Defining functional distances over Gene Ontology
 Angela del Pozo^{1}Email author,
 Florencio Pazos^{2} and
 Alfonso Valencia^{1}
DOI: 10.1186/14712105950
© del Pozo et al; licensee BioMed Central Ltd. 2008
Received: 11 July 2007
Accepted: 25 January 2008
Published: 25 January 2008
Abstract
Background
A fundamental problem when trying to define the functional relationships between proteins is the difficulty in quantifying functional similarities, even when wellstructured ontologies exist regarding the activity of proteins (i.e. 'gene ontology' GO). However, functional metrics can overcome the problems in the comparing and evaluating functional assignments and predictions. As a reference of proximity, previous approaches to compare GO terms considered linkage in terms of ontology weighted by a probability distribution that balances the nonuniform 'richness' of different parts of the Direct Acyclic Graph. Here, we have followed a different approach to quantify functional similarities between GO terms.
Results
We propose a new method to derive 'functional distances' between GO terms that is based on the simultaneous occurrence of terms in the same set of Interpro entries, instead of relying on the structure of the GO. The coincidence of GO terms reveals natural biological links between the GO functions and defines a distance model D_{ f }which fulfils the properties of a Metric Space. The distances obtained in this way can be represented as a hierarchical 'Functional Tree'.
Conclusion
The method proposed provides a new definition of distance that enables the similarity between GO terms to be quantified. Additionally, the 'Functional Tree' defines groups with biological meaning enhancing its utility for protein function comparison and prediction. Finally, this approach could be for functionbased protein searches in databases, and for analysing the gene clusters produced by DNA array experiments.
Background
Current genome sequencing projects are producing a wealth of data in the form of sequences of biological polymers. For this data to be useful, it has to be interpreted in functional terms. Thus, efficient systems to describe and classify protein function are needed, as well as tools to predict the function of the huge number of new sequences.
There is much evidence for the need of welldefined and structured functional descriptions [1–4]. However, the main difficulty encountered is that 'function' is not a well defined concept and it is not as unequivocal as 'sequence' or 'structure'. Indeed, protein function is a very complex and multidimensional phenomenon.
In many cases, functional descriptors are based on the available experimental techniques or are due to historical reasons. However, they do not necessarily have any meaning in biological terms (evolution, molecular mechanism). The methods we use to study biological systems require conceptualization and categorization, which are sometimes taken beyond their role as mere tools of the scientific method and are 'imposed' on the cell. One example is the artificial distinction between processes such as 'transmission of information' (for example DNA/RNA processing), 'metabolism' (of small compounds) and 'transport' (communication with the environment). Such disjointed classifications, as used in the first schemes to describe protein function, clearly do not extend to the molecular or evolutionary level. These schemes have been used in the past for classifying proteins into functional classes and for developing systems to assign newly sequenced proteins to them [5, 6].
The current tendency is to use vocabularies and ontologies that allow complex functional descriptions beyond disjointed classes. Among these, the important effort of the Open Biomedical Ontologies (OBO) [7] in developing controlled vocabularies for a wide scope of applications in a biological and medical context must be recognised. The OBO ontologies are designed as graphic architectures formed by univocal concepts (terms) that are linked together by relationships that satisfy some prefixed and formal rules [8]. The Gene Ontology (GO) project [9] has become the 'defacto' standard in biomedical ontologies. Formally, GO is designed as a Direct Acyclic Graph (DAG) based on two unconstrained relationships ('isa' and 'partof') that link a vocabulary of functional terms [2]. This graph structure, together with the simple conceptualization, permits comparisons between any two GO terms to assess their functional similarity. However, certain problems, such as the functionbased search for potential genes/proteins of interest across multiple annotated databases and the analysis of high throughput microarray data, have led to the in depth exploration of ontology in order to propose models and criteria to measure the functional relationships between the terms.
