Biomedical Engineering Reference
In-Depth Information
Several attempts have been made to incorporate Eq. (
30
) into the Michaelis-
Menten equation to describe concentration-dependent reactions that occur in spa-
tially constrained conditions. Berry [
7
] made a substitution producing the formula
v
max
C
K
M
0
t
h
=
C
.
v
(33)
+
Using Monte Carlo simulations on a 2D lattice to model enzyme reactions in
low-dimensional media, it was found [
7
]that
h
increases independently with in-
creasing obstacle density on the lattice and increasing initial substrate concentration.
Kosmidis et al. [
35
] also performed Monte Carlo simulations and found that Eq.
(
33
) holds mainly when the initial substrate concentration is high, either through an
intravenous bolus administration or a high rate of absorption. In addition, Eq. (
33
)
was incorporated into a one-compartment model [
35
]. Simulations performed by
Aranda et al. [
4
] also confirm these results but suggest that
K
M
0
is characterized by
multifractality and hence a set of fractal exponents.
2.7
Steady State Fractal Kinetics
As seen above, the effect of complex geometry on the rate of transient reactions
produces an anomalous kinetic rate coefficient. Anacker and Kopelman [
2
] demon-
strated that under steady state conditions, however, the effect of the geometry is
manifested as an anomalous reaction order. They showed that Eq. (
19
) should be
replaced by the effective rate equation
kC
X
v
=
,
(34)
where
X
is a fractal reaction order related to the spectral dimension of the random
walk. For example,
⎧
⎨
2
d
s
1
+
for
A
+
A
reactions,
X
=
(35)
⎩
4
d
s
1
+
for
A
+
B
reactions.
These
equations
have
been
confirmed
using
Monte
Carlo
simulations.
Anacker et al. [
2
] found that
X
01
(as expected) for the homogeneous cubic lattice. Klymko and Kopleman [
32
] found
that for bimolecular reactions in solids,
X
ranged from the homogeneous value of
2 up to a value of 30. Newhouse and Kopelman [
50
] found values of
X
=
2
.
44 for the 2D Sierpinski gasket and
X
=
2
.
≈
5for
ensembles of 10
5 islands. Therefore,
as available space becomes more finely divided, as in the example a fractal dust like
the Cantor set [
44
],
d
s
→
×
10 islands and
X
≈
15 for ensembles of 5
×
0 and therefore
X
→
∞
.
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