Constitutive Models for the Force-Extension Behavior of Biological Filaments

Biopolymer filaments form the molecular backbone of biological structures throughout the body. The biomechanical response of single filaments yields insight into their individual behavior at the molecular level as well as their concerted networked behavior at the cellular and tissue scales. This paper focuses on modeling approaches for axial force vs. extension behavior of single biopolymer filaments within three stiffness regimes: flexible, semiflexible, and stiff. The end-to-end force-extension behaviors of flexible and semiflexible filaments arise as a result of a reduction in configurational space as the filament is straightened and are captured with entropic models including the freely jointed chain model and the worm-like chain model. As the filament is straightened and the end-to-end distance approaches the filament contour length, the contour length is directly axially extended and an internal energy contribution governs the force-extension behavior in this limiting extension regime. On the other hand, for stiff filaments in originally crimped or kinked configurations, the end-to-end force vs. extension behavior results from the unbending (straightening) of the crimped configuration as governed by an internal energy based elastica approximation which is also complemented by an axial stretching contribution once the end-to-end distance approaches the contour length of the filament. Simplified, analytical force-extension relationships are developed for the worm-like chain model for semiflexible filaments, and for the Euler elastica model for stiffer, wavy fibers. For the case of flexible molecules containing modular folded domains, the influence of force-induced unfolding on the force-extension behavior of single molecules and assemblies of multiple molecules is also presented.