Biomedical Engineering Reference
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to structural properties of binding molecules. This point will be rapidly considered
in the next section.
In conclusion
, while it might appear for some years that Equation 1.1 provided
a tractable way of describing the force dependence of molecular bonds [32], more
recent work showed that (1)
k
off
was often more complicated than suggested by
Bell's law due to the existence of multiple barriers and possibly multiple dissociation
pathways, as suggested to interpret catch bond behavior [143], and (2) a function
such as
k
off
(
F
)
may not exist, even with a more complicated form than Equation 1.1
due to the effect of history and dependence on the properties of linker molecules.
These points will be discussed with more detail in a following chapter of this topic.
(
F
)
1.4.2.2 Distance-Dependent Association Rates
The importance and significance of association rates (i.e.,
k
on
parameter) will now
be discussed.
First, there are many important examples supporting the prominent biologi-
cal importance of association rates. As indicated above, the efficiency of selectin
molecules was ascribed to their capacity to tether rapidly flowing leukocytes to
endothelial cells, which required a particularly high association rate. Also, experi-
mental data supported the view that the association rate of antibodies progressively
increased during the so-called maturation of immune responses, a finding that was
intepreted as
a premium on the capacity to bind target rapidly
[73]. Finally, the cell
capacity to probe its environment is dependent on the capacity of membrane recep-
tors to bind to their ligand during a transient approach of a receptor-bearing mem-
brane protrusion toward a ligand-bearing surface. In all these case, it seems that
the efficiency of bond formation should be calculable if we knew a function
k
on
)
defined as the frequency (per unit of time) of bond formation between a ligand and
a receptor molecules located at distance
d
. Such a function would include sufficient
information to account for interactions between soluble molecules (i.e., 3D condi-
tions) and surface-attached molecules (i.e., 2D conditions). Unfortunately, the deter-
mination and even the very definition of such a function are fraught with difficulties
for at least two complementary reasons.
(1) If the association between molecules
A
and
B
is a multiphasic reaction
involving a high number of interaction states, the discrimination between free and
bound states may be somewhat arbitrary. Indeed, if bond formation is not an all-
or-none phenomenon but require a progressive strengthening, it is not obvious to
chose a threshold to discriminate between free and bound states. Thus, while the
streptavidin-biotin interaction might have been considered as strong enough to allow
easy detection of bound states, several investigators reported on the time-dependent
maturation of this interaction [159] and existence of a number of weak association
states [153]. Indeed, if the number of intermediate states is high, the concept of
association constant becomes meaningless. This point was recently demonstrated
in a quantitative study made on the binding efficiency of antibody-coated micro-
spheres encountering antigen-bearing surfaces in a laminar flow chamber [169]. The
probability of bond formation scaled as a power of encounter duration that was
(
d
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