In recent years, many studies have addressed this matter [10–14], although Lord was the first to establish a semantic distance for any two terms in GO [10], adjusting the ideas of Resnik [15] for general taxonomies. In the model proposed by Lord, the similarity of any two GO terms is determined as a function of the information content of common ancestors that are calculated from corpus statistics. Recently, further efforts to identify functionally related gene products in annotated databases based on the distances calculated by Lord [11] have been shown to produce a good agreement with homology searches [12]. Nevertheless, using the more informative common ancestors as a proximity reference presents some restrictions. First, the depth of the shared parent nodes is not a suitable criteria for some limited cases in which the terms to be compared are close to the root. Furthermore, the information content (i.e. probability) of a node is highly dependent on the annotated database selected and its release version.
Models have been developed to overcome these limitations that take into account other aspects of the ontology structure. For example, the distance between two terms may also integrate the density of the terms and the path that links them [13]. Alternatively, a new definition has been used that considers the local relationships in the subgraph generated by the terms, rather than their global positions in the DAG [14].
A common feature of these different approaches is that they rely mainly on the semantic links of the DAG. Unfortunately, there are inherent problems in this approach due to the nonhomogeneity and the uneven distribution of the biological knowledge. As a result some regions of the DAG are more densely populated than others, so that the connections between terms are not comparable. In addition, the depth of a node (which is related to its specificity) can not be assigned in an unequivocal way. This type of problem is especially relevant for nodes that are profusely connected to the root by various paths of different lengths.
In this work, we propose a novel method that associates the Molecular Function GO (MFGO) terms based on their cooccurrences in a 'curated' set of proteins and enriched by the semantic relationships from the ontology. Interpro is used as a curated database as it integrates protein information from other databases that describe protein families, domains and functional sites, such as PROSITE, PRINTS, Pfam, ProDom, SMART and TIGRFAMs [16].
Conceptually, the method is, to some extend, similar to the way in which similarities between aminoacids are 'learnt' from examples (structural curated alignments) rather than obtained from the raw chemical properties of the aminoacids. Methodologically, it shares aspects of the algorithm used in the DAVID tool [17] for clustering heterogeneous annotation contents from different resources into annotation groups based on the coassociation of the annotated genes in the databases.
The method analyses the mutual occurrences of the MFGO terms across the Interpro entries. The occurrences are used as the basis of the comparison of the terms on the assumption that the persistent coincidence of two terms describes its 'relation' in the general functional space. The analysis of the occurrences provides a useful mathematical tool to quantify the functional similarity between terms. A hierarchical tree linking the MFGO terms is built from the similarity matrix. We termed this tree the 'Functional Tree' and it formally constitutes a Distance Model since it satisfies the ultrametric triangle inequality. In this context, the Functional Distance for a pair of terms, D_{ f }, is defined as the height of their least common ancestor in the 'Functional Tree'. In addition, the tree allows the GO terms to be clustered into compact and homogeneous groups with biological meaning.
We describe here how the Functional Tree was built, how the tree is clustered and the groups generated are analyzed in terms of the functions they describe. The Functional Distance D_{ f }derived from the Functional Tree was used to calculate the distances between pairs of yeast proteins to assess the reliability of the tree. We also compare this new metric with another based on semantic similarities.
Results
Algorithm
First, for each Molecular Function term we create a profile vector that represents its presence/absence in different Intepro entries (Figure 1, box 1). These vectors resemble the 'phylogenetic profiles' used to encode the proteins present in different organisms and to detect protein relationships [18, 19].
Initially, we started with 1532 MFGO terms present in 5535 Intepro entries. Additionally, we included the semantic relationships represented by the Gene Ontology DAG by assigning the same Interpro domain to the parent(s) of a given GO term. The profiles were checked to detect the terms that were associated exclusively to one Interpro entry and to ensure that this entry was not annotated with any other term. Any such profiles were removed because they do not help to extract relationships between the terms. After filtering, we obtained a matrix of 1778 Interpro entries with 1392 MFGO terms. In a second step (Figure 1, box 2), we built a matrix of cooccurrences of GO terms in Interpro entries. The occurrences were accumulated through all the profiles and we obtained the total mutual occurrences in the universe of the 1392 terms. Each cooccurrence vector describes a MFGO term in relation to the rest of the MFGO terms, which enables it to be used as a feature vector in the application of statistical learning techniques.
