Multi-symmetric close packings of equal spheres on the spherical surface

Tarnai, Tibor; Gáspár, Zsolt

doi:10.1107/s0108767387098842

Cited by 36 publications

(32 citation statements)

References 4 publications

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“…The observed minimum angular distance between neighbouring nuclei ( i. e. the circle diameter divided by the sphere radius, the quantity measured in degrees and independent of the egg size) for the best tetrahedral pattern was d T = 27.17864°. This value was greater than known for octahedral ( d O = 26.32837°) or icosahedral ( d I = 26.82139°) nearly optimal packings of 60 circles [30]. This analysis confirmed the advantage of the regular tetrahedral pattern of IFCs with respect to all possible symmetric cell configurations in the considered system.…”

Section: Resultssupporting

confidence: 59%

Topological Control of Life and Death in Non-Proliferative Epithelia

et al. 2009

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Programmed cell death is one of the most fascinating demonstrations of the plasticity of biological systems. It is classically described to act upstream of and govern major developmental patterning processes (e.g. inter-digitations in vertebrates, ommatidia in Drosophila). We show here the first evidence that massive apoptosis can also be controlled and coordinated by a pre-established pattern of a specific ‘master cell’ population. This new concept is supported by the development and validation of an original model of cell patterning. Ciona intestinalis eggs are surrounded by a three-layered follicular organization composed of 60 elongated floating extensions made of as many outer and inner cells, and indirectly spread through an extracellular matrix over 1200 test cells. Experimental and selective ablation of outer and inner cells results in the abrogation of apoptosis in respective remaining neighbouring test cells. In addition incubation of outer/inner follicular cell-depleted eggs with a soluble extract of apoptotic outer/inner cells partially restores apoptosis to apoptotic-defective test cells. The 60 inner follicular cells were thus identified as ‘apoptotic master’ cells which collectively are induction sites for programmed cell death of the underlying test cells. The position of apoptotic master cells is controlled by topological constraints exhibiting a tetrahedral symmetry, and each cell spreads over and can control the destiny of 20 smaller test cells, which leads to optimized apoptosis signalling.

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Section: Resultssupporting

confidence: 59%

Topological Control of Life and Death in Non-Proliferative Epithelia

et al. 2009

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“…Evidently, the oxygen around oxygen mean coordination number is not a monotonically increasing function of pressure along an isotherm. At high pressure the oxygen superlattice assumes the form of an (irregular) icosahedron, OO [12] , a structure with an especially high packing efficiency (Tarnai and Gáspár, 1987;Clare and Kepert, 1991;Kottwitz, 1991). Both the OG and M liquids approach this asymptotic state of CN $ 12 at high pressure.…”

Section: Coordination Statistics At 5000 Kmentioning

confidence: 99%

Structure, thermodynamic and transport properties of liquid MgSiO3: Comparison of molecular models and laboratory results

Spera

Ghiorso

Nevins

2011

Geochimica et Cosmochimica Acta

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“…Quasi-equivalence postulates that the best capsid structure is one arranged so that the necessary deviations from some one optimal bonding pattern cost minimum energy; this is the assumption of quasi-equivalent bonding. Tarnai and Gaspar (1987), referring to the quasiequivalence theory, mention its "local extrema arrangements." Because the nearest neighbor interactions of hexavalent subunits are more similar (quasi-equivalent) to nearest neighbor interactions of pentavalent subunits than they would be to those of tetravalent or trivalent subunits, the constraint of quasi-equivalent bonding selects the icosahedron.…”

Section: Introductionmentioning

confidence: 99%

Pattern formation in icosahedral virus capsids: the papova viruses and Nudaurelia capensis beta virus

Marzec¹,

Day²

1993

Biophysical Journal

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The capsids of the spherical viruses all show underlying icosahedral symmetry, yet they differ markedly in capsomere shape and in capsomere position and orientation. The capsid patterns presented by the capsomere shapes, positions, and orientations of three viruses (papilloma, SV40, and N beta V) have been generated dynamically through a bottom-up procedure which provides a basis for understanding the patterns. A capsomere shape is represented in two-dimensional cross-section by a mass or charge density on the surface of a sphere, given by an expansion in spherical harmonics, and referred to herein as a morphological unit (MU). A capsid pattern is represented by an icosahedrally symmetrical superposition of such densities, determined by the positions and orientations of its MUs on the spherical surface. The fitness of an arrangement of MUs is measured by an interaction integral through which all capsid elements interact with each other via an arbitrary function of distance. A capsid pattern is generated by allowing the correct number of approximately shaped MUs to move dynamically on the sphere, positioning themselves until an extremum of the fitness function is attained. The resulting patterns are largely independent of the details of both the capsomere representation and the interaction function; thus the patterns produced are generic. The simplest useful fitness function is sigma 2, the average square of the mass (or charge) density, a minimum of which corresponds to a "uniformly spaced" MU distribution; to good approximation, the electrostatic free energy of charged capsomeres, calculated from the linearized Poisson-Boltzmann equation, is proportional to sigma 2. With disks as MUs, the model generates the coordinated lattices familiar from the quasi-equivalence theory, indexed by triangulation numbers. Using fivefold MUs, the model generates the patterns observed at different radii within the T = 7 capsid of papilloma and at the surface of SV40; threefold MUs give the T = 4 pattern of Nudaurelia capensis beta virus. In all cases examined so far, the MU orientations are correctly found.

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Multi-symmetric close packings of equal spheres on the spherical surface

Cited by 36 publications

References 4 publications

Topological Control of Life and Death in Non-Proliferative Epithelia

Topological Control of Life and Death in Non-Proliferative Epithelia

Structure, thermodynamic and transport properties of liquid MgSiO3: Comparison of molecular models and laboratory results

Pattern formation in icosahedral virus capsids: the papova viruses and Nudaurelia capensis beta virus

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