A certain enzyme has an ionizable group at the active site such that the enzyme is active...

A certain enzyme has an ionizable group at the active site such that the enzyme is active only when this ionizable group is protonated:

Assume that the conversion of H+ES to products and H+E is slow so that equilibria are established for the four reversible steps shown.

a. What is the relation among the equilibrium constants of the four steps forming a cyclic path?

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b. Show that the enzyme obeys Michaelis-Menten kinetics and express KM and Vmax in terms of the equilibrium constants of part (a) in addition to k, [H+] and [E]0.

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c. How does the rate depend on pH at high substrate concentrations?