Since equilibrium properties are searched one can write
From Michaelis–Menten kinetics the rate at which Z is dephosphorylated is known to be  and the rate at which Z is phosphorylated is . Here the K stand for the Michaelis–Menten constant which describes how well the enzymes X and Y bind and catalyze the conversion whereas the kinetic parameters k and k denote the rate constants for the catalyzed reactions. Assuming that the total concentration of Z is constant one can additionally write that [Z] = [Z] + [Z] and one thus gets:
with the constants
If we thus solve the quadratic equation (1) for z we get:
Thus (3) is a solution to the initial equilibrium problem and describes the equilibrium concentration of [Z] and [Z] as a function of the kinetic parameters of the phosphorylation and dephosphorylation reaction and the concentrations of the kinase and phosphatase. The solution is the Goldbeter–Koshland function with the constants from (2):