Neighborhoods, sequence space

Figure 13. Neighborhood relations in sequence space. In A, B, and C we show neighborhoods of pair of sequences (/[, 12) with Hamming distances d(l, 2) of 1, 2, or 3, respectively (v = 6). Part drawn in thick lines is general connections in thin lines depend on chain length v. To give example, for v = 5 one connection of equivalent set has to be eliminated. (D) Assignment of numbers to individual sequences of sequence space for v = 5 shown as used in Figures 14-30.

The conformation an amino acid adopts depends on the residues in its neighborhood. This was made clear from a study of identical pentapeptides in unrelated proteins which were observed to adopt the same conformation only 20% of the time (24). The effect that residues neighboring in sequence or in space have on the conformation is not well understood. 

Next, we have infinite-dimensional spaces. The simplest examples are the set of infinite sequences of real numbers, TV°, and similarly C°° (but in a sense, they are actually the same space). We also have spaces which, although not identical to 7L°°, can be continuously and invertibly mapped onto TV°, at least in some neighborhood around any given point. 

The CNF procedure consists of a sequence of canonical (or equivalenfly symplectic) fransformations changing fhe phase space coordinafes z = (q, p) in a phase space neighborhood of fhe equilibrium poinf Zq. This is accomplished order by order according to the sequence... 

Figure 6 shows examples of the above operations and their impact on the packing. The above neighborhoods described are simple perturbations of the sequence pair, but they result in remarkably different packing when compared visually. Due to the non-linear space cost, we observe that shifting of vessels between different berths may provide improvement in the solution. This motivated us to define the next class of neighborhood structure. 

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