From three-dimensional weavings to swollen corneocytes

Abstract

A novel technique to generate three-dimensional Euclidean weavings, composed of closepacked, periodic arrays of one-dimensional fibres, is described. Some of these weavings are shown to dilate by simple shape changes of the constituent fibres (such as fibre straightening). The free volume within a chiral cubic example of a dilatant weaving, the ideal conformation of the G 129 weaving related to the S þ rod packing, expands more than fivefold on filament straightening. This remarkable three-dimensional weaving, therefore, allows an unprecedented variation of packing density without loss of structural rigidity and is an attractive design target for materials. We propose that the G 129 weaving (ideal S þ weaving) is formed by keratin fibres in the outermost layer of mammalian skin, probably templated by a folded membrane. Keywords: intermediate filaments; entangled weavings; dilatant materialsWe have developed a technique to generate a variety of crystalline close-packed arrays of one-dimensional filaments, via projection of line arrays in two-dimensional hyperbolic space into conventional three-dimensional space. The procedure relies on mapping of the hyperbolic patterns onto three-periodic minimal surfaces (TPMS), generalizing a technique used to enumerate systematically crystalline nets [1]. A rich catalogue of three-dimensional weavings can be constructed using this technique, with varying degrees of entanglement between filaments. Indeed, the entanglement can be tuned by adjusting the orientation of the line arrays in the hyperbolic plane, thereby varying the pitch of filament windings around channels of the TPMS. Generic examples are woven from curved and twisted filaments, whose detailed form depends on their trajectories on the TPMS [2]. Since these weavings contain close-packed fibres that can unwind at constant fibre volume without unjamming, they offer novel geometries for dilatant structures. An intriguing natural material that can sustain significant variations in fibre-packing densities is corneocytes, which make up the bulk of the outermost layer of mammalian skin, the stratum corneum. Corneocytes contain predominantly packed keratin fibrils, whose arrangement remains uncertain. We have found a dilatant chiral weaving with cubic symmetry that admits a sevenfold increase in the free volume on straightening the fibres cooperatively. The porosity variations exhibited by this weaving match closely the observed variations in human stratum corneum water contents. Further, the geometry of keratin intermediate filaments (IFs) deduced from X-ray crystallography matches well with data for fibres within this weaving in its canonical configuration, corresponding to unhydrated corneocytes.Since the specific curvilinear form of component filaments within a weaving can be varied without changing its entanglement, it is helpful to formulate a canonical geometry for each weaving. We have adapted algorithms developed to form canonical 'ideal' or 'tight' embeddings of knots [3,4] to arrive at these canonical embeddings...

Engineering Gyroid‐Structured Functional Materials via Templates Discovered in Nature and in the Lab

“…Besides, G‐structured hydrogel scaffolds fabricated by stereolithography around several hundreds of micrometers are also demonstrated to perform well in static cell culturing due to their excellent ability in cell transferring, nutrient substance diffusivity, and mechanical robustness . Interestingly, a precisely ordered and chiral arrangement of keratin intermediate filaments in mammalian skin, closely related to DG patterns (Figure e,f), is proposed to interpret its highly variable permeability properties and swelling with complete reversibility . This model comprehensively contributes to better demonstration of the superior performance of DG morphology in mass transport.…”

In a Material World. Hyperbolische Geometrie in biologischen Materialien

“…Die Hautzellen können dabei um ein Vielfaches ihrer ursprünglichen Größe anschwellen. Moderne Mikroskopiemethoden erlauben einen Einblick in den Aspekt der Hautstruktur, der für diese Fähig-keit der Haut verantwortlich ist: Ein hochsymmetrisches hierarchisches und poröses Arrangement von Keratinfilament wird sichtbar, dessen Zwischenräume mit Wasser gefüllt sind [4,22]. Wir argumentieren in diesem Artikel, dass die besondere Schwelleigenschaft der HautVervielfachung des Volumens ohne Verlust der mechanischen Integrität -zu einem großen Teil mittels geometrischer Studien zu verstehen ist: erstens das Problem, Spiralen im Raum möglichst dicht zu packen, und zweitens das Dilemma, dass sich die hyperbolische Ebene À 2 (oder eine Sattelfläche konstanter Gaußkrümmung) nicht im Euklidischen Raum einbetten lässt.…”

“…1 gezeigt [4]. Diese kubische Helix-Packung ist dreifach-periodisch, d. h. invariant unter drei zueinander orthogonalen Translationen, und eng verwandt mit einer Packung unendlich langer Stäbe, die in der Strukturchemie wohlbekannt ist [24].…”

“…In einem geeignet orientierten Gyroid, besetzen die verwobenenen Keratinfilamente der Haut gerade eine der beiden LabyrinthDomänen, wohingegen die zweite Domäne mit Wasser gefüllt ist [4,6] …”

Ideal geometry of periodic entanglements

“…Straightening of helical rods, and consequent loosening of the tight packing, is accompanied by variations of packing fraction dependent on the particular packing. Those loosening modes are often auxetic (with negative Poisson's ratio), as explored in more detail elsewhere [74,75]. …”