limiting Model species [ES] and [ESI] complexes are in instantaneous equilibrium. Based on the second assumption stated above, the ODE equations above can be set to zero since the values are not changing: Since we know the following expression is equal to zero: We can cross out the following part in the [ES] rate equation: In order to derive the Michaelis-Menten equation, we must re-write [ES] in terms of only [S]. To start we will modify the above equation to isolate [ES]: Solving for [ESI] in an equation above yields where we can plug in the expression found for [ES]: The total enzyme concentration depends on the amount of free enzyme [E], enzyme bound to complex [ES], and enzyme-substrate-inhibitor complex [ESI]. Plugging in the expressions for [ESI] and [ES] will yield: Isolated [E] to only show up on a single side of the above equation: From the above derivation of [ES], we can further modify the equation by replacing [E] with the equation directly above: is known as the Michaelis constant and is commonly written as: , , and Finally, to get the Michaelis-Menten equation for uncompetitive inhibition: Where: References Enzyme kinetics Enzyme inhibitors