 Research article
 Open Access
The distanceprofile representation and its application to detection of distantly related protein families
 ChinJen Ku^{1} and
 Golan Yona^{1}Email author
https://doi.org/10.1186/147121056282
© Ku and Yona; licensee BioMed Central Ltd. 2005
 Received: 25 May 2005
 Accepted: 29 November 2005
 Published: 29 November 2005
Abstract
Background
Detecting homology between remotely related protein families is an important problem in computational biology since the biological properties of uncharacterized proteins can often be inferred from those of homologous proteins. Many existing approaches address this problem by measuring the similarity between proteins through sequence or structural alignment. However, these methods do not exploit collective aspects of the protein space and the computed scores are often noisy and frequently fail to recognize distantly related protein families.
Results
We describe an algorithm that improves over the state of the art in homology detection by utilizing global information on the proximity of entities in the protein space. Our method relies on a vectorial representation of proteins and protein families and uses structurespecific association measures between proteins and template structures to form a highdimensional feature vector for each query protein. These vectors are then processed and transformed to sparse feature vectors that are treated as statistical fingerprints of the query proteins. The new representation induces a new metric between proteins measured by the statistical difference between their corresponding probability distributions.
Conclusion
Using several performance measures we show that the new tool considerably improves the performance in recognizing distant homologies compared to existing approaches such as PSIBLAST and FUGUE.
Keywords
 Feature Vector
 Query Protein
 Association Measure
 Homology Detection
 Remote Homology
Background
The ongoing sequencing efforts continue to discover the sequences of many new proteins, whose function is unknown. Currently, protein databases contain the sequences of about 1,800,000 proteins, of which more than half are partially or completely uncharacterized [1]. Typically, proteins are analyzed by searching for homologous proteins that have already been characterized. Homology establishes the evolutionary relationship among different organisms, and the biological properties of uncharacterized proteins can often be inferred from those of homologous proteins. However, detecting homology between proteins can be a difficult task.
Our ability to detect subtle similarities between proteins depends strongly on the representations we employ for proteins. Sequence and structure are two possible representations of proteins that hinge directly on molecular information. The essential difference between the representation of a protein as a sequence of amino acids and its representation as a 3D structure traditionally dictated different methodologies, different similarity or distance measures and different comparison algorithms. The power of these representations in detecting remote homologies differ markedly. Despite extensive efforts, current methods for sequence analysis often fail to detect remote homologies for sequences that have diverged greatly. In contrast, structure is often conserved more than sequence [2–4], and detecting structural similarity may help infer function beyond what is possible with sequence analysis. However, structural information is sparse and available for only a small part of the protein space.
One may argue that the weakness of sequence based methods is rooted in the underlying representation of proteins, the model used for comparison and/or the comparison algorithm, since in principle, according to the central dogma of molecular biology, (almost) all the information that is needed to form the 3D structure is encoded in the sequence. Indeed, in recent years better sequencebased methods were developed [5–8]. These methods utilize the information in groups of related sequences (a protein or domain family) to build specific statistical models associated to different groups of proteins (i.e. generative models) that can be used to search and detect subtle similarities with remotely related proteins. Such generative models assume a statistical source that generates instances according to some underlying distributions, and model the process that generates samples and the corresponding distributions.
When seeking a new similarity measure for proteins that departs from sequence and structure, a natural question is what is the "correct" encoding of proteins? Several works studied the mathematical representation of protein sequences based on sequence properties such as amino acid composition or chemical features [9–12]. However, these representations had limited success, since they did not capture the essence of proteins as ordered sequences of amino acids.
Recently, alternative representations of protein sequences based on the socalled kernel methods were proposed. These methods are drawn from the field of machine learning and strive to find an adequate mapping of the protein space onto the Euclidean space where classification techniques such as support vector machines (SVM) or artificial neural networks (ANN) can be applied. Under the kernel representation, each protein is typically mapped to a vector in a (highdimensional) feature space, and the resulting vector is termed feature vector. Subsequently, an inner product is defined in the feature space in order to estimate the (dis)similarity among different proteins. A major advantage of the kernel methods is that with an adequate choice of the kernel function, the feature vectors need not be computed explicitly in order to evaluate the similarity relationships. In addition, the users may build a specific feature space such that the kernel function directly estimates these relationships. However, string kernels do not easily lend themselves to this property, and therefore they need to be computed explicitly. The main difference between the different kernel methods reside in the definition of feature elements which are either related to the parameters of some generative process for each group of related proteins or some measure of similarity among the protein sequences.
For instance, the SVMFisher algorithm [13] uses the Fisher kernel which is based on hidden Markov models (HMMs). The components of the feature vector are the derivatives of the loglikelihood score of the sequence with respect to the parameters of a HMM that has been trained for a particular protein family. Tsuda et. al. [14] implemented another representation based on marginalized and joint kernels and showed that the Fisher kernel is in fact a special case of marginalized kernel. They also experimented with the marginalized count kernels of different orders, that are similar to the spectrum kernel which was first introduced by [15]. The spectrum kernel is evaluated by counting the number of times each possible klong subsequence of amino acids (kmer) occurs in one given protein. The marginalized count kernel takes into account both the observed frequency of different subsequences and the context (e.g. exon or intron for DNA sequences). The mismatchspectrum kernel [16] is a generalization of the spectrum kernel that considers also mutation probabilities between kmers which differ by no more than m characters. The homology kernel [17] is another biologically motivated sequence embedding process that measures the similarity between two proteins by taking into account their respective groups of homologous sequences. It can be thought of as an extension of the mismatchspectrum kernel by adding a wildcard character and distinguishing the mismatch penalty between two substrings depending on whether the sequences are grouped together or not.
The covariance kernel is another type of kernel that uses a framework similar to the one employed by our method. The covariance kernel approach is probabilistic and much work is focused on the implementation of the generative models. In [18], the covariance kernel is based on the mutual information kernel which measures the similarity between data samples and a certain generative process. The generative process is characterized by a mediator distribution defined between the (usually vague) prior and the posterior distribution. On the other hand, [19] focuses on the representation of biological sequences using the probabilistic suffix tree [20] as the generative model for different groups of related proteins. The proposed kernel generates a feature vector for protein sequences, where each feature corresponds to a different generative model and its value is the likelihood of the sequence based on that model. Finally, we mention another related work [21] that uses the notion of pairwise kernel. Under this framework, each protein is represented by a vector that consists of pairwise sequence similarities with respect to the set of input sequences. It is also worth mentioning the many related studies in the field of natural language processing and text analysis. For example, an approach that in some ways is similar to the pairwise and covariance kernels and in other ways is related to the spectrum kernel approach, is used in [22] to represent verbs and nouns in English texts, with nouns represented as frequency vectors over the set of verbs (based on their association with different verbs) and vice versa. The nouns and verbs are then clustered based on this representation. Studies that attempted to devise automatic methods for text categorization and webpage classification often use similar techniques, where the text is represented as a histogram vector over a vocabulary of words (e.g. [23]).
