Research Papers

Mechanics of Fibrous Biological Materials With Hierarchical Chirality

[+] Author and Article Information
Huijuan Zhu

Department of Mechanics,
Tianjin University,
Yaguan Road No.135,
Tianjin 300054, China
e-mail: zh_huijuan@foxmail.com

Takahiro Shimada

Department of Mechanical
Engineering and Science,
Kyoto University,
Kyoto 615-8540, Japan
e-mail: shimada@me.kyoto-u.ac.jp

Jianshan Wang

Department of Mechanics,
Tianjin University,
Yaguan Road No.135,
Tianjin 300054, China
e-mail: wangjs@tju.edu.cn

Takayuki Kitamura

Department of Mechanical
Engineering and Science,
Kyoto University,
Kyoto 615-8540, Japan
e-mail: kitamura@kues.kyoto-u.ac.jp

Xiqiao Feng

Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China
e-mail: fengxq@tsinghua.edu.cn

1Corresponding author.

Manuscript received February 27, 2016; final manuscript received June 25, 2016; published online August 18, 2016. Assoc. Editor: Kyung-Suk Kim.

J. Appl. Mech 83(10), 101010 (Aug 18, 2016) (7 pages) Paper No: JAM-16-1114; doi: 10.1115/1.4034225 History: Received February 27, 2016; Revised June 25, 2016

Chirality simultaneously exists at different length scales in many biological materials, e.g., climbing tendrils and bacterial flagella. It can transfer from lower structural levels to higher structural levels, which is tightly associated with the growth and assembly of biological materials. In this paper, a continuum mechanics model is presented for understanding the bottom–up transfer of chirality in fibrous biological materials. Basic physical mechanisms underlying the chirality transfer in biological world are revealed. It is demonstrated that the chirality of constituent elements at the microscale can induce the twisting of higher-level structures, which may further transfer into the macroscopic morphology in different manners, rendering the formation of hierarchically chiral structures in tissues or organs. The bottom–up transfer mechanism of chirality may provide a limit to the macroscopic size of biological materials through the accumulative contribution of twisting.

