Key: {{{ = delete; <<< = add
Julia

 Assessing Annotation Transfer: Revisiting the relationships between protein sequence, structure, and function by comparing traditional and probabilistic scores
Abstract
Measuring in a quantitative, statistical sense the degree to which structural and functional information can be “transferred” between pairs of protein sequences at various levels of similarity is an essential prerequisite for robust genome annotation.  To perform such measurements we performed pairwise sequence, structure, and function comparisons on ~30000 pairs of protein domains.  Our domain pairs, which are constructed from the known structures according to the  SCOP protein fold classification, achieve a range of similarities from sharing the same fold to <<<being nearly identical>>> {{{near identity}}}. Our results show that traditional scores for sequence and structural similarity (percent identity and RMS) have the same basic exponential relationship as found earlier. However, as the scale of our survey is much larger than any previous investigations, our results have greater statistical weight and precision. We have been able to express the relationship of sequence and structural similarity using more modern scores, such as probabilistic P-values for both sequence and structure comparison. {{{These overcome some}}} <<<We have thus addressed some of the >>> problems with traditional scores such as determining a conserved core, <<<ascertaining>>> length dependency, etc. We found that the basic exponential sequence-structure relationship is very general; the same relation is observed across all the scoring schemes and between the different classes of secondary structural composition.  To relate function to sequence and structure we assigned various levels of functional similarity to the domain pairs, based on a simple functional classification scheme derived by merging and extending the annotations in the ENZYME database and FLYBASE.  We found sigmoidal relations between similarity in function and sequence, with clear thresholds for different levels of functional conservation.  For pairs whose domains share the same fold, specific function appears to be conserved down to 35% sequence identity, whereas functional class is conserved to 25%. Results of all the pairwise comparisons can be accessed from a web database at http://bioinfo.mbb.yale.edu/align/scop
Introduction 
The problem of genome annotation
Perhaps the most valuable information to be gained from a genome analysis is functional annotation of all the gene products studied.  Unfortunately, of all the proteins whose sequences are known, functions have been experimentally determined for only a very small number (Andrade & Sander, 1997).  Given the current size and accessibility of sequence and structure data, homologues of a newly sequenced gene’s product can be identified via database searches, and likely structure and function assigned to the gene product (Bork et al. 1998).  This is based on the concept that sequence similarity implies structural and functional similarity (Andrade & Sander, 1997; Rychlewski et al. 1999).  However, structural and functional annotations should be transferred with caution.  If a protein is assigned an incorrect function in a database, the error could carry over to other proteins for which structure or function is inferred by homology to the errant protein (Brenner, 1999; Karp, 1998; Karp et al. 1996a; Karp, 1996; Karp et al. 1996b).  In large databases such an error can propagate out of control, presenting a serious quality-control issue as we move to larger multicellular genomes.
Benchmarking fold and function recognition
In this investigation we used manually <<<??? Curated???>>> curated structural and functional classifications as standards {{{to use}}} in analyzing to what degree annotations of a protein’s structure and function can be transferred to a similar sequence.  The knowledge gained from the study can be used to establish confidence levels for structure and function prediction, improving our understanding of how long it will take to annotate accurately an entire genome.
Our simultaneous analysis of relations between sequence and structure, sequence and function, and structure and function (Figure 1) may provide insight into paradigms for functional prediction other than that based on sequence similarity alone.  For example, the relation of distant sequences via transitive intermediates (Gerstein, 1998) has proved useful in detecting <<<the >>> evolutionary relationship between sequences with increased sensitivity (Park et al. 1997), and in finding more distant homologues of a given sequence (Park et al. 1998; Salamov et al. 1999).  Given the relations we examine here, perhaps this strategy might be extended to incorporate structures as intermediates in place of sequences.
Past results
(i) sequence-structure. The transfer of structural annotation is well characterized.  Chothia & Lesk (1986) found that structural divergence, when expressed as the RMS separation of matching alpha-carbons, could be expressed as an exponential function of sequence divergence, {{{when}}} expressed as the fraction of residues that differed between sequences. <<<[preceding sentence not clear--too many “when” clauses]>>> The reliability of structural annotation transferred by homology, then, depends on the sequence identity of the homologous proteins (Chothia & Lesk, 1986).  Flores et al. (1993), Russell and Barton (1994), and Russell et al. (1997) observed the same general trend, and also characterized the conservation of structural features other than the Ca backbone, such as secondary structure, accessibility, and torsion angles.
Others have focused on the limits of sequence comparison<<<,>>> {{{ — }}}specifically around the “twilight zone” (Rost, 1999; [additional references]), <<<i.e., >>> the region of sequence similarities which do not reliably imply structural homology<<<,>>> {{{ — }}}and in establishing cutoffs for significant sequence similarity.  Brenner et al. (1995, 1996, 1998) assessed the effectiveness of the popular FASTA and BLASTP programs and their probabilistic scoring schemes (i.e. the e-value) (Pearson & Lipman, 1988; Pearson, 1996; Altschul et al., 1990, 1994; Karlin & Altschul, 1993). They found that the FASTA e-value closely tracked the number of false positives, i.e. the error rate, and that at a conservative e-value cutoff of .001, the FASTA program could detect nearly all the relationships that <<<would be detected by >>> a full Smith-Waterman comparison {{{would}}} (Smith & Waterman, 1981). Specifically, they found that FASTA with a .001 threshold would find 16% more of the structural relationships in SCOP than would be found by standard sequence comparison with a 40% identity threshold. 

