King-Altman kinetics

Multiple-substrate/product King-Altman kinetics... 

ENZYME ENERGETICS ISQTQPIC PERTURBATION KINETIC PARAMETERS KINETIC RESOLUTION KING-ALTMAN METHOD... 

Unfortunately, the form of Equation (8.53) is a little way off the form of the Michaelis-Menten equation. For this reason, the King-Altman approach is usually supplemented by an approach developed by Cleland. The Cleland approach seeks to group kinetic rate constants together into numbers (num), coefficients (Coef) and constants (const) that themselves can be collectively defined as experimental steady-state kinetic parameters equivalent to fccat> Umax and fCm of the original Michaelis-Menten equation. After such substitutions, the result is that equations may be algebraically manipulated to reproduce the form of the Michaelis-Menten equation (8.8). Use of the Cleland approach is illustrated as follows. 

The King-Altman approach described here can be summarised as a process in which an original rate equation (such as Equation (8.53)) is customarily developed, converted into a coefficient form (such as Equation (8.54)) and from there simplified to steady state kinetic forms (for example Equations (8.62) and (8.63)) by algebraic manipulation and Haldane simplification. This King-Altman approach is an approach that can be generalised for the derivation of most steady state kinetic equations based upon most complex kinetic schemes. Clearly these derivations can be substantial, but we shall not bother to reproduce these here except to cover a few important examples of particular relevance to the biocatalyst examples described in Section 8.1. 

Lam CF, Priest DG (1972) Enzyme kinetics. Systematic generation of valid King-Altman patterns. 

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... 

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). 

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... 

Even when forward reactions proceed rapidly at laboratory conditions, as is observed with Se(IV) and Cr(VI) reduction, evidence exists that chemical and isotopic equilibrium are not approached rapidly. Altman and King (1961) studied the kinetics of equilibration between Cr(III) and Cr(Vt) at pH = 2.0 to 2.5 and 94.8°C. Radioactive Cr was used to determine exchange rates, and Cr concentrations were greater than 1 mmol/L. Time scales for equilibration were found to be days to weeks. The mechanism of the reaction was inferred to involve unstable, ephemeral Cr(V) and Cr(IV) intermediates. Altman and King (1961) stated that the slowness of the equilibration was expected because the overall Cr(VI)-Cr(III) transformation involves transfer of three electrons and a change in cooordination (tetrahedral to octahedral). Se redox reactions also involve multiple electron transfers and changes in coordination. 

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 ... 

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... 

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 use of graphs in kinetics was pioneered by King and Altman [15] and later studied in depth by Temkin and Bonchev [16] and others (see, for example, Ref [16-18]). This field is more familiar to the enzymatic community [19-21] and sadly rarely used by the rest of the homogeneous catalysis researchers. In this section, we will make use of some tools and jargon of this theory (Scheme 9.2). 

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