2.5 Kinetics of Enzyme-Catalyzed Reactions 119Under steady-state conditions the rates of break-
down and formation of EA are equal (cf. Equation
2.31):
k 1 (E)(A 0 )=(k− 1 ∗k 2 )(EA) (2.35)
Also, the concentration of free enzyme, [E], can
not be readily determined experimentally. Hence,
free enzyme concentration from the above re-
lationship ([E]=[E 0 ]−[EA]) is substituted in
Equation 2.35:
k 1 [(E 0 )−(EA)](A 0 )=(k− 1 ∗k 2 )(EA) (2.36)
Solving Equation 2.36 for the concentration of the
enzyme-substrate complex, [EA], yields:
(E A)=
(E 0 )(A 0 )
k− 1 +k 2
k 1 +(A^0 )(2.37)The quotient of the rate constants in Equa-
tion 2.37 can be simplified by defining a new
constant, Km, called theMichaelisconstant:
(EA)=
(E 0 )(A 0 )
Km+(A 0 )(2.38)Substituting the value of [EA] from Equation 2.38
in Equation 2.32 gives theMichaelis–Menten
equation for v 0 (initial reaction rate):
υ 0 =
k 2 (E 0 )(A 0 )
Km+(A 0 )(2.39)Equation 2.39 contains a quantity, [E 0 ], which
can be determined only when the enzyme is
present in purified form. In order to be able
to make kinetic measurements using impure
enzymes,MichaelisandMentenintroduced an
approximation for Equation 2.39 as follows.
In the presence of a large excess of substrate,
[A 0 ] Kmin the denominator of Equation 2.39.
Therefore, Km can be neglected compared
to [A 0 ]:
υ 0 =
k 2 (E 0 )(A 0 )
(A 0 )=V (2.40)Thus, a zero order reaction rate is obtained. It is
characterized by a rate of substrate breakdown or
product formation which is independent of sub-
strate concentration, i. e. the reaction rate, V, is
dependent only on enzyme concentration. This
rate, V, is denoted as the maximum velocity.
From Equation 2.40 it is obvious that the catalytic
activity of the enzyme must be measured in the
presence of a large excess of substrate.
To eliminate the [E 0 ] term, V is introduced into
Equation 2.39 to yield:v 0 =V(A 0 )
Km∗(A 0 )(2.41)If[A 0 ]=Km, the following is derived from Equa-
tion 2.41:υ 0 =V
2(2.42)Thus, theMichaelisconstant, Km, is equal to
the substrate concentration at which the reaction
rate is half of its maximal value. Km is inde-
pendent of enzyme concentration. The lower the
value of Km, the higher the affinity of the en-
zyme for the substrate, i. e. the substrate will be
bound more tightly by the enzyme and most prob-
ably will be more efficiently converted to product.
Usually, the values of Km, are within the range
of 10−^2 to 10−^5 mol· 1 −^1. From the definition
of Km:Km=k− 1 +k 2
k 1(2.43)it follows that Kmapproaches the enzymesub-
strate dissociation constant, Ks, only ifk+ 2 k− 1.k 2 k− 1 Km≈k− 1
k 1=KS (2.44)Some values for the constants k+ 1 ,k− 1 ,andk 0
are compiled in Table 2.9. In cases in which the
catalysis proceeds over more steps than shown in
Equation 2.30 the constant k+ 2 is replaced by k 0.
The rate constant, k+ 1 , for the formation of the
enzyme-substrate complex has values in the or-
der of 10^6 to 10^8 : in a few cases it approaches
the maximum velocity (∼ 1091 ·mol−^1 s−^1 ), espe-
cially when small molecules of substrate readily
diffuse through the solution to the active site of
the enzyme. The values for k− 1 are substantially
lower in most cases, whereas k 0 values are in the
range of 10^1 to 106
s−^1.