Natural topology

Consider a finite but structurally arbitrary lattice C, with geometry completely specified by the (NxN) adjacency matrix L  

Defining a distance function d i,j) = minpat/ii of links (i,j path) and a d -size neighborhood of H as = fc such that d i, k) d, we wish to study 

Given any lattice C and arbitrary transition rule we define a natural topology 

We are effectively tran.sforming functional identity into topological information he. different choices of rules correspond to different topologies and all rules are taken to Vre simple functions of local sums. 

The general form of the global transition function may then be written 

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Fig. 5.3 Natural topologies C,p for three one-dimensional additive rules ( , s sum modulo q). For an arbitrary acting on a given lattice C, is always (Bq aneigMoTsY...

The above-described compounds may already be considered as natural topological polymers. Synthetic compounds of a similar type (polycatenanes, polyrotaxancs) have as well been described in the literature however, not all the results may be re-... 

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

Knots nd Ca.tena.nes, Closed-circular DNA hehces can cross over one another three or more times to form topological knots. These stmctures are not common, but have been found to occur naturally in some bacteriophage DNAs. 

PMD color or the nature of the electron transitions produces the widest appHcation for PMDs. Depending on the polymethine chain length, the end-group topology, and the electron shell occupation, polymethines can absorb light in uv, visible, and near-ir spectral regions. 

Richardson, J.S. p-sheet topology and the relatedness of proteins. Nature 268 495-500, 1977. 

Figure S.7 The subunit structure of the neuraminidase headpiece (residues 84-469) from influenza virus is built up from six similar, consecutive motifs of four up-and-down antiparallel fi strands (Figure 5.6). Each such motif has been called a propeller blade and the whole subunit stmcture a six-blade propeller. The motifs are connected by loop regions from p strand 4 in one motif to p strand 1 in the next motif. The schematic diagram (a) is viewed down an approximate sixfold axis that relates the centers of the motifs. Four such six-blade propeller subunits are present in each complete neuraminidase molecule (see Figure 5.8). In the topological diagram (b) the yellow loop that connects the N-terminal P strand to the first P strand of motif 1 is not to scale. In the folded structure it is about the same length as the other loops that connect the motifs. (Adapted from J. Varghese et al.. Nature 303 35-40, 1983.)...

Figure S.IS Schematic diagram (a) and topology diagram (b) for the y-crystallin molecule. The two domains of the complete molecule have the same topology each is composed of two Greek key motifs that are joined by a short loop region, [(a) Adapted from T. Blundell et ah. Nature 289 171-777, 1981.]...

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