Biomolecular Recognition: The Current Challenge - Dynamic Force Spectroscopy and Biomolecular Recognition - page 28

orientations leading to a decreased free energy are favored. After integrating over
all orientations and weighting with Boltzmann's factor, one obtains

p 1 p 2 24π 2 ε 2 k B T r 6

F K = − ( / ) / (1.11)

For two water molecules, the numerator is about 4300 k B T when r is expressed
in Angstr om. The timescale of molecular rotations ranges between picoseconds
and nanoseconds depending on molecule size. The effective dielectric constant is
expected to decrease according to rotation velocity.
The interaction between a dipole and polarizable molecule (see Equation 1.9) is
called Debye interaction. While this is dependent on temperature, there is a nonzero
high temperature limit since the net interaction between a dipole and a polarizable
molecule is attractive whatever the dipole orientation. This limit is proportional to
r − 6 , similar to Keesom interaction.
Finally, it is known that even in the absence of permanent dipole moments, two
polarizable molecules located at distance r exert a mutual attraction. This is called
dispersion or London force since the first calculation of the dispersion interaction
between hydrogen atoms was calculated by London (1930). This is proportional to
r − 6 . The numerical coefficient for two water molecules is about 740 k B T ,when r is
expressed in Angstrom.
Thus, there is some theoretical support for the concept that two freely interacting
molecular groups will exert a mutual attraction with an energy proportional to r − 6 .
This is often denominated as van der Waals attraction. Further, it is well known that
in addition a short-distance repulsion will prevent a collapse of molecule pairs. This
repulsion was sometimes called Born repulsion . It displays very rapid variation with
distance, which led to represent it empirically either as a step function (this is the
simplest “hard wall” model), or as a r −

12

function, leading to the empirical 6-12 or

Lennard Jones potential:

r 6 r 12 F LJ = − 4ε [( σ / ) − ( σ / )] (1.12)

1.5.1.4 Using the Formalism of Surface Physical-Chemistry:
Hydrophobic Bonds
The sharp energy distance relationship illustrated by Equation 1.12 is an incentive
to view molecular interactions as contact forces occurring at the interface between
rigid bodies. Clearly, this is not a rigorous model. A major problem is that molecular
interactions are not fully additive [122] [198]. However, this view is simple enough
to be felt useful, at least as a first approximation leading us to describe biomolecule
interactions as a set of contact interactions between contacting groups. This con-
cept provides a convenient way of accounting for the so-called hydrophobic bond.
The formalism of surface chemistry provides a convenient framework to discuss this
point. As already indicated, the free energy of interaction between two molecules
numbered 1 and 2 embedded in a medium 3 results from a balance between bond
formation and bond rupture as follows: