algae/notes/mm_derivation.html

Derivation of the Michaelis-Menten equation

For the enzyme catalyzed reaction

v = k₃*[ES]

Rate of formation of ES = k₁* [E]*[S]

Rate of breakdown of ES = (k₂ + k₃) * [ES]

At steady state, the formation and breakdown are equal. This steady state would be temporary in any enzyme assay or nutrient uptake experiment.

k₁* [E]*[S] = (k₂ + k₃) * [ES]

rearranging:

[ES] = [E]*[S]/((k₂ + k₃)/k₁)

We can lump these kinetic constants to make a new constant called Km = (k₂ + k₃)/k₁)

[ES] = [E]*[S]/Km

The total enzyme is the amount bound and unbound to the substrate

[E_tot] = [E] + [ES]

Substituting [E_tot] - [ES] for [E]

[ES] = ([E_tot] - [ES]) *[S]/Km

Solving for [ES]

[ES] = ([E_tot] *(( [S]/Km)/(1+[S]/Km))

Which simplifies to

[ES] = [E_tot] * [S] / ([S] + Km)

Multiplying both sides by k₃ gives the velocity of the reaction

v = k₃*[ES] = k₃* [E_tot] * [S] / ([S] + Km)

substituting Vmax for k₃* [E_tot] gives the familar form of the MM equation

v = Vmax * [S] / ([S] + Km)