Copyright © 2016 by ASME
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Barry, E. , Hensel, Z. , Dogic, Z. , Shribak, M. , and Oldenbourg, R. , 2006, “ Entropy-Driven Formation of a Chiral Liquid-Crystalline Phase of Helical Filaments,” Phys. Rev. Lett., 96(1), p. 018305. [CrossRef] [PubMed]
Upmanyu, M. , Wang, H. L. , Liang, H. Y. , and Mahajan, R. , 2008, “ Strain-Dependent Twist-Stretch Elasticity in Chiral Filaments,” J. R. Soc. Interface, 5(20), pp. 303–310. [CrossRef] [PubMed]
Gerbode, S. J. , Puzey, J. R. , McCormick, A. G. , and Mahadevan, L. , 2012, “ How the Cucumber Tendril Coils and Overwinds,” Science, 337(6098), pp. 1087–1091. [CrossRef] [PubMed]
Ghafouri, R. , and Bruinsma, R. , 2005, “ Helicoid to Spiral Ribbon Transition,” Phys. Rev. Lett., 94(13), p. 138101. [CrossRef] [PubMed]
Schulgasser, K. , and Witztum, A. , 2004, “ The Hierarchy of Chirality,” J. Theor. Biol., 230(2), pp. 281–288. [CrossRef] [PubMed]
Aggeli, A. , Nyrkova, I. A. , Bell, M. , Harding, R. , Carrick, L. , McLeish, T. C. B. , Semenov, A. N. , and Boden, N. , 2001, “ Hierarchical Self-Assembly of Chiral Rod-Like Molecules as a Model for Petide β-Sheet Tapes Ribbons, Fibrils, and Fibers,” Proc. Natl. Acad. Sci. U. S. A., 98(21), pp. 11857–11862. [CrossRef] [PubMed]
Zhao, Z. L. , Zhao, H. P. , Wang, J. S. , Zhang, Z. , and Feng, X. Q. , 2014, “ Mechanical Properties of Carbon Nanotube Ropes With Hierarchical Helical Structures,” J. Mech. Phys. Solids, 71, pp. 64–83. [CrossRef]
Qin, Z. , and Buehler, M. J. , 2010, “ Molecular Dynamics Simulation of the α-Helix to β-Sheet Transition in Coiled Protein Filaments: Evidence for a Critical Filament Length Scale,” Phys. Rev. Lett., 104(19), p. 198304. [CrossRef] [PubMed]
Srigiriraju, S. V. , and Powers, T. R. , 2006, “ Model for Polymorphic Transitions in Bacterial Flagella,” Phys. Rev. E, 73(1), p. 011902. [CrossRef]
Zhao, Z. L. , Zhao, H. P. , Li, B. W. , Nie, B. D. , Feng, X. Q. , and Gao, H. J. , 2015, “ Biomechanical Tactics of Chiral Growth in Emergent Aquatic Macrophytes,” Sci. Rep., 5, p. 12610. [CrossRef] [PubMed]
Ye, H. M. , Wang, J. S. , Tang, S. , Xu, J. , Feng, X. Q. , Guo, B. H. , Xie, X. M. , Zhou, J. J. , Li, L. , Wu, Q. , and Chen, G. Q. , 2010, “ Surface Stress Effects on the Bending Direction and Twisting Chirality of Lamellar Crystals of Chiral Polymer,” Macromolecules, 43(13), pp. 5762–5770. [CrossRef]
Wada, H. , 2012, “ Hierarchical Helical Order in the Twisted Growth of Plant Organs,” Phys. Rev. Lett., 109(12), p. 128104. [CrossRef] [PubMed]
Wang, J. S. , Wang, G. , Feng, X. Q. , Kitamura, T. , Kang, Y. L. , Yu, S. W. , and Qin, Q. H. , 2013, “ Hierarchical Chirality Transfer in the Growth of Towel Gourd Tendrils,” Sci. Rep., 3, p. 3102. [PubMed]
Hattne, J. , and Lamzin, V. S. , 2011, “ A Moment Invariant for Evaluating the Chirality of Three-Dimensional Objects,” J. R. Soc. Interface, 8(54), pp. 144–150. [CrossRef] [PubMed]
Wang, J. S. , Cui, Y. H. , Shimada, T. , Wu, H. P. , and Kitamura, T. , 2014, “ Unusual Winding of Helices Under Tension,” App. Phys. Lett., 105(4), p. 043702. [CrossRef]
Wang, J. S. , Feng, X. Q. , Xu, J. , Qin, Q. H. , and Yu, S. W. , 2011, “ Chirality Transfer From Molecular to Morphological Scales in Quasi-One-Dimensional Nanomaterials: A Continuum Model,” J. Comput. Theor. Nanosci., 8(7), pp. 1278–1287. [CrossRef]
Claessens, M. M. A. E. , Semmrich, C. , Ramos, L. , and Bausch, A. R. , 2008, “ Helical Twist Controls the Thickness of F-Actin Bundles,” Proc. Natl. Acad. Sci. U. S. A., 105(26), pp. 8819–8822. [CrossRef] [PubMed]
Grason, G. M. , and Bruinsma, R. F. , 2007, “ Chirality and Equilibrium Biopolymer Bundles,” Phys. Rev. Lett., 99(9), p. 098101. [CrossRef] [PubMed]
Makowski, L. , and Magdoff-Fairchild, B. , 1986, “ Polymorphism of Sickle Cell Hemoglobin Aggregates: Structural Basis of Limited Radial Growth,” Science, 234(4781), pp. 1228–1231. [CrossRef] [PubMed]
Turner, M. S. , Briehl, R. W. , Ferrone, F. A. , and Josephs, R. , “ Twisted Protein Aggregates and Disease: The Stability of Sickle Hemoglobin Fibers,” Phys. Rev. Lett., 90(12), p. 128103. [CrossRef] [PubMed]
Fratzl, P. , Misof, K. , and Zizak, I. , 1998, “ Fibrillar Structure and Mechanical Properties of Collagen,” J. Struct. Biol., 122(1–2), pp. 119–122. [CrossRef] [PubMed]
Harris, A. B. , Kamien, R. D. , and Lubensky, T. C. , 1999, “ Molecular Chirality and Chiral Parameters,” Rev. Mod. Phys., 71(5), pp. 1745–1757. [CrossRef]
Grason, G. M. , 2009, “ Braided Bundles and Compact Coils: The Structures and Thermodynamics of Hexagonally Packed Chiral Filament Assemblies,” Phys. Rev. E, 79(4), p. 041919. [CrossRef]
Guo, Q. , Chen, Z. , Li, W. , Dai, P. , Ren, K. , Lin, J. , Taber, L. A. , and Chen, W. , 2014, “ Mechanics of Tunable Helices and Geometric Frustration in Biomimetic Seashells,” EPL, 105(6), p. 64005. [CrossRef]
Helfrich, W. , 1991, “ Elastic Theory of Helical Fibers,” Langmuir, 7(3), pp. 567–568. [CrossRef]
Neville, A. C. , 1993, Biology of Fibrous Composites: Development Beyond the Cell Membrane, Cambridge University Press, Cambridge, UK.
Emons, A. M. C. , and Mulde, B. M. , 1998, “ The Making of the Architecture of the Plant Cell Wall: How Cells Exploit Geometry,” Proc. Natl. Acad. Sci. U. S. A., 95(12), pp. 7215–7219. [CrossRef] [PubMed]
Marklund, E. , and Varna, J. , 2009, “ Modeling the Effect of Helical Fiber,” Appl. Compos. Mater., 16(4), pp. 245–262. [CrossRef]
Yang, Y. S. , Meyer, R. B. , and Hagan, M. F. , 2010, “ Self-Limited Self-Assembly of Chiral Filaments,” Phys. Rev. Lett., 104(25), p. 258102. [CrossRef] [PubMed]
Lloyd, C. , and Chan, J. , 2004, “ Microtubules and the Shape of Plants to Come,” Nat. Rev. Mol. Cell Biol., 5(1), pp. 13–23. [CrossRef] [PubMed]
Li, Q. , Kang, Y. L. , Qiu, W. , Li, Y. L. , Huang, G. Y. , Guo, J. G. , Deng, W. L. , and Zhong, X. H. , 2011, “ Deformation Mechanisms of Carbon Nanotubes Fibres Under Tensile Loading by In Situ Raman Spectroscopy Analysis,” Nanotechnology, 22(22), p. 225704. [CrossRef] [PubMed]
Cox, H. L. ,1952, “ The Elasticity and Strength of Paper and Other Fibrous Materials,” Br. J. Appl. Phys., 3(3), p. 72. [CrossRef]
Wu, X. F. , and Dzenis, Y. A. , 2005, “ Elasticity of Planar Fiber Network,” J. Appl. Phys., 98(9), p. 093501. [CrossRef]
Fu, S. Y. , and Lauke, B. , 1996, “ Effects of Fiber Length and Fiber Orientation Distributions on the Tensile Strength of Short-Fiber-Reinforced Polymers,” Compos. Sci. Technol., 56(10), pp. 1179–1190. [CrossRef]
Tu, Z. C. , Li, Q. X. , and Hu, X. , 2006, “ Theoretical Determination of the Necessary Condition for the Formation of ZnO Nanorings and Nanohelices,” Phys. Rev. B, 73(11), p. 115402. [CrossRef]
Wang, J. S. , Ye, H. M. , Qin, Q. H. , Xu, J. , and Feng, X. Q. , 2012, “ Anisotropic Surface Effects on the Formation of Chiral Morphologies of Nanomaterials,” Proc. R. Soc. A, 468(2139), pp. 609–633. [CrossRef]
Gray, D. G. , 1989, “ Chirality and Curl of Paper Sheets,” J. Pulp Pap. Sci., 15(3), pp. J105–J109.
Dionne, I. , Werbowyj, R. S. , and Gray, D. G. , 1991, “ Chiral Twisting Curl in Newsprint Sheets,” J. Pulp Pap. Sci., 17(4), pp. J123–J127.
Alvaa, M. , and Niskanen, K. , 2006, “ The Physics of Paper,” Rep. Prog. Phys., 69(3), pp. 669–723. [CrossRef]
Lakes, R. , 2015, “ Third-Rank Piezoelectricity in Isotropic Chiral Solids,” Appl. Phys. Lett., 106(21), p. 212905. [CrossRef]