Spectral Clustering considers S as the Adjacency Matrix of a normalized weighted graph G, where the nodes stand for the MFGO terms linked by the similarity values. Thus, the clustering problem is transformed into a partitioning graph problem. We only considered the graph comprised of terms that were connected with significant relationships, that is those connected by a pairwise similarity greater than a manually selected threshold value (see Methods, 'Similarity Matrix' subsection). After imposing this constraint, we obtained 995 MFGO terms from the total of approximately 7500 terms integrated in the released version of this work.
Finally, from the vector projections of the MFGO terms, we built a dendrogram with an Agglomerative Hierarchical Clustering algorithm [22] (Figure 1, box 5). The tree obtained ('Functional Tree') defines a distance D_{ f }between any two MFGO terms from the set of 995 (see Additional file 1). The distance for two terms was the minimum height of their common nodes. From a mathematical point of view, D_{ f }satisfies the topological properties that induces a metric space (see 'Properties of a Metric Space' from the appendix). So, the metric generated by the Functional Tree establishes a 'distance scheme' that provides a measure of the closeness of any two MFGO terms within the tree.
Testing
Functional Groups
Some of these groups were very specific, like the group containing the 21 amino acyltRNA ligase activities. This group includes all the tRNA ligases and no other GO term, and hence the automatic clustering algorithm achieved a perfect segregation of this functional group (group 2).
Another big group mostly composed by activities related to hydrolysis (hydrolases, peptidases, nucleases, lipases) was labelled as group 1. Although this group was homogeneous for this activity, the coverage was not perfect since other hydrolases lay outside of this group. For example, group 7, which was far from group 2 in the tree, was mainly comprised of peptidase activities.
Interestingly, many different activities associated with DNA processing tended to cluster together despite the fact that they were apparently unrelated (i.e.: transcription factors and enzymes involved in DNA metabolism, DNA ligases, topoisomerases, etc... – group 3). As for the hydrolases case commented above, although this group contained only DNArelated activities and other DNArelated functional terms were not included in this group.
Most of the kinases of small metabolic compounds were clustered in a large group (group 15), while protein kinases were more widespread even though some of them clustered together in group 6. Many membrane transporters of apparently different nature (transporters for inorganic ions, drugs, proteins, etc...) were also clustered together in homogeneous groups.
All the 'protein inhibitor' activities within the dataset were clustered together in a homogeneous group (group 20), which is interesting given that the proteins they inhibit are of a very different nature (phosphatases, ribonucleases, proteases, etc...).
Functional clustering was also evident for many other GO terms: methyltransferases, phosphorybosyltransferases, peptidases, some peptidic hormones, neurotransmitter receptors, phosphatehydrolases, hydratases/dehydrateses, adenylyltransferases, etc....
For other clusters, this functional 'homogeneity' was not so evident. For example, oxidoreductases were spread across many groups even though some groups contained oxidoreductase activities only (group 19). This dispersion could be explained by the fact that this function is present in proteins with a very different evolutionary origin. Similarly, activities related to RNA metabolism were spread among the different clusters, except the tRNAligases discussed above.
In general, the clustering represented by the tree makes sense, meaning that GO molecular functions that are intuitively 'similar' were close in the tree and vice versa. This emphasise that the metric represented by the tree can be used to quantify functional similarities.
The Functional Tree as Metric Model
The functional distance D_{ f }defined from the Functional Tree allows a new quantitative analysis of the functional relationship between gene products. To assess D_{ f }as a functional similarity measure we correlated sequence and annotated function similarity over a set of aligned pairs of yeast proteins. The benchmark set has been selected by applying a very restrictive criterion to obtain a high reliable set of annotated proteins. The selection process (see Methods section) takes as quality assay the evidence codes in GOA. In this work, we picked only those sequences that had been functionally characterized either by experimental assay (IDA evidence code) or by traceable published works (TAS evidence code) and whose GO terms were included in our functional tree.
