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We use PDB-SELECT databank [36] to construct our CLs, and use FSSP database [38] to derive the substitu- tion matrix CLESUM. Particularly, PDB-SELECT con- tains 1544 non-membrane proteins from PDB [37] with amino acid identity less than 25%. FSSP is based on ex- haustive all-against-all structure comparison of the rep- resentative protein structures, where the representative set contains no pair which has more than 25% sequence identity. A tree for the fold classification of the 2,860 rep- resentative set is constructed by a hierarchical clustering method based on the structural similarities. Family in- dices of the FSSP are obtained by cutting the tree at levels of 2, 4, 8, 16, 32 and 64 standard deviations above the database average. 

Four contiguous C$_{α}$ atoms, say i − 2, i − 1, i and i + 1, determine two bending angles θ, θ ′ and a torsion angle τ which is the dihedral angle between the two planes of tri- angles i − 2 , i − 1 , i and i − 1 , i, i +1 (see Fig. 1). By using a mixture model for the density distribution of the three 

## II. MATERIALS AND METHOD 

## B. Conformational letter and its substitution matrix 

## A. Datasets 

angles, the local structural states from PDB-SELECT have been clustered as 17 discrete states (see our previ- ous work [14] for more details). To use our SAs directly for the structural comparison, a score matrix similar as BLOSUM [32] for AAs is desired. In details, we first con- vert the structures of the representative set from FSSP to their CL strings; then collect all the pair alignments with the same first three family indices (DALI Z-Score ≥ 8) (see Fig. 2); finally count all ungapped aligned pairs of CLs to generate the substitution matrix, say CLESUM (Table I). The total number of structures is 10,047 pairs, consisting of 175,723 fragment pairs and 1,284,750 code pairs. 

From FSSP, which contains also the AA information, it is possible to construct a substitution matrix in the joint space of the structure and sequence. However, such matrix would have about (17 × 20) × (17 × 20) parameters (Fig. 3(A)). If we group the 20 AAs into two clus- ters, then the parameters of the matrix are reduced to (17 × 2) × (17 × 2). 

Generally, the mutual information I of two discrete 

## C. Entropic clustering 