Here we study a general framework of protein representation called the distanceprofile representation, that utilizes the global information in the protein space in search of statistical regularities. The representation draws on an association measure between input samples (e.g. proteins and protein families) and can use existing measures of similarity, distance or probability (even if limited to a subset of the input space for which the measure can be applied). This representation induces a new measure of similarity for all protein pairs based on their vectorial representations. Our representation is closely related to the covariance and pairwise kernels described above. However, it is the estimation of pvalues through statistical modeling of the background process, coupled with the transformation to probability distributions, the noise reduction protocols and the choice of the distance function that result in a substantial impact on the performance, and we demonstrate how an adequate choice of the score transformation and the distance metric achieves a considerable improvement in detection of remote homologies.
This paper is organized as follows. We first introduce the notion of distanceprofile. We describe how to process the feature vectors through noise reduction, and pvalue transformation followed by normalization. We compare the performance of our new method against several standard algorithms by testing them on a large set of protein families.
Results
The distanceprofile representation
Our goal is to seek a faithful representation of the protein space that will reflect evolutionary distances even if undetectable by means of existing methods of comparison. We explore a technique based on the distanceprofile technique described in [25] and its derivative as applied to protein sequences in [26]. The power of the representation stems from its ability to recover structure in noisy data and boost weak signals [25, 26].
The distanceprofile representation is simple and can be applied to arbitrary spaces $X$, $Y$ if there exist an association measure between instances of $X$ and instances of $Y$ such as a distance function, similarity function or a probability measure. Given an instance X in the input space $X$, a reference set {Y_{1}, Y_{2}, ... Y_{ n }} of entities in $Y$ (e.g. proteins or protein families, sequences or generative models) and an underlying association measure, we associate with the instance X a position in a high dimensional space, where every coordinate is associated with one member of the reference set and its value is the similarity with that particular reference object. I.e., we map X to a vector of dimension n in the host space
This simple representation leads to the definition of a new distance or similarity measure among samples based on their vectorial representation, and when the reference set is identical to the input space ($X$ = $Y$), an iterative application of this representation can be used to form hierarchical clustering over the input samples [25]. In this paper we demonstrate the application of this method to the problem of homology detection between distantly related proteins.
One might observe the resemblance of our method with pairwise kernels and covariance kernels mentioned in the 'Background' section. However, it is the processing of the feature vectors and the choice of the metric, as is laid out next, which are the crucial ingredients that differentiate our method from the previous studies. As exemplified in this paper, under the proper transformations the distanceprofile representation has mathematical and statistical interpretations that have other implications, and it is these transformations that deem this method very effective for homology detection, database search and clustering.
The reference set
Remotely related proteins usually share little sequence similarity, however, they are expected to have similar structures. Therefore, as a reference set for our experiments we chose a nonredundant structure library consisting of domain structures that represent the current protein structure space. The set is derived from the SCOP database [27], release 1.57. Specifically, we used the Genetic Domain Sequence dataset that we downloaded from the Astral webpage [28]. The dataset is obtained from 14,729 entries in the Protein Data Bank (PDB) [29] and contains only protein domains with less than 40% identity between pairs of sequences. Our library consists of 3,964 distinct SCOP domains covering in total 644 folds, 997 superfamilies, and 1,678 families. For notation purpose, we denote this library of template proteins by SCOPDB.
The association measure
We rely on "structureaided" sequence alignment to bridge the gap between the sequence space and structure space. Given the library of structures $Y$ = {Y_{1}, Y_{2}, Y_{3}, ..., Y_{ n }}, every protein sequence P is mapped to a structurespecific ndimensional feature vector, as is described above, where the association measure S(P, Y_{ i }) is the similarity score of the sequencestructure alignment between the sequence P and the template structure Y_{ i }, computed with the FUGUE threading algorithm [30] (see Appendix). From this point on we discuss the application of the distanceprofile representation with threadingbased association measures. However, we note that the methods described in this paper can be used to process feature vectors using other association measures. In the 'Discussion' section we show that a similar approach applied to sequenceprofile alignment scores also produces a considerable improvement in detecting remote homologies.
Processing feature vectors: pvalue conversion and normalization
The choice of the underlying association function S(P, Y_{ i }) can have a drastic impact on the effectiveness of the representation and we tested several variations.
The score association measure
A possible choice is obviously the score reported by the algorithm that compares entities of $X$ with entities of $Y$. (for example, the zscore reported by the FUGUE threading algorithm). We denote feature vectors that are based on the score association measure by P^{ score }.
The pvalue association measure
If the association measure is distributed over a wide range, the most significant scores will inevitably shadow other numerically less important but still significant matches, thus reducing the sensitivity of the representation. This is the case with most types of similarity scores, including the threading zscores assigned by the FUGUE program.
F(x) = Prob(x' ≤ x) = e^{φ(x)} where φ(x) = e^{λ(xμ)}. (2)
Based on this background distribution we replace the original zscores with a new association measure such that S'(P, Y_{ i }) = F(S(P, Y_{ i })) where S(P, X_{ i }) is the similarity zscore reported by FUGUE. With this transformation, all coordinates are bounded between 0 and 1, with high zscores transformed to values close to 1. Note that the pvalue of a given zscore x is pvalue(x) = 1  F(x). We denote feature vectors that are based on the F(x) association measure by P^{ pvalue }. It should also be noted that in practice F(x) = 1  pvalue(x) = 1 for large x because of machine precision limitations. Therefore, the pvalues that are associated to significant zscores (typically above 5) are approximated by their empirical distribution, thus allowing distinction between a pair of highly significant yet numerically disparate zscores, e.g. 10 versus 60.
The probability association measure
the third variation we tested is based on a simple normalization of each feature vector to form a probability distribution. This transformation enables us to explore distance measures that are suited for probability vectors, as described in section 'Metrics and score functions'. Indeed, in this representation, the normalized vector entries can be considered as the coefficients of a mixture model where the components models are the protein structures, each one inducing a different probability distribution over the protein sequence space. This interpretation emphasizes the similarity with covariance methods which also resort to a probabilistic representation of different protein families as described in the 'Background' section. We denote feature vectors that are based on this association measure by P^{ prob }.
Reducing noise: sparse feature vectors
Our reference set is composed of proteins that belong to different protein families and folds (see section 'The reference set'). Within that data set no two proteins share more than 40% sequence identity. Therefore, for a given query protein we expect to observe only a few significant similarity values in the vector P. That is, the entries that correspond to the structural templates of protein families that are related to the query. In other words, the feature vectors contain many entries that are essentially random and meaningless. These random numbers will contribute to the differences between feature vectors, thus masking possibly significant similarities. To reduce noise due to unrelated proteins we eliminate all entries with zscore below a certain threshold τ, or pvalue above a certain threshold τ', to reflect the fact that the corresponding sequencestructure pair is considered irrelevant (Another alternative is to weight the differences by the significance of the measurements. However, to speed up the processing and comparison of feature vectors we adopted the threshold approach.) The parameter τ(or τ') is optimized to maximize performance, as described in section 'Parameter optimization'. Note that entries with low zscores that are filtered in this step (assigned 0 zscore) remain zero under the transformation to the cdf pvalue as described above. The processed feature vector is denoted by ${\widehat{P}}^{score}$.
Illustration of noise reduction, pvalue conversion and normalization on the feature vectors associated with proteins d 1qmva_ (Thioredoxin peroxidase 2) (denoted by c.47.1.10.3) and d 1hd 2a_ (Peroxiredoxin 5) (denoted by c.47.1.10.4). We display the 2163th up to the 2170th entries of the different feature vectors. The zscore cutoff value τ is set at 3.5. The feature vector for c.47.1.10.3 reaches its maximum at the 2165th position which corresponds to the selfalignment zscore.
Representation  Sequence  2163th to 2170th entries of the feature vectors  

