Biomedical Engineering Reference
In-Depth Information
Here
D
f,diss
is the fractal dimension of the surface for the dissociation step.
t
diss
represents
the start of the dissociation step. This corresponds to the highest concentration of the
analyte
receptor complex on the surface. Henceforth, its concentration only decreases.
D
f,bind
may or may not be equal to
D
f,diss
.
k
d
and
k
diss
, and
D
f,d
and
D
f,diss
are used interchangeably
in this chapter and in this topic.
One may obtain a unit for the dissociation rate coefficient,
k
d
,inasimilarmannerasdonefor
the binding rate coefficient. In this case, the units for the binding and the dissociation rate coeffi-
cient are the same. The unit for the dissociation rate coefficient,
k
d
is (pg)(mm)
2
(sec)
(
D
f,diss
3)/2
.
Once again, note that the unit dependence on time exhibited by
k
d
changes slightly due to the
dependence on
D
f,diss
.
2.2.3 Dual-Fractal Analysis
Binding Rate Coefficient
The single-fractal analysis we have just presented is extended to include two fractal
dimensions. At present, the time (
t
t
1
) at which the first fractal dimension “changes” to
the second fractal dimension is arbitrary and empirical. For the most part it is dictated by
the data analyzed and the experience gained by handling a single-fractal analysis. The
r
2
(regression coefficient) value obtained is also used to determine if a single-fractal analysis
is sufficient, or one needs to use a dual-fractal analysis to provide an adequate fit. Only if
the
r
2
value is less than 0.97 for a single-fractal analysis, do we use a dual-fractal model.
In this case, the analyte-receptor complex is given by:
¼
8
<
:
t
ð
3
D
f1
,
bind
Þ=
2
t
p
1
,
¼
t
<
t
1
t
ð
3
D
f2
,
bind
Þ=
2
ð
Analyte
Receptor
Þ
t
p
2
,
ð
2
:
5c
Þ
¼
t
1
<
t
<
t
2
¼
t
c
t
1
=
2
,
t
>
t
c
In analyte-receptor binding, the analyte-receptor binds with the active site on the surface and
the product is released. In this sense the catalytic surface exhibits an unchanging fractal
surface to the reactant in the absence of fouling and other complications. In the case of
analyte-receptor association, the biosensor surface exhibits a changing fractal surface to the
analyte in solution. This occurs because as each association reaction takes place, smaller
and smaller amounts of “association” sites or receptors are available on the biosensor surface
to which the analyte may bind. Furthermore, as the reaction proceeds, there is an increasing
degree of heterogeneity on the biosensor surface for some reaction systems. This is
manifested by two degrees of heterogeneity or two fractal dimensions on the biosensor sur-
face. In the theoretical limit one might envisage a temporal fractal dimension wherein there
is a continuous change in the degree of heterogeneity on the surface; though of course, such
situations would be very rare, if at all.
