Experimental setup

Several benchmark data sets of non-homologous protein structures have been developed in the last few years [16–19]. In this study, we have chosen the 36 protein domains of [16] and the 86 prototype protein domains of [17]. The Chew-Kedem data set, which consists of 36 protein domains drawn from PDB entries of three classes (alpha-beta, mainly-alpha, mainly-beta), was introduced in [16] and further studied in [13]. The Sierk-Pearson data set, which consists of a non-redundant subset of 2771 protein families and 86 non-homologous protein families from the CATH protein domain database [20], was introduced in [17].

For both the Chew-Kedem and the Sierk-Pearson data sets, we have considered several alternative representations. Besides the standard representation of amino acid sequences in FASTA format [21], we have also used a text file consisting of the ATOM lines in the PDB entry for the protein domain, the topological description of the protein domain as a TOPS string of secondary structure elements [22–25], and the complete TOPS string with the contact map. The TOPS model is based on secondary structure elements derived using DSSP [26], plus the chirality algorithm of [25]. For instance, the various representations of PDB protein domain 1hlm00, a globin from the sea cucumber Caudina arenicola [27, 28], are illustrated in Figure 1.

We have also included in this study a benchmark data set of 15 complete unaligned mitochondrial genomes, referred to as the Apostolico data set [29].

In summary, the six data sets with the acronyms used in this paper are as follows:

AA-15-DNA: Apostolico data set of 15 species, mitochondrial DNA complete genomes.

CK-36-PDB: Chew-Kedem data set of 36 protein domains, amino acid sequences in FASTA format.

CK-36-REL: Chew-Kedem data set of 36 protein domains, complete TOPS strings with contact map.

CK-36-SEQ: Chew-Kedem data set of 36 protein domains, TOPS strings of secondary structure elements.

SP-86-ATOM: Sierk-Pearson data set of 86 protein domains, ATOM lines from PDB entries.

SP-86-PDB Sierk-Pearson data set of 86 protein domains, amino acid sequences.

We considered twenty different compression algorithms and, for some of them, we tested up to three variants. The choice of the compression algorithms reflects quite well the spectrum of data compressors available today, as outlined in the Methods section. Finally, dissimilarity matrices, both corresponding to the USM methodology and to other well established techniques, were computed as described in the Methods section.

Intrinsic classification abilities: the ROC analysis

This set of experiments aims at establishing the intrinsic classification ability of the dissimilarity matrices obtained via each data compressor and each of UCD, NCD and CD. In order to measure how well each dissimilarity matrix separates the objects in the data set at the level of CATH class, architecture, and topology, we have taken the similarity of two protein domains as the score of a binary classifier putting them into the same class, architecture, or topology, as follows.

We first converted each of the 36 × 36 dissimilarity matrices (for the CK-36 data set) and each of the 86 × 86 dissimilarity matrices (for the SP-86 data set) to similarity vectors of length 1,296 and 7,398, respectively, and used each of these vectors, in turn, as predictions. Also, for each classification task, we obtained a corresponding symmetric matrix with entries 1 if the two protein domains belong to the same CATH class, architecture, or topology (depending on the classification task) and 0 otherwise, and we converted these matrices to vectors of length 1,296 and 7,398, respectively, and used them as class labels. We have performed the ROC analysis using the ROCR package [30]. The relevant features about ROC analysis are provided in the Methods section.

The result of these experiments is a total of 5 × 3 × 3 × 24 = 1, 080 AUC (Area under the ROC curve) values, one for each of the five alternative representations of the CK-36 and SP-86 data sets, three dissimilarity measures, three classification tasks, and 25 compression algorithms, together with 5 × 3 × 3 = 45 ROC plots, which are summarized in Figures 2, 3, 4, 5, 6.

Classification via algorithms: UPGMA and NJ

This set of experiments aims at establishing how well the dissimilarity matrices computed via UCD, NCD and CD, with different compressors, can be used for classification by two well known and widely available classification algorithms. We have chosen UPGMA [31] and NJ [32], two classic tree construction algorithms, as implemented in BioPerl [33]. In relation to the clustering literature [34], both can be considered as Hierarchical Methods. In fact, UPGMA is also known as Average Link Hierarchical Clustering. NJ does not seem to have a counterpart in the clustering literature, although it certainly belongs to the Hierarchical Clustering algorithmic paradigm.

