A Random Model with Two Dead-End Complexes

In the most general case, A and B or P and Q can occupy the enzyme simultaneously and aU four ligands can bind by themselves. In addition, the dead-end complexes EAP and EBQ are formed. However, in this case, the EAP conplex often forms with elevated dissociation constant, relative to those of the binary complexes, because of the overlap of the common pieces in reactants. The reaction will proceed in both directions when all four ligands are present. 

The general rate equations, if all four substrates and products are present, and when interconverted with the Haldane relationship (8.43) are 

In rapid equilibrium systems, all inhibition constants represent the tme dissociation constants of respective enzyme complexes. The nomenclature of Cleland, described in Section 8.1, is sufficient to describe most kinetic constants. However, for those constants that are leading to the formation of dead-end complexes (Xas. Khq, Kha, /Cjip), novel or extra descriptions are necessary (Cleland, 1967). Since all enzyme forms in reaction (8.35) are in the thermodynamic equilibrium, the novel constants are mutually related by the following relationships  

the last two denominator terms in Eq. (8.37) can be written in yet another form  

The rate Eq. (8.37) has nine denominator terms, each one for one of the enz5mie forms EA, EB, EP, EQ, EAB, EPQ, EBQ, and EAP the unity represents the free enz5mie. The rate equations for the forward reaction, in the presence of product Q or product P, are now identical with similar equations derived for cases with single dead-end complexes. 

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