To compare the models the mean distance values for each bin of sequence identity are superposed in figure 5C. In average, both approaches correlate well with sequence similarity and exhibit a similar trend for homologous pairs. This is partly due to the homologybased mechanism of annotation that transfers directly a source set of MFGO terms to many homologous sequences.
In consequence, more than 84% of the alignments with sequence similarity values greater than 80% share the same annotations. This lack of richness in the annotations limits further analysis of the methods. However, we can observe that D_{ f }and D_{ s }show a different behavior. The distance space is discretized into three welldefined groups (Figure 5A) whereas the semantic similarity values produced a great spread.
These natural 'cutoffs' allow classifying the pairs into three categories with biological meaning that can be roughly labelled as 'closely functionally related' (distances less than 0.1), 'not related at all' (more than 0.9) and 'divergence in functionality' (in the intermediate interval with distances between 0.5 and 0.7). This partition results from the structure of the clusters (Figure 6B) showing small intragroup and large intergroup distances. This is in part due to 'biological' reasons but is also affected by the function transfer by sequence homology.
The benchmark set includes some example pairs in which D_{ f }assigns close distances to functionally related pairs while D_{ s }does not. One is the pair formed by [Uniprot:P20134] and [Uniprot:P10961] that shares 50% sequence identity. The first is a transcriptional repressor and activator annotated with the term GO:0016566 ('specific transcriptional repressor activity'). The second is a trimeric heat shock transcription factor annotated with GO:0003700 ('transcription factor activity'). Both are characterized by HSFtype DNAbinding Pfam domain. The relative posititions of their annotated terms in the DAG can be checked in Figure 7. D_{ s }is 0.76 indicating a weak relation between the proteins. However, D_{ f }situates the pair into the 'closely functionally related' region because the terms belong to the cluster 3 described before.
Other similar example is the pair formed by the protein kinases [Uniprot:P32801] and [Uniprot:P41808]. The proteins are characterized by protein kinase pfam domain and are annotated respectively with GO:0004674 ('protein serine/threonine kinase activity') and GO:0004707 ('MAP kinase activity'). Both terms belong to group 6 (Functional Groups subsection). So, as in the example before, the distance D_{ f }is 0 whereas D_{ s }is 0.6. Although these terms are close in the ontology ('protein serine/threonine kinase activity' is ancestor of 'MAP kinase activity' and separated only by two depth levels), the Lord model assigns such a value distance because the shared parent ('protein serine/threonine kinase activity') is referred many times in the gene association.goa human file. According to Lord definition, high probable terms carry low information content producing high distance values in the comparison of terms. In the case of the kinases pair, the probability introduces a bias that shifts the semantic distance value to a region that indicates, as the example before, a weak relation between the proteins.
Finally, the Functional Distance model is sensitive enough to detect subtle differences in the pairs [Uniprot:P15700]/[Uniprot:P07170] and [Uniprot:P15700]/[Uniprot:P26364] that are explained by sequence analyses. In both cases, the members of the pair were annotated with GO:0004849 ('uridine kinase activity') and GO:0004017 ('adenylate kinase activity') respectively. Lord's Semantic Model produced a distance of 0.37 between these proteins, indicating semantical relation. In fact, the aforementioned terms are close in the GO hierarchy, and the deepest common parent shared by both GO terms, two levels above, is 'nucleobase, nucleoside, nucleotide kinase activity' (GO:0019205). However, our Functional Distance located that pair of GO terms within the intermediate interval at a distance of 0.65. A thorough analysis of the sequences revealed that [Uniprot:P15700] has the 'ADK' Pfam domain. Adenylate kinases are phosphotransferases with well conserved ADK domains that include an important arginine which inactivates the enzyme if mutated, and an aspartate that is located in the catalytic cleft and that forms a crucial salt bridge. However, in the particular case of [Uniprot:P07170] and [Uniprot:P26364], the putative ADK domain is interrupted by another PFAM domain, the ADK lid. Looking at the sequence of this particular region, the ADK domain boundaries were not clearly delineated due to a high degree of divergence in the active site. So, in this example our metric is able to capture the 'functional difference' between these two proteins due to the inserted domain.