In order to assess the performance of compression-based classification, via UPGMA and NJ under various compression algorithms, we have computed two external measures [34], the F-measure and the partition distance, against a gold standard. The relevant features of those two measures is presented in the Methods section.

For the classification of protein domains, we have taken as the gold standard the CATH classification [20], although we might have adopted the SCOP classification [35] instead and, as a matter of fact, there are ongoing efforts to combine both classifications of protein domains [36, 37]. In order to obtain a partitional clustering solution from the tree computed by UPGMA, we place in the same cluster all proteins that descend from the same level one ancestor in the UPGMA tree. Then, we can compute the F-measure by using this clustering solution and the gold standard division of proteins in groups according to CATH class. The same procedure is used for NJ.

On the other hand, the classification of species we have taken as the gold standard is the NCBI taxonomy [38]. In this case, we can simply compare the two trees computed by UPGMA and NJ against the NCBI taxonomy, via the partition distance. Again, we might have adopted any other classification of species instead such as, for instance, the global phylogeny of fully sequenced organisms of [39].

Compression performance

Kolmogorov Complexity and Information Theory: the Universal Similarity Metric and its compression-based approximations

The conditional Kolmogorov complexity K(x|y) of two strings x and y is the length of the shortest binary program P that computes x with input y [8, 40]. Thus, K(x|y) represents the minimal amount of information required to generate x by any effective computation when y is given as an input to the computation. The Kolmogorov complexity K (x) of a string x is defined as K(x|λ), where λ stands for the empty string. Given a string x, let x* denote the shortest binary program that produces x on an empty input; if there is more than one shortest program, x* is the first one in enumeration order. The Kolmogorov complexity K (x, y) of a pair of objects x and y is the length of the shortest binary program that produces x and y and a way to tell them apart. It is then possible to define the information distance ID (x, y) between two objects satisfying the following identity, up to logarithmic additive terms:

ID (x, y) = max {K (x|y), K (y|x)}

has been defined and properly denoted as the universal similarity metric. In fact, it is a metric, it is normalized and it is universal: a lower bound to, and therefore a refinement of, any distance function that one can define and compute.

It is well known that there is a relationship between Kolmogorov complexity of sequences and Shannon information theory [42]: the expected Kolmogorov complexity of a sequence x is asymptotically close to the entropy of the information source emitting x.

Such a fact is of great use in defining workable distance and similarity functions stemming from the theoretic setting outlined above. Indeed, Kolmogorov complexities are non-computable in the Turing sense, so the universal similarity metric must be approximated, via the entropy of the information source emitting x. However, it is very hard to infer the information source from the data. So, in order to approximate K (x), one resorts to compressive estimates of entropy.

Let C be a compression algorithm and C (x) (usually a binary string) its output on a string x. Let |C (x)|/|x| be its compression rate, i.e., the ratio of the lengths of the two strings. Usually, the compression rate of good compressors approaches the entropy of the information source emitting x [42]. While entropy establishes a lower bound on compression rates, it is not straightforward to measure entropy itself, as already pointed out. One empirical method inverts the relationship and estimates entropy by applying a provably good compressor to a sufficiently long, representative string. That is, the compression rate becomes a compressive estimate of entropy.

Based on it, we consider

NCD (x, y) = min {NCD₁ (x, y), NCD₁ (y, x)}

Data sets

The Chew-Kedem data set consists of the following proteins:

alpha-beta 1aa900, 1chrA1, 1ct9A1, 1gnp00, 1qraA0, 2mnr01, 4enl01, 5p2100, 6q21A0, 6xia00.

mainly beta 1cd800, 1cdb00, 1ci5A0, 1hnf01, 1neu00, 1qa9A0, 1qfoA0.

mainly alpha 1ash00, 1babA0, 1babB0, 1cnpA0, 1eca00, 1flp00, 1hlb00, 1hlm00, 1ithA0, 1jhgA0, 1lh200, 1mba00, 1myt00, 2hbg00, 2lhb00, 2vhb00, 2vhbA0, 3sdhA0, 5mbn00.

