Triangulation numbers

Can any number of identical subunits be accommodated in the asymmetric unit while preserving specificity of interactions within an icosahedral arrangement This question was answered by Don Caspar then at Children s Hospital, Boston, and Aaron Klug in Cambridge, England, who showed in a classical paper in 1962 that only certain multiples (1, 3, 4, 7...) of 60 subunits are likely to occur. They called these multiples triangulation numbers, T. Icosahedral virus structures are frequently referred to in terms of their trian-gulation numbers a T = 3 virus structure therefore implies that the number of subunits in the icosahedral shell is 3 x 60 = 180. 

Electron microscopic studies have suggested that the alphavirus particle has icosahedral symmetry (see below). The triangulation number is not certain, however (Murphy, 1980). Previous estimates for the molecular weight were compatible with 240 subunits per virus particle, and electron micrographs appear to show a T = 4 surface lattice (von Bons-dorff and Harrison, 1975). More information is now needed to determine the surface organization, since compositional data show fewer than 240 subunits. 

In Virology, the number t(k, l) = k2 + kl + l2 (used for icosahedral fullerenes) is called triangulation number. In terms of Buckminster Fuller, the number k 4-1 is called frequency, the case l = 0 is called Alternate, and the case l = k is called... 

Figure 1. Triangulation numbers T = (fp+hk + k2) represented on an equilateral triangular net. An icosadeltahedron (see figure 2) with a five-fold vertex at the origin of this net and a neighbouring five-fold vertex at a lattice point of index h, k will have A = 207 triangular facets, V6 = lOfT1— 1) six-connected vertices, and V5 = 12 five-vertices.

Figure 2. Models of icosadeltahedra for the first five possible triangulation numbers (built from...

Fig. 14.16. Nanoscaled systems of the biosphere and of chemistry show the same topology schematic representation of the icosahedron spanned by the 12 centres of the (Mo)IVIo5 units of the M0132 type cluster (a) and of the satellite tobacco necrosis virus (STNV) with the triangulation number F = 1...

The Web site http //www.ncbi.nlm.nih.gov/ICTVdb/fr-fst-g.htm offers color images of the experimentally determined three-dimensional structures of many different viruses. The pictures are of great quality. In addition, the Web site offers an excellent introduction to the topology of icosahedra and triangulation numbers. 

Fig. 2 An icosahedron with each face showing triangulation numbers of, clockwise from top, T=, T= 4. T=9, 7= 16, and 7=25, respectively. (View this art in color at www. dekker.com.)...

It may therefore be concluded that the asymmetric structure unit of the picornaviral capsid is composed of three polypeptides and conforms to the prediction of Finch and Klug. It has a molec]jlar mass of approximately 86,000 daltons, a diameter of about 68 A (35) and is repeated 60 times in the virus capsid. This corresponds to the simplest icosahedral lattice, with a triangulation number of 1 (57) Considering the I4S pentamers as capsomeres, there would be 12 capsomeres per virion. The location of the 6 (7P4) polypeptides in the capsid has not yet been established, but it is possible that they are distributed over the internal surface and in direct contact with the virion RNA (5Q, 39I see below). 

Icosahedral structures with triangulation numbers of T = 7 and 3 are indicated by three-dimensional image reconstruction of human wart virus and tomato bushy stunt virus, and a three-dimensional X-ray analysis at 30 A resolution of the bushy stunt virus is consistent with this interpretation. ... 

The numerical accuracy of simulations performed using this model is affected by several factors. These include a) the degree of triangulation, b) the number of marching steps taken along the flow direction and c) the order of the polynomial basis function. Numerical accuracy improves as a, b and c increase, however the computational time can become excessive. Therefore, it was necessary to quantitatively determine the effects of these variables on numerical accuracy. 

This method essentially requires triangulation of the surface and gives only approximate value of the curvatures due to the fitting character of their determination. In order to increase the accuracy a large number of parallel shifts is usually made which is sometimes computationally exhausting. 

The approach taken to come up with ATS production estimates is one of triangulation, estimating production based on reported seizures of the end products in combination with some assumptions of law enforcement effectiveness, seizure data of precursor chemicals and estimates based on the number of consumers and their likely levels of per capita consumption. The average of these three estimates is then used to arrive at UNODCs global estimates for amphetamine, methamphetamine and ecstasy production. The estimation procedure remained largely unchanged from the one used since the... 

Regardless of how the simulation is performed, the results must be analyzed. At present, only a limited number of cybertools are available for measuring the properties of simulations beyond merely recording the trajectories. Examples of such cybertools found routinely in many packages involve the calculation of correlation functions, triangulation of structure, Fourier transforms, clustering metrics, and informatics-based metrics. 

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