Calculations of the percolation thresholds of a three-dimensional (icosahedral) Penrose tiling by the cubic approximant method

Zakalyukin, R. M.; Chizhikov, V. A.

doi:10.1134/1.2132400

Cited by 3 publications

(10 citation statements)

References 15 publications

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“…Small graphs are expected to have relatively broad percolation transitions, as shown for both a capsid and random 120-subunit graphs in Figure 4. An examination of percolation in larger icosahedra saw a similar effect [45]. For very large graphs, one expects a more abrupt phase transition.…”

Section: Discussionmentioning

confidence: 84%

Molecular jenga: the percolation phase transition (collapse) in virus capsids

et al. 2018

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Virus capsids are polymeric protein shells that protect the viral cargo. About half of known virus families have icosahedral capsids that self-assemble from tens to thousands of subunits. Capsid disassembly is critical to the lifecycles of many viruses yet is poorly understood. Here, we apply a graph and percolation theory to examine the effect of removing capsid subunits on capsid stability and fragmentation. Based on the structure of the icosahedral capsid of hepatitis B virus (HBV), we constructed a graph of rhombic subunits arranged with icosahedral symmetry. Though our approach neglects dependence on energetics, time, and molecular detail, it quantitatively predicts a percolation phase transition consistent with recent in vitro studies of HBV capsid dissociation. While the stability of the capsid graph followed a gradual quadratic decay, the rhombic tiling abruptly fragmented when we removed more than 25% of the 120 subunits, near the percolation threshold observed experimentally. This threshold may also affect results of capsid assembly, which also experimentally produces a preponderance of 90 mer intermediates, as the intermediate steps in these reactions are reversible and can thus resemble dissociation. Application of percolation theory to understanding capsid association and dissociation may prove a general approach to relating virus biology to the underlying biophysics of the virus particle.

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Section: Discussionmentioning

confidence: 84%

Molecular jenga: the percolation phase transition (collapse) in virus capsids

et al. 2018

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“…Note that the icosahedral Penrose lattice has a range of coordination numbers with an average also of 6, and has site threshold of 0.285 and a bond threshold of 0.225 [21]; these thresholds were found by an approximate method and the expected errors are note clear.…”

Section: Discussionmentioning

confidence: 98%

“…In Table I we compare the thresholds found here with other 3D systems with a coordination number of 6, both regular lattices (the kagome stack and simple cubic) and more disordered ones (the dice stack and the icosahedral Penrose lattice [21]). Having a distribution of coordination number z is seen to lower the threshold, but this does not apply to the case of site percolation on the packed spheres.…”

Section: Discussionmentioning

confidence: 99%

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Percolation of disordered jammed sphere packings

Ziff

Torquato

2017

J. Phys. A: Math. Theor.

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We determine the site and bond percolation thresholds for a system of disordered jammed sphere packings in the maximally random jammed state, generated by the Torquato-Jiao algorithm. For the site threshold, which gives the fraction of conducting vs. non-conducting spheres necessary for percolation, we find p c = 0.3116(3), consistent with the 1979 value of Powell 0.310(5) and identical within errors to the threshold for the simple-cubic lattice, 0.311608, which shares the same average coordination number of 6. In terms of the volume fraction φ, the threshold corresponds to a critical value φ c = 0.199. For the bond threshold, which apparently was not measured before, we find p c = 0.2424(3). To find these thresholds, we considered two shape-dependent universal ratios involving the size of the largest cluster, fluctuations in that size, and the second moment of the size distribution; we confirmed the ratios' universality by also studying the simple-cubic lattice with a similar cubic boundary. The results are applicable to many problems including conductivity in random mixtures, glass formation, and drug loading in pharmaceutical tablets.

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“…1, eq. ( 25) (solid line) captures the main trend as a function of 1/Z of the calculated p c values for several periodic lattices with coordination radius including first and beyond first-nearest neighbours [9][10][11][12][13], and for periodic and aperiodic systems with distributed values of the coordination number Z i [14][15][16]. The present two-site EMA gives thus reasonable estimates of p c for both periodic and topologically disordered lattices already at the two-site approximation level, which explains the 25).…”

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confidence: 98%

A complete graph effective medium approximation for lattice and continuum percolation

Grimaldi

2011

EPL

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In this paper we introduce an effective medium theory that is capable of describing site percolation in both lattice and continuum three-dimensional systems. By exploiting selfconsistency with a complete graph or network of identical conductors, the resulting effective medium equation allows for a straightforward estimate of the percolation threshold, which compares well with the available numerical results. The present formalism is easily generalized to describe transport in conductor-insulator composites with different microstructure properties.

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Calculations of the percolation thresholds of a three-dimensional (icosahedral) Penrose tiling by the cubic approximant method

Cited by 3 publications

References 15 publications

Molecular jenga: the percolation phase transition (collapse) in virus capsids

Molecular jenga: the percolation phase transition (collapse) in virus capsids

Percolation of disordered jammed sphere packings

A complete graph effective medium approximation for lattice and continuum percolation

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