Biomedical Engineering Reference
In-Depth Information
Dissociation Rate Coefficient
In this case the dissociation rate coefficient is given by:
t ð 3 D f1 , diss Þ= 2 ,
Þ
t diss <
t
<
t d1
ð
Ab
Ag
ð
2
:
5d
Þ
t ð 3 D f2 , diss Þ= 2 ,
t d1 <
t
<
t d2
Here D f,diss is the fractal dimension of the surface for the dissociation step. t diss represents the
start of the dissociation step. This corresponds to the highest concentration of the analyte-
receptor on the surface. Henceforth, its concentration only decreases. D f,bind or D f,assoc may
or may not be equal to D f,diss . The dissociation rate coefficients, k d1 and k d2 in the dual-fractal
analysis have the same units (pg)(mm) 2 (sec) ( D fd1 3)/2 and (pg)(mm) 2 (sec) ( D fd2 3)/2 , respec-
tively, as the dissociation rate coefficient, k d , in the single-fractal analysis.
2.2.4 Triple-Fractal Analysis
As will be shown later in the topic, one resorts to a triple-fractal analysis when the dual-
fractal analysis does not provide an adequate fit. The equation for fractal analysis is generic
in nature, and one may easily extend the single- and the dual-fractal analysis equations
( Equations 2.5a and 2.5c ) to describe the binding (and/or the dissociation) kinetics for a triple
fractal analysis. In fact, in the extreme case, n fractal dimensions may be present. In this case,
the degree of heterogeneity, D f , or the fractal dimension is continuously changing on the
biosensor surface, and the surface needs to be represented by D f i where i goes from 1 to n.
Similarly, we have n binding rate coefficients on the biosensor surface. A similar representa-
tion may also be made for the dissociation phase.
It is perhaps appropriate here to at least mention one more approach that has been used
to model the binding kinetics on surfaces.
2.2.5 Pfeifer's Fractal Binding Rate Theory
Pfeifer and Obert (1989) have suggested an alternate form of the binding rate theory. In the
equation given in this reference, N is the number of complexes, N 0 is the number of receptors
on the solid surface, D is the diffusion coefficient of the analyte, L is the receptor diameter,
and
is the mean distance between two neighboring receptors. This equation may also be
used to analyze the analyte-receptor binding kinetics. The problem, however, is that it may
not be possible in all instances to estimate a priori all the parameters described in the equa-
tion (not given here). In that case, one may have to approximate or assume certain values,
and this will affect the accuracy and reliability of the analysis. The suggested equation does
have an advantage compared to the fractal analysis described above in that it does include
a prefactor necessary to convert the time interval over which fractal scaling is observed
into a length interval. It also provides an expression for t c (
l
L 2 / D ), which separates the
¼
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