Figure 1A portrays the dependence of the average cluster size <S > (Eq. 3) on the number K_c of clusters. Actually, K_c determines the spatial resolution to view the universe of the 50-residue segments: With decreasing K_c, <S > increases because structurally different segments are fused into a cluster. The change of <S > was rapid for small K_c and slow for larger K_c.

The segments were generated by sliding a 50-residue window one residue by one residue along the domain sequences (see Methods). Consequently, two segments taken from the same protein domain with mutual adjacency in the sequence might have similar structures and might therefore be involved in a cluster. We did the following analysis to verify this possibility quantitatively: Presume that a cluster u involves n_m segments originated in a protein m. Subsequently, we introduced a quantity: , where the summation is taken over proteins that supply segment(s) to the cluster u, and N_p is the number of those proteins. Figure 1B presents a plot of the average of O_u as a function of K_c: . For K_c = 1000, <O > converged to 2.2. Consequently, a protein supplies only two or three segments to a cluster on average: i.e., a cluster does not contain excessive segments derived from a single protein for K_c ≥ 1000.

Figure 2 depicts the number (n_u) of segments involved in a cluster as a function of the cluster ordinal number for K_c = 1000. The decay of n_u is non-exponential. It is particularly interesting that even cluster #950 involves more than 100 segments, which means that the cluster comprises more than 40 (= 100/2.5) different proteins (<O > ≈ 2.5 for K_c = 1000). In the last 50 clusters, n_u decreased quickly. These clusters consist of randomly structured segments. Although segments were taken from all-α, all-β, α/β, and α+β SCOP classes, the structures can be random.

Figure 3 depicts <f >_Kc (Eq. 9) depending on K_c. The value of <f >_Kc was 0.60–0.65 for K_c ≥ 1000. The similarity threshold f₀ for assigning the inter-cluster linkage (Eq. 7) was 0.7. Figure 3 presents that the inter-residue similarity is compatible with the intra-cluster similarity.

The inter-cluster (inter-node) links were assigned to the K_c clusters according to the adjacency matrix a_uv. Directly connected clusters have mutually similar inter-residue contact patterns. Internal architectures of the networks were investigated by dividing the networks into communities (sub-networks) using Newman's method 44. In parallel, we projected the networks into a 3D space to obtain positions in the conformational space (see Additional file 1 for details). Although the clusters were embedded in the 3D space, the inter-cluster links were given to clusters that are mutually close in the full-dimensional space.

Each community was characterized by five biophysical structural features: the α, β, αβ secondary-structure elements, the radius of gyration, and the number of inter-residue contacts, denoted respectively as n_α, n_β, n_αβ, R_g, and N_contact. Then, the communities were classified into four types (α, β, αβ, and randomly structured communities) depending on the five structural features (see Methods for details).

Figure 4 portrays the 3D cluster distributions at K_c = 1000, 2000, and 3000, where a single color was assigned to a community depending on secondary-structure elements n_α, n_β, and n_αβ (see Additional file 1 for details). This figure clearly illustrates that the 3D cluster network is partitioned into four fold-regions (mainly α, mainly β, αβ, and randomly structured regions) independent of K_c, which respectively consist of α, β, αβ, and randomly structured communities. We termed this partitioning as "main partitioning". Figure 5 shows that the overall shape of the network adopted a three-leaf clover shape (mainly α, mainly β, and αβ regions surrounding the randomly structured region). We checked quantitatively whether the 3D distribution reflected the original full-dimensional distribution by calculating F-measure (see Additional file 1 for the definition of ). The value of was, respectively, 0.804 for K_c = 1000, 0.673 for K_c = 2000, and 0.593 for K_c = 3000. The large value of for K_c = 1000 indicates that the 3D cluster distribution fairly reflects the full-dimensional distribution. The value decreased concomitantly with increasing K_c. However, the three-leaf clover shape of the distribution was conserved at various K_c, which strongly suggests that the main partitioning exists in the 50-residue segments universe.

