Deriving the Michaelis-Menten Equation:

      (1)

Let v₀ be the initial velocity of the reaction, i.e. the appearance of the product P in solution (d[P]/dt) whose first-order rate equation is:

v₀ = k2[ES] (2)

containing an experimentally measurable variable, v₀, a kinetic parameter, k2, and another unknown variable, [ES].

So in the steady state:

k_-1[ES] + k2[ES] = k₁[E][S] (3)

Collect together the kinetic constants, and the concentrations (variables):

Group the kinetic constants, defining them as K_m:

K_m = (k_-1 + k2)/k₁ (5)

Express [E] in terms of [ES] and [E]_total, to limit the number of unknowns:

[E] = [E]_total - [ES] (6)

Substitute (5) and (6) into (4):

K_m = ([E]_total - [ES]) [S]/[ES] (7)

Solve for [ES]: 

First multiply both sides by [ES]:  

[ES] K_m = [E]_total[S] - [ES][S]  

Then collect terms containing [ES] on the left:  

[ES] K_m + [ES][S] = [E]_total[S]

Factor [ES] from the left-hand terms:

[ES](K_m + [S]) = [E]_total[S]

and divide both sides by (K_m + [S]):

[ES] = [E]_total [S]/(K_m + [S]) (8)

Substitute (8) into (2):

v₀ = k2[E]_total [S]/(K_m + [S]) (9)

The maximum velocity V_max occurs when the enzyme is saturated - thus:

V_max = k2 [E]_total (10)

Substitute V_max into (9) for k2 [E]_total: