Icosadeltahedral Geometry of Geodesic Domes, Fullerenes and Viruses: A Tutorial on the T-Number

Šiber, Antonio

doi:10.3390/sym12040556

Cited by 17 publications

(11 citation statements)

References 64 publications

(137 reference statements)

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“…We also emphasize that, as shown in Fig. 16, the elementary subunits of many viral-capsid proteins can be asymmetric in form, heterogeneous in protein composition, and nontrivial in their mutual interactions [3,9,28,29,31,97]. These features per se may severely impact the icosahedral appearance of some capsids (with planar tilings [222,223], e.g., being impossible to construct from these asymmetric structural elements).…”

Section: B Biological and Biomedical Relevancementioning

confidence: 97%

“…Note that the stable assembly of icosahedral viral capsids of different sizes (or, of different triangulation numbers T [3,9,35,105]) from the same protein subunits may not always be possible [219]. The concept of triangulation also limits the geometrically allowed number of protein subunits on a capsid to a discrete set of numbers…”

Section: B Biological and Biomedical Relevancementioning

confidence: 99%

“…Geometrically ideal closed three-dimensional surfaces, starting from the Platonic solids [1], have inspired many thinkers and scientists over centuries and millennia. The inherent stability of such polyhedral shells constructed from jointed triangular subunits has influenced some famous architects, such as Buckminster Fuller with his visionary geodesic domes [2,3]. The name "Buckminster fullerene" was coined for C 60 "buckyballs" discovered by Kroto et al [4,5] and composed of n C = 60 carbon atoms connected into a shell with 60 vertices and 32 faces (with 12 pentagonal and 20 hexagonal faces) [8].…”

Section: Introductionmentioning

confidence: 99%

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Buckling transitions and soft-phase invasion of two-component icosahedral shells

et al. 2020

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What is the optimal distribution of two types of crystalline phases on the surface of icosahedral shells, such as of many viral capsids? We here investigate the distribution of a thin layer of soft material on a crystalline convex icosahedral shell. We demonstrate how the shapes of spherical viruses can be understood from the perspective of elasticity theory of thin two-component shells. We develop a theory of shape transformations of an icosahedral shell upon addition of a softer, but still crystalline, material onto its surface. We show how the soft component "invades" the regions with the highest elastic energy and stress imposed by the 12 topological defects on the surface. We explore the phase diagram as a function of the surface fraction of the soft material, the shell size, and the incommensurability of the elastic moduli of the rigid and soft phases. We find that, as expected, progressive filling of the rigid shell by the soft phase starts from the most deformed regions of the icosahedron. With a progressively increasing soft-phase coverage, the spherical segments of domes are filled first (12 vertices of the shell), then the cylindrical segments connecting the domes (30 edges) are invaded, and, ultimately, the 20 flat faces of the icosahedral shell tend to be occupied by the soft material. We present a detailed theoretical investigation of the first two stages of this invasion process and develop a model of morphological changes of the cone structure that permits noncircular cross sections. In conclusion, we discuss the biological relevance of some structures predicted from our calculations, in particular for the shape of viral capsids.

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Section: B Biological and Biomedical Relevancementioning

confidence: 97%

Section: B Biological and Biomedical Relevancementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Buckling transitions and soft-phase invasion of two-component icosahedral shells

et al. 2020

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“…For the results shown in this work, the stress-and strain-free state is a spherical shell of radius R 0 = 38.4a, where a is the mean length of the mesh edge. The mesh has V = 21644 vertices, E = 64926 edges, and F = 43284 faces, and V −E +F = 2, as it must be according to the Euler formula for polyhedra [50].…”

Section: Mesh Triangulationmentioning

confidence: 99%

Mechanics of inactive swelling and bursting of porate pollen grains

Božič

Šiber²

2021

Preprint

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The mechanical structure of pollen grains, typically characterized by soft apertures in an otherwise stiff exine shell, guides their response to changes in the humidity of the environment. These changes can lead both to infolding but also to excessive swelling and even bursting of pollen grains. We use an elastic model to explore the mechanics of pollen grain swelling and the role that soft, circular apertures (pores) play in this process. We identify and explore a mechanical weakness of the pores, which are prone to a rapid inflation once the grain swells to a critical extent. This transition leads to the bursting of the grain and the release of its content. Our results shed light on the inactive part of the mechanical response of pollen grains to hydration once they land on a stigma as well as on bursting of airborne pollen grains during rapid changes in air humidity.

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“…Further details of all these geometrical construction schemes are summarized and analyzed in recent reviews by Zandi et al ( 13 ) and Šiber. 14 …”

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confidence: 99%

Minimal Design Principles for Icosahedral Virus Capsids

et al. 2021

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The geometrical structures of single- and multiple-shell icosahedral virus capsids are reproduced as the targets that minimize the cost corresponding to relatively simple design functions. Capsid subunits are first identified as building blocks at a given coarse-grained scale and then represented in these functions as point particles located on an appropriate number of concentric spherical surfaces. Minimal design cost is assigned to optimal spherical packings of the particles. The cost functions are inspired by the packings favored for the Thomson problem, which minimize the electrostatic potential energy between identical charged particles. In some cases, icosahedral symmetry constraints are incorporated as external fields acting on the particles. The simplest cost functions can be obtained by separating particles in disjoint nonequivalent sets with distinct interactions, or by introducing interacting holes (the absence of particles). These functions can be adapted to reproduce any capsid structure found in real viruses. Structures absent in Nature require significantly more complex designs. Measures of information content and complexity are assigned to both the cost functions and the capsid geometries. In terms of these measures, icosahedral structures and the corresponding cost functions are the simplest solutions.

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Icosadeltahedral Geometry of Geodesic Domes, Fullerenes and Viruses: A Tutorial on the T-Number

Cited by 17 publications

References 64 publications

Buckling transitions and soft-phase invasion of two-component icosahedral shells

Buckling transitions and soft-phase invasion of two-component icosahedral shells

Mechanics of inactive swelling and bursting of porate pollen grains

Minimal Design Principles for Icosahedral Virus Capsids

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