- Research article
- Open Access

# Towards an automatic classification of protein structural domains based on structural similarity

- Vichetra Sam
^{1}, - Chin-Hsien Tai
^{2}, - Jean Garnier
^{1, 3}, - Jean-Francois Gibrat
^{3}, - Byungkook Lee
^{2}and - Peter J Munson
^{1}Email author

**9**:74

https://doi.org/10.1186/1471-2105-9-74

© Sam et al; licensee BioMed Central Ltd. 2008

**Received: **18 July 2007

**Accepted: **31 January 2008

**Published: **31 January 2008

## Abstract

### Background

Formal classification of a large collection of protein structures aids the understanding of evolutionary relationships among them. Classifications involving manual steps, such as SCOP and CATH, face the challenge of increasing volume of available structures. Automatic methods such as FSSP or Dali Domain Dictionary, yield divergent classifications, for reasons not yet fully investigated. One possible reason is that the pairwise similarity scores used in automatic classification do not adequately reflect the judgments made in manual classification. Another possibility is the difference between manual and automatic classification procedures. We explore the degree to which these two factors might affect the final classification.

### Results

We use DALI, SHEBA and VAST pairwise scores on the SCOP C class domains, to investigate a variety of hierarchical clustering procedures. The constructed dendrogram is cut in a variety of ways to produce a partition, which is compared to the SCOP fold classification.

Ward's method dendrograms led to partitions closest to the SCOP fold classification. Dendrogram- or tree-cutting strategies fell into four categories according to the similarity of resulting partitions to the SCOP fold partition. Two strategies which optimize similarity to SCOP, gave an average of 72% true positives rate (TPR), at a 1% false positive rate. Cutting the largest size cluster at each step gave an average of 61% TPR which was one of the best strategies not making use of prior knowledge of SCOP. Cutting the longest branch at each step produced one of the worst strategies.

We also developed a method to detect irreducible differences between the best possible automatic partitions and SCOP, regardless of the cutting strategy. These differences are substantial. Visual examination of hard-to-classify proteins confirms our previous finding, that global structural similarity of domains is not the only criterion used in the SCOP classification.

### Conclusion

Different clustering procedures give rise to different levels of agreement between automatic and manual protein classifications. None of the tested procedures completely eliminates the divergence between automatic and manual protein classifications. Achieving full agreement between these two approaches would apparently require additional information.

## Keywords

- True Positive Rate
- Pairwise Similarity
- Hierarchical Cluster Method
- Homogeneous Cluster
- Span Cluster

## Background

This work investigates a large number of different methods for producing an automatic classification of structural domains of proteins based on all-against-all pairwise structural similarity scores. We produce candidate classifications and compare them to the human expert-curated classification SCOP [1]. Perhaps the earliest attempt at automatic classification of protein structures was by Holm and Sander [2], who produced the FSSP database, consisting primarily of a tree obtained by applying hierarchical clustering using the pairwise structural similarity scores among a sequence representative set [3]. Later, Dietmann et al [4] refined FSSP by introducing the notion of domain [5]. More recent work by Gewehr *et* al. [6] and others [7–10], focused on the problem of assigning new, unclassified domains into their correct pre-established fold classes. Other authors [11–15] compared a set of pairwise similarity measures computed by various structure comparison methods to the SCOP (or other) pre-existing classification, to investigate the causes of divergence between automatically determined pairwise structural similarity and expert-curated classification. In particular, the comparison between FSSP, SCOP and CATH, by Hadley and Jones [12], makes use of FSSP pairwise scores between protein chains which are present in all three databases, as FSSP does not use domain as the classification unit. Although these studies advanced the understanding of divergence between automatic similarity and expert-curated classification, improvement of automatic classification procedures was not their primary objective.

Our earlier work [15] pointed to structural variations within SCOP folds as a main cause of divergence between automatic and expert-curated classifications. Intra-fold structural variation affects the measured structural similarity both within and between folds. The measured similarities are often not uniform among domains within a fold and the average similarity among domains within one fold can be different from that within another fold. We investigate different partitioning strategies in an attempt to accommodate such uneven similarity distributions. In particular, we investigate dendrogram cutting strategies as a potential means of isolating both tightly clustered, homogeneous folds and more heterogeneous ones, using a single procedure.

One source of potential confusion in the literature is the failure to explicitly consider the fundamental difference between mathematical properties of a similarity score matrix compared to a classification or partition. Mathematically, each can be represented as an M by M matrix where M is the number of protein domains. Unlike a pairwise similarity score matrix which can be any square matrix, a "partition matrix" is made of 0 or 1 elements indicating when two domains are in the same (1) or different (0) clusters of the partition. The partition matrix can always be transformed into a block-diagonal form by sorting the rows and columns appropriately, reflecting a property related to the definition of a partition itself. A partition of a set of domains is comprised of non-overlapping clusters, meaning that if A and B are in the same cluster, while B and C are also in the same cluster, then A and C are necessarily in that same cluster, a type of transitivity here called the "partition constraint". This property may be absent in a pairwise similarity matrix.

Building a classification by clustering based on pairwise similarity scores is essentially the same thing as transforming the pairwise similarity matrix into one which satisfies the partition constraint. Clearly, this process may force some domain pairs originally considered as dissimilar into the same cluster while other, similar pairs would be forced into different clusters. In many studies comparing the automatically determined pairwise similarities and a structural classification database [11–15], the effect of the presence of the partition constraint was not explicitly considered. Some of the reported discrepancy between similarity scores and classifications may potentially result from the failure of the similarity scores to satisfy this constraint. In the present study, we eliminate this factor by first converting the pairwise similarity dataset into a partition (via clustering) and then comparing the two partitions.

The derivation of a partition from a set of pairwise similarities is not a trivial process, but involves several distinct steps of computations, requiring careful analysis. Here, particular attention will be given to hierarchical clustering methods and dendrogram cutting strategies. We obtain a partition from pair-wise similarity scores by first transforming the similarity measure into a distance measure, applying various standard hierarchical clustering methods, obtaining a dendrogram, or binary cluster joining tree, and in the final step, applying a strategy which removes the root node and successively lower level nodes leaving behind a set of trees corresponding to the clusters in the partition. The pairwise similarity scores are provided by VAST [15, 16], SHEBA [17] and DALI [18, 19], three distinctive and efficient approaches for measuring structural similarity.

## Results

### Performance of hierarchical clustering as tree building methods

_{01}) and the number of clusters in each partition, resulting from these various partitioning approaches, using the 1% FPR termination criterion, are reported in Tables 1 and 2.

TPR values at 1% FPR (TPR_{01}) for Hierarchical Cluster Methods and Tree Cutting Strategies.

Group | Tree Cutting Strategy | Hierarchical Cluster Method | VAST Nres | VAST Pcli | SHEBA Nres | SHEBA Zscore | DALI Nres | DALI Zscore | Average | Range |
---|---|---|---|---|---|---|---|---|---|---|

2 | Level | Average | 28 | 24 | 23 | 33 | 28 | 38 | 29 | 15 |

2 | Level | Complete | 28 | 31 | 29 | 40 | 39 | 27 | 32 | 13 |

2 | Level | Single | 14 | 10 | 12 | 21 | 16 | 8 | 13 | 13 |

2 | Level | Ward | 50 | 48 | 48 | 54 | 50 | 49 | 50 | 6 |

3 | Largest Size | Average | 54 | 50 | 48 | 59 | 55 | 41 | 51 | 18 |

3 | Largest Size | Complete | 44 | 44 | 43 | 43 | 50 | 51 | 46 | 8 |

3 | Largest Size | Single | 29 | 22 | 27 | 39 | 34 | 13 | 27 | 26 |

3 | Largest Size | Ward | 61 | 59 | 61 | 58 | 62 | 65 | 61 | 7 |

3 | Tree Completeness | Ward | 57 | 59 | 61 | 56 | 56 | 63 | 58 | 7 |

3 | Highest tree | Ward | 57 | 61 | 62 | 56 | 57 | 63 | 59 | 7 |

1 | Tree skewness | Ward | 25 | 15 | 26 | 22 | 31 | 21 | 23 | 10 |

1 | Longest branch | Ward | 29 | 20 | 26 | 22 | 27 | 13 | 23 | 16 |

2 | Maximum Entropy | Ward | 45 | 40 | 43 | 46 | 58 | 48 | 46 | 18 |

1 | BestTPR | Ward | 12 | 13 | 13 | 10 | 8 | 16 | 12 | 8 |

4 | Best ratio TPR &FPR | Ward | 73 | 72 | 73 | 71 | 75 | 75 | 73 | 4 |

4 | MI | Ward | 73 | 71 | 72 | 71 | 76 | 76 | 73 | 5 |

Direct pw | none | 50 | 52 | 60 | 61 | 52 | 67 | 57 | 17 |

Number of Clusters for Hierarchical Cluster Methods and Tree Cutting Strategies at 1%FPR.

Group | Tree Cutting Strategy | Hierarchical Cluster Method | VAST Nres | VAST Pcli | SHEBA Nres | SHEBA Zscore | DALI Nres | DALI Zscore |
---|---|---|---|---|---|---|---|---|

2 | Level | Average | 460 | 573 | 576 | 402 | 444 | 389 |

2 | Level | Complete | 307 | 250 | 332 | 195 | 221 | 396 |

2 | Level | Single | 810 | 1023 | 921 | 686 | 787 | 1041 |

2 | Level | Ward | 110 | 120 | 138 | 106 | 113 | 121 |

3 | Largest Size | Average | 183 | 210 | 222 | 190 | 169 | 256 |

3 | Largest Size | Complete | 81 | 84 | 88 | 94 | 69 | 82 |

3 | Largest Size | Single | 515 | 658 | 528 | 420 | 439 | 792 |

3 | Largest Size | Ward | 71 | 79 | 81 | 71 | 67 | 72 |

3 | Tree Completeness | Ward | 95 | 89 | 87 | 88 | 101 | 96 |

3 | Highest tree | Ward | 88 | 80 | 84 | 81 | 92 | 91 |

1 | Tree skewness | Ward | 603 | 701 | 603 | 592 | 606 | 641 |

1 | Longest node | Ward | 409 | 704 | 627 | 837 | 646 | 991 |

2 | Maximum Entropy | Ward | 143 | 146 | 109 | 101 | 81 | 109 |

1 | BestTPR | Ward | 1121 | 1112 | 1054 | 1106 | 1185 | 1029 |

4 | Best ratio TPR &FPR | Ward | 59 | 65 | 70 | 61 | 57 | 63 |

4 | MI | Ward | 105 | 76 | 72 | 68 | 55 | 77 |

Trees generated by Ward's method result in better performance. The TPR_{01} values obtained for Ward's clustering method are, with one exception, always higher than for the Average, Complete or Single linkage method. The simplest, level cut strategy with Ward's method achieves an average TPR_{01} of 50% across all six structure comparison scores. Complete or Average linkage clustering constitute the next best alternatives to Ward's method, with average TPR_{01} across all similarity scores and structure comparison methods, of 32% and 29% respectively. Single linkage clustering shows uniformly the most divergence from SCOP. Not only does Ward's method achieve the highest average TPR_{01} value, but the values vary less across different similarity scores and structure comparison methods, than do those for Complete, Average and Single Linkage, suggesting that Ward's is more satisfactory for this application.

With other tree cutting strategies the trend among various clustering methods is the same. For the largest size cut strategy, Ward's method gives an average TPR_{01} value of 61%, at least 10% better than for the three other clustering methods. Again, Complete and Average Linkage give similar average TPR_{01} values, while Single Linkage is very low in comparison. In view of the clearly superior results with Ward's method, we investigate tree cutting strategies using that method alone.

### Performance of tree cutting strategies

_{01}values (Figure 1 and Table 1), and are represented by the longest branch cut (1), level cut (2), largest size cut (3) and mutual information cut (4) strategies. Figure 1 illustrates the comparative performance among these representative tree cutting strategies, for the VAST number of matched residues similarity score.

Among the four SCOP-independent tree cutting strategies shown, "largest size cut" achieves the best performance with its ROC curve dominating all other SCOP-independent ROC curves. Moreover, "largest size cut" ROC curve is the closest to that for Mutual Information cut, a strategy which attempts to maximize the agreement of the partition with SCOP at each step. Similar results (data not shown) were seen for SHEBA and DALI similarity scores.

