Elastic moduli of biological fibers in a coarse-grained model: crystalline cellulose and β-amyloids

We study the mechanical response of cellulose and beta-amyloid microfibrils to three types of deformation: tensile, indentational, and shear. The cellulose microfibrils correspond to the allomorphs I alpha or I beta whereas the beta-amyloid microfibrils correspond to the polymorphs of either two- or three-fold symmetry. This response can be characterized by three elastic moduli, namely, Y-L, Y-T, and S. We use a structure-based coarse-grained model to analyze the deformations in a unified manner. We find that each of the moduli is almost the same for the two allomorphs of cellulose but Y-L is about 20 times larger than Y-T (140 GPa vs. 7 GPa), indicating the existence of significant anisotropy. For cellulose we note that the anisotropy results from the involvement of covalent bonds in stretching. For beta-amyloid, the sense of anisotropy is opposite to that of cellulose. In the three-fold symmetry case, Y-L is about half of Y-T (3 vs. 7) whereas for two-fold symmetry the anisotropy is much larger (1.6 vs. 21 GPa). The S modulus is derived to be 1.2 GPa for three-fold symmetry and one half of it for the other symmetry and 3.0 GPa for cellulose. The values of the moduli reflect deformations in the hydrogen-bond network. Unlike in our theoretical approach, no experiment can measure all three elastic moduli with the same apparatus. However, our theoretical results are consistent with various measured values: typical Y-L for cellulose I beta ranges from 133 to 155 GPa, Y-T from 2 to 25 GPa, and S from 1.8 to 3.8 GPa. For beta-amyloid, the experimental values of S and Y-T are about 0.3 GPa and 3.3 GPa respectively, while the value of Y-L has not been reported.