Biomedical Engineering Reference
In-Depth Information
A regression coefficient at this stage could be beneficial in understanding the efficacy of this
metric. However, an easier method, without the use of the required log-log plots, is presented
below.
This is the equation developed by
Havlin (1989)
for diffusion of analytes toward fractal
surfaces.
2.2.2 Single-Fractal Analysis
In the literature some authors refer to binding as comprising of two phases, an association
phase and a dissociation phase. In this chapter and in the topic, we will refer to binding as
just binding. The dissociation phase is separate.
Binding Rate Coefficient
Havlin (1989)
indicates that the diffusion of a particle (analyte) from a homogeneous solution
to a solid surface (e.g., receptor-coated surface) on which it reacts to form a product (analyte-
receptor complex) is given by:
t
ð
3
D
f
,
bind
Þ=
2
t
p
,
¼
t
<
t
c
ð
Analyte
Receptor
Þ
ð
2
:
5a
Þ
t
1
=
2
,
t
>
t
c
where the analyte-receptor represents the association (or binding) complex formed on the
surface. Here
p
-
b
, and
D
f
is the fractal dimension of the surface.
Havlin (1989)
states
that the crossover value may be determined by
r
c
¼
t
c
. Above the characteristic length,
r
c
,
the self-similarity of the surface is lost and the surface may be considered homogeneous.
Equation (2.5a)
indicates that the concentration of the product [analyte-receptor] on a solid
fractal surface scales at short and intermediate times as analyte-receptor
t
p
with the coeffi-
cient
p
1/2 at intermediate time scales. Note that
D
f
,
D
f,assoc
, and
D
f,bind
are used interchangeably. This equation is associated with the short-term
diffusional properties of a random walk on a fractal surface. Note that, in perfectly stirred
kinetics on a regular (nonfractal) structure (or surface), the binding rate coefficient,
k
1
,isa
constant, that is, it is independent of time. In other words, the limit of regular structures
(or surfaces) and the absence of diffusion-limited kinetics leads to
k
1
being independent of
time. In all other situations, one would expect a scaling behavior given by
k
1
¼
(3-
D
f
)/2 at short time scales and
p
¼
k
0
t
b
with
-
b
0 is the consequence
of two different phenomena, that is, the heterogeneity (fractality) of the surface and the imper-
fect mixing (diffusion-limited) condition. Finally, for a homogeneous surface where
D
f
is equal
to 2, and when only diffusional limitations are present,
p
¼
p
<
0. Also, the appearance of the coefficient,
p
different from
p
¼
¼
½ as it should be. Another way of
looking at the
p ¼
1/2 case (where
D
f,bind
is equal to 2) 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 association process, the diffusion of the analyte from the solution to the
