Chapter 9: Reaction Mechanisms, Pathways, Bioreactions and Bioreactors

Derive the Rate Law for Uncompetitive Inhibition

Uncompetitive Inhibition

Inhibition only has affinity for enzyme-substrate complex

\( \text{E} + \text{S} \underset{k_2}{\overset{k_1}{\rightleftharpoons}} \text{E} \cdot \text{S} \xrightarrow{k_{\text{cat}}} \text{P} \)

\( \text{I} + \text{E} \cdot \text{S} \underset{k_5}{\overset{k_4}{\rightleftharpoons}} \text{I} \cdot \text{E} \cdot \text{S} \)

Developing the rate law

\( r_P = -r_S = k_{\text{cat}} (E \cdot S) \)

\( (1) \quad r_{E \cdot S} = 0 = k_1 (E)(S) - k_2 (E \cdot S) - k_{\text{cat}} (E \cdot S) - k_4 (I)(E \cdot S) + k_5 (I \cdot E \cdot S) \)

\( (2) \quad r_{I \cdot E \cdot S} = 0 = k_4 (I)(E \cdot S) - k_5 (I \cdot E \cdot S) \)

Adding (1) and (2)

\( k_1 (E)(S) - k_2 (E \cdot S) - k_{\text{cat}} (E \cdot S) = 0 \)

\( (E \cdot S) = \frac{k_1 (E)(S)}{k_2 + k_{\text{cat}}} = \frac{(E)(S)}{K_M} \)

From (2)

\( (I \cdot E \cdot S) = \frac{k_4}{k_5} (I)(E \cdot S) = \frac{(I)(E \cdot S)}{K_I} = \frac{(I)(E)(S)}{K_I K_M} \)

\( K_I = \frac{k_5}{k_4} \)

\( r_p = k_{\text{cat}} (E \cdot S) = \frac{k_{\text{cat}} (E)(S)}{K_M} \)

Total enzyme

\( E_t = (E) + (E \cdot S) + (I \cdot E \cdot S) \)

\( = (E) \left( 1 + \frac{(S)}{K_M} + \frac{(I)(S)}{K_I K_M} \right) \)

\( r_p = \frac{k_{\text{cat}} E_t (S)}{K_M \left( 1 + \frac{(S)}{K_M} + \frac{(I)(S)}{K_I K_M} \right)} \)

\( \boxed{ -r_S = r_p = \frac{V_{\max} (S)}{K_M + (S) \left( 1 + \frac{(I)}{K_I} \right)} } \)

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