Implementation
The Spectral Clustering algorithm is implemented in Matlab 7.4.0 using the clustering functions available in the Statistics Toolbox. Lord's model is implemented in Python 2.5, and Python was also used to calculate the functional and semantic distances.
Discussion and Conclusion
Here, we propose a new method to derive 'functional distances' between GO terms based on the cooccurrence of them in the same set of proteins. The simultaneous occurrence of terms in Interpro entries provides a natural biological link between the GO functions. The relationship between terms in the GO structure provides additional semantic information that helps to refine the metric model.
In this method, an initial profile is constructed for each GO term representing its association with a set of Interpro domains (after expanding the Interpro annotations with the parenthood relationships of the GO terms). These profiles are used to generate a matrix of cooccurrence between GO terms. A graph is constructed where the nodes are the GO terms and the edges are weighted according to the distances extracted from this cooccurrence matrix. Spectral clustering is applied to this graph in order to obtain an optimal number of groups of functionally similar GO terms. The distances derived in this way provide a hierarchical clustering of GO terms (functional tree) where the groups of terms with similar biological meaning tend to be close. Additionally, this 'Functional Tree' represents a metric model D_{ f }whereby the distances between the terms fulfil the mathematical properties of a metric space.
The main difference of this method from previous approaches [10–14] is that D_{ f }is learned from examples. Hence, in contrast to other proposed methods that derive the distances from the semantic relationships within the GO ontology, our method provides new associations between the terms that enables a different way to compare proteins in functional terms. We have selected some cases to illustrate this point, for which the functional similarity between related proteins is better estimated with this metric model than with currently available algorithms, such as Lord's 'Semantic Similarity Model' [10]. Moreover, we also tried to qualitatively assess some of the groups automatically extracted from the distances by a clustering algorithm. Over and above the comparison with these examples and the qualitative assessment of the functional tree, it is actually difficult to assess the general quality of any functional metric.
The overall representation of the functional distances in the Functional Tree originates very compact groups of terms separated by well defined intervals as it is shown in Figure 6B. This structure of the clusters produces a not uniform distribution of distances because the values tend to concentrate in three regions (low, intermediate and high), which is obviously a problem since it makes the metric to some extent 'qualitative'. On the other hand, this categorisation produces natural 'cutoffs' and functional similarities that are naturally classified by the method in three categories with biological meaning in a totally unsupervised way as been discussed in Results section.
This discretization is in part due to 'biological' reasons but is also affected by the homologybased transfer that causes that only a reduced set of terms is used for annotation purposes. In consequence, the coverage with respect to the total number of terms in the Molecular Function ontology is low (around 20%). In addition, only clear relations are selected resulting a set of 995 MFGO terms that are considered in the Functional Distance Model.
It is important to note that the relationships between the GO terms obtained and the relationships in the GO ontology represent different elements. The GO DAG represents qualitative semantic relationships ('isa' and 'partof') while our relationships represent quantitative 'functional distances'.
Thus, the metric proposed here provides a way of quantifying how similar two functional annotations are. This can be very useful for training systems for function prediction, for functionbased protein searches in databases, or to assess the accuracy of a functional prediction (comparing the predicted set of annotations with the real one). This metric could also be useful for analysing the gene clusters produced by DNA array experiments. We think it may also provide insights into how functions evolved and the relationships between sequence, structure and functional spaces.
Methods
Similarity Matrix
The GO annotations for a given Interpro entry are retrieved from the mapping of Interpro to Gene Ontology [23] (interpro2go file, release May 2006). Only the GO terms belonging to the MFGO are considered.
Note that the cosine distance generates values ranking between 0 and 1. The similarity value can be considered as a description of the functional relationship between these terms, whereby similarities equal to 0 stand for unrelated terms and 1 stands for strongly related. The similarity matrix S is plotted in Figure 2A together with its histogram (Figure 2B).
