Biomedical Engineering Reference
In-Depth Information
It is of interest to note that for a dual-fractal analysis as the fractal dimension increases by a
factor of 2 from a value of D f1 equal to 1.3706 to D f2 equal to 2.7478, the binding rate coeffi-
cient increases by a factor of 51.70 from a value of k 1 equal to 0.00161 to k 2 equal to 0.08324.
Figure 9.2a shows for a dual-fractal analysis the increase in the binding rate coefficient, k 1
with an increase in the bradykinin concentration in solution in the 8-128 nM range. For the
data shown in Figure 9.2a , the binding rate coefficient, k 1 is given by:
0 : 934 0 : 174
k 1
¼ð
0
:
000175
0
:
000081
Þ½
Bradykinin, nM
ð
9
:
4a
Þ
The fit is very good. Only five data points are available. The availability of more data points
would lead to a more reliable fit. The binding rate coefficient, k 1 exhibits close to a first
(equal to 0.934) order of dependence on the bradykinin concentration in solution. The
noninteger order of dependence exhibited lends support to the fractal nature of the system.
Figure 9.2b and Tables 9.1 and 9.2 show the increase in the binding rate coefficient, k 2 , with
an increase in the fractal dimension, D f2 , in the binding phase for a dual-fractal analysis. For
the data shown in Figure 9.2b the binding rate coefficient, k 2 is given by:
D 6 : 571 3 : 429
k 2 ¼ð
0
:
000231
0
:
000230
Þ
ð
9
:
4b
Þ
f2
There is scatter in the data. Only five data points are available. The availability of more data
points would lead to a more reliable fit. The binding rate coefficient, k 2 , is extremely sensi-
tive to the fractal dimension, D f2 , or the degree of heterogeneity that exists on the biosensor
chip surface as noted by the close to six and a half order of dependence exhibited.
Figure 9.2c shows the increase in the affinity, K 2 (
k 2 / k d ), with an increase in the fractal
dimension ratio, D f2 / D fd . For the data shown in Figure 9.2c , the affinity, K 2 is given by:
¼
1 : 44 0 : 664
K 2
ð¼
k 2
=
k d
Þ¼ð
0
:
001667
þ
0
:
002999
Þð
D f2
=
D fd
Þ
ð
9
:
4c
Þ
There is scatter in the data. Only four data points are available. The availability of more data
points would lead to a more reliable fit. Only the positive value of the error is given since the
affinity cannot have a negative value. The affinity, K 2 exhibits close to a one and a half
(equal to 1.44) order of dependence on the ratio of the fractal dimensions in the binding
and in the dissociation phases, ( ¼ D f2 / D fd ), respectively.
Fang et al. (2006) also analyzed the binding and the dissociation of different Bradykinin con-
centrations (16-128 nM) in solution to internalized receptors through endocytes or diffusion into
the cytoplasm of cells. These internalized receptors were immobilized on a RWG biosensor sur-
face. Figure 9.3a shows the binding and the dissociation of 128 nM Bradykinin in solution to the
internalized receptor immobilized on a RWG biosensor surface. A dual-fractal analysis is
required to adequately describe the binding and the dissociation kinetics. The values of (a) the
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