Figure 6 displays segment tertiary structures picked from clusters. This figure portrays that the structure classification by the five structural features correlates well with the visual secondary-structure constitution. Most segments originating in the all-α SCOP fold class were assigned to the α communities (see a-1 and a-2 in Figure 6). Those that originated in the all-β SCOP fold class were assigned to the β communities (see b-1 – b-3). The majority of segments taken from the α/β SCOP fold class were assigned to the αβ communities (see c-1 – c-4), although some were involved in other fold regions. In contrast, segments from the α+β SCOP fold class scattered to all the fold regions because the α+β proteins are a mixture of helices, strands, and randomly structured fragments, where the α and β secondary-structure elements are not necessarily neighbors to each other in the sequence. Consequently, the 50-residue segments from the α+β proteins can involve various structural features. The randomly structured region contained clusters with a few secondary-structure elements (see r-1 – r-4 in Figure 6). However, its polypeptide packing was loose, as portrayed in Figure 7, where the randomly structured clusters had large R_g.

The protein-domain universe is known to be an extremely biased distribution 845. Many studies have suggested a power-law statistic to represent the relation between the number of families and the number of folds 94647. For instance, Shakhnovich and co-workers created a protein-domain universe graph (PDUG) with adoption of a DALI Z-score for the similarity score, and showed that the domain universe followed a power-law distribution 9. Consequently, it is interesting to check if the currently produced network of the 50-residue segments follows the power law distribution.

First, we calculated the number (n_seg) of segments involved in each cluster. Figures 8A, B, and 8C portray the relation between n_seg and the number of clusters that respectively involve n_seg segments at K_c = 1000, 2000, and 3000. The distributions were symmetric (the value of skewness was 0.138 for K_c = 1000, 0.006 for K_c = 2000, and -0.066 for K_c = 3000) on the X-axis, log(n_seg), and far from the power-law statistics. Therefore, the currently obtained universe differs from those that have ever been reported. Additionally, we calculated the number (n'_seg) of segments involved in each community, and showed the relation between n'_seg and the number of communities involved n'_seg fragments for K_c = 1000, 2000, and 3000. We again obtained non-power-law statistics in the relation (data not shown).

Next, we calculated a connectivity distribution, P(k), of the networks to investigate details of the cluster network 48. The P(k) is defined as a distribution function of clusters that have k links to other clusters. Figures 9A, B, and 9C respectively present P(k) at K_c = 1000, 2000, and 3000. Subsequently, P(k) decays exponentially with increasing k. Therefore, these distributions are exponential ones (or possibly truncated power-law distributions). Consequently, non-power-law networks (i.e., non-scale-free networks) are again observed for the current networks.

We conducted modularity analysis to study cluster networks from another perspective. First, the networks were divided into communities (see Methods). A modularity Q_mod is an index to assess how well the network is divided into communities 49: 0 ≤ Q_mod ≤ 1. A network with a large Q_mod is characterized by numerous intra-community links and a few inter-community links. Figure 10A portrays the K_c dependence of Q_mod, which has the maximum at K_c = 200, indicating that the communities were highly isolated at K_c = 200. For K_c > 200, the communities were connected gradually by links, thereby decreasing Q_mod. For K_c ≥ 1000, Q_mod converged to a value (0.63), which indicates that the 50-residue segment network is characterized by high modularity.

We next calculated the number of communities at various K_c. We classified the communities into major and minor communities. Major ones are communities consisting of more than three clusters. Then, minor ones are small isolated communities consisting of only one or two clusters without links to other communities. No community involves only one cluster linked to another community. The K_c dependence of the number (N_com) of the major communities is presented in Figure 10B. The minor communities do not characterize the overall property of the network because only 10% of clusters belong to the minor communities at any K_c. The increment of N_com with increasing K_c was rapid for 100 ≤ K_c ≤ 1000 and slow for K_c ≥ 1000. The values of N_com were, respectively, 36, 38, and 38 at K_c = 1000, 2000, and 3000. This result shows that the number of communities was conserved for K_c ≥ 1000.

In addition to the analysis presented above, we checked to determine whether the contents (i.e., segments) involved in the communities are conserved with variation of K_c. Subsequently, we assigned a single color to communities common to the universes at K_c = 1000 (Figure 11A), 2000 (Figure 11B), and 3000 (Figure 11C). For instance, the majority of segments in the orange community of Figure 11A were involved in the orange ones in Figures 11B and 11C. Consequently, the communities are conserved well in the universes at different K_c. In other words, the network partitioning into communities is universal, independent of the spatial resolution (i.e., K_c). We termed this inter-community partitioning as "sub-partitioning", whereas the main partitioning is inter-regional partitioning (Figure 5).