The distribution of the similarity values shows that almost 90 per cent of the pairs of terms are only weakly related. This structure of the relationships reflects the presence of welldefined groups of terms. However, the search space has been limited as the cosine distance assigns a nonzero value even to pairs of terms that rarely share the same Interpro entry. Thus, based on the inspection of the histogram, we set a threshold of 0.8 to select strong functional links. In total there are 995 MFGO terms that are significantly connected.
Additionally, we reduced the dimension of S by applying a NJW Spectral Clustering (NJWSC) algorithm [21]. The similarity matrix in spectral space is shown in 6A, while the details of the algorithm are outlined in the 'Spectral Clustering Algorithm' subsection from the appendix.
Optimization Approach
As the number of clusters K is not initially known, an optimization approach is used, such as the multiway normalized cut value MNCut, (see 'Multiway Cuts' subsection from the appendix). The MNCut was calculated from the normalized matrix P (Transition Probability matrix). P represents the total probability of transit between any two clusters C_{ i }and C_{ j }for a given partition C = {C_{1} ... C_{ K }} of the graph and MNCut and represents the total sum of the transition probabilities between the clusters. The goal of optimization is to find the eigenvalue cutoff that generates a partition C* (K) that minimizes the MNCut value. In particular, we addressed optimization by exploiting the minimization of the gap value over the spectra of S.
Benchmark Dataset of aligned proteins
To compare our metric with others developed previously and to evaluate its relationship with sequence similarity, we took a set of proteins with reliable annotations taken from the Saccharomyces Genome Database (SGD). As an annotation source we used the file gene association.sgd (release October 2006).
The confidence of the Gene Ontology Annotations (GOA) is represented by the Evidence Code (EVC). Although there is no consensus rule that establishes a standard order of annotations based on the EVCs, the Gene Ontology Consortium has outlined a rank of EVCs as a guide [24]. The hierarchy of confidence establishes that the TAS (Traceable Author Statement) and IDA (Inferred from Direct Assay) tagged annotations offer the highest confidence.
Despite the efforts of the GOA project to improve the general reliability of its databases, the bulk of GO assignments are still made by automatic techniques with no expert curation. This homologybased transfer generates highly redundant sets of GO annotations. Moreover, as our method to derive the similarity between GO terms is to some extent affected by sequence similarity (given that it uses Interpro domains), we decided to exclude GO annotations derived from sequence relationships.
In gene association.sgd, there were 1264 yeast proteins annotated with GO terms with EVCs unrelated by homology (TAS and IDA) that also appeared in our functional tree. After filtering this set for sequence redundancy with CDhit [25] at 95%, we obtained a final set of 1193 yeast proteins. We then perform fast alignments of allagainstall using BLAST, having chosen a permissive evalue (0.1) to permit alignments between distant sequences. Nevertheless, alignments covering less than 50 residues and/or with less than 10% similarity were excluded. The final set comprised 1426 protein pairs and the distribution of sequence similarity for these pairs is shown in Figure 5D.
Functional Comparison between Gene Products
Usually the Hausdorff distance is evaluated over the Euclidean space, although in this work we applied equation 3 using two distances: (A) the distance D_{ f }obtained from our Functional Tree and (B) the distance proposed by Lord et al. [10]
We implemented Lord's semantic similarity using as a reference the annotated database gene association.goa human [26] (released version 45.0), and we normalised the values between 0 and 1 to compare it with the metric derived in this work.
Appendix
Spectral Clustering
Spectral clustering has its origin in spectral graph partitioning [27] and is intended to efficiently identify good discrete partitions of a graph based on the eigenvalues and eigenvectors of the Laplacian matrix of the graph.
Spectral clustering belongs to a collection of techniques that are designed to overcome the problems of previous approaches by using new ideas such as the eigenvectors of the generalised/normalized Laplacian or the multiway spectral cut. A systematic comparison between the existing published algorithms can be found in the work of Verma and Meila [20]. Therein, the authors present a clear description of the basic steps of the algorithms and their general classification based on three different strategies: (I) recursive spectral; (II) multiway spectral; and (III) nonspectral.
