Elasticity of a polymer chain with deformable bonds under fixed extension and constant force

The force-extension response of a polymer chain provides a direct link between molecular conformations and mechanical behavior. Motivated by directional control conditions commonly encountered in single-molecule stretching, we investigate the elasticity of a polymer chain with deformable bond lengths and bond angles in fixed-extension (FE) and constant-force (CF) ensembles, where the extension is defined as the chain's end-to-end distance projected along the pulling direction. We formulate both ensembles in terms of this projected extension and express their partition functions in transfer-matrix form. Exploiting the resulting structural similarity, we develop a unified computational framework that enables a direct comparison of their finite-chain responses. We show that the FE-CF difference is a finite-chain effect that decreases with increasing force and chain length. We further establish quantitative criteria for the onset of practical ensemble equivalence over finite force ranges and examine their dependence on bond stretching and bond-angle coupling. Applied to realistic carbon-backbone chains, we show that FE-like and CF-like responses become practically equivalent at experimentally relevant chain lengths in the higher-force regime probed by atomic force microscopy, whereas finite-chain ensemble differences remain appreciable in the lower-force regimes accessed by optical and magnetic tweezers. In the long-chain regime of practical equivalence, the semianalytical deformable freely rotating chain (dFRC) model proposed in our previous work [J. Zhu and L. Brassart, Phys. Rev. Lett. 134, 218101 (2025)] provides an accurate reduced description of the full statistical-mechanical response. These results clarify how ensemble choice, finite-chain effects, and local bond deformations jointly shape single-chain elasticity and provide a framework for interpreting and modeling single-chain stretching experiments.