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):

(the total enzyme is either bound to substrate or free)[E]tot = [E]free + [ES] (2)

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].

rearranging, solving for [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)

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.

shanec@mit.edu