The longest branch cut group (Table 1, Group 1) contains the strategies of lowest performance, e.g. tree skewness strategy, as indicated by their respective mean TPR_{01} values. Longest branch cut strategy results in one of the lowest agreements with SCOP, in particular for FPR values below 30% (Figure 1). Above this FPR level, its performance becomes more acceptable relatively to the upper bound given by the mutual information cut. This strategy seems to make more sense when inter-cluster distance is high, i.e. for highest nodes of the tree. But it behaves much worse than others when the inter-cluster distance is reduced, as its ROC curve is closer to the main diagonal than the ROC for other strategies. In the lowest part of the tree, inter-cluster distances vary only slightly so that the longest branch criterion for choosing among the many possibilities, the next sub-tree to be cut might not be discriminative enough.

Level cut and maximum entropy cut strategies form another group (Table 1, Group 2) characterized by middling performance. Their grouping is explained by the fact that these strategies perform comparably at low FPR values and their average TPR_{01} values are comparable. Higher variability among the TPR values for maximum entropy cut strategy is noted, however. Indeed, for VAST and SHEBA, level cut strategy is better than maximum entropy cut, as a rule, with DALI number of matched residues (Nres) score the only exception. The highest TPR_{01} for the maximum entropy cut strategy (58%) is obtained with the Nres metric, while the lowest value (40%) is obtained by VAST Pcli.

The third, largest size cut strategy group (Table 1, Group 3) also includes the tree completeness cut, and highest tree cut strategies and corresponds to SCOP-independent strategies of highest performance, with average TPR_{01} values ranging from 58 to 61%. The tree height and tree completeness are closely related to the number of leaves, i.e. the size of the tree. These results indicate that of these three topological properties, the size is a better measure, for the purpose of creating partitions that agree with SCOP. This is true for partitions of small size (the large FPR values), but also when the number of clusters is large (the small FPR values).

The fourth group (Table 1, Group 4) is made up of mutual information cut and Best TPR/FPR Ratio cut strategies, all of which make direct use of the SCOP fold partition and thus are not useful for an independent classification. The poor performance of the best TPR cut strategy (Table 1, Group 1) indicates that strategies which ignore the false positives will not generate SCOP-like partitions.

The size of each partition is reported in Table 2, and shows a large variation in the number of clusters depending on the hierarchical clustering method. Single linkage always leads to partitions with a high number of clusters, in fact close to the total number of domains M, when FPR is kept low at 1%.

### Comparison of partitions

Distance between partitions*, (Δ-distance in 1000s)

Partition | VNS | SNS | DNS | VPS | SZS | DZS | VNL | SNL | DNL | VPL | SZL | DZL | SCOP |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

VNS | 0 | 30 | 24 | 31 | 30 | 28 | 27 | 46 | 37 | 47 | 41 | 51 | 65 |

SNS | 30 | 0 | 24 | 35 | 25 | 23 | 44 | 32 | 38 | 51 | 38 | 51 | 65 |

DNS | 24 | 24 | 0 | 31 | 21 | 22 | 39 | 42 | 25 | 48 | 37 | 48 | 66 |

VPS | 31 | 35 | 31 | 0 | 37 | 32 | 43 | 50 | 43 | 28 | 47 | 56 | 71 |

SZS | 30 | 25 | 21 | 37 | 0 | 27 | 42 | 43 | 37 | 50 | 28 | 51 | 69 |

DZS | 28 | 23 | 22 | 32 | 27 | 0 | 44 | 47 | 36 | 49 | 42 | 41 | 63 |

VNL | 27 | 44 | 39 | 43 | 42 | 44 | 0 | 48 | 37 | 49 | 37 | 51 | 75 |

SNL | 46 | 32 | 42 | 50 | 43 | 47 | 48 | 0 | 41 | 56 | 47 | 49 | 84 |

DNL | 37 | 38 | 25 | 43 | 37 | 36 | 37 | 41 | 0 | 52 | 39 | 53 | 73 |

VPL | 47 | 51 | 48 | 28 | 50 | 49 | 49 | 56 | 52 | 0 | 54 | 56 | 83 |

SZL | 41 | 38 | 37 | 47 | 28 | 42 | 37 | 47 | 39 | 54 | 0 | 56 | 70 |

DZL | 51 | 51 | 48 | 56 | 51 | 41 | 51 | 49 | 53 | 56 | 56 | 0 | 91 |

SCOP | 65 | 65 | 66 | 71 | 69 | 63 | 75 | 84 | 73 | 83 | 70 | 91 | 0 |

Δ-distance*, in 1000s, between partitions within each method and minimum Δ-distance from each method to SCOP

Method | Average Distance◊ | Maximum Distance◊ | Average Distance to SCOP | Minimum Distance to SCOP |
---|---|---|---|---|

1 VAST | 43 | 56 | 73 | 65 |

2 SHEBA | 35 | 47 | 72 | 65 |

3 DALI | 37 | 53 | 73 | 63 |

4 Size | 28 | 37 | 66 | 63 |

5 Level | 48 | 59 | 80 | 70 |

Distance between Automatic Partitions and SCOP fold Partition

Automatic Partition | Maximum Δ-distance to Others* | Δ-distance to SCOP* | FPR (%) | TPR (%) |
---|---|---|---|---|

| 51 | 65 | 0.5 | 53 |

| 51 | 65 | 0.7 | 59 |

| 48 | 66 | 0.5 | 54 |

| 56 | 71 | 0.8 | 55 |

| 51 | 69 | 0.5 | 51 |

| 49 | 63 | 0.6 | 60 |

| 51 | 75 | 1.5 | 54 |

| 56 | 84 | 1.6 | 55 |

| 53 | 73 | 1.1 | 56 |

| 56 | 83 | 1.6 | 53 |

| 56 | 70 | 1.1 | 56 |

| 56 | 91 | 1.9 | 54 |

### Comparison of cluster size distributions

We observe first that all partitions, including SCOP, have lower median cluster size and greater spread of size than for random. There is evidently sufficient signal strength within the similarity score matrix to influence the size distribution. Second, the 75th percentiles for automated methods tend to be larger than for SCOP, while the SCOP distribution shows larger positive skewness, with a greater number of unusually large clusters. Third, there is some uniformity in the size distribution within tree cutting strategies, with largest size cut showing somewhat higher 75th percentiles and less skewness than do the level cut distributions. Level cut distributions are closer the SCOP distribution in terms of median, 75th percentile size and the larger number of outliers, than the largest size cut distributions.

