Biomedical Engineering Reference
In-Depth Information
that use evanescent waves have been used effectively in research. The resonant waveguide
grating (RWG) biosensor has been used to help determine affinities and the kinetics of target
analytes in a simple binding to the biological receptors immobilized on a sensor surface.
These authors report that there is increasing interest in the activities of living cells including
cell adhesion and spreading, toxicity, and proliferation (
Ramsden et al., 1994; Voros et al.,
2000; Quinn et al., 2000; Hide et al., 2002
).
In this chapter we use fractal analysis to analyze the binding and dissociation (if applicable)
kinetics for (a) the binding and dissociation of different concentrations of bradykinin to a bra-
dykinin B
2
receptor on a RWG biosensor, (
Fang et al., 2006
), (b) the binding and dissociation
of m
b
CD cholesterol to HeLa cells cultivated on a gold-plated prism (
Ziblat et al., 2006
), and
(c) binding and dissociation (if applicable) of calcium
FRET-based calcium biosensor
employing troponin (
Mank et al., 2006
) for the binding and dissociation of TXNL in solution
to the sensor-chip surface.
þ
9.2 Theory
9.2.1 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
9
:
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
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 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 receptor surface creates a deple-
tion layer of width (
Ðt
)
½
where
Ð
is the diffusion constant. This gives rise to the fractal
power law, (Analyte
¼
½ as it should be. Another way of looking at the
p
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
Receptor)
