26
2 MET-Type Bioelectrocatalysis
where k cat denotes the catalytic constant in solution, and K M(X) is the Michaelis
constant of X. In the presence of an excess amount of S (i.e. c S >> K M(S) ), k M is
given by
k M =
n S
n M
k cat c E
c M ox + K M(M)
.
(2.10)
Furthermore, when c
0
M K M(M) , k M becomes independent of c M ox and the
enzyme reaction becomes linear to c M ox :
k M =
n S
n M
k cat
K M(M)
c E ≡ k 2 c E ,
(2.11)
where k 2 [≡(n S /n M ) k cat /K M(M) ] denotes the second-order reaction rate constant
between E and M, and it is independent of S. Under the limiting condition: E E
◦◦
M ,
c M ox (0) becomes c
0
M , and the limiting value of the steady-state current (i
lim
S ) is given
by
i
lim
s
= F Ac
0
M
n S n M D M k 2 c E ,
(2.12)
where A denotes the electrode surface area. Identical equation (but with sign −) holds
for the reduction of S.
The parameter k 2 to be obtained in this system is very important for characterizing
the enzymatic reaction between E and M and can be easily evaluated from the slope of
the linear dependence of i
lim
s on c
0
M (the linear line in Fig. 2.2B). Successive addition
of a solution containing M is acceptable in the electrochemical measurements of
i
lim
s , as shown in Fig. 2.2A. This technique has a contrastive benefit for evaluating
k 2 compared to conventional photometric measurements.
When we consider Eq. (2.10), i
lim
S is given by
i
lim
S
n M F A
(n S /n M )D M k cat K M(M) c E
=
2
c
0
M
K M(M)
− ln
1 +
c
0
M
K M(M)
. (2.13)
The experimental data in Fig. 2.2B is well reproduced by Eq. (2.13). In the case:
c
0
M >> K M(M) , i
lim
S depends on the square root of c
0
M as follows [3]:
i
lim
S = F A
2n S n M D M k cat c E c
0
M .
(2.14)
Under the conditions, the reaction thickness μ is given by
μ =
D M c
0
M
2(n S /n M )k cat c E
.
(2.15)