The largest SCOP fold (c.1), is in fact, split by all strategies and methods as none include a cluster with 182 domains. Largest size cut strategy intentionally eliminates outliers of large size, thereby creating more clusters of intermediate sizes, with greater spread (inter-quartile range) than for level cut. The level cut generates a few large size outliers but the clusters are smaller, typically. This is consistent with the observation made earlier that the level cut behaves like the maximum entropy cut at small FPR ranges. Using partitions of 94 clusters, we find FPR values higher than 1% for level cut strategy but lower than 1% for largest size cut strategy (Table 5), indicating that a trade-off must be made in matching the size distribution as well as maximizing the TPR in relationship to the SCOP partition.

### Identification of dispersed folds

Excess Span and Largest Homogeneous Cluster Size for SCOP C Class folds.

Excess Span Size | Largest Homog. Cluster (%) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Row | fold | SCOP fold description | Ndom* | Nfam | VAST | SHEBA | DALI | INT◊ | VAST | SHEBA | DALI | Nc |

CONSISTENT FOLDS | ||||||||||||

1 | c.100 | Thiamin pyrophosphokinase, catalytic domain | 2 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

2 | c.101 | Undecaprenyl diphosphate synthase | 2 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

3 | c.106 | SurE-like | 2 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

4 | c.109 | PEP carboxykinase N-terminal domain | 4 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

5 | c.114 | YchN-like | 2 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

6 | c.117 | Amidase signature (AS) enzymes | 3 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

7 | c.16 | Lumazine synthase | 3 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

8 | c.17 | Caspase-like | 3 | 2 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

9 | c.22 | Ribosomal protein L4 | 2 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

10 | c.27 | Nucleoside phosphorylase/phosphoribosyltransferase catalytic domain | 3 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

11 | c.28 | Cryptochrome/photolyase, N-terminal domain | 4 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

12 | c.32 | Tubulin, GTPase domain | 2 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

13 | c.33 | Cysteine hydrolase | 3 | 2 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

14 | c.34 | DFP DNA/pantothenate metabolism flavoprotein | 2 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

15 | c.36 | Thiamin diphosphate-binding fold (THDP-binding) | 19 | 4 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

16 | c.39 | Nicotinate mononucleotide:5,6-dimethylbenzimidazole phosphoribosyltransferase (CobT) | 2 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

17 | c.42 | Arginase/deacetylase | 3 | 2 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

18 | c.43 | CoA-dependent acyltransferases | 7 | 3 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

19 | c.5 | MurCD N-terminal domain | 2 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

20 | c.50 | Leucine aminopeptidase (Aminopeptidase A), N-terminal domain | 2 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

21 | c.59 | MurD-like peptide ligases, peptide-binding domain | 5 | 2 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

22 | c.6 | Cellulases | 3 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

23 | c.62 | vWA-like | 8 | 2 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

24 | c.65 | Formyltransferase | 3 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

25 | c.67 | PLP-dependent transferases | 35 | 6 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

26 | c.68 | Nucleotide-diphospho-sugar transferases | 15 | 12 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

27 | c.7 | PFL-like glycyl radical enzymes | 4 | 4 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

28 | c.70 | Nucleoside hydrolase | 2 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

29 | c.71 | Dihydrofolate reductases | 8 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

30 | c.73 | Carbamate kinase-like | 2 | 2 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

31 | c.74 | AraD-like aldolase/epimerase | 3 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

32 | c.76 | Alkaline phosphatase-like | 6 | 3 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

33 | c.77 | Isocitrate/Isopropylmalate dehydrogenases | 4 | 2 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

34 | c.79 | Tryptophan synthase beta subunit-like PLP-dependent enzymes | 6 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

35 | c.81 | Formate dehydrogenase/DMSO reductase, domains 1–3 | 5 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

36 | c.83 | Aconitase iron-sulfur domain | 2 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

37 | c.86 | Phosphoglycerate kinase | 2 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

38 | c.89 | Phosphofructokinase | 2 | 1 | 0 | 0 | 0 | 0 | 100 | 100 | 100 | 1 |

MODERATELY CONSISTENT FOLDS | ||||||||||||

39 | c.25 | Ferredoxin reductase-like, C-terminal NADP-linked domain | 12 | 5 | 643 | 0 | 0 | 0 | 92 | 100 | 100 | 1 |

40 | c.61 | PRTase-like | 17 | 2 | 0 | 0 | 68 | 0 | 100 | 100 | 88 | 1 |

41 | c.82 | ALDH-like | 7 | 2 | 0 | 954 | 0 | 0 | 100 | 86 | 100 | 1 |

42 | c.87 | UDP-Glycosyltransferase/glycogen phosphorylase | 6 | 6 | 0 | 937 | 0 | 0 | 100 | 83 | 100 | 1 |

43 | c.116 | alpha/beta knot | 5 | 3 | 3 | 0 | 0 | 0 | 80 | 100 | 100 | 1 |

44 | c.41 | Subtilisin-like | 4 | 2 | 0 | 0 | 1056 | 0 | 100 | 100 | 75 | 1 |

45 | c.98 | MurF and HprK N-domain-like | 4 | 2 | 8 | 0 | 0 | 0 | 75 | 100 | 100 | 1 |

46 | c.46 | Rhodanese/Cell cycle control phosphatase | 7 | 3 | 130 | 54 | 0 | 0 | 86 | 86 | 100 | 1 |

47 | c.57 | Molybdenum cofactor biosynthesis proteins | 3 | 2 | 1056 | 0 | 0 | 0 | 67 | 100 | 100 | 1 |

48 | c.93 | Periplasmic binding protein-like I | 13 | 1 | 3 | 0 | 0 | 0 | 62 | 100 | 100 | 1 |

49 | c.45 | (Phosphotyrosine protein) phosphatases II | 12 | 2 | 0 | 0 | 267 | 0 | 100 | 100 | 58 | 1 |

50 | c.107 | DHH phosphoesterases | 2 | 2 | 0 | 0 | 1058 | 0 | 100 | 100 | 50 | 1 |

51 | c.18 | DNA glycosylase | 2 | 2 | 1057 | 0 | 0 | 0 | 50 | 100 | 100 | 1 |

52 | c.108 | HAD-like | 11 | 10 | 1048 | 932 | 0 | 0 | 55 | 91 | 100 | 1 |

53 | c.10 | Leucine-rich repeat, LRR (right-handed beta-alpha superhelix) | 18 | 10 | 304 | 0 | 0 | 0 | 44 | 100 | 100 | 1 |

54 | c.48 | TK C-terminal domain-like | 5 | 3 | 50 | 0 | 47 | 0 | 60 | 100 | 80 | 1 |

