Modeling and Theory - Biosensors and Biosensor Kinetics - page 18

diffusion process cannot be characterized by the classical Fick's law. In this range, anoma-
lous diffusion applies. Fatin-Rouge et al. (2004) emphasize that at larger distances than in
the above window range, the effects of disorder on diffusion may be very small due to statis-
tical effects, and may cancel each other.

Prior to presenting the Havlin (1989) analysis modified for the analyte-receptor binding
occurring on biosensor surfaces, it is appropriate to discuss briefly the analysis presented
by Fatin-Rouge et al. (2004) on size effects on diffusion processes within agarose gels, and
apply it to analyte-receptor binding and dissociation for biosensor kinetics. This analysis
provides some insights into general fractal-related processes. Fatin-Rogue et al. (2004) have
considered diffusion within a fractal network of pores. They indicate that fractal networks
such as percolating clusters may be characterized by a power law distribution ( Havlin, 1989 ):

D f

M ð L Þ ð 2 : 1 Þ

Here M is the average number of empty holes in the (gel) space characterized by a linear
size, L . The exponent, D f is the mass fraction dimension. Fatin-Rogue et al. (2004) emphasize
that in the general case of fractals, D f is smaller than the dimension of space of interest.
Furthermore, the independence of D f on scale is also referred to as self-similarity, and is
an important property of rigorous fractals.

Havlin and Ben-Avraham (1987) point out that the diffusion behavior of a particle within a
medium can be characterized by its mean-square displacement, r 2 ( t ) versus time, G t , which

is written as:

r 2 t ð 2 =D w Þ

ð t Þ¼ ð 2 : 2a Þ

Here Ð is the transport coefficient, and D w is the fractal dimension for diffusion. Normal or
regular diffusion occurs when D w is equal to 2. In this case, r 2 ( t ) is equal to t . In other words,

r 2 ð t Þ¼ 2 dDt ð 2 : 2b Þ

Here d is the dimensionality of space, and D is the diffusion coefficient.

Harder et al. (1987) and Havlin (1989) describe anomalous diffusion where the particles sense
obstructions to their movement. This is within the fractal matrix, or in our case due to hetero-
geneities on the biosensor surface, perhaps due to irregularities on the biosensor surface.
Fatin-Rogue et al. (2004) are careful to point out that anomalous diffusion may also occur
due to nonelastic interactions between the network and the diffusing particles in a gel matrix
( Saxton, 2001 ). Furthermore, Fatin-Rouge et al. (2004) indicate that anomalous diffusion is dif-
ferent from trapped diffusion where the particles are permanently trapped in holes, and are
unable to come out of these holes. When the particles (analytes in our case) are in these trapped