Cooperative conformational transitions in macromolecules under mechanical stretching: Exact solution and validation of a two-state model against single-molecule experiments

The stretching behavior of linear macromolecules undergoing conformational transitions is investigated using a two-state Ising-like model capable of describing the mechanical response of polymers with widely varying structures and compositions. This minimal framework incorporates the essential parameters governing such transitions: two characteristic lengths, two elastic force constants, a free-energy difference between states, and a nearest-neighbor interaction energy that accounts for cooperativity. Chain stiffness is introduced through a bending potential. In the long-chain limit, exact equations are derived for both the molecular extension and the state populations as functions of the applied force. In the absence of bending rigidity, the model reduces to the two-state Elastic Freely Jointed Chain (EFJC). Within this approximation, it accurately reproduces the experimental force-extension behavior of poly(ethylene glycol) (PEG) and hyaluronic acid (HA), revealing the absence of cooperativity in PEG and the presence of negative cooperativity in HA. The model also captures the positive cooperativity of the B-DNA-to-S-DNA conformational transition when chain semiflexibility is taken into account. Within the EFJC framework, we further examine the mathematical conditions necessary for the occurrence of a transition and identify two fundamental driving mechanisms: differences in Kuhn lengths and differences in elastic force constants. Possible extensions to systems with more than two conformational states are also considered. The results apply equally to transitions intrinsic to the macromolecular architecture and to those induced by ligand-receptor interactions, providing a unified and thermodynamically consistent framework for describing both phenomena.