P ^{score}  c.47.1.10.3  0  5.170  63.210  6.420  15.790  4.150  0  1.590 
c.47.1.10.4  1.980  1.730  7.070  53.530  6.380  3.290  0  0  
${\widehat{P}}^{score}$  c.47.1.10.3  0  5.170  63.210  6.420  15.790  4.150  0  0 
c.47.1.10.4  0  0  7.070  53.530  6.380  0  0  0  
P ^{pvalue}  c.47.1.10.3  0  0.998916  0.999921  0.999163  0.999619  0.996863  0  0 
c.47.1.10.4  0  0  0.999234  0.999888  0.999158  0  0  0  
P ^{prob}  c.47.1.10.3  0  0.052803  0.052856  0.052816  0.052840  0.052694  0  0 
c.47.1.10.4  0  0  0.077210  0.077260  0.077204  0  0  0 
Metrics and score functions
Under the distanceprofile representation, the similarity (distance) between two protein sequences P and P' is defined as the similarity (distance) of their corresponding feature vectors
S(P, P') = f(P, P')
The function f can be a similarity function or a distance function and we considered several different variants. We tested the L_{2} norm (the Euclidean metric) and the L_{1} norm (the Manhattan distance). For probability distributions we also tested the JensenShannon (JS) measure of divergence [34]. Given two probability distributions p and q, for every 0 ≤ λ ≤ 1, their λJensenShannon divergence is defined as
where D^{ KL }[pq] is the KullbackLeibler (KL) divergence [35] defined as ${D}^{KL}[pq]={\displaystyle {\sum}_{i}{p}_{i}{\mathrm{log}}_{2}\frac{{p}_{i}}{{q}_{i}}}$ and r = λ p + (1  λ)q can be considered as the most likely common source distribution of both distributions p and q, with λ as a prior weight (without a priori information, a natural choice is λ = 1/2). Unlike the KullbackLeibler measure, the JS measure is symmetric and bounded. It ranges between 0 and 1, where the divergence for identical distributions is 0. This measure has been used successfully in [7, 36, 37] to detect subtle similarities between statistical models of protein families and in [38] for automatic domain prediction from sequence information.
Formally, consider the two pvalue feature vectors ${P}_{1}^{pvalue}=({p}_{1}^{1}\dots {p}_{n}^{1})$ and ${P}_{2}^{pvalue}=({p}_{1}^{2}\dots {p}_{n}^{2})$ that are obtained by mapping each zscore z_{ i }to its pvalue p_{ i }using the EVD background distribution as described in section 'Processing feature vectors'. Their pvalue distance is defined as
In practice, the PD score is evaluated through its logarithm:
(It should be noted that here the measure pvalue(z) is used directly to define the feature values, as opposed to F(z) = 1  pvalue(z) that was used before when compiling the feature vectors P^{ pvalue }. However, to simplify notation we also refer to these feature vectors as P^{pvalue.)}
To distinguish all the measures discussed in this section from the association measures discussed in section 'Processing feature vectors', we refer to all of them from now on as distance metrics, although they are not necessarily metrics or distance functions.
Discussion
Dataset preparation
We use the SCOP classification of protein structures [27] as our benchmark. The SCOP database is built mostly based on manual analysis of protein structures and is characterized by a hierarchy with four main levels: class, fold, superfamily and family. Proteins that belong to the same family display significant sequence similarity that indicates homology. At the next level (superfamily), families are grouped into superfamilies based on structural similarity and weak sequence similarity (e.g. conserved functional residues). Proteins that belong to different families within the same superfamily are considered remotely related. It is this level that has been used in many studies to evaluate sequence comparison algorithms (e.g. [7, 39, 40]). The challenge is to automatically detect similarities between families within the same superfamily, that were established manually by the SCOP experts.
To determine the optimal parameters for the distanceprofile representation and compare its performance to other algorithms, we split the library SCOPDB into a training set and a test set. Since our purpose is to test the ability to find remotely related proteins at the superfamily and fold levels, we first discard all proteins that have fewer than 5 remote homologs in SCOPDB (i.e. are in superfamilies of size 5 or less). From the remaining 2,570 sequences we randomly select 100 for the training set, and the rest (2,470 sequences) are compiled into the test set.
Performance indices
To evaluate the performance of a given method for a specific query protein we compare the protein against SCOPDB, sort the results and assess the correlation of the sorted list with established homology relations. In our experiments we consider two proteins to be related (a positive pair) if they belong to the same SCOP superfamily. All other pairs are treated as negatives. We also consider a more relaxed definition, where proteins are deemed related if they belong to the same SCOP fold.
A popular measure of performance used in signal detection and classification is the ROCk measure [41]. This is the cumulative count of positive samples detected until k negative samples are met in the sorted list of results. We use four different indices to assess performance, all are variations on commonly used sensitivity and accuracy measures. These indices measure the ability of a given algorithm to recognize different levels of structural similarity between protein sequences and within neighborhoods of varying sizes:

