Biomedical Engineering Reference
In-Depth Information
Then, the solution is
1
S
c
=
+
c
-
G
0
0
V
S
æ
S
ö
æ
ö
æ
ö
-
k
G -
c
t
-
k
G -
c
t
1
1
ç
÷
ç
÷
on
0
0
on
0
0
è
ø
è
ø
V
ç
V
÷
e
-
1
-
e
S
æ
S
ö
ç
÷
G
c
-
G
è
ø
ç
÷
0
0
0
V
è
ø
V
V
c
S
As for the classic Langmuir kinetics, the kinetic curve approaches an asymptote
when time is sufficiently important. The value of the asymptote is obtained by tak-
ing
t =
¥ in (7.70)
By letting
t
®
¥ in the preceding equation, one finds
c
®
0 and
G ®
< G
0
0
-
c
¥
= +
c
ˆ
c
After substitution of the value of
c
, the asymptotic value of the concentration is
+
c
¥
=
c
(7.72)
This result shows that the asymptote is simply given by the second root of the poly-
nomial. This result can be derived directly from (7.67): when a permanent regime is
attained, the time derivative of
c
vanishes, and the asymptotic value for the concen-
tration then verifies the zero right-hand side of (7.67). Thus, the positive root (
c
+
)
of the characteristic polynomial is the value of the asymptote. The asymptotic value
of the surface concentration—if there is no saturation—is then given by
V
c
+
G =
(
-
c
)
(7.73)
0
S
The kinetics at
t =
0
is the same as that of the Langmuir model
d
G
=
k c
G
t
=
0
on
0
0
dt
As we have seen, the ratio between the surface of capture and the volume of the
chamber is an important parameter. Targeted concentration levels in biochips are
usually in the range
c
0
= 10
-9
to 10
-12
mole/l. Depending if the targets are DNA
strands or antibodies, the surface density of ligands lies in the range G
0
= 500 to
1,000 molecules/
μ
m
2
. Typical range of values of the ratio S/V is obtained by con-
sidering two types of microsystems: the first one is that of Figure 7.34 with a round
functionalized spot, the surface of the functionalized spot is approximately
S =
p R
2
,
R
= 2 mm and
V
is of the order of 10 x 10 x 1 mm
3
, so that
S/V =
0.13 m
-1
.
The second case is that of a capillary tube of radius
R =
100
μ
m, functionalized
along one-tenth of its length. The ratio S/V is then equal to
S/V =
2
/R
= 2,000 m
-1
.
The depletion model shows that precautions should be taken when the detection
device is operated under a closed volume condition (no circulation of fluid) or under
a closed-loop condition. In such cases, depletion in targets/analytes has to be taken
into account in the kinetics. Langmuir kinetics and (7.70) do not agree if the
S/V
