Biomedical Engineering Reference
In-Depth Information
In (7.90), the subscripts
n
or
n +
1 refer to the time step. Note that the velocity term
must be discretized following the flow direction. Next, using an implicit scheme,
(7.89) becomes
n
n
+
1
G +
k c
G D
t
i
on
i
0
,0
i
+
1
G
=
(7.91)
n
+
1
1 (
)
+
k c
+
k
D
t
on
off
i
,0
where the notation
c
i,
0
refers to the concentration at the wall. Fick's law can be
discretized by
é
n
+
1
n
+
1
n
n
ù
n
+
1
n
c
-
c
c
-
c
G
- G
D
i
i
i
,0
i
,1
i
,0
i
,1
ê
ú
= -
+
(7.92)
D
t
D
ê
y
2
2
ú
ë
û
n
+ 1
After substitution of (7.92) in (7.91), we obtain the linear relation between
c
i
,0
n
+ 1
, and the whole system can be cast under the matrix form
and
c
i
,1
n
+
1
n
[
A c
]{
}
=
{
s
}
(7.93)
where the vector {
s
n
} depends on the concentrations at the preceding time step. By
using the relevant boundary conditions, and by inversing the system [22], one ob-
tains the concentration distribution at the new time step
n +
1.
Example of Advection-Diffusion-Reaction Kinetics
In this example, we show how experimental records of hybridization kinetics com-
bined with the numerical model of the preceding section can be used to find the ki-
netics constants of different DNA strands. The experiment setup corresponds to that
of Figure 7.44. In this experiment, a constant buffer fluid flow carries different types
of DNA strands—with different sequences and length. The average flow velocity is
1 mm/s (10
μ
l/mn) and the dimensions of the microchamber are 10 × 10 × 1 mm.
The flow is turned on during 50 minutes, then it is stopped during 310 minutes and
it is again turned on for the rest of the experimental time. Hybridization kinetics is
monitored by fluorescence. Four different kinetics are obtained for the four differ-
ent types of oligonucleotides (Figure 7.46).
The approach is to use the numerical model of the preceding section and to fit
of the kinetics curves by varying the parameters
c
0
(bulk concentration),
D
(diffu-
sion coefficient),
k
on
(constant of hybridization), and
k
off
(constant of desorption).
At the experiment temperature,
k
off
can be considered negligible, so that the fit is
performed by varying three parameters only (Figure 7.47). A few trials are enough
to find the values of
c
0
,
D,
and
k
on
.
Table 7.1 shows the typical values for DNA strands of different length obtained
by this approach. One of the conclusions of this analysis is that the adsorption con-
stant depends not only on the length of the DNA strand but also on the nature of
the basis pairs (A,C,G,T).
7.4.2.4 Displacement (Competition) Reactions
Introduction and Principles
In the preceding sections, we dealt with a class of heterogeneous reactions that may
be called
sandwich reactions
. Up to now, sandwich reactions have been the most
