While every amino acid backbone can be specified by the same three dihedral angles, and , there is a far greater diversity among sidechain dihedrals, . At the extremes are glycine, which has no sidechain, and tryptophan, which has 12 dihedral angles. Our simulations do not modify dihedral angles which affect only hydrogen positions, or those involved in rings, so the number of dihedrals is significantly reduced. Both alanine and glycine have zero PGMC sidechain dihedrals (), while tryptophan, tyrosine, phenylalanine and histidine have only two, despite being very large sidechains. The values of for the common amino acids, excluding alanine and glycine, are given in Table . Although proline is a ring, we allow to vary while holding the atom fixed. This enables reasonable conformations of - to be sampled by modifying only a single dihedral, .
Table also lists the number of occurrences of each amino acid in the H64 dataset as well as the number of populated (non-zero) gridpoints at each spacing level. These numbers can be compared to the total number of gridpoints at each spacing level, listed in Table . It is clear that there is insufficient data for the multidimensional grids () at the fine spacings. For these cases, the number of populated gridpoints approaches the sample size. In other words, almost every conformation occupies a different gridpoint and the probability grid is extremely flat. This extreme variability is due primarily to the enormous number of possible conformations available for these structures (Table ), rather than unusual flexibility in these particular sidechains. The , distributions of these residues makes this more clear (see Table ): only lysine has an unusually large number of populated conformations when only and are considered.
One- and two-dimensional probability grids are shown in Figures and , respectively. It is not possible to show the higher dimensional grids in their entirety. These figures make it clear that there is a great deal of variety even among residues with the same number of significant dihedrals. One caveat about these grids: some of the 's actually have a period of 180, rather than 360 as shown. This arises when two branches are the same, as in Asp, where the two carboxylate oxygens are chemically identical but only one is labeled O and is used to specify . This labeling is not always done the same way, hence there are separate peaks at 150 and -30. Note that this does not affect our Monte Carlo simulations, since the orientations will be simulated identically with a total probability equal to the sum of the individual probabilities. The great variety in sidechain conformations can also be seen in Table , which lists the highest probability sidechain conformation for each amino acid.