55 | c.14 | ClpP/crotonase | 11 | 3 | 240 | 335 | 0 | 0 | 73 | 64 | 100 | 1 |

56 | c.24 | Methylglyoxal synthase-like | 3 | 3 | 450 | 0 | 27 | 0 | 67 | 100 | 67 | 1 |

57 | c.8 | The "swivelling" beta/beta/alpha domain | 10 | 8 | 0 | 28 | 4 | 0 | 100 | 80 | 40 | 1 |

58 | c.63 | CoA transferase | 5 | 2 | 0 | 1 | 1 | 0 | 100 | 60 | 60 | 1 |

59 | c.55 | Ribonuclease H-like motif | 53 | 19 | 400 | 362 | 0 | 0 | 94 | 21 | 100 | 1 |

60 | c.91 | PEP carboxykinase-like | 4 | 2 | 449 | 0 | 80 | 0 | 50 | 100 | 50 | 1 |

61 | c.95 | Thiolase-like | 13 | 2 | 4 | 0 | 4 | 0 | 46 | 100 | 46 | 1 |

DISPERSED FOLDS | ||||||||||||

62 | c.47 | Thioredoxin fold | 51 | 13 | 1 | 1 | 1 | 0 | 45 | 82 | 82 | 2 |

63 | c.12 | Ribosomal proteins L15p and L18e | 2 | 1 | 1 | 3 | 1 | 0 | 50 | 50 | 50 | 2 |

64 | c.31 | DHS-like NAD/FAD-binding domain | 10 | 5 | 4 | 3 | 4 | 3 | 70 | 70 | 70 | 2 |

65 | c.1 | TIM beta/alpha-barrel | 182 | 70 | 3 | 1148 | 1148 | 3 | 53 | 43 | 62 | 6 |

66 | c.15 | BRCT domain | 7 | 4 | 130 | 12 | 4 | 2 | 71 | 86 | 86 | 2 |

67 | c.13 | SpoIIaa-like | 3 | 2 | 10 | 4 | 7 | 3 | 67 | 67 | 67 | 2 |

68 | c.30 | PreATP-grasp domain | 11 | 5 | 4 | 6 | 7 | 3 | 64 | 36 | 45 | 3 |

69 | c.51 | Anticodon-binding domain-like | 13 | 6 | 6 | 35 | 7 | 0 | 54 | 69 | 77 | 2 |

70 | c.3 | FAD/NAD(P)-binding domain | 46 | 5 | 7 | 1015 | 1014 | 7 | 41 | 41 | 41 | 8 |

71 | c.84 | Phosphoglucomutase, first 3 domains | 6 | 1 | 10 | 94 | 266 | 10 | 67 | 67 | 67 | 2 |

72 | c.4 | Nucleotide-binding domain | 7 | 3 | 46 | 18 | 14 | 14 | 57 | 29 | 29 | 3 |

73 | c.9 | Barstar-like | 2 | 2 | 18 | 413 | 593 | 17 | 50 | 50 | 50 | 2 |

74 | c.19 | FabD/lysophospholipase-like | 3 | 2 | 1056 | 940 | 40 | 40 | 67 | 67 | 67 | 2 |

75 | c.53 | Resolvase-like | 9 | 3 | 444 | 44 | 65 | 16 | 44 | 44 | 44 | 3 |

76 | c.97 | Cytidine deaminase-like | 4 | 2 | 45 | 411 | 139 | 28 | 75 | 75 | 75 | 2 |

77 | c.44 | Phosphotyrosine protein phosphatases I-like | 3 | 2 | 52 | 57 | 120 | 46 | 67 | 67 | 67 | 2 |

78 | c.58 | Aminoacid dehydrogenase-like, N-terminal domain | 11 | 5 | 72 | 404 | 59 | 32 | 55 | 55 | 55 | 3 |

79 | c.49 | Pyruvate kinase C-terminal domain-like | 6 | 2 | 237 | 94 | 182 | 61 | 67 | 67 | 67 | 2 |

80 | c.52 | Restriction endonuclease-like | 23 | 21 | 1036 | 216 | 141 | 128 | 91 | 57 | 91 | 3 |

81 | c.72 | Ribokinase-like | 15 | 7 | 185 | 372 | 165 | 151 | 67 | 67 | 67 | 2 |

82 | c.80 | SIS domain | 6 | 3 | 198 | 420 | 1054 | 23 | 67 | 67 | 67 | 2 |

83 | c.92 | Chelatase-like | 11 | 5 | 311 | 932 | 221 | 200 | 36 | 36 | 36 | 3 |

84 | c.60 | Phosphoglycerate mutase-like | 7 | 4 | 446 | 415 | 280 | 237 | 57 | 57 | 57 | 2 |

85 | c.26 | Adenine nucleotide alpha hydrolase-like | 42 | 11 | 1017 | 913 | 460 | 459 | 29 | 29 | 29 | 7 |

86 | c.94 | Periplasmic binding protein-like II | 25 | 2 | 654 | 970 | 592 | 511 | 96 | 84 | 88 | 2 |

87 | c.66 | S-adenosyl-L-methionine-dependent methyltransferases | 35 | 23 | 620 | 908 | 1025 | 602 | 23 | 80 | 83 | 3 |

88 | c.78 | ATC-like | 12 | 2 | 643 | 931 | 1048 | 625 | 42 | 33 | 58 | 3 |

89 | c.35 | Phosphosugar isomerase | 3 | 2 | 652 | 940 | 1057 | 634 | 67 | 67 | 67 | 2 |

90 | c.37 | P-loop containing nucleotide triphosphate hydrolases | 122 | 20 | 937 | 821 | 938 | 814 | 25 | 40 | 28 | 9 |

91 | c.23 | Flavodoxin-like | 65 | 31 | 1045 | 878 | 1046 | 873 | 14 | 18 | 18 | 18 |

92 | c.56 | Phosphorylase/hydrolase-like | 24 | 11 | 1035 | 919 | 1036 | 912 | 75 | 83 | 75 | 4 |

93 | c.2 | NAD(P)-binding Rossmann-fold domains | 99 | 10 | 960 | 928 | 961 | 921 | 24 | 32 | 32 | 9 |

94 | c.69 | alpha/beta-Hydrolases | 51 | 26 | 1059 | 1064 | 1060 | 1059 | 35 | 96 | 35 | 2 |

The first 38 folds reported in Table 6 are in perfect agreement with dendrograms of all three structure comparison methods VAST, SHEBA and DALI. Together they comprise 187 of 1330 domains in the C-class. The next 23 folds (Table 6, rows 39–61, comprising 227 domains) agree perfectly with the dendrogram of at least one structure comparison method. Thus 61 out of 94 SCOP folds in the C class are consistent with the dendrogram built from at least one pairwise structural similarity measure.

