Chapter 9: Reaction Mechanisms, Pathways, Bioreactions and Bioreactors

Derive: Uncompetitive Substrate Inhibition Rate Law

\( r_p = k_5 (E \cdot S) \)

\( r_{S.ES} = 0 = k_1 (E)(S) - k_2 (E \cdot S) - k_3 (S)(E \cdot S) + k_4 (S \cdot E \cdot S) - k_5 (E \cdot S) \)

\( r_{S.ES} = 0 = k_3 (S)(E \cdot S) - k_4 (S \cdot E \cdot S) \)

\( (S \cdot E \cdot S) = \frac{k_3}{k_4} (S)(E \cdot S) \)

\( (E \cdot S) = \frac{k_1 (E)(S)}{k_2 + k_5} \)

\( (S \cdot E \cdot S) = \frac{k_3 k_1 (S)^2 E}{k_4 (k_2 + k_5)} \)

\( E_T = E + \frac{k_1}{k_2 + k_5} (E)(S) + \frac{k_3 k_1 S^2 E}{k_4 (k_2 + k_5)} \)

\( E = \frac{E_T}{1 + \frac{k_1}{k_2 + k_5} S + \frac{k_1 k_3 S^2}{k_4 (k_2 + k_5)}} \)

\( r_p = k_5 (E \cdot S) = \frac{k_1 k_5 (E)(S)}{k_2 + k_5} \)

\( r_p = \frac{k_1 k_5 E}{k_2 + k_5} \frac{S}{1 + \frac{k_1}{k_2 + k_5} S + \frac{k_1 k_3 S^2}{(k_2 + k_5) k_4}} \)

\( r_p = \frac{V_{\max} K_I S}{S^2 + K_I S + K_M K_I} \)

where \( K_I = \frac{k_4}{k_3} \), \( K_M = \frac{k_2 + k_5}{k_1} \), \( V_{\max} = k_5 E_T \)