Biomedical Engineering Reference
In-Depth Information
increases by a factor of 1.66 from a value of
D
f1
equal to 1.7038 to
D
f2
equal to 2.8228, the
binding rate coefficient increases by a factor of 6.19 from a value of
k
1
equal to 15.644 to
k
2
equal to 97.008 for a dual-fractal analysis. Note, once again, that an increase in the degree of
heterogeneity or the fractal dimension on the sensor chip surface leads to an increase in the
binding rate coefficient.
Figure 4.5c
shows the binding of 5.0 micromole inorganic phosphate ion (P
i
) in solution to a
rhodamine-PBP phosphate biosensor (
Okoh et al., 2006
). A dual-fractal analysis is required
to adequately describe the binding kinetics. The values of (a) the binding rate coefficient,
k
, and the fractal dimension,
D
f
, for a single-fractal analysis, and (b) the binding rate
coefficients,
k
1
and
k
2
, and the fractal dimensions,
D
f1
and
D
f2
, for a dual-fractal analysis
are given in
Table 4.5
. It is of interest to note, once again, that as the fractal dimension
increases by a factor of 1.76 from a value of
D
f1
equal to 1.70 to
D
f2
equal to 3.0, the binding
rate coefficient increases by a factor of 5.93 from a value of
k
1
equal to 8.425 to
k
2
equal
to 50 for a dual-fractal analysis. Note, once again, that an increase in the degree of hetero-
geneity or the fractal dimension on the sensor chip surface leads to an increase in the binding
rate coefficient.
Figure 4.6a
and
Table 4.5
show the increase in the fractal dimension,
D
f1
, with an increase in
the phosphate ion, P
i
in solution in the 0.2-5.0 micromole concentration range for a dual-
fractal analysis. For the data shown in
Figure 4.6a
, the fractal dimension,
D
f1
, is given by:
0
:
1800
0
:
1580
D
f1
¼ð
1
:
354
0
:
361
Þ½
P
i
ð
4
:
6a
Þ
The fit is reasonable. Only three data points are available. The availability of more data points
would lead to a more reliable fit. The fractal dimension,
D
f1
, exhibits only a very mild depen-
dence on the P
i
concentration in solution as noted by the 0.1800 order exhibited. The fractal
dimension is based on a log scale, and even very small changes in the fractal dimension value
indicate significant changes in the degree of heterogeneity on the biosensor chip surface.
Figure 4.6b
and
Table 4.5
show the increase in the fractal dimension,
D
f2
, with an increase in
the phosphate ion, P
i
in solution in the 0.2-5.0 micromole concentration range for a dual-
fractal analysis. For the data shown in
Figure 4.6b
, the fractal dimension,
D
f2
, is given by:
0
:
06656
0
:
0197
D
f2
¼ð
2
:
716
0
:
081
Þ½
P
i
ð
4
:
6b
Þ
The fit is good. Only three data points are available. The availability of more data points would
lead to a more reliable fit. The fractal dimension,
D
f2
, exhibits only a very mild dependence
on the P
i
concentration in solution as noted by the 0.06656 order exhibited. Once again,
the fractal dimension is based on a log scale, and even very small changes in the fractal dimen-
sion value indicate significant changes in the degree of heterogeneity on the biosensor chip
surface.
