Conjecture on the lateral growth of type I collagen fibrils

• Published in: arXiv.org 2013-10
• Main Authors: Charvolin, Jean, Jean-François Sadoc
• Format: Article
• Language: English
• Subjects: Algorithms; Chirality; Collagen; Configurations; Edge dislocations; Elastic limit; Elastic properties; Helices; Isotropy
• ISSN: 2331-8422

Type I collagen fibrils have circular cross sections with radii mostly distributed in between 50 and 100 nm and are characterized by an axial banding pattern with a period of 67 nm. The constituent long molecules of those fibrils, the so-called triple helices, are densely packed but their nature is such that their assembly must conciliate two conflicting requirements : a double-twist around the axis of the fibril induced by their chirality and a periodic layered organization, corresponding to the axial banding, built by specific lateral interactions. We examine here how such a conflict could contribute to the control of the radius of a fibril. We develop our analysis with the help of two geometrical archetypes : the Hopf fibration and the algorithm of phyllotaxis. The first one provides an ideal template for a twisted bundle of fibres and the second ensures the best homogeneity and local isotropy possible for a twisted dense packing with circular symmetry. This approach shows that, as the radius of a fibril with constant double-twist increases, the periodic layered organization can not be preserved without moving from planar to helicoidal configurations. Such changes of configurations are indeed made possible by the edge dislocations naturally present in the phyllotactic pattern where their distribution is such that the lateral growth of a fibril should stay limited in the observed range. Because of our limited knowledge about the elastic constants involved, this purely geometrical development stays at a quite conjectural level.