Derivation of Complicated Steady-State Equations

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  

Draw the reaction scheme (the master pattern) with the required reaction arrows interconnecting all relevant enzyme species (free and complexed forms). 

Annotate all reaction arrows with the corresponding unitary rate constant. For forward reactions where a substrate is involved, place its letter of designation next to the rate constant in the scheme do the same for reverse reactions involving a product. 

For the n enzyme forms (one of free enzyme and the rest complexes or isomeric forms of the enzyme), draw reaction patterns that have n - 1 arrows and yield a continuous path or paths that lead to each enzyme form. In addition, no closed loops of steps are allowed in the pattern. 

The expression for [ES,] [E]r, where [ES,] is any enzyme form and [E]7 is the total enzyme concentration, is given by the summed products of concentrations and rate constants from each pattern. 

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