In this section we will introduce the notation and the basic steps for the NJW spectral clustering algorithm [21] and the ideas behind multiway spectral cuts as a criterion of optimisation to find the best partition of the data. Here we implemented a modified version of NJW algorithm suggested in [20].
NJW Spectral Clustering Algorithm
Consider a dataset U formed by N points to be clustered. For each pair of points within U, a similarity value can be defined as s_{ ij }= s_{ ji }≥ 0 by any similarity measure. U can be represented by a weighted directed graph G = (V, E) where the S = [s_{ ij }] matrix plays the role of the adjacency matrix of the graph. A clustering C = C_{1}, C_{2},..., C_{ K }is a partitioning of U into nonempty disjointed subsets C_{1}, C_{2},..., C_{ K }.
The outdegree of a node j is defined as ${d}_{i}={\displaystyle \sum _{j=1}^{N}{s}_{ij}}$. We represent D for a diagonal matrix of outdegrees as: D = diag(d_{1},..., d_{ N }).
 1.
Compute the transition probability matrix P = D^{1} S.
 2.
Select the first K eigenvalues of P and form the X matrix by stacking the eigenvectors in columns: X = [v^{1}v^{2}...v^{ K }]
 3.Normalize each row of X to unit length to form the Y matrix:${Y}_{ij}=\frac{{X}_{ij}}{\sqrt{{\displaystyle \sum _{j}{X}_{ij}^{2}}}}$
 4.
Treat each row of Y as a point in K dimensions. The points can be grouped by any standard clustering technique. In this work, the points are organized in a hierarchical tree.
Multiway Cuts
The majority of approaches in spectral clustering deal with partitioning the graph in two optimal parts by using one eigenvector at a time and applying this approach reiteratively until K clusters are found.
A way to use the K first eigenvectors simultaneously to find the optimum partition of the graph has been proposed, minimizing the cut of two partitions over all possible partitions in U [28]. Most of the approaches assume that the number of clusters K is known in advance, but in many problems related to clustering there is not indirect evidence that reveals the optimal number of groups.
Here we expose the basics of the multiway normalized cut (MNCut) concept that has been applied to find the optimum number K:
D denotes the diagonal matrix formed by D_{ i }. The volume of a subset A ⊆ U is $VolA={\displaystyle \sum _{i\in A}{D}_{i}}$, (we assume that no node has volume 0).
Given the partition C = {C_{1},...C_{ K }}, over U the multiway normalized cut clustering criteria introduced in [28] is defined as:
The multiway normalized cut represents the total sum of the transition probabilities between the clusters of C. If MNCut(C) is small then the probability of evading C_{ k }in a random walk is also small.
It has been also shown that gap(C) is 0 if P has piecewise constant eigenvectors, that is if P is an ideal block stochastic matrix [29].
Therefore, from a set of M different partition solutions of the data [C_{ i }]_{i∈M}, the optimal C* is that which minimizes the gap measure.
Properties of a Metric Space
By construction, the hierarchical clustering procedure over the Gene Ontology terms defines a generalized distance D between any two terms go_{ a }and go_{ b }that satisfies the mathematical properties of a metric space [22]. That is, any set of elements (terms) of the space x_{1}, x_{2} and x_{3} fulfil:

Nonnegativity: D(x_{1}, x_{2}) ≥ 0

Reflexivity: D(x_{1}, x_{2}) = 0 if and only if x_{1} = x_{2}

Symmetry: D(x_{1}, x_{2}) = D(x_{2}, x_{1})

Triangle Inequality: D(x_{1}, x_{2}) + D(x_{2}, x_{3}) ≥ D(x_{1}, x_{3})
Declarations
Acknowledgements
The authors want to acknowledge the members of the Structural Bioinformatics Group (CNIO), especially Dr. Ana Rojas for her help analysing the biological examples and Dr. Michael Tress for critical reading of the manuscript. This work has been partially funded by the GeneFun EU project (LSGCT2004503567).
Authors’ Affiliations
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