The ROC1 superfamily index (ROC1S).

The ROC1fold index (ROC1F).

The topsuperfamilysuperfamily index (TSS).

The topfoldfold index (TFF).
Given the sorted list of results for a query protein p, the ROC1S index totals the number of proteins in the same superfamily as p that are observed from the top of the sorted list until the first false match (i.e. different superfamily) appears. Likewise, the ROC1F index is defined by counting the number of proteins in the same fold as p from the top of the sorted list until the first match that involves two proteins with different folds. The last two indices are characterized by the following generic definition: the topXY index for a protein p counts the total number of proteins sharing the same Y SCOP denomination among the n_{ X }closest sequences of p, where n_{ X }is the total number of sequences in the library that have the same X SCOP denomination as p itself (For example, to compute the topfoldfold index for a query protein that belongs to a SCOP fold containing n proteins, we look at the top n proteins in the sorted list and count how many of them are actually in the same fold as the query protein). Selfsimilarity is ignored in the assessment of all performance indices.
Note that all these indices are closely related to sensitivity measures at different levels, however, the relevant neighborhood is calibrated on a persuperfamily/fold basis. And while the two ROC indices stop as soon as one false match is encountered, the topXY indices credit a method that detects many true positives at the top even if mixed with a few false positives. Therefore, a method yielding a lower ROC1 index but higher TSS (or TFF) should be still considered successful since it clusters the query close to a larger number of related objects.
To obtain the overall performance of a method with respect to a set of queries, we simply take the sum of all their corresponding performance indices as the global result. I.e. given a query set Q = {q_{1}, ..., q_{ n }}, the performance of a method M, using index I is
where I(M, q_{ i }) is the performance of the method M for the protein q_{ i }(for example, TFF(FUGUE, q_{ i }) is the TFF performance of FUGUE on protein q_{ i }).
Normalized performance indices
The global performance indices might be affected by the specific make up of the query set, since superfamilies and folds vary greatly in size. In order to reduce a potential bias due to large superfamilies/folds that perform very well, we use also normalized performance indices. To compute these, we divide each index by its upper bound, i.e. the total number of proteins in the SCOPDB library that are classified to the same superfamily/fold as the query (except itself). For example, for a query protein q that belongs to a fold F of size n_{ F }the normalized TFF measure is given by
TFF^{ N }(M, q) = TFF (M, q)/(n_{ F } 1)
This ratio is essentially the sensitivity of the method M on the query q, over a match list of size n_{ F }. The size of the relevant match list changes with each query.
The resulting ratios are then averaged at the superfamily level so as to obtain a representative average performance index per protein superfamily that is bounded between 0 and 1. Finally, the final index is computed by averaging over all the representative indices. I.e. given a query set Q = {q_{1}, ..., q_{ n }} that are classified to k different superfamilies F_{1}, ..., F_{ k }with n_{ i }queries in superfamily F_{ i }, then the overall performance of a method M, using the normalized index I^{ N }is given by
Parameter optimization
Our method depends on three parameters: the noise reduction level (the zscore cutoff threshold), the association measure and the distance metric. We consider all possible combinations among five zscore cutoff values (τ= 2.5 to 4.5 by increment of 0.5), three association measures that are P^{ score }(zscore), P^{ pvalue }(pvalue) and P^{ prob }(prob), and four distance metrics: L_{1}, L_{2}, the JS divergence measure and the new pvaluedistance (PD) measure. Note that the JS measure is only applicable to normalized vectors since it requires the input vector to represent a probability distribution and the PD measure is only applicable to the pvalue vectors.
Parameter optimization of the distanceprofile method: performance indices based on the training set. The normalized performance indices are given in parentheses and expressed in percentages. For instance, the ROC1S and TFF indices obtained under the P^{pvalue} representation with zscore threshold τ= 4 and L_{2} distance metric amount to 267 and 363, respectively. The PD measure was evaluated only for τ= 3 to 4.5.
Index  τ  Association measure/Distance metric  

zscore  pvalue  prob  
L _{1}  L _{2}  L _{1}  L _{2}  PD  L _{1}  L _{2}  JS  
ROC1S  2.5  30 (2.38)  133 (13.87)  10 (0.14)  12 (0.16)  309 (21.67)  75 (1.11)  333 (22.62)  
3  90 (7.95)  156 (16.88)  59 (2.14)  59 (2.14)  453 (41.60)  365 (29.91)  146 (4.75)  375 (30.00)  
3.5  134 (11.98)  164 (18.25)  184 (13.23)  188 (13.55)  479 (44.63)  442 (40.04)  262 (16.33)  438 (39.95)  
4  167 (16.82)  164 (18.71)  262 (28.28)  267 (29.00)  475 (44.53)  449 (42.32)  373 (33.58)  438 (42.09)  
4.5  180 (19.75)  159 (18.31)  290 (34.20)  299 (35.35)  467 (42.65)  443 (41.23)  373 (35.86)  427 (41.02)  
ROCIF  2.5  31 (2.38)  138 (11.99)  11 (0.16)  13 (0.17)  426 (18.78)  149 (1.31)  460 (20.16)  
3  92 (7.16)  160 (14.43)  59 (2.03)  59 (2.03)  585 (34.69)  478 (26.15)  246 (4.79)  497 (26.30)  
3.5  137 (10.80)  166 (15.23)  185 (12.00)  189 (12.30)  594 (36.60)  530 (33.44)  386 (14.49)  536 (33.27)  
4  168 (14.06)  165 (15.39)  264 (23.75)  272 (24.47)  575 (36.40)  477 (34.70)  514 (27.89)  455 (34.04)  
4.5  181 (16.43)  160 (15.24)  295 (28.41)  306 (29.52)  540 (34.97)  461 (33.36)  492 (29.56)  446 (32.89)  
TSS  2.5  96 (5.38)  183 (17.06)  68 (1.62)  73 (1.78)  433 (29.71)  156 (3.49)  449 (31.46)  
3  144 (9.91)  193 (19.18)  119 (4.10)  119 (4.10)  545 (46.70)  503 (40.85)  209 (7.03)  506 (40.30)  
3.5  174 (13.55)  193 (19.78)  232 (14.84)  234 (15.07)  546 (47.81)  526 (45.53)  294 (18.12)  531 (45.95)  
4  195 (18.07)  180 (19.46)  306 (31.34)  313 (31.77)  545 (48.11)  530 (47.40)  382 (33.80)  530 (47.18)  
4.5  203 (20.74)  172 (18.96)  325 (36.22)  343 (37.64)  509 (45.21)  511 (45.29)  396 (36.67)  509 (45.73)  
TFF  2.5  171 (5.75)  301 (15.48)  126 (2.00)  136 (2.31)  892 (28.59)  383 (4.41)  915 (30.44)  
3  220 (9.66)  301 (17.02)  150 (4.15)  151 (4.16)  963 (41.52)  908 (36.98)  448 (7.52)  913 (36.47)  
3.5  260 (13.22)  302 (17.42)  263 (14.08)  267 (14.32)  875 (40.64)  818 (38.86)  556 (16.84)  831 (39.42)  
4  277 (15.98)  297 (17.40)  350 (26.28)  363 (26.94)  762 (40.05)  751 (39.60)  633 (28.87)  777 (39.38)  
4.5  295 (18.42)  288 (17.14)  401 (30.29)  395 (31.26)  613 (37.55)  616 (37.51)  618 (30.84)  664 (38.03) 
In conclusion, these results suggest that the best performance is achieved with the pvalue association measure, the pvaluedistance (PD) metric and a zscore cutoff threshold τ= 3.5. The same conclusions are reached when using the normalized performance indices. It should be noted that although the prob association measure is not as good as the pvalue measure (combined with the PD metric), it still produces very good performance overall (with the JS or the L_{1} metrics), further justifying the statistical interpretation of our representation and its equivalence with the coefficients of a mixture model over independent sources (as discussed in section 'Conclusions').
Figure 3a,b illustrate the distribution of the pairwise L_{1}distances between proteins under the representations P^{ score }and P^{ prob }, respectively. We observe that the pairwise distance between feature vectors P^{ score }spreads over a very large range and it is difficult to set a natural threshold below which feature vectors can be considered similar. In contrast, for P^{ prob }, about 95% of the pairwise distances are equal to 2, the maximum L_{1} distance. This is the distance between pairs of normalized feature vectors (probability distributions) whose set of nonzero features do not overlap. The distribution shown in Figure 3b only focuses on those pairwise distances smaller than 2. The combination of noise reduction, pvalue conversion and normalization procedures effectively delimits the range of the pairwise distances, and any distance smaller than 2 indicates common features between the feature vectors.
Performance Comparison
Comparison between BLAST, PSIBLAST, FUGUE, DPFUGUE (τ= 3.5, PD, pvalue) and URMS on the training set. The normalized indices (in percentages) are given in parentheses. PSIBLAST's parameters were set to h = 1e^{5}, e = 100 and j = 10 (although no improvement was observed after the fourth iteration). FUGUE was run using the default parameters.
Index  BLAST  PSIBLAST (2 to 4 iterations)  FUGUE  DPFUGUE  URMS  

2 iterations  3 iterations  4 iterations  
ROC1S  212 (25.64)  265 (29.80)  281 (30.91)  279 (30.68)  296 (35.03)  479 (44.63)  610 (60.20) 
ROC1F  212 (21.65)  265 (24.58)  281 (25.23)  279 (25.12)  306 (28.64)  594 (36.60)  719 (51.58) 
TSS  278 (29.32)  315 (33.48)  335 (34.54)  335 (34.54)  422 (40.20)  546 (47.81)  797 (69.07) 
TFF  344 (24.76)  386 (28.08)  410 (28.84)  409 (28.81)  659 (34.72)  875 (40.64)  1681 (67.30) 
Comparison between BLAST, PSIBLAST, FUGUE and DPFUGUE (τ= 3.5, PD, pvalue) on the test set. The normalized indices (in percentages) are given in parentheses.
Index  BLAST  PSIBLAST  FUGUE  DPFUGUE 

ROC1S  6658 (24.42)  9266 (28.75)  12143 (35.10)  23307 (45.55) 
ROC1F  6694 (18.86)  9319 (21.93)  12620 (26.91)  24693 (35.21) 
TSS  11379 (28.51)  15149 (32.62)  21176 (41.03)  30238 (49.62) 
TFF  13986 (22.72)  18172 (25.85)  28473 (33.68)  40376 (40.66) 
The distanceprofile representation over sequenceprofile metrics
To test the effectiveness of the distanceprofile method on other types of input we applied it to feature vectors that were generated with PSIBLAST, a sequence to profile alignment algorithm [5]. The feature values are set to the log (evalue) of the similarity score, as reported by PSIBLAST after four iterations, unless the program converged before (as Table 3 demonstrates, the performance plateaus after four iterations). The PSIBLAST evaluebased feature vectors are processed in a similar fashion to the FUGUE zscorebased feature vectors. Each evalue e is mapped to its corresponding pvalue pvalue(e) = 1  exp(e) as in [44] and the value of the corresponding feature is defined as 1  pvalue(e) = exp(e). If the evalue is greater than a given threshold τ' then we reset the value of the feature. Finally, the normalization converts the resulting feature vectors to probability distributions.
The parameters are optimized using a similar procedure to the one described in section 'Parameter optimization'. The optimal parameters are sought among the combinations of 4 evalue cutoff thresholds (τ' = 0.01, 0.1, 1, 10), two association measures (P^{ pvalue }(pvalue), P^{ prob }(prob)) and three possible metrics (L_{1}, L_{2} and JS divergence measure). In this case, the best combination based on the training set is (τ' = 10, L_{1}, prob).
Comparison between PSIBLAST_{4} (PSIBLAST with 4 iterations) and DPPSIBLAST (τ' = 10, L_{1}, prob) based on the training set. The normalized indices (in percentages) are given in parentheses.
Index  PSIBLAST_{4}  DPPSIBLAST 

ROC1S  279 (30.68)  339 (35.16) 
ROC1F  279 (25.12)  339 (28.86) 
TSS  335 (34.54)  462 (39.63) 
TFF  409 (28.81)  510 (33.06) 
The effect of multiple sequence alignment on the performance
All our experiments were performed in a singlequery mode. I.e. in each test case a single query sequence is compared against the database. However, there are multiple reports [40, 45, 46] that suggest that significant performance gain can be obtained when using a multiple sequence alignment (MSA) or a sequence profile as a query.
To test the effect of MSA on the performance we had the change our experimental setup. We tested the impact of the new setting on PSIBLAST and FUGUE based on 7 SCOP families that were randomly chosen from all families for which the performance of PSIBLAST and FUGUE was poor. For each family we generated a MSA of all sequences in the family using CLUSTALW [47]. Each MSA was also converted to a position specific scoring matrix (profile). These MSA were used as queries for FUGUE (instead of the individual sequences) and the profiles as input for PSIBLAST, in search for related sequences in SCOPDB. To fairly compare the results of PSIBLAST and FUGUE in the MSA mode to the DP method we had to run our method under a similar setup. The "MSA" mode of the DP method utilizes the information from all the sequences in a protein family in the same spirit a MSA does so, by combining the distance profiles associated to each member of the SCOP family. For each sequence in SCOPDB, we take the average of its distance versus each member of the family in question in order to compute the "familyspecific" distances with respect to SCOPDB.
Performance indices of FUGUE, PSIBLAST and DPFUGUE under the single and the MSA query modes. The counts exclude those sequences in the SCOP family in question (that were used to build the MSA). In single query mode we report the average performance. Results are reported using the ROC1S and the TSS indices. Similar trends were observed with the ROC1F and the TFF indices.
Family  Mode  ROC1S  TSS  

FUGUE  PSIBLAST  DPFUGUE  FUGUE  PSIBLAST  DPFUGUE  
a.3.1.1  single  0.4  0.4  2.6  2.8  1.7  3.5 
MSA  0  2  4  2  3  4  
a.3.1.4  single  2.3  1.6  11  6.3  5.3  12.3 
MSA  1  4  12  4  7  12  
a.39.1.5  single  2.8  5  12  7.7  9.5  13.4 
MSA  15  5  14  16  9  15  
b.47.1.4  single  1.6  0  22.3  8.6  1.6  22.3 
MSA  3  1  23  17  5  23  
c.2.1.3  single  0.2  0.1  3.1  7.6  5.4  15.4 
MSA  2  0  0  14  6  26  
c.3.1.2  single  1.8  0.7  9  8.7  5.5  14.1 
MSA  1  3  12  20  8  19  
c.47.1.2  single  4.1  2.3  16.1  8.8  4.8  19.3 
MSA  12  5  24  19  7  25 
We should comment that the MSA setup differs from our original idea of using the DP representation to perform unsupervised clustering of objects based on their distance profile. In the unsupervised learning mode we do not have information on the family association of each sequence, and therefore it is difficult to define the exact set of related sequences from which to generate a multiple sequence alignment.
Superfamily and fold prediction with the distanceprofile method
We tested the power of our method on a new set of protein sequences that were added to SCOP after we compiled our benchmark. Our goal was to test if the method can classify new sequences to their correct class. The new set consists of proteins in release 1.67 of SCOP that were not included in our SCOPDB dataset (based on release 1.57 of SCOP) and either belong to new families within existing superfamilies, new superfamilies within known folds or completely new folds. For instance, the family b.2.3.4 did not exist in the SCOP 1.57 database. This family is part of the b.2.3 superfamily that in release 1.57 contains the families b.2.3.1, b.2.3.2, b.2.3.3 and a total of 5 representatives in our reference set.
In total we found 624 sequences belonging to 453 new families within known superfamilies, 267 sequences associated to 182 new superfamilies within known folds and 375 sequences in 245 new folds, all with less than 40% identity between pairs of sequences. Each one of these new sequences was compared against all the sequences in SCOPDB using all of the methods evaluated in this paper, and the matches were sorted based on the score or distance, as before. To apply the distanceprofile method the sequences were first processed and mapped to feature vectors as described in section 'Results'.
Class prediction for new SCOP sequences. Comparison between BLAST, PSIBLAST, FUGUE, DPFUGUE (τ= 3.5, PD, pvalue). The normalized indices (%) are given in parentheses.
Index  BLAST  PSIBLAST  FUGUE  DPFUGUE  

New families  ROC1S  187 (7.03)  211 (7.71)  380 (16.35)  847 (22.56) 
ROC1F  216 (5.68)  239 (6.34)  450 (11.65)  1185 (17.98)  
TSS  419 (8.97)  458 (10.31)  1014 (24.66)  1412 (27.58)  
TFF  715 (7.78)  827 (8.79)  2518 (19.61)  3512 (22.69)  
New superfamilies  ROC1F  8 (0.51)  11 (0.56)  42 (0.89)  187 (1.24) 
TFF  156 (1.19)  169 (1.25)  853 (3.58)  1097 (3.33) 
Class prediction for new SCOP sequences with little sequence identity. In this case the analysis is limited to new SCOP sequences with less than 20% sequence identity with respect to the reference set.
Index  BLAST  PSIBLAST  FUGUE  DPFUGUE  

New families  ROC1S  41 (2.52)  43 (3.14)  73 (8.44)  156 (11.77) 
ROC1F  43 (1.95)  45 (2.57)  75 (6.41)  181 (9.89)  
TSS  94 (4.85)  101 (5.36)  208 (15.80)  255 (17.11)  
TFF  142 (3.90)  161 (4.49)  364 (12.42)  602 (13.77)  
New Superfamilies  ROC1F  1(0.09)  1 (0.09)  6 (0.36)  50 (0.738) 
TFF  20 (0.80)  23 (0.92)  119 (2.39)  108 (2.22) 
Conclusion
We study a new method for remote homology detection that utilizes global information on the proximity of entities in the protein space. Our method relies on the distanceprofile representation of proteins and protein families that maps each query protein to a highdimensional feature space, where the coordinates are determined by some association measures with respect to a reference set. These vectors are then processed and transformed to sparse feature vectors that are treated as statistical fingerprints of the query proteins. We experimented with several different types of association measures and demonstrated how an adequate choice of distance metric combined with a proper transformation of the feature vectors through noise reduction, pvalue conversion and normalization can greatly increase the performance of homology recognition (or prediction) compared to the existing approaches.
Interestingly, excellent performance is obtained with normalized feature vectors that correspond to probability distributions. The success of the distanceprofile method in general and especially when using probability distributions suggests a relation to mixture models [48]. Specifically, one can consider this representation as the coefficients of a mixture model or of a functional expansion, similar to the Taylor polynomial expansion. Given a set of basis functions such as polynomial functions one can span the complete space of continuous wellbehaved functions with the right coefficients. The same principle applies here as our reference set essentially defines a set of basis functions. In statistical terms, each element of the reference set induces a different probability distribution over the protein sequence space. In our experiments the reference set is composed of protein structures, each one can be perceived as a different generative model. The likelihood of generating a sequence according to a model can be estimated by computing the probability that the sequence will fold into the corresponding structure, as measured with the pvalue association measure over the threading similarity score. Although these probability distributions (that correspond to different elements in the reference set) do not necessarily meet the requirement of orthogonality to be considered "basis functions", a sufficiently diverged set of proteins is expected to have the desired properties. It has yet to be defined more precisely what sufficiently diverged means and the minimal required diversity.
One intriguing aspect that has not been fully addressed is the interaction between the association function, the distance metric and the zscore cutoff value. In some cases the coupling is not surprising. For example, when the prob association function is used, the L_{2} metric clearly underperforms compared to L_{1} and JS metrics, as expected, since the vectors compared correspond to probability distributions. With the other association measures, L_{1} and L_{2} perform similarly. Study of the theoretical aspect of this phenomenon will help understand how to take further advantage of these feature vectors to cluster the protein sequences more accurately.
A word of caution is in order here regarding the evaluation. While SCOP is considered the gold standard, it is not perfect and often one can find misclassifications [7, 24, 49]. While it is hard to estimate the exact rate of errors, it is unlikely that they exceed thousands and we do not anticipate the results to change drastically even if these misclassification were corrected.
We should also mention that the DP representation is not effective for detailed, atomresolution prediction of 3D structure or for sitespecific functional annotation, since it cannot produce alignments. Another weakness is that the distanceprofile representation and the new pvaluedistance measure may fail to distinguish two proteins if they have almost identical "preferences" for the known structures but different preferences for other, unknown structures that are yet to be determined. However, given the current size of the protein structure space, it is expected that for most proteins the available structural information (as embodied in our reference set) is sufficient to estimate their proximity. Indeed, only 20% of the new SCOP 1.67 sequences were assigned to new folds, Clearly, as more structures are determined and integrated into the reference set, the distanceprofile representation is expected to improve.
One potential contribution of this work is the possibility of combining the transformations described in section 'Processing feature vectors' to the feature vectors used by existing kernel methods. These techniques can effectively reduce noise and increase the accuracy of classification of the feature space. In addition, since the feature vectors are typically sparse after noise reduction, an efficient computation of the kernel function can be implemented in a highdimensional feature space.
Finally, a major advantage of the distanceprofile representation is in its great flexibility. The underlying association measure can be based on sequence, structure, predicted function, threading, or any other similarity measure. Clearly, more distinctive association measures will create better representations. Indeed, the FUGUE zscore is clearly a better choice than the PSIBLAST evalue since FUGUE exploits sequencestructure alignment information rather than just sequence alignment information. If the association measure can report a significance value (such as zscore or evalue) that emphasizes extremes and pinpoint the interesting cases, the statistical measure will be preferred over the raw score. This is especially useful when the raw score is meaningless by itself. In these cases one needs a yardstick or a scale to tell what is close and what is far and the statistical estimates provide such a scale. Nevertheless, it is important to note that the distanceprofile representation and the induced similarity measure are quite robust and work well even with raw association scores, noisy or corrupted data, and weak signaltonoise ratio [25]. All our data, including the FUGUE results, the PSIBLAST results and the feature vectors are available at [50].
Appendix
The sequencestructure association measure
Historically, sequencestructure threading [4] was proposed as an alternative approach to predict the structural fold of polypeptide chains. In contrast to ab initio strategies that exploit secondary structure prediction, energy minimization and molecular dynamics to predict the structure of a protein sequence, sequencestructure threading consists of finding nativelike folding structure(s) for a query protein from a database of known structures. Its motivation originates from the observation that proteins adopt a limited number of spatial architectures and that larger proteins are frequently composed of modules that can be found in other proteins.
To perform sequencestructure threading of a query sequence, the process typically starts by obtaining a set of structural conformations from a database of known structures. The amino acid sequence of the query protein is aligned to each conformation in search of an alignment that would produce the minimal total energy (that depends on the structural environment of each reside, or the types of neighboring residues as determined by the sequencestructure alignment). The most likely candidate conformations are the ones yielding the lowest energy. If the energy values are significantly low compared for example to those obtained for shuffled sequences, then these structural conformations can be considered as compatible with the query sequence.
Many different approaches and implementations of threading algorithms have been proposed in the past [30, 51–54]. The exact details of the alignment algorithm and the computation of the total energy vary from one method to another. Unfortunately, most of them are not publicly available to allow an extensive comparison of sequences and structures. Of the few that are available, we chose FUGUE for our study. FUGUE [30] is a sequencestructure alignment algorithm that uses environmentspecific substitution tables and structuredependent gap penalties to evaluate the alignment score. It switches between local and global alignment based on the ratio between the length of the query sequence and that of the structural profile. Previous experimental results have shown that FUGUE outperforms other methods in fold recognition such as PSIBLAST [5], SAMPSIBLAST [32], HMMERPSIBLAST [33], THREADER [51] and GenTHREADER [53]. It is worth mentioning that FUGUE should be considered as a structurebased sequence alignment algorithm rather than a sequencestructure alignment algorithm per se, since it does not involve the definition and the minimization of any energy function to measure the goodness of fit of the query sequence to the template structure. However, its use of substitution tables and structuredependent parameters provides additional information which is unavailable to pure sequencebased methods. In FUGUE, the compatibility of each sequencestructure pair is assessed through the zscore, which measures the departure of the observed threading score value from its mean, normalized by the standard deviation (where the mean and standard deviation are computed based on the distribution of alignment scores over shuffled sequences. Zscores of meaningful matches are then shifted such that the minimal zscore starts at 0. It is claimed that a zscore less than 2 typically implies an uncertain match between the template structure and the query amino acid sequence, and a zscore larger than six implies an almost certain match between the protein and the template folding structure [30].
It should be noted that threading measures are asymmetric. Given two proteins, A and B with known structures, then S(B, A) does not necessarily equal to S(A, B). Actually, for most protein pairs the equality S(B, A) = S(A, B) does not hold, as different protein structures have different "sequence capacities". These capacities affect the ability of an arbitrary sequence to conform with the given structure, and therefore also the probability that the structure will be energetically favorable for the given sequence. However, the asymmetry may be a fundamental feature of the protein space that we may want to preserve and study later on in our analysis.
The statistical significance of the PD measure
In Figure 6b, we plot the distribution of the PD measure over its full range (up to 900) in log scale. We observe that the frequency of occurrence of large PD values (typically above 100) decreases exponentially, i.e. linearly in log scale (the increasingly dispersed pattern at large PD value > 600 is mainly due to data sparsity). This phenomenon can be explained by examining again the definition of the PD measure given in (4). If we consider a zscore as a random variable, its associated pvalue is a random variable uniformly distributed between 0 and 1. Based on the assumption that ${p}_{i}^{1}$ and ${p}_{i}^{2}$ are independent (which is true if the two feature vectors are randomly drawn from our database), max(${p}_{i}^{1}$, ${p}_{i}^{2}$) follows a triangular distribution and log max(${p}_{i}^{1}$, ${p}_{i}^{2}$) is an exponential random variable. Therefore, log PD can be viewed as a sum of exponential random variables. From the statistical literature, we know that the sum of k independent and identically distributed exponential random variables with parameter α can be modeled by the Gamma function.
For large x, we have
where ε is an additive constant. Hence, log f_{ G }(x; k, α) approximately decreases with x in a linear fashion. In our case, clearly the summands in (4) are not independent of each other since the reference set is composed of groups of related sequences. Nonetheless, one may expect that the dependency is not too strong because these sequences share less than 40% sequence identity. The plot indeed suggests that the distribution of the PD measure follows this trend.
Examples of homology detection
Homology detection for a few example query proteins. For each query and method we report the results using the performance indices described in the 'Discussion' section.
Protein  Index  BLAST  PSIBLAST  FUGUE  DPPSIBLAST  DPFUGUE 

