cha provides for a condensation of King-Altman patterns and shortens considerably the procedure of derivatioa... 

For multistep intermediate complex models and those with multiple substrates, rate equations are most rehably written down by using the method of Cha (1968). Cha s method is a modification of the entire King-Altman schematic method of analysis in which simplification is achieved by treating some of the steps as quasi-equihbria (Chapter 4). It is particularly useful in the analysis of pH dependence, because rapid-equilibrium assumptions may be justified by the high rates of proton-transfer steps in aqueous media (Knowles, 1976). 

Altman and King made the first detailed study of this system, using Cr(ril) solutions containing only the monomeric species and Cr(VI) solutions which had been allowed to age. The isotopic method and lead chromate precipitation separation were used to obtain kinetic data at a temperature of 94.8 °C. Over the range of concentrations, Cr(VI) 2.3x10 to 8.4x10 A/, Cr(III) 1.8x10 to... 

The additional E El branch is present in all the diagrams. Thus, in calculating the number of valid King-Altman patterns, only the closed loops need be considered. The determinants of E, EA, EAB, and El can be obtained by the method just described ... 

E. L. King and C. Altman. A schematic method of deriving the rate laws for enzyme-catalyzed reactions. J. Phys. Chem., 60 1375-1378, 1956. 

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. 

Let us consider the Ordered Bi Bi mechanism in reaction (16.3) and derive the initial rate equation for the A-Q isotope exchange with the aid of the King-Altman method. The basic King-Altman figure for this case is shown in Fig. 1. 

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). 

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... 

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... 

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