Theoretical Models in Force Spectroscopy - Dynamic Force Spectroscopy and Biomolecular Recognition - page 123

the measurement. Hence, increasing the degeneracy of the bound state enhances the
stability of the cluster-surface interaction, and thus increases the equilibrium rupture
force F N , eq .
Equation 3.93 reveals a detail about tethered single bonds not considered in the
previous sections on single-bond systems. If only a single ligand is linked to the
force probe by a flexible linker, then the average bond valency is obviously N 1.

But if the density of the receptors is high enough such that more than one is available
M

=

1, then Equation 3.93 does not reduce to the equilibrium force for a single-
ligand single-receptor bond F eq

>

= 2 k poly Δ G 1 , but more generally, it retains the

entropic term and reduces to

2 k poly

F 1 , eq = ( Δ G 1 + k B T ln M ) (3.95)

which suggests an approach to determining the surface density of receptor sites using
mono-valent force spectroscopy, when the single-ligand/single-receptor free energy
Δ G 1 is known.

3.5.5 C OUNTING B OND V ALENCY IN F ORCE M EASUREMENTS

When multiple molecules are present on the contacting surfaces, intentionally or oth-
erwise, it is ultimately desirable to know how many participate in the measurement.
For molecules linked rigidly to the contacting surfaces, their rupture is most likely
correlated such that their failure is highly cooperative. The appearance of peaks in
the rupture force histogram may indicate the force quanta for each valency, how-
ever, this is only well-resolved if the individual distributions for each valency are
small compared with the mean. Otherwise, the resulting distribution is broadened by
the presence of multiple bonds with no direct distinction of each valency in the his-
togram. Below is two approaches toward dealing with the bond valency problem in
force measurements. The first is based on a statistical approach, which is appropri-
ate for molecules whose discrete rupture events are indistinguishable. This is usually
the case for molecules linked closely to the probe whereby rupture of the bonds is
correlated. The second is based on the linear scaling of stiffness with the number
of parallel polymer tethers, which can be used to deduce bond valency when the
molecules of interest are linked via flexible tethers.

3.5.5.1 Poisson Analysis of Rupture Events
Evans & Williams (2002) showed how the fraction of occurrences of adhesion events
within a total sample of attempted force measurements can be related to the Poisson
distribution to estimate the mean number of bonds formed in the sample, and the
fraction of those which are single bonds. If one prepares the experiment such that the
probability of bond formation between the contacting surfaces is low, then one can
assume Poisson statistics govern the number of bonds formed on each contact. To
make proper use of such an analysis, the same protocol should be used for each force
measurement, such as contact time and force. For a large sample size, the discrete