b.1.1.1.14  ROC1S  14  26  30  78  66 
ROC1F  14  26  30  78  66  
TSS  31  54  64  86  99  
TFF  39  62  78  90  127  
g.3.11.1.8  ROC1S  4  4  7  6  10 
ROC1F  4  4  7  6  10  
TSS  5  5  11  6  20  
TFF  7  7  24  8  24  
c.2.1.2.26  ROC1S  2  2  5  2  28 
ROC1F  2  2  5  2  28  
TSS  6  6  25  28  42  
TFF  6  6  25  28  42  
c.37.1.13.6  ROC1S  3  4  3  4  3 
ROC1F  3  4  3  4  3  
TSS  7  7  9  7  4  
TFF  7  7  9  7  4 
Closest neighbors of d 2tgf __ (Transforming growth factor alpha) (g.3.11.1.8). For each method we report the top 16 neighbors and their distance/similarity. To highlight relations within families and superfamilies we represent each SCOP domain by its family designation and append a serial number to create a unique identifier. The complete list of SCOP IDs and their numeric designations is available at http://biozon.org/ftp/data/papers/distanceprofile/
BLAST  PSIBLAST  FUGUE  DPPSIBLAST  DPFUGUE  

g.3.11.1.8  1e18  g.3.11.1.8  1e19  g.3.11.1.8  26.970  g.3.11.1.8  0.0000  g.3.11.1.8   
g.3.11.1.9  8e04  g.3.11.1.9  3e04  g.3.11.1.9  9.830  g.3.11.1.9  0.2066  g.3.11.1.10  101.76 
g.3.11.1.10  0.067  g.3.11.1.10  0.028  g.3.11.1.10  9.390  g.3.11.1.7  0.4665  g.3.11.1.9  96.05 
g.3.11.1.7  0.11  g.3.11.1.7  0.057  g.3.11.1.5  5.920  g.3.11.1.10  0.5266  g.3.11.1.3  88.64 
d.158.1.1.1  1.9  d.158.1.1.1  1.2  g.3.11.1.7  5.430  g.3.11.1.3  1.2862  g.3.11.1.16  75.32 
c.69.1.19.1  4.6  c.69.1.19.1  2.0  g.3.11.1.3  5.350  g.3.11.1.1  1.6956  g.3.11.1.1  74.53 
b.40.2.1.5  7.3  b.40.2.1.5  6.3  g.3.11.1.16  5.310  d.158.1.1.1  1.8567  g.3.11.1.7  73.60 
d.159.1.3.3  7.9  d.159.1.3.3  8.3  g.27.1.1.5  5.050  a.102.1.1.2  1.8617  g.3.11.1.14  70.52 
a.118.8.1.5  11  a.118.8.1.5  12  g.3.11.1.14  4.760  d.10.1.3.2  1.8617  g.3.11.1.5  68.21 
b.77.3.1.2  17  b.68.1.1.2  14  g.3.11.1.1  4.040  d.13.1.1.2  1.8617  g.3.11.1.6  61.55 
b.68.1.1.2  21  b.29.1.3.1  15  a.4.10.1.1  3.990  d.15.9.1.1  1.8617  g.27.1.1.4  37.84 
c.1.10.2.1  21  a.138.1.3.1  16  d.158.1.1.1  3.870  d.5.1.1.2  1.8617  g.3.11.1.13  35.63 
b.45.1.2.1  23  b.77.3.1.2  17  g.3.11.1.2  3.790  a.45.1.1.1  1.8719  g.3.11.1.18  30.36 
b.29.1.3.1  24  c.1.10.2.1  22  g.3.11.1.6  3.600  d.169.1.2.1  1.8987  g.3.11.1.17  28.75 
d.58.3.1.2  25  b.45.1.2.1  24  g.26.1.1.1  3.570  c.3.1.2.7  1.9085  g.3.11.1.11  25.26 
Closest neighbors of d 1hdoa_ (Biliverdin IX beta reductase) (c.2.1.2.26). For each method we report the top 33 neighbors and their distance/similarity.
BLAST  PSIBLAST  FUGUE  DPPSIBLAST  DPFUGUE  

