Enzymes - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 337

In this case, the maximum forward rate of the reaction is r max . The value of r max changes if
more enzyme is added, but the addition of more substrate has no influence on r max .K m is
often called the Michaelis e Menten constant. A low value of K m suggests that the enzyme
has a high affinity for the substrate. Also, K m corresponds to the substrate concentration at
which the reaction rate is half of the maximal reaction rate.
An equation of exactly the same form as Eqn (8.9) can be derived with a different, more
general assumption applied to the reaction scheme in Eqn (8.2) .

8.2.2.2. The Pseudosteady-State Hypothesis
In many cases, the assumption of rapid equilibrium following mass-action kinetics is not
valid, although the ES reaction still shows saturation-type kinetics.
G. E. Briggs and J. B. S. Haldane first proposed using the pseudosteady-state assumption
in 1925. In most experimental systems, a closed system (batch reactor) is used in which the
initial substrate concentration greatly exceeds the initial enzyme concentration. They suggest
that since [E] 0 was small, r ES z

0. (This logic is flawed. Do you see why?) Exact solutions of
the actual time course represented by Eqns (8.3), (8.4), and (8.5) have shown that in a closed
system, the pseudosteady-state hypothesis (PSSH) holds after a brief transient if [S] 0 >> [E] 0 .

Figure 8.9 displays one such time course.
By applying PSSH to Eqn (8.4) , we find

k 1 þ k 2

k 1 ½ E ½ S

½ ES ¼ (8.10)

Substituting the enzyme conservation Eqn (8.3) in Eqn (8.8) and solve for [ES], we obtain

½ 0 ½

E S

½ ES ¼ (8.11)

k 1 þ k 2
k 1

þ½ S

Substituting Eqn (8.11) into Eqn (8.3) yields

k 2 ½ E 0 ½ S

r P ¼ k 2 ½ ¼ (8.12)

ES

k 1 þ k 2
k 1

þ½

S

or

K m þ½

r max ½ S

r P ¼ (8.13)

S

which is identical to Eqn (8.9) . However,

k 1 þ k 2
k 1

K m ¼ (8.14)

which is not the equilibrium constant as that in Eqn (8.7) . Under most circumstances
(simple experiments), it is impossible to determine whether Eqn (8.7) or Eqn (8.14) is more
suitable. Since K m results from the more general derivation, we will use it in the rest of
our discussions.