  A statistical mechanical distance constraint model (DCM) is presented that
explicitly accounts for network rigidity among constraints present within a
system. Constraints are characterized by local microscopic free energy
functions. Topological re-arrangements of thermally fluctuating constraints are
permitted. The partition function is obtained by combining microscopic free
energies of individual constraints using network rigidity as an underlying
long-range mechanical interaction -- giving a quantitative explanation for the
non-additivity in component entropies exhibited in molecular systems. Two
exactly solved 2-dimensional toy models representing flexible molecules that
can undergo conformational change are presented to elucidate concepts, and to
outline a DCM calculation scheme applicable to many types of physical systems.
It is proposed that network rigidity plays a central role in balancing the
energetic and entropic contributions to the free energy of bio-polymers, such
as proteins. As a demonstration, the distance constraint model is solved
exactly for the alpha-helix to coil transition in homogeneous peptides.
Temperature and size independent model parameters are fitted to Monte Carlo
simulation data, which includes peptides of length 10 for gas phase, and
lengths 10, 15, 20 and 30 in water. The DCM is compared to the Lifson-Roig
model. It is found that network rigidity provides a mechanism for cooperativity
in molecular structures including their ability to spontaneously self-organize.
In particular, the formation of a characteristic topological arrangement of
constraints is associated with the most probable microstates changing under
different thermodynamic conditions.
