Binding and Dissociation Kinetics During Hybridization on Biosensor Surfaces - Biosensors and Biosensor Kinetics - page 447

often adequate to describe the binding and the dissociation kinetics. Peculiarities in the
values of the binding and the dissociation rate coefficients, in the systems being analyzed will
be carefully noted, if applicable.

16.2.1 Single-Fractal Analysis

Binding Rate Coefficient

Havlin (1989) reports 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 16 : 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 as it should be. Another way of looking at the 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 recep-
tor-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 depletion layer of
width ( Ðt ) ½ where Ð is the diffusion constant. This gives rise to the fractal power law,
(Analyte t (3 D f,bind )/2 . For the present analysis, t c is arbitrarily chosen and we assume

that the value of the t c is not reached. One may consider the approach as an intermediate
“heuristic” approach that may be used in the future to develop an autonomous (and not
time-dependent) model for diffusion-controlled kinetics.

Receptor)

Dissociation Rate Coefficient

The diffusion of the dissociated particle (receptor [Ab] or analyte [Ag]) from the solid sur-
face (e.g., analyte [Ag]-receptor [Ab] complex coated surface) into solution may be given,
as a first approximation by:

t ð 3 D f , bind Þ= 2 t p , ð Ab Ag Þ ¼ t > t diss ð 16 : 2 Þ

Here D f,diss is the fractal dimension of the surface for the dissociation step. This corresponds
to the highest concentration of the analyte-receptor complex on the surface. Henceforth,
its concentration only decreases. The dissociation kinetics may be analyzed in a manner
“similar” to the binding kinetics.