For the purposes of our analysis we will assume "steady state kinetic
conditions". That is, [S] and [P] are changing, but [ES] does not change
(a constant flux of S "through" the enzyme). Mathematically, this can
be written as:
Also (from conservation of matter):
[E]tot = [E]free + [ES] (2)(the total enzyme is either bound to substrate or free)
Divide Vo = k2[ES] (eq.1 on the previous page) by [E]tot:
Now, since d[ES]/dt = 0, we know that the rate (velocity) of formation of [ES] must equal the rate of breakdown of [ES].
Vformation = k1 [E]free[S] (2nd order rate equation) Vbreakdown = k2 [ES] + k-1 [ES] = (k2 + k-1) [ES] (Two 1st order rate equations) k1 [E]free[S] = (k2 + k-1) [ES] (Rates must be equal)rearranging, solving for [ES]:
If we define the Michaelis-Menten Constant, Km (The "m" stands for Michaelis-Menten - these equations were formulated by Leonor Michaelis and Maud Menten in 1913.):
and substitute it into (3), we get:
This is an important equation.
Now, let us rearrange (2) to [E]free = [E]tot - [ES]
and then substitute into (4), giving:
Then, solving for [ES] gives:
Then multiply top and bottom by Km to get:
Finally, substitute into (1) to get:
This is the expression for Vo in terms of known quantities.