Theoretical Models in Force Spectroscopy - Dynamic Force Spectroscopy and Biomolecular Recognition - page 126

with force-dependent contour length

N k

l f l u

L C ( F tot / N p )=

e Δ G ( F tot / N p ) / k B T +

(3.103)

e −

Δ G ( F tot / N p ) / k B T

+ +

1 1

Assuming the contour length may vary slightly between measurements, the only free
parameters in the model are the number of Kuhn segments N k and number of teth-
ers N p , which are completely uncoupled parameters. This approach for determining
bond valency was first employed by Sulchek et al. (2005, 2006) to determine bond
valency in PEG-tethered Mucin-1/antibody binding, which led to direct verification
of the multivalent model of Williams (2003) for parallel independent bonds (see
Equation 3.86 and Figure 3.8).

3.6 THERMODYNAMICS, WORK, AND FLUCTUATION
THEOREMS
We've seen in the preceding sections that, under increasing load, the force at which
a bond breaks is not a unique property of the bond alone, but it also depends on
how rapidly we increase the load. Thus, the observed rupture force is acquired under
nonequilibrium conditions. However, we are typically interested in probing systems
to determine their characteristic properties when the system is in thermodynamic
equilibrium. Of fundamental importance is the equilibrium free energy of the system,
which defines the work exchanged by the system with its surroundings when the
work done on or by the system is completely reversible. There are varying definitions
of free energy, the most prominent being Helmholtz A (common to physicists) and
Gibbs G (common to chemists). However, in the case of single-molecule events, the
difference between the two is subtle:

G = A + pV (3.104)

where p is the constant pressure and V the final volume of the system. For single-
molecule transformations, we assume the volume change will be negligible and
therefore Gibbs and Helmholtz free energies can be used interchangeably:

=

G A = − k B T ln Z (3.105)

where Z is the partition function of the system. Throughout this chapter, we chose to
use G to define the free energy of a system and Δ G to designate the change in free
energy over a reversible transformation.

3.6.1 F LUCTUATION T HEOREMS

The second law of thermodynamics states that the ensemble average work W done

on a system cannot be less than the equilibrium free energy:

W ≥ Δ G (3.106)

However, for small systems, such as individual molecules, fluctuations can signifi-
cantly broaden the distribution of workρ

( W ) for a given process. Although the mean