Grahic Jump Location
Fig. 1

Hierarchy of chirality in biological materials: (a) sugar unit, (b) cellulose molecule, (c) cellulose fibril, (d) single cell with helical winding of cellulose fibrils, (e) fiber bundle, (f) fiber network, (g) twisted belt, and (f) macroscopic helix

Grahic Jump Location
Fig. 2

Physical mechanism of intrinsic twist. (a) A fully swelled cell with the lowest value of helical angle α=α0, (b) a partially swelled cell with helical angle α=α1>α0, and (c) a fully deswelled cell with the largest helical angle α=α2>α1. During the deswelling process from (a) to (c), the helical angle of the cellulose fibril helix increases with decreasing cell wall radius. The shape changes of cellulose fibril helices induce an equivalent torque acting on the cross section of cell, which leads to the intrinsic torque acting on the cell. The directions and values of the equivalent torque are determined by the helical angle change. Furthermore, the cross-sectional radius of the cell wall is limited by cellulose winding.

Grahic Jump Location
Fig. 3

The formed twisted belts of fully deswelled paper sheets

Grahic Jump Location
Fig. 4

Variations of the twisting angle per unit length with the water content for different paper sheets. The water content is defined as X=(m2−m1)/m1×100%, where m1 and m2 are the weight of paper sheet before and after swelling, respectively.

Grahic Jump Location
Fig. 5

Variations of the twisting angle per unit length with thewidth for different paper sheets. Fitting curves: Yblack=2.484x−1.1,Yblue=2.903x−0.964 and Yred=2.904x−0.725.

Grahic Jump Location
Fig. 6

The formation of the torsion and curvature of the fibernetwork with the helical angle changes of cellulose fibrilhelices. Here, b=2 mm and κ0=1000. In the calculation, we take H=2r and the cross-sectional radius of cellulose fibrilsr=r0(1+βTΔC), where r0=2.0×10−6 m, Q11=3.798 GPa, Q22=91.37 GPa, Q12=9.154 GPa, Q33=1.2 GPa, Q44=Q55=1.008 GPa, Q66=0.998 GPa,Q12=Q23=0 GPa [28], p=1 and q=2.

Grahic Jump Location
Fig. 7

The evolution of the torsion and curvature of the fiber network with the helical angle changes of cellulose fibril helices. The helical shape variations of the fiber network with the helical angles of cellulose fibril helices at the subcellular level. We take the initial curvature κ0=125.



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