Network Rigidity At Finite Temperature: Relationships Between Thermodynamic Stability, The Nonadditivity Of Entropy, And Cooperativity In Molecular Systems

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 rearrangements 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 nonadditivity in component entropies exhibited in molecular systems. Two exactly solved two-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 biopolymers, such as proteins. As a demonstration, the distance constraint model is solved exactly for the ?-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 1 for gas phase, and lengths 1, 15, 2, and 3 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.