Biomedical Engineering Reference
In-Depth Information
phases for analyte-receptor binding are available (
Sadana, 2001
) in the literature. The details
are not repeated here except that the equations are given to permit easier reading. These
equations have been applied to other biosensor systems also (
Ramakrishnan and Sadana,
2001
;
Sadana, 2001; Sadana, 2005
). For most applications, a single- or a dual-fractal analysis
is often adequate to describe the binding and the dissociation kinetics. Peculiarities in the
values of the binding and dissociation rate coefficients, as well as in the values of the fractal
dimensions with regard to the dilute analyte systems being analyzed will be carefully noted,
if applicable.
7.2.1 Single-Fractal Analysis
Binding Rate Coefficient
Havlin (1989)
reports 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
7
:
1
Þ
t
1
=
2
,
t > t
c
Here
D
f,bind
or
D
f
(as it is used later on in the Chapter) 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
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
¼
½ as it should be. Another way of
looking at the
p
½ case (where
D
f,bind
is equal to 2) is that the analyte in solution views
the fractal object, in this case, the receptor-coated biosensor surface, from a “large distance.”
In essence, in the association 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, (Analyte
¼
t
(3
D
f,bind
)/2
. For the present
analysis,
t
c
is arbitrarily chosen and we assume that the value of
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 autonomous (and not time-dependent) model for diffusion-controlled kinetics.
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:
