Derive Michaelis-Menten equation for an uncompetitive inhibitor.
The Michaelis-Menten equation for an uncompetitive inhibitor can be derived by considering the interactions between the enzyme (E), the substrate (S), and the inhibitor (I). Uncompetitive inhibition occurs when the inhibitor binds only to the enzyme-substrate complex (ES) and not to the free enzyme. This binding effectively reduces the concentration of the enzyme-substrate complex available for catalysis.
Step 1: Define the reactions
For an enzyme-catalyzed reaction without an inhibitor, the following equilibrium exists:
E+S⇌ES→k2E+PE + S \rightleftharpoons ES \xrightarrow{k_2} E + P
Where:
- EE is the enzyme,
- SS is the substrate,
- ESES is the enzyme-substrate complex,
- PP is the product,
- k2k_2 is the rate constant for the conversion of ESES to E+PE + P.
Now, in the presence of an uncompetitive inhibitor (I), the inhibitor binds only to the enzyme-substrate complex (ES) to form the enzyme-substrate-inhibitor complex (ESI). This is represented as:
ES+I⇌ESIES + I \rightleftharpoons ESI
Thus,
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