Structure comparison methods differ in the consistency of their associated dendrogram with the SCOP folds. Dendrogram derived from each similarity score method disagree with SCOP for various folds. Forty-eight folds (comprising 1080 domains) disagree for the VAST tree, 41(1026 domains) for the SHEBA tree, and 43 (991 domains) for the DALI tree, according to this criterion.

We consider a fold to be highly dispersed if it disagreed with trees of all three structure comparison methods. There are 33 such folds (comprising 916 domains) and they are reported in Table 6, row 62–94. None of these 33 folds could be obtained as a homogeneous cluster, thus each contributes to the loss of agreement between any automatic partition and the SCOP fold partition. For these 33 folds, the intersection of their excess span was computed, and reported in Table 4, column labelled INT. INT is a measure of the disagreement with SCOP that remains even if all three dendrograms were combined. Thirty-one out of 33 folds give rise to a positive INT, meaning that same domains contributed to their dispersion within trees. These 31 folds therefore contribute to the remaining distance between the automatic and expert-curated partitions, regardless of tree cutting strategy or structure comparison method used.

### Dispersion caused by low structural similarity within folds

In Figure 6, the structural superposition (d) of domains from different folds produces even a better alignment than the superposition (c) of domains from two different clusters within the same c.58 fold. The disagreement between automatic and expert-curated classification, arises directly from the low similarity within fold c.58, and the substantial similarity to domains of another fold. It becomes impossible to merge the two distinct homogeneous clusters of fold c.58 without including domains from c.78 as well.

## Discussion

There are two approaches for comparing results from automatic structure comparison methods to an expert-curated reference classification such as SCOP. One can either directly compare the pairwise structural similarity measures to a similar measure derived from the reference partition, or produce a partition from the pairwise structural similarity score and compare it to the reference partition. We adopted the latter approach in this study, as we expected that enforcing partitioning constraints would introduce a new element that is not present in the pairwise similarity measures alone.

We explored a variety of methods for obtaining automatic partitions from pairwise structural similarity measures computed by VAST, SHEBA and DALI. Specifically, four different hierarchical clustering methods were used to construct dendrograms or binary trees, which were then cut into sub-trees by ten different tree-cutting strategies, to produce partitions. The results show that the combination of Ward's method with the largest size cut strategy has best agreement with SCOP among all combinations of clustering methods and tree-cutting strategies explored so far.

Trees generated by Ward's method result in partitions agreeing better with SCOP folds, than those generated by Single, Complete or Average linkage, regardless of the tree cutting strategy applied. Clusters formed by Ward's method tend to be highly concentrated around a mean, as they are formed so that the variance within the cluster is minimized. This suggests that SCOP folds are constructed based on the cohesiveness of the group as a whole rather than on similarity of individual pairs. This may be a more appropriate view of folds than as complete sub-graphs where every structure is related to every other structure as suggested by Complete linkage clustering. Although Average linkage produces clusters which are organized around a constructed mean, it does not minimize the variance around this mean, in contrast to Ward's method. Its lower performance relative to Ward's method strengthens the view of folds as structurally cohesive groups of domains. The mediocre results obtained by Single linkage emphasize even more the suitability of the notion of structural cohesiveness in modelling folds. Indeed Single linkage folds can be seen as minimum spanning trees, which do not necessarily require structural cohesiveness.

Largest size tree cutting strategy and those associated with it, such as highest tree cut and tree completeness cut, achieve much better agreement with SCOP, than other SCOP independent strategies. In particular they out-perform the "standard" level cut strategy. This is an unexpected result and suggests that the fold size was perhaps an implicit criterion, among others, in the formation of SCOP folds and that large groups of protein structures were split even when their internal similarity was higher than the internal similarity of smaller sized group of proteins. The small spread of the sizes of SCOP folds shown in Figure 3 supports this notion. On the other hand, a highly structurally distinctive fold such as TIM barrel (c.1) which is highly populated relative to the majority of folds and easily, visually recognizable, form an exceptionally large cluster.

The best combination of hierarchical clustering and SCOP independent tree cutting strategy e.g. Ward's method with either largest size, tree completeness or highest tree cut, resulted in an average TPR_{01} values (61%, 58% and 59% respectively) that are only slightly above the average agreement of 57% achieved by comparing pairwise similarities directly, without clustering, to SCOP folds partition, using the same set of SCOP C class domains. This suggests the idea that the persistent discrepancy between automatically determined similarity and SCOP is most likely not due to the partitioning constraint, even though all possible partitioning strategies, such as most recently developed clustering techniques in [21, 22] for example, have not been examined. Figure 2 provides additional evidence supporting this view. It shows that automatic partitions produced by introducing partitioning constraints into pairwise structural similarity measures, cluster together in the space of partitions, far away from the expert-curated classification SCOP implying that automatic structure comparison methods agree well with each other, and that there might be irreducible differences between them and SCOP.

Automatic methods VAST, SHEBA and DALI obtained 71%–76% TPR_{01} using the combination of Ward's method and the MI cut strategy which maximizes the agreement with SCOP. This is higher than the maximum 65% and 67% obtained by the largest size cut and the direct pairwise comparison scheme, respectively, on the same set of the SCOP C class protein domains. As the MI cut strategy proceeds with prior knowledge of SCOP folds information, this range of agreement is close to the maximum obtainable between automatic classification and manually curated SCOP dataset, regardless of the partitioning procedure. The inherent level of disagreement can be seen from the number of SCOP folds for which the spanning cluster exceeds the size of the fold, in Table 6.

