J-shaped stress-strain diagram of collagen fibers: Frame tension of triangulated surfaces with fixed boundaries

Abstract

We present Monte Carlo data of the stress-strain diagrams obtained using two different triangulated surface models. The first is the canonical surface model of Helfrich and Polyakov (HP), and the second is a Finsler geometry (FG) model. The shape of the experimentally observed stressstrain diagram is called J-shaped. Indeed, the diagram has a plateau for the small strain region and becomes linear in the relatively large strain region. Because of this highly non-linear behavior, the J-shaped diagram is far beyond the scope of the ordinary theory of elasticity. Therefore, the mechanism behind the J-shaped diagram still remains to be clarified, although it is commonly believed that the collagen degrees of freedom play an essential role. We find that the FG modeling technique provides a coarse-grained picture for the interaction between the collagen and the bulk material. The role of the directional degrees of freedom of collagen molecules or fibers can be understood in the context of FG modeling. We also discuss the reason for why the J-shaped diagram cannot (can) be explained by the HP (FG) model. *

Mathematical modeling and simulations for large-strain J-shaped diagrams of soft biological tissues

“…In fact, almost the same results are obtained for some limited range of κ; however, the stress-strain diagram obtained from the canonical 2D model for a certain finite value of κ is not J-shaped, as reported in Ref. [28]. Another reason is that only the FG model reproduces the diagrams that are convex upwards for the large-strain region.…”

“…In Refs. [28,29], we studied J-shaped curves by the Finsler geometry (FG) modeling technique and obtained Monte Carlo (MC) data consistent with previously reported experimental results, in which the toe length is up to 40% ∼ 50% on the strain axis. For these experimental J-shaped curves of small toe length, FG modeling successfully describes the diagrams.…”

“…Although this 2D model is the same as the one in Ref. [28], we summarize the Hamiltonian here in a self-contained manner. The Hamiltonian is given by a linear combina- tion of four terms such that…”

“…(14) with the experimental data, we have to change simulation units to physical units. For this purpose, we explicitly use k B T and the lattice spacing a [28,47], which are suppressed by k B T = 1 and a = 1 in the expression for τ in Eq. (14).…”

“…In fact, the results of the canonical model are always linear for the large-strain region, and they cannot be fit to the experimental curve with the convex part. This is a non-trivial difference between the FG and canonical models, as is the fact that the diagram of the canonical model becomes linear at a certain value of κ, while the diagram of the FG model is always J-shaped independent of κ [28].…”