Modeling and Theory - Biosensors and Biosensor Kinetics - page 22

receptor surface creates a depletion layer of width (Ð t ) ½ where Ð is the diffusion constant.
This gives rise to the fractal power law, (Analyte Receptor)

t ð 3 D f , bind Þ= 2 .

The values of the parameters k (binding rate coefficient), p , and D f in Equation (2.5a) may be
obtained for analyte-receptor association kinetics data. This may be done by a regression
analysis using, for example, Corel Quattro Pro (1997) along with Equation (2.5a) where
(analyte

kt p ( Sadana and Beelaram, 1994; Sadana et al., 1995 ). The fractal
dimension may be obtained from the parameter p . Since p

receptor) ¼

(3- D f,bind )/2, D f,bind is equal
to (3-2 p ). In general, low values of p would lead to higher values of the fractal dimension,
D f,bind . Higher values of the fractal dimension would indicate higher degrees of “disorder”
or heterogeneity or inhomogeneity on the surface.

¼

Another way of looking at the diffusive process is that it inherently involves fluctuations at
the molecular level that may be described by a random walk ( Weiss, 1994 ). This author
points out that the kinetics of transport on disordered (or heterogeneous) media needs to
be described by a random-walk model. When both of these are present, that is the diffusion
phenomena as well as a fractal surface, then one needs to analyze the interplay of both these
fluctuations. In essence, the disorder on the surface (or a higher fractal dimension, D f ) tends
to slow down the motion of a particle (analyte, in our case) moving in such a medium. Basi-
cally, according to Weiss (1994) , the particle (random walker analyte) is trapped in regions in
space as it oscillates for a long time before resuming its motion.

Havlin (1989) indicates that the crossover value may be determined by r c t c . Above the

characteristic length, r c , the self-similarity of the surface is lost. Above t c , the surface may
be considered homogeneous, and “regular” diffusion is now present. One may consider the
analysis to be presented as an intermediate “heuristic” approach in that in the future one
may also be able to develop an autonomous (and not time-dependent) model of diffusion-
limited kinetics in disordered media.

It is worthwhile commenting on the units of the association and the dissociation rate coeffi-
cient(s) obtained for the fractal analysis. In general, for SPR biosensor analysis, the unit for
the analyte receptor complex on the biosensor surface is RU (resonance unit). One thousand

resonance units is generally 1 ng/(mm) 2 (of surface), or 1 RU is 1 pg/(mm) 2 . Here, ng and pg
are nanogram and picogram, respectively. Then, to help determine the units for the binding
coefficient, k , from Equation (2.5a) :

kt ð 3 D f , bind Þ= 2

2

kt p

ð analyte receptor Þ ,pg =ð mm Þ ¼ ¼

This yields a unit for the binding rate coefficient, k as (pg)(mm) 2 (sec) ( D f,bind 3)/2 . Note that
the unit of dependence in time exhibited by the association (or binding) rate coefficient, k ,
changes slightly depending on the corresponding fractal dimension obtained in the binding
phase, D f,bind . The fractal dimension value is less than or equal to three. Three is the highest