Modeling and Theory - Biosensors and Biosensor Kinetics - page 19

holes, then as time t !1 , the mean-square displacement, r 2 ( t ) tends to a constant value. Fatin-
Rouge et al. (2004) emphasize that in real heterogeneous porous media anomalous diffusion
of particles occurs over a limited length- or time-scale since the structure is only fractal over
a limited size scale. In other words, there is a lower bound and an upper bound beyond which
the fractal structure applies. Similarly, in our case, the anomalous diffusion of the analyte on
the biosensor surface occurs over a limited range of length- or time-scale.

For anomalous diffusion, one may combine the right-hand side of Equations (2.2a) and
(2.2b) . Then, the diffusion coefficient, D is given by ( Fatin-Rouge et al., 2004 ):

DðtÞ¼ð 1 = 4 Þt ½ð 2 =D w Þ 1 ð 2 : 3 Þ

Due to the temporal nature of D ( t ), it is better to characterize the diffusion of the analyte in
our case by D w . If we were still talking about the medium and gels, then D w would refer to
the diffusing medium.

We will now develop the theory for the analyte-receptor binding and dissociation on bio-
sensor surfaces. We will use the Havlin (1989) approach.

2.2 Theory

We present now a method of estimating fractal dimension values for analyte-receptor binding
and dissociation kinetics observed in biosensor applications. The following chapters will
present the different examples of data that have been modeled using the fractal analysis.
The selection of the binding and dissociation data to be analyzed in the later chapters is
constrained by whatever is available in the literature.

2.2.1 Variable Rate Coefficient

Kopelman (1988) points out that classical reaction kinetics are sometimes unsatisfactory when
the reactants are spatially constrained at the microscopic level by either walls, phase boundaries,
or force fields. Such heterogeneous reactions, for example, bioenzymatic reactions, that occur at
interfaces of different phases, exhibit fractal orders for elementary reactions and rate coefficients
with temporal memories. In such reactions, the rate coefficient exhibits a form given by:

k 0 t b k 1 ¼ 0 b 1 ð t 1 Þ ð 2 : 4 Þ

In general, k 1 depends on time whereas k 0 ¼ k 1 ( t ¼ 1) does not. Kopelman (1988) points out

that in three dimensions (homogeneous space) b 0. This is in agreement with the results

obtained in classical kinetics. Also, with vigorous stirring, the system is made homogeneous
and b again equals zero. However, for diffusion-limited reactions occurring in fractal spaces,
b > 0; this yields a time-dependent rate coefficient.

¼