Biomedical Engineering Reference
In-Depth Information
E
K
c
1
K
S
*
V
*
K
c
2
E
*
E
*
S
*
ES
*
+
+
P
cccc
cccc
E
*
C
cccc
cccc
cccc
cccc
cccc
cccc
Figure 14.2
Enzymatic reactions involving water-insoluble substrates.
constants of micelle and micelle-enzyme complexes);
s
is the concentration of monomeric
substrate in the water phase (it can be calculated from the total concentration
s
0
using the
expression
s
0
= s
n·s
n
/K
n
);
n
is the average number of
S
molecules in micelle. The concen-
tration of free monomers (
s
) is frequently substituted by the total concentration
s
0
= s
+
s
,
though this assumption is valid only at a small
s
0
. At higher substrate concentrations,
increasing amounts of
S
* are produced due to the association
n·S
+
n s
*
≈
↔
S
n
, and the expression
s
0
n s
* becomes more appropriate.
Reactions with water-immiscible substrates are even more complex (Figure 14.2). Thus,
an emulsifier is required to form the interface (active capturing surface
C
), where the sur-
face-bound forms of the enzyme (
E
*) and the substrate (
S
*) react. Some emulsifiers, such
as bile salts, might occlude lipase and form the inactive complex
E
*
C
(Lowe, 2002 , Pan and
Bahnson, 2007). Other surfactants, for example Triton X-100, do not demonstrate such
inhibiting effects (Salis
et al
., 2003). The initial velocity of the reaction at interface can be
written as:
≈
V
V
*
*
v
=
=
(14.10)
K
K
⎛
K c
cK
⎞
*
*
S
c
1
mS
1
+
1
+
+
1
+
⎜
⎟
*
*
s
⎝
⎠
s
c
2
where
s
* is the concentration of substrate at the interface (usually treated as its bulk
concentration);
V
* is the maximal rate of forward reaction,
c
is the capturing surface per
volume unit (it is often equated to the bulk concentration of emulsifier),
K
c
1
stands for the
equilibrium dissociation constant of
E
* from the surface
CCC
C (Figure 14.2 ),
K
c
2
describes
dissociation of the encapsulated inactive complex E*C. If
c
is kept constant, it can be
incorporated into the apparent constant
K
mS
*
= K
s
*
(1
c/K
c
2
)
, whereupon
v
becomes
a function of one variable,
s
* . Dependence on another variable
c
(capturing surface = emulsifier)
at a constant
s
* should be analysed using another form of the same equation:
+
K
c
1
/c
+
V
*
V
k
s
*
*
*
c
1
v
=
;
V
=
;
K
=
;
K
=
K
(1
+
)
(14.11)
*
K
c
1
c
2
c
2
*
K
K
c
*
s
*
1
+
S
1
++
c
1
1
+
s
*
*
c
s
Ks
*
K
c
2
This function reaches its maximum at
c = (K
c
1
*
·K
c
2
*
)
½
and approaches zero at c
→
∞
(in the
),
C
acts as an
essential activator. Hence, the dependence of
v
on
c
becomes a simple hyperbole (Martin
et al
.,
1994). Presence of residual enzymatic activity without emulsifier means that the capturing
case of
C
-caused inhibition). If the emulsifier causes no inhibition (
K
c
2
→
∞
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