We generated a structure library of 50-residue segments with reference to the all-α, all-β, α/β, and α+β fold classes defined in the SCOP database (release 1.69) 5. The SCOP database presents a list that provides a representative for each protein family. We selected tertiary structures of the representative domains from the PDB database 54 with elimination of multi-chain domains, those involving structurally undetermined regions, and those shorter than 50 residues. Furthermore, we eliminated domains consisting of 400 residues or more, which might involve structurally repeating units. Then we obtained 1803 domains (456 from all-α, 393 from all-β, 393 from α/β, and 561 from α+β). A domain that is n_r amino-acid residues long produces n_r - 49 segments from sliding a 50-residue window along the sequence one residue-by-one residue. Finally, we obtained an ensemble of 186 821 segments (32 040 from all-α, 39 375 from all-β, 63 177 from α/β, and 52 229 from α+β). The residue site of each segment was re-numbered from 1 to 50 in our study.

We classify the collected segments into clusters as follows: First, the inter-C_α atomic distances were calculated for segment s, where the distance between residues i and j is denoted as r_s(i, j). We eliminated residue pairs |i - j| < 3 because the distances for these pairs are similar for all segments. In other words, those distances have less sensitivity to discriminate the structural differences of segments. Then, the number (N_pair) of the C_α-atomic pairs in a 50-residue segment is 1128: N_pair = 1128. The set of distances is expressed as a N_pair-dimensional vector: = [r_s(1, 4), r_s(1, 5), ..., r_s(47, 50)]. We define the root mean square distance (rmsd_st) between and as in the N_pair-dimensional Cartesian space: .

For classifying the 186 821 segments into K_c clusters, we applied Lloyd's K-means algorithm 55 to the set of rmsd_st values, where s, t = 1, ..., 186821. One should set K_c in advance in the K-means algorithm. We examined various values for K_c (K_c ≤ 5000). In Lloyd's method, the K_c clusters are set randomly at the beginning. The finally converged clusters are output. We have checked that the main results are independent of the initial set of clusters.

We calculated the center () of a cluster u in the N_pair-dimensional space as , where the element is given as

The n_u is the number of constituent segments of the cluster u.

We defined a size S_u of the cluster u as

This equation simply quantifies the average distance from the cluster center to segments belonging to the cluster u in the N_pair-dimensional space. The average cluster size is defined simply as

where the summation is taken over all the K_c clusters.

In this subsection, we present computation of the inter-cluster and intra-cluster structural similarity based on the inter-residue contact patterns. The inter-residue contacts in segment s were defined as follows: Calculating all the inter-heavy atomic distances between residues i and j for the segment, their minimum distance was registered as the inter-residue distance q_s(i, j). Then, if q_s(i, j) < 6.0 Å, we judged that the residues i and j were contacting and set a quantity c_s(i, j) to 1 (otherwise, c_s(i, j) = 0). Here, we again eliminated residue pairs of |i - j| < 3 in the calculation of c_s(i, j). The set of c_s(i, j) constructs a matrix C_s, where element (i, j) is c_s(i, j).

The upper limit (6.0 Å) for q_s(i, j) allows no penetration of a water molecule between residues i and j: At q_s(i, j) = 6.0 Å, the substantial space for water penetration between the residues is approximately 2.0 Å (= 6.0 - 2 × 2.0) assuming that radii of segment heavy atoms are 2.0 Å. This space of 2.0 Å is smaller than the diameter of a water molecule (2.8 Å).

A structural similarity between segments s and t might be counted by comparing C_s and C_t. However, a strict comparison engenders an oversight of the similarity in the following case: Presume that c_s(i, j) = 1 and c_t(i,+ 1, j) = 0 in the segment s, and c_s(i, j) = 0 and c_t(i,+ 1, j) = 1 in segment t. The inter-residue contacts in these segments differ but they are similar. The strict comparison does not count such a similarity. To incorporate such similarity, smoothing of C_s was performed as

This smoothing (see Figure 13) was done only when residues i' and j' are not contacting and the residues i and j are contacting in the segment. If Eq. 4 produces a negative value, then c_s(i', j') is set to zero. If a non-contacting residue pair (i', j') has multiple values for c_s(i', j') attributable to contributions of some contacting pairs around (i', j'), then the largest value is assigned to the non-contacting pair. As described in this paper, the inter-residue contact matrix C_s indicates that after the smoothing.