Examination of two such folds corroborates the findings of our previous paper [15] that some SCOP folds includes structural variation causing measured similarity between members of the same fold to fall below that between members of related, but different folds. Indeed, the distance matrix in Figure 5 shows that fold c.58 displays this pattern, due to low similarity among domains from its distinct homogeneous clusters, and a high similarity with domains from a different fold c.78. The superpositions in Figure 6 (a), (b) and 6 (c), indicate that in all cases, be it intra or inter-cluster, all three structure comparison methods aligned the structural feature defining the fold, and yet intra-cluster structural similarity is higher than inter-cluster structural similarity. Regarding the TIM barrel fold, a detailed study by Nagano *et* al. [23] also characterized the fold as highly diverse, although using the CATH [24] database. Thus, structural variation within folds, shown in [15] as the main cause of divergence between automatic and expert-curated classifications, does not disappear even after satisfying the partitioning constraint. This strengthens our previous findings and emphasizes the importance of properly handling the structural variation in order to reduce the gap between automatic and expert-curated partitions. We also have underlined some relationships between distinct folds. The domains which correspond to the excess span of a dispersed fold could be seen as evidence suggestive of a evolutionary relationship among folds, as discussed by Lupas *et* al [25]. Domains from fold c.6, which are excess span of TIM barrel fold, are actually variants of TIM barrels.

Finally, in the light of this analysis, we think that future improvements to automatic protein structures classification would likely come by explicitly identifying common structural cores. Pairwise similarity scores alone appear to be limited in that regard, so techniques involving multiple structural alignment will likely be needed.

## Conclusion

The level of agreement between manual and automatic classifications varies with clustering methods and tree partitioning strategies. However, the best agreement reaches similar upper bound than when structural pairwise similarity is compared directly to the manual classification. Therefore divergence between automatic and manual classifications is not eliminated by the introduction of partitioning constraints.

Our observations are based on SCOP C class, but are likely valid for other classes as well, as C class domains contain both types of secondary structure elements, alpha helix and beta strand, and are the most difficult to classify due to higher confusion among them compared to the all alpha or all beta classes [15]. Our exploration of potential classification procedures of proteins based on structural similarity is complementary to the analysis done in our previous paper [15]. The modular structure of the proteins has been accounted for by using a database of structural domains defined in SCOP although this domain parsing may be at variance with other domain parsing such as that in CATH [24], or those based on amino acid sequences in CDD[26], ProDOM [27] or Pfam [28]. We did not address this issue here. Notwithstanding efforts made in this present work, there is still a discrepancy between the results of the automatic structure comparison methods and the SCOP classification. Based on the maximum attainable TPR of about 76%, roughly one quarter of the C class domains are not classified by SCOP according to measured global structural similarity. As we previously suggested, the main reason is that global structure similarity cannot entirely account for the characteristic local structural features on which the SCOP classification is based. Future research should be aimed at finding algorithms able to automatically extract such evolutionarily conserved, common local structural features of domains. As observed by Chothia and Lesk [29], in proteins having low sequence identity and the same biochemical function, only about half of secondary structures are conserved. The remaining challenge is to identify the conserved half!

## Methods

We consider a set *E* of *M =* 1330 domains in the C class selected from the set of SCOP domains with less than 40% pairwise sequence identity in ASTRAL [30] version 1.63. Their associated *N* = 94 folds constitute a partition, i.e. a collection of disjoint folds or subsets ${\left\{{F}_{j}\right\}}_{j=1}^{N}$ of *E* whose union is *E*.

Our objective is to compare this fold partition, an expert-curated classification, with a partition of the same set *E* of domains, obtained automatically as follows.

First, all *M*^{2} = *1,768,900* pairs of domains drawn from this set *E* were compared by VAST, SHEBA and DALI. The calculations were made using the high-performance computational capabilities of the Biowulf PC/Linux cluster at the National Institutes of Health, Bethesda, MD [31].

Data resulting from the computation were used to produce matrices of similarity scores [*S*_{
qt
}], where *S*_{
qt
}is either the *Nres* (*Number of Matched Residues*) between domains *q* and *t*, or the *Zscore* for SHEBA and DALI and the *Pcli* score for VAST.

Second, these similarity matrices were transformed into distance matrices and symmetrized by replacing the upper diagonal values with the lower diagonal values. There exists many ways of obtaining a pseudo-distance measure from a similarity measure [32], even though the triangular inequality is not always guaranteed. We choose a simple approach which guarantees that self-distance is zero and all distances are non-negative. With similarity defined as the number of matched residues, the distance satisfies the triangular inequality. For domains *q* and *t* within *E*, and similarity measure *S*_{
qt
}, the distance *D*_{
qt
}is defined as follow:*D*_{
qt
}= *S*_{
qq
}+ *S*_{
tt
}- *2 S*_{
qt
}

These matrices of pairwise distances were then entered into hierarchical clustering algorithms to obtain dendrograms or binary trees. We considered Matlab implementation [33] of four representative hierarchical clustering methods [20]: Single, Average, Complete linkage and Ward's method.

### Tree cutting algorithm

A dendrogram resulting from a hierarchical clustering is a binary tree, where each node is associated with an inter-cluster distance defined by the joining distance between its left and right children. The inter-cluster distance for the leaves of the tree is defined to be zero. To obtain a partition from a dendrogram, one ordinarily chooses a level, i.e. an inter-cluster distance value, falling within the dendrogram, then removes or "cuts" all the join-nodes above the chosen level, leaving behind a set of trees whose roots or join-nodes fall below that specified level. Here, we generalize this approach and find partitions made up of a set of sub-trees which join with root nodes at various levels, by introducing a panel of tree cutting criteria making use of a variety of tree characteristics, in addition to the inter-cluster distance. Our recursive algorithm starts with the original tree. When its root is cut or deleted, two trees remain, representing the two clusters in the growing current partition. Depending on the number of clusters desired and various topological features of each of the remaining trees, one tree is selected and its root is cut, leaving behind two smaller trees, each again representing a cluster in the refined partition. The procedure is terminated when the desired stopping criterion is reached.

More formally the tree cutting algorithm starts from the initial partition consisting of one cluster containing all elements of *E*. Given a partition $K={\displaystyle \underset{i=1}{\overset{\left|K\right|}{\cup}}{K}_{i}}$ of size |*K*| > 1, a partition of size |*K*| + 1 is obtained from *K* by splitting one of the clusters *K*_{
i
}into 2 distinct clusters. The splitting of a cluster is represented by removing the root node of its associated tree.

At the initial step of this process, there is no choice, but cutting the root node of the tree. Further partitioning involves a choice of which cluster to split. A tree cutting strategy determines this choice. We present below several strategies which have been developed here and used in this analysis. We consider three different ways of stopping the tree partitioning process. First, the partitioning process stops when no further partition can be obtained, corresponding to the situation where all clusters of the partition are singletons. Second, the partitioning process stops when a partition with a given number of clusters is obtained. We will be mainly interested by partitions of size 94 clusters corresponding to the number of SCOP C class folds in the dataset. Third the partitioning process stops based on reaching < = 1% FPR (see below) when comparing the automatic partition with SCOP.

