By Ross Kindermann

The learn of Markov random fields has introduced interesting new difficulties to chance idea that are being constructed in parallel with simple research in different disciplines, such a lot particularly physics. The mathematical and actual literature is frequently particularly technical. This e-book goals at a extra light creation to those new components of study.

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**Sample text**

These assumptions limit the number of possible, roughly spherical shell structures, each one containing twelve pentagonal units and a certain number of hexagonal units. To catalog the possible geometries, Caspar and Klug defined a number, T, which corresponds to the number of coat proteins at each corner of a triangular face of the shell. Thus, T = 1 for the shell of the satellite tobacco necrosis virus, and T = 3 for the poliovirus shell. In this virus shell model, the only T numbers allowed are 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, and so on.

One can also imagine the structure as being made up of five proteins gathered at each of the twelve corners, or vertices, of an icosahedron, as is shown in the figure below. In effect, its surface can be thought of as consisting of twelve protein pentagons. Larger shells have additional protein units at the corners of their triangular faces. For example, the poliovirus shell consists of 180 coat proteins, with three proteins in each corner, for a total of nine on each face. This structure can also be pictured as consisting of twelve groups of five proteins each at the twelve vertices of an icosahedron and twenty groups of six proteins each at the center of each of the faces.

In recent years, a small group of mathematicians has pioneered a novel perspective on virus self-assembly— how structural order emerges out of randomness in the microcellular realm. This research suggests that sets of simple rules, which define the way proteins stick together, automatically lead to the kinds of virus structures that biologists observe under their electron microscopes. 3 micrometers in size, viruses have highly regular structures; often they look like mineral crystals with flat faces, distinct angles, and definite edges.