Biomedical Engineering Reference
In-Depth Information
Here
D
f,diss
is the fractal dimension of the surface for the dissociation step. This corresponds
to the highest concentration of the analyte-receptor complex on the surface. Henceforth,
its concentration only decreases. The dissociation kinetics may be analyzed in a manner
“similar” to the binding kinetics.
11.2.2 Dual-Fractal Analysis
Binding Rate Coefficient
Sometimes, the binding curve exhibits complexities and two parameters (
k
,
D
f
) are not sufficient
to adequately describe the binding kinetics. This is further corroborated by low values of the
r
2
factor (goodness-of-fit). In that case, one resorts to a dual-fractal analysis (four parameters;
k
1
,
k
2
,
D
f1
,and
D
f2
) to adequately describe the binding kinetics. The single-fractal analysis presented
above is thus 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 empiri-
cal. For the most part, it is dictated by the data analyzed and experience gained by handling a
single-fractal analysis. A smoother curve is obtained in the “transition” region,if care is taken
to select the correct number of points for the two regions. In this case, the product (antibody-
antigen; or analyte-receptor complex, Ab
¼
Ag or analyte
receptor) is given by:
8
<
t
ð
3
D
f1
,
bind
Þ=
2
t
p
1
,
¼
t
<
t
1
t
ð
3
D
f2
,
bind
Þ=
2
¼ t
p
2
,
ð
Ab
Ag
Þ
t
1
< t < t
2
¼ t
c
ð
11
:
3
Þ
:
t
1
=
2
,
t
>
t
c
In some cases, as mentioned above, a triple-fractal analysis with six parameters (
k
1
,
k
2
,
k
3
,
D
f1
,
D
f2
, and
D
f3
) may be required to adequately model the binding kinetics. This is when the bind-
ing curve exhibits convolutions and complexities in its shape due perhaps to the very dilute
nature of the analyte (in some of the cases to be presented) or for some other reasons. Also,
in some cases, a dual-fractal analysis may be required to describe the dissociation kinetics.
11.3 Results
We will use fractal analysis to analyze the binding (hybridization) and dissociation kinetics
exhibited by different analyte-receptor reactions occurring on biosensor surfaces. This is just
one possible method of analyzing the kinetics of the different analyte-recptor (hybridization
reactions) presented in this chapter. Alternative expressions for fitting the data are available
that include saturation, first-order reaction, and no diffusion limitations, but these expressions
are apparently deficient in describing the heterogeneity that inherently exists on the surface.
One might justifiably argue that the appropriate modeling may be achieved by using a
Langmuirian or other approach. The Langmuirian approach may be used to model the data
presented if one assumes the presence of discrete classes of sites (for example, double
