Biomedical Engineering Reference
In-Depth Information
immobilized on an IS-FET-based biosensor (
Uno et al., 2007
), (h) binding and dissociation of
RNA synthesized on a (i) 42 nM template and a (ii) 420 nM template (
Blair et al., 2007
),
and (i) binding (hybridization) of different concentrations of ss DNA in solution preincubated
with prehybridized 22-nt FQ duplex to a “broken beacon” immobilized on a sensor surface
(
Blair et al., 2007
). One may consider the fractal analysis as an alternate method of analyzing
the kinetics of binding and dissociation during hybridization in these types of analyte-receptor
reactions occurring on biosensor surfaces.
11.2 Theory
11.2.1 Single-Fractal Analysis
Binding Rate Coefficient
Havlin (1989)
points out that the diffusion of a particle (analyte [Ag]) from a homogeneous
solution to a solid surface (e.g., receptor [Ab]-coated surface) on which it reacts to form a
product (analyte-receptor complex; Ab
Ag) is given by:
t
ð
3
D
f
:
bind
Þ=
2
t
p
,
¼
t
<
t
c
ð
Ab
Ag
Þ
ð
11
:
1
Þ
t
1
=
2
,
t
>
t
c
Here
D
f,bind
or
D
f
is the fractal dimension of the surface during the binding step.
t
c
is the
cross-over value.
Havlin (1989)
points out that the cross-over value may be determined by
r
c
2
t
c
. Above the characteristic length,
r
c
, the self-similarity of the surface is lost and the
surface may be considered homogeneous. Above time,
t
c
the surface may be considered
homogeneous, since the self-similarity property disappears, and “regular” diffusion is now
present. For a homogeneous surface where
D
f
is equal to 2, and when only diffusional
limitations are present,
p
½ case
(where
D
f,bind
is equal to two) is that the analyte in solution views the fractal object, in our
case, the receptor-coated biosensor surface, from a “large distance.” In essence, in the asso-
ciation process, the diffusion of the analyte from the solution to the receptor surface creates a
depletion layer of width (
Ðt
)
½
where Ð is the diffusion constant. This gives rise to the fractal
power law,
¼
½ as it should be. Another way of looking at the
p
¼
t
ð
3
D
f
,
bind
Þ=
2
. For the present analysis,
t
c
is arbitrarily cho-
sen and we assume that the value of the
t
c
is not reached. One may consider the approach as
an intermediate “heuristic” approach that may be used in the future to develop an autono-
mous (and not time-dependent) model for diffusion-controlled kinetics.
ð
Þ
Analyte
Receptor
Dissociation Rate Coefficient
The diffusion of the dissociated particle (receptor [Ab] or analyte [Ag]) from the solid sur-
face (e.g., analyte [Ag]-receptor [Ab]) complex coated surface) into solution may be given,
as a first approximation by:
t
ð
3
D
f
,
diss
Þ=
2
t
p
,
ð
Þ
¼
t
>
t
diss
ð
:
Þ
Ab
Ag
11
2
