Theoretical Models in Force Spectroscopy - Dynamic Force Spectroscopy and Biomolecular Recognition - page 125

3.5.5.2 Valency from Polymer Tethers
The use of polymer linkers provides a direct approach to determining the true number
of bonds involved in a multivalent force measurements. The force-extension behav-
ior of a variety of polymers have been well-characterized ill date. In most cases,
polymers under tension in solution are described by the worm-like-chain, the freely-
jointed-chain (FJC), or variations on these models. The primary disadvantage of
polymer tethers is the nonlinear force with extension that does not directly lend the
data to simple dynamic models where linearly applied forces F r f t are assumed.

However, accounting for the nonlinearity can be done numerically, or one can assume
the force in the region around the unbinding event is linear and use the compliance
of the tether in this region for the effective loading rate (Friedsam et al. 2003).
Once the parameters of the single polymer force-extension model are determined,
a model to estimate the number of multiple bonds is straightforward. Assuming the
polymer dynamics remain at equilibrium throughout the pulling process (i.e., the F
vs. z curve is independent of pulling speed), then the force due to N p identical parallel
polymers, at any extension z ,isjust N p times the force of one polymer. The model
that describes the single polymer tether is simply recast using F tot

( t )=

N p in place of the
force F ,where F tot is the total force applied to the multivalent cluster (see Figure
3.10). For example, referring to the discussion above for the commonly used PEG
polymer tethers (see Equations 3.75 through 3.81), the extension of N p parallel PEG
tethers is

/

coth F tot L k
N p k B T

L C (3.102)

N p k B T
F tot L k

N k F tot
N p κ s

−

z ( F tot / N p )= ( F tot / N p )+

700

600

Num. of tethers
1
2
3
4
5

500

400

300

200

100

0

0 10 20 30 40 50

Extension/nm

FIGURE 3.10 (See color insert.) Theoretical force-extension curves of multiple, parallel
PEG polymers from numerical solution of Equation 3.102. (Adapted from Sulchek, T. et al.
2006. Biophys. J. , 90 (12), 4686-4691.)