c.2.1.2.26  1e110  c.2.1.2.26  1e120  c.2.1.2.26  62.720  c.2.1.2.26  0.0000  c.2.1.2.26   
c.2.1.2.1  0.12  c.2.1.2.1  0.069  c.2.1.2.27  6.310  c.2.1.2.4  1.4380  c.2.1.2.14  102.46 
c.69.1.1.3  0.73  c.69.1.1.3  0.34  c.2.1.3.5  6.270  d.108.1.1.1  1.5288  c.2.1.2.13  91.26 
c.31.1.5.1  1.3  d.108.1.1.1  0.65  c.2.1.2.13  6.250  b.7.1.1.5  1.6281  c.2.1.2.18  90.81 
d.108.1.1.1  1.3  c.31.1.5.1  1.1  c.2.1.2.7  5.810  b.82.2.2.1  1.6281  c.2.1.2.17  85.56 
a.93.1.1.4  2.4  a.93.1.1.4  2.2  c.78.2.1.1  5.620  c.81.1.1.2  1.6326  c.2.1.2.23  84.11 
c.1.2.4.6  3.7  c.4.1.2.2  2.5  c.2.1.2.16  5.570  c.2.1.2.3  1.6598  c.2.1.2.15  84.01 
d.144.1.1.12  3.9  c.1.2.4.6  2.6  c.2.1.3.6  4.930  c.66.1.13.1  1.6834  c.2.1.2.7  80.02 
d.127.1.1.4  5.2  d.144.1.1.12  3.6  c.23.5.1.1  4.740  c.2.1.2.5  1.6908  c.2.1.2.25  80.00 
c.4.1.2.2  5.3  d.127.1.1.4  4.2  c.2.1.6.5  4.520  c.2.1.2.1  1.6957  c.2.1.2.19  73.76 
c.37.1.8.1  6.1  c.37.1.8.1  4.8  c.2.1.6.6  4.410  c.69.1.1.1  1.7249  c.2.1.2.11  73.52 
b.77.2.1.1  7.4  a.118.2.1.5  6.2  d.142.1.2.3  4.370  c.1.8.7.1  1.7272  c.2.1.2.8  72.24 
c.3.1.2.7  7.9  a.104.1.1.4  6.3  c.2.1.2.9  4.330  c.55.7.1.3  1.7273  c.2.1.2.21  71.62 
a.118.2.1.5  8.3  b.77.2.1.1  6.4  c.34.1.1.1  4.210  d.108.1.1.2  1.7273  c.2.1.2.16  69.59 
d.126.1.3.1  10  a.79.1.1.1  6.7  c.4.1.2.2  4.200  c.69.1.1.2  1.7277  c.2.1.2.10  69.55 
c.1.10.1.4  12  c.3.1.2.7  6.8  c.93.1.1.1  4.080  c.69.1.17.4  1.7374  c.2.1.6.12  69.55 
a.104.1.1.4  15  c.60.1.3.1  7.5  c.2.1.5.2  3.970  c.69.1.2.1  1.7412  c.2.1.2.20  69.55 
c.60.1.3.1  15  c.59.1.1.3  7.7  c.2.1.2.15  3.940  c.2.1.2.2  1.7443  c.2.1.2.24  69.55 
a.56.1.1.3  16  c.1.10.1.4  7.8  c.3.1.2.2  3.910  c.69.1.2.2  1.7502  c.2.1.2.27  69.55 
d.104.1.1.13  17  a.118.2.1.2  7.9  c.37.1.2.1  3.900  d.108.1.1.4  1.7548  c.2.1.2.9  69.16 
d.15.9.1.1  18  d.126.1.3.1  8.0  c.37.1.8.4  3.890  b.93.1.1.1  1.7658  c.2.1.2.28  68.02 
a.79.1.1.1  19  c.37.1.10.7  8.8  c.2.1.2.19  3.850  c.2.1.2.6  1.7731  c.2.1.2.2  63.80 
b.60.1.1.7  19  c.31.1.3.3  8.9  a.146.1.1.1  3.820  e.6.1.1.2  1.7799  c.2.1.2.22  61.33 
c.45.1.2.4  19  d.104.1.1.13  9.4  c.2.1.2.28  3.820  b.40.5.1.2  1.8001  c.2.1.6.3  51.80 
b.1.1.5.24  20  c.45.1.2.4  11  c.3.1.5.14  3.770  c.69.1.1.4  1.8074  c.2.1.6.6  48.27 
c.31.1.3.3  20  b.43.3.2.1  12  c.2.1.2.1  3.760  c.69.1.1.3  1.8135  c.2.1.5.7  45.83 
e.28.1.1.1  20  c.93.1.1.10  12  c.2.1.2.17  3.750  c.78.1.1.6  1.8210  c.2.1.2.1  45.75 
b.1.1.1.5  22  b.1.1.5.24  13  c.2.1.3.7  3.740  g.17.1.3.1  1.8259  c.2.1.5.9  42.88 
c.59.1.1.3  22  d.15.9.1.1  13  d.95.1.1.1  3.700  a.138.1.3.5  1.8261  c.4.1.2.2  37.63 
c.93.1.1.10  22  c.29.1.1.2  14  g.18.1.1.12  3.700  a.144.1.1.2  1.8261  c.3.1.2.3  37.57 
e.29.1.1.1  22  c.93.1.1.1  14  c.2.1.9.1  3.680  c.31.1.2.1  1.8261  c.3.1.4.1  37.09 
b.60.1.1.4  23  e.29.1.1.1  14  c.68.1.3.1  3.650  c.31.1.5.1  1.8261  c.2.1.3.  36.66 
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Declarations
Acknowledgements
We thank Dr. Kenji Mizuguchi for providing us with the FUGUE program for our study, and the Astral team for making their data available. We also thank the anonymous reviewers for their very helpful comments and suggestions. The work is supported by the National Science Foundation under Grant No. 0133311 to Golan Yona.
Authors’ Affiliations
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