Protein chain 2vhb00 appears twice in this data set, as 2vhb00 and 2vhbA0, in order to test whether the two chains are detected by compression-based classification methods to be identical (and thus clustered together) or not.

The Sierk-Pearson protein data set is as follows:

alpha-beta 1a1mA1, 1a2vA2, 1akn00, 1aqzB0, 1asyA2, 1atiA2, 1auq00, 1ax4A1, 1b0pA6, 1b2rA2, 1bcg00, 1bcmA1, 1bf5A4, 1bkcE0, 1bp7A0, 1c4kA2, 1cd2A0, 1cdg01, 1d0nA4, 1d4oA0, 1d7oA0, 1doi00, 1dy0A0, 1e2kB0, 1eccA1, 1fbnA0, 1gsoA3, 1mpyA2, 1obr00, 1p3801, 1pty00, 1qb7A0, 1qmvA0, 1urnA0, 1zfjA0, 2acy00, 2drpA1, 2nmtA2, 2reb01, 4mdhA2.

mainly beta 1a8d02, 1a8h02, 1aozA3, 1b8mB0, 1bf203, 1bjqB0, 1bqyA2, 1btkB0, 1c1zA5, 1cl7H0, 1d3sA0, 1danU0, 1dsyA0, 1dxmA0, 1et6A2, 1extB1, 1nfiC1, 1nukA0, 1otcA1, 1qdmA2, 1qe6D0, 1qfkL2, 1que01, 1rmg00, 1tmo04, 2tbvC0.

mainly alpha 1ad600, 1ao6A5, 1bbhA0, 1cnsA1, 1d2zD0, 1dat00, 1e12A0, 1eqzE0, 1gwxA0, 1hgu00, 1hlm00, 1jnk02, 1mmoD0, 1nubA0, 1quuA1, 1repC1, 1sw6A0, 1trrA0, 2hpdA0, 2mtaC0.

The Apostolico data set consists of the mitochondrial DNA complete genome of the following species:

Laurasiatheria blue whale (B. musculus), finback whale (B. physalus), white rhino (C. simum), horse (E. caballus), gray seal (H. grypus), harbor seal (P. vitulina).

Murinae house mouse (M. musculus), rat (R. norvegicus).

Hominoidea gorilla (G. gorilla), human (H. sapiens), gibbon (H. lar), pigmy chimpanzee (P. paniscus), chimpanzee (P. troglodytes), sumatran orangutan (P. pygmaeus abelii), orangutan (P. pygmaeus).

Algorithms

ROC curves and external measures

A ROC curve [58] is a plot of the true positive rate (the sensitivity) versus the false positive rate (one minus the specificity) for a binary classifier as the discrimination threshold changes. The AUC is a value ranging from 1 for a perfect classification, with 100% sensitivity (all true positives are found) to 0 for the worst possible classification, with 0% sensitivity (no true positive is found). A random classifier has an AUC value around 0.5. When plotted, the better the ROC curve follows the ordinate and then the abscissa line, the better the classification. The ROC curve of a random classifier, when plotted, is close to the diagonal. The F-measure [59] takes in input two classifications of objects, that is, two partitions of the same set of objects, and returns a value ranging from 0 for highest dissimilarity to 1 for identical classifications. The partition (or symmetric difference) distance [60, 61], takes in input the tree topologies of two alternative classifications of n species and returns a value ranging from 0 to 4n - 10. It is the number of clades in the two rooted trees that do not match and it is increasing with dissimilarity. When zero, it indicates isomorphic trees.

Experimental set-up and availability

All the experiments have been performed on a 64-bit AMD Athlon 2.2 GHz processor, 1 GB of main memory running both Windows XP and Linux. All software and data sets involved in our experimentation is available at the supplementary web site [1]. The software is given both as C++ code and Perl scripts and as executable code, tested on Linux (various versions – see supplementary material website), BSD Unix, Mac OS X and Windows operating systems.