Here, we calculate the contact patterns which are specific to a cluster. For this purpose, we averaged C over the entire segment library and over all segments in cluster u. We denote these averaged matrices as and , respectively. Then, we defined a quantity , where element (i, j) is denoted as . The similarity between clusters u and v was measured using the following correlation coefficient:

where

The term in Eq. 5 is defined by setting u = v in Eq. 6, and the term by setting = 1. A large correlation coefficient indicates similar inter-residue contact patterns between the clusters.

The coefficient is useful as a distance between clusters u and v in a multi dimensional space. Consequently, the set of coefficients define a multi-dimensional weighted graph (i.e., weighted network). In this work, we must convert this weighted graph into an un-weighted one to perform community analysis, which only deals with the un-weighted graph. Therefore, we introduce an adjacency matrix a_uv in which element (u, v) is given as follows.

The inter-residue contact patterns are similar between clusters u and v only when . Herein, we set f₀ to 0.7. The meaning of 0.7 is explained in the Results section.

We next assessed the intra-cluster similarity. First, we defined a quantity for a segment s, where element (i, j) of ΔC_s is denoted as ΔC_s(i, j). Then, we averaged ΔC_s(i, j) over the segments in cluster u:

We define a matrix G_u for that the element (i, j) as g_u(i, j). Then, we calculated the correlation coefficient f(G_u, ΔC_s) between G_u and ΔC_s for segments in cluster u, using the same definition as that in Eq. 5. Subsequently, we calculated an averaged correlation coefficient <f >_u over f(G_u,ΔC_s) of the segments in the cluster u. This quantity is a measure to express the similarity of the inter-residue contact patterns among the segments in cluster u. Finally, <f >_u was averaged over all clusters.

The larger the value of , the more similar the inter-residue contact patterns in each cluster are, on average.

We constructed a distribution (i.e., fold universe) of K_c clusters in a 3D conformational space with embedding clusters into the 3D. Details are presented in Additional file 1. As explained in the Introduction, lowering of the space dimensionality hides the internal architecture of the fold universe. To compensate the full-dimensional information to the 3D distribution, links were assigned to clusters with similar inter-residue contact patterns (a_uv = 1). The generated networks were subjected to the modularity analysis described in the next subsection.

To investigate a property of the cluster network, we divided the network into communities (i.e., sub-networks) using an efficient method 44. An example of a network is presented in Figure 14, where two communities (Com 1 and Com 2) exist. A modularity Q_mod is an index to assess how well the network is divided into communities 49:

where I_w is the number of links connecting clusters within a community w, N_com is the number of communities existing in the entire network, and I is the number of links existing in the entire network. The quantity d_w is called the "total degree", which is defined for each community as d_w = 2I_w + I_w-other, where I_w-other is the number of links connecting clusters in the community w and clusters outside the community. The value of Q_mod is 0–1: Q_mod approaches 1 when the number of links connecting different communities decreases. For instance, the network in Figure 14A has Q_mod of 0.466 (I = 34, I₁ = 18, I₂ = 15, d₁ = 37, and d₂ = 31). That of Figure 14B has Q_mod of 0.388 (I = 37, I₁ = 18, I₂ = 15, d₁ = 40, and d₂ = 34). The two networks are equivalent except for the inter-community links.

The manner of differentiating the communities is important. Herein, we characterize the communities depending on five biophysical structural features: radius of gyration (R_g), number of inter-residue contacts ( with removal of pairs of |i - j| < 3), number of α-helical residues (n_α), number of β-helical residues (n_β), and the sum of n_α and n_β (i.e., n_αβ = n_α + n_β).

First, we calculate the five quantities for each segment. The secondary-structure assignment to each residue in a segment is done using software available at the STRIDE web site http://webclu.bio.wzw.tum.de/stride/56. Next, we took the average for each of the five quantities over segments in a community. We designate the average quantities in a community w as R_g(w), N_contact(w), n_α(w), n_β(w), and n_αβ(w). Then, we classify the communities into α, β, αβ, and randomly structured ones according to the five quantities: Randomly structured communities are those with R_g > 14 Å and N_contact(w) < 100 or those with n_αβ(w) < 15. In the remaining communities, α communities are those with n_α(w) > 0.7 × n_αβ(w). In the remaining communities, β communities are those with n_α(w) > 0.7 × n_αβ(w). The finally remaining communities are classified as αβ communities. Each segment in the αβ communities significantly involves both an α helix and a β strand.