Biomedical Engineering Reference
In-Depth Information
be often more convenient than homogeneous reactions, because, for one thing,
the ligand at the wall can be reused after washing of the reaction chamber. Also,
it is easier to proceed in two steps: immobilization of the ligands on a “reaction”
surface at the wall, and introduction of the analytes by a carrier fluid, rather than
designing a complex micofluidic system where the targets and ligands merge and
mix at the same time in the reaction chamber. Finally, it is also often more conve-
nient to detect the binded couples (targets-ligands) when they are immobilized on
a wall surface.
In this section, we give some examples of how the kinetics of heterogeneous
reactions is calculated. The first example is that of a diffusion-reaction problem of
the Langmuir type with concentration depletion.
7.4.2.1 Example of Concentration Depletion
There are two general trends in the treatment of biochemical reactions in biotech-
nology. First, the volumes are getting smaller and the ratio between the reaction
surface (functionalized or labeled surface) and the volume is increasing. Second, the
number of target molecules or particles is getting smaller in order to increase the
specificity and efficiency of the biochip. It follows from these two considerations
that the concentration in targets may be affected by depletion during the reaction
[17]. It is no more a constant as was supposed when we produced the solution of
the Langmuir equation.
We investigate here the solution to the Langmuir equation in the case of a uni-
form concentration in the liquid volume but decreasing with time and we show that
there exists a closed form solution to the Langmuir equation in the case of depletion
assuming the concentration is spatially homogeneous. Consider the schematic case
of Figure 7.33.
The mathematical formulation of the problem is obtained by replacing the con-
stant concentration
c
0
by a variable concentration
c
in the Langmuir equation.
d
G
=
k c
(
G - G -
)
k
G
(7.63)
on
0
off
dt
Figure 7.33
Schematic view of the microchamber. Because of depletion, the concentration
c
in the
chamber decreases.