### Tree cutting strategies independent of SCOP

We define seven SCOP independent tree cutting strategies. Such a strategy determines the next sub-tree to cut, without using a prior knowledge of SCOP fold partition. They all have been implemented for this analysis.

#### Level cut

#### Largest size cut

At a given step of the tree cutting strategy, resulting trees are available candidates for cutting. The candidate tree which contains the largest number of domains or leaves, is cut. This strategy counts the exact number of domains in the subtree evaluated for cutting.

#### Highest tree cut

The topological height *h(K*_{
i
}*)* of the node *K*_{
i
}is the number of intermediate nodes along the longest path from *K*_{
i
}to the leaves. The leaves are of topological height zero. For each cluster *K*_{
i
}, the topological height of its associated sub-tree is computed. The tree with the topologically highest root node is cut.

#### Tree completeness cut

Given the height *h(K*_{
i
}*)* of the sub-tree with root node *K*_{
i
}, the number of nodes in this sub-tree if it were complete would be 2^{h(Ki)+1}-1. If the sub-tree is not complete, its actual number of nodes, including the root and the leaves, is less than 2^{h(Ki)+1}-1. Tree completeness is defined as the ratio of the actual number of nodes of the sub-tree, and the number of nodes if it were complete. The smaller the ratio, the less complete is the tree. The tree with the smallest ratio (least complete) is cut.

#### Tree skewness cut

Given a tree, its skewness is defined as the ratio of the number of nodes in its left and right children. The ratio is defined such that the number of nodes of the smallest child, left or right, is divided by the number of nodes of the largest child. The tree of greatest skewness is chosen for cutting.

#### Longest branch cut

The branch length of a tree is the difference between the intercluster distance of the root node and the smaller of the inter-cluster distances of its two children. The tree with the longest branch value is cut.

#### Maximum entropy cut

*H(K)*of an entire partition

*K*is defined by[34]:

where ${P}_{i}=\frac{\left|{K}_{i}\right|}{M}$, the probability of the cluster *K*_{
i
}, is ratio of the number of domains within *K*_{
i
}to the total number *M* of domains being partitioned. The tree which would result in the greatest increase in entropy value for the current partition is chosen for cutting.

### Tree cutting strategies which refer to SCOP

We define here three tree cutting strategies which use the prior knowledge of SCOP fold partition to determine the next sub-tree to cut:

#### Mutual Information (MI) cut

*I(K:F)*between a partition

*K*resulting from the cutting of a binary tree and the SCOP fold partition

*F*, is defined as follow:

with *P*_{
i
}as defined for entropy, and ${P}_{ij}=\frac{\left|{K}_{i}\cap {F}_{j}\right|}{M}$ is ratio of the number of domains in common to cluster *K*_{
i
}and fold *F*_{
j
}to *M*. Cutting a particular tree results in a new partition, as well as a new value of *I(K:F)*. The tree which would result in the greatest increase of the mutual information is cut.

#### Best TPR/FPR Ratio cut

TPR and FPR values (defined below) refer to the SCOP fold partition. The tree whose cutting would results in the greatest TPR/FPR ratio, is cut.

#### Best TPR cut

The tree which would result in the greatest TPR value is cut.

### ROC curves for comparing partitions

The ROC curves are constructed using the definition of true and false positive rates introduced in [15], but with the notion of membership in a cluster replacing the notion of similarity cutoff value. The following definitions reflect this variation.

*TPR*

_{ j }is defined by:

where *n*_{
jk
}is the number of domains common to cluster *k* and SCOP fold *j*, and *n*_{
j
}is the number of domains in fold *j*.

when *i* and *j* referring to SCOP folds *i* and *j*.

### Distance between partitions

*K*and

*P*, we define the distance between these two partitions as follow:

where *K* and *P* are binary incidence matrices of the two partitions whose elements take on the value 1 when the domains i and j are in the same cluster, and 0 otherwise.

### Multidimensional scaling (MDS) plot

We use classical multidimensional scaling to plot a set of points in Euclidean space, such that each point represents a partition and the Euclidean distance between two points in the plot approximates the value of the Δ-distance between partitions they represent. Thirteen partitions are represented in the plot, including the expert-curated fold partition SCOP. Automatic partitions were chosen to have as many clusters as folds in the C class, by requiring the partitioning algorithm to stop when a partition of size 94 clusters is reached. The computation of pairwise Δ-distance between the 13 partitions resulted in a 13 by 13 matrix (see Table 3) which is then used as input to the classical MDS procedure in Matlab [35]. A two dimensional plot of the 13 partitions is obtained by projecting the points onto the two X and Y axes where the X axis is the eigenvector corresponding to the largest positive eigenvalue, and the Y axis the eigenvector corresponding to the second largest eigenvalue.

### Measure of fold dispersion

We define a measure of the dispersion of a SCOP fold within a binary tree, based on two metrics: the size of the excess span of the fold, and the size of the largest homogeneous cluster of the fold. The spanning cluster of a SCOP fold is the smallest cluster in the tree including all its domains. Proceeding upward from the leaves (domains) of that fold and stopping at the first (lowest) node common to all its leaves, we find the spanning cluster of the fold. The size of this spanning cluster, i.e. the number of leaves or domains it includes, is always greater or equal to the size of the associated fold. The set theoretic difference between the spanning cluster and the associated fold, is called excess span. The intersection of spanning clusters from distinct trees formed using distinct methods, identifies domains which might reasonably be considered as similar to domains in that fold. The size of the intersection less the members of that fold is a measure of how much larger the fold should be if it were to include all reasonably similar domains.

A homogeneous cluster is one containing domains of only one fold. Folds may be comprised of more than one homogeneous cluster. The size of the largest homogeneous cluster as a fraction of the size of a fold measures the fraction of the fold which can be easily recognized by the similarity measure and clustering algorithm. Figure 8 illustrates the notion of homogeneous and spanning clusters for a schematic binary tree.

## Declarations

### Acknowledgements

Molecular graphics images were produced using the UCSF Chimera package [36] from the Resource for Biocomputing, Visualization, and Informatics at the University of California, San Francisco (supported by NIH P41 RR-01081).

This research was supported in part by the Intramural Research Program of the NIH, Center for Information Technology and the National Cancer Institute.

## Authors’ Affiliations

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