Biomedical Engineering Reference
In-Depth Information
On the other hand, at the functionalized wall, the Langmuir model for binding
yields
d
G
=
k c
(
G - G -
)
k
G
(7.76)
on w
0
off
d t
where G is the concentration in immobilized analytes, G
0
the initial concentration
in available hybridization sites,
k
on
the adsorption coefficient at the wall,
k
off
the
desorption coefficient at the wall—also called elution—and
c
w
the concentration at
the wall.
Equations (7.76) and (7.75) are not independent. They are coupled by the Fick's
law
d
G
= - Ñ
D c
(7.77)
w
d t
Equations (7.76) can be substituted in (7.77) and we obtain the value of the wall
concentration as a function of the wall concentration (and its derivative)
D c
Ñ
+
k c
G
w
on w
0
G =
(7.78)
(
k c
+
k
)
on w
off
Equation (7.78) shows that there is some kind of equilibrium between the value of
the concentration near the wall and the surface concentration in hybridized targets.
If the concentration near the functionalized surface decreases—for any reason, such
as an interruption in the arrival of targets—there will be a temporary depletion of
targets near the wall, and the immobilized DNA strands will start to dissociate. On
the other hand, if there is a large supply of targets near the wall, the rate of hybrid-
ization will increase.
Numerical Approach
Numerical methods must be set up to solve such problems. If one has access to a
finite element software, the numerical approach is straightforward. If not, and if
the geometry of the microchamber is simple, a numerical formulation based on a
finite difference approach can be set up using the following discretization based on
the grid of Figure 7.37.
First, using a Crank-Nicholson semi-implicit scheme [19], the advection-diffusion
equation (7.75) can be discretized under the form
n
+
1
n
+
1
n
+
1
n
+
1
n
+
1
n
+
1
é
ù
n
+
1
n
c
-
2
c
+
c
c
-
2
c
+
c
c
-
c
D
i j
,
i j
,
i j
,
i j
,
i
+
1,
j
i
-
1,
j
i j
, 1
+
i j
, 1
-
=
ê
+
ú
2
2
D
t
2
(
D
x
)
(
D
y
)
ê
ú
ë
û
(7.79)
n
n
n
n
n
n
é
ù
c
-
2
c
+
c
c
-
2
c
+
c
D
i j
,
i j
,
i
+
1,
j
i
-
1,
j
i j
, 1
+
i j
, 1
-
+
ê
+
ú
2
2
2
(
D
x
)
(
D
y
)
ê
ú
ë
û