(ii) sequence-function.  Although the exact dependence of functional similarity on sequence and structural similarity is not completely clear, initial indications of a gene product’s function are most often based on simple sequence similarity (Bork et al. 1994; Bork et al. 1998).  However, possibilities for more robust annotation transfer are increasingly becoming available.  The rate at which new protein structures are being solved has increased such that more families of structures are available and can be analyzed to recognize active sites{{{,}}} and<<<,>>> in some cases, catalytic mechanisms (Koonin et al. 1998).  In fact, there are some instances in which the structure of a protein is known before the function is fully understood.  In such cases structural homology can be used to aid functional prediction (Russell & Sternberg, 1997).

(iii) sequence-structure-function. One {{{idea for applying}}} <<<application of >>> the more well-defined sequence-structure relationship to assist function prediction is to {{{initially}}} predict the structure of an uncharacterized sequence and to what degree it is conserved, then predict the function based on the limited repertoire of functions known to occur with that structure.  In that vein, Hegyi & Gerstein (1999) investigated the relationship of fold to function.  By finding Swissprot entries corresponding to SCOP domains, they mapped functions to folds.  Fetrow & Skolnick (1998) have investigated a related approach.  They predicted structural profiles based on threading and ab initio methods, then searched those intermediates against profiles of known structures in order to predict function.
In this analysis we look at pairwise comparisons of protein sequence, structure, and function among proteins that share the same fold.  We assess the trends relating sequence, structure, and function and consider the implications for structural and functional annotation transfer.
Probabilistic scoring methods
The past studies regarding sequence, structure, and function relationships use RMS separation and percent sequence identity (or a linear variant of it, such as the fraction of mutated residues) to express similarities in structure and in sequence, respectively.  However, it has become increasingly common to use probabilistic scoring schemes (P-values) to express the {{{goodness}}} <<<quality>>>  of a match in terms of statistical significance rather than an arbitrary raw score such as percent identity (Pearson, 1996; Karlin et al. 1991; Altschul et al. 1994; Bryant & Altschul, 1995).  With P-values<<<,>>>  scores from different investigations can be compared in a common framework.  Recently, it was found that sequence and structure similarity significance{{{s}}} can be expressed as P-values in the same unified statistical framework (Levitt & Gerstein, 1998).  In this investigation we use such probabilistic scoring methods to overcome limitations of the more traditional scores.
Growth of Data
Another <<<recent>>> development {{{that has occurred during this time}}}  is the tremendous growth of solved protein structures.  The Protein Data Bank (Bernstein et al. 1977) contains 9,248 protein structures as of this writing.  These structures are broken into domains, and then domains which share a fold are paired up with each other for comparison (Figure 1B).  Here we survey ~30,000 pairs of protein domains that are known to have the same fold, approximately 1,000 times the number compared by Chothia & Lesk (1986).  The large scale of this comparison affords greater statistical weight to the results.
Alignment of 30,000 pairs from SCOP
The basic unit of comparison: a pair of protein domains
The protein domains that we studied were classified by SCOP, Structural Classification of Proteins (Murzin et al. 1995; Hubbard et al. 1997), a hierarchy with five levels:
(i)	class—domains that have the same secondary structural content: all-a, all-b, a/b, or a+b;
(ii)	fold—domains that geometrically share the same tertiary fold;
(iii)	superfamily—domains descended from the same ancestor (but which lack measurable sequence similarity);
(iv)	family—domains in the same protein sequence family (which have appreciable sequence similarity); and
(v)	species and protein.
Pairs of protein domains that are grouped together at the fold, superfamily, or family level form the basic unit of our comparisons. 
Selection of pairs
There are{{{ potentially}}} over fifty million pairs of domains that can be constructed out of SCOP.  In order to keep the number of pairs manageable we used a straightforward clustering scheme<<<,>>> described in the caption to Figure 1.  {{{With the SCOP classification and the clustering we}}} <<<We>>> selected 29,454 representative{{{s}}} pairs from{{{ this total}}} <<<the total achieved via the SCOP classification and clustering>>>.  To achieve a wide range of similarities, we constructed the pairs on three levels of the SCOP hierarchy: 
(i)	family pairs, 19,542 pairs of domains in the same family;
(ii)	superfamily pairs, 4,220 pairs of domains in the same superfamily but different families; and
(iii)	fold pairs, 5,692 pairs of domains in the same fold but different superfamilies.
We selected domains that were at least 50 residues in length from the four major SCOP secondary structural classes: all-a, all-b, a/b, and a+b (Figure 1C).
We automatically aligned each of our selected domain pairs twice{{{:}}} <<< --- >>> once by global Needleman-Wunsch sequence comparison (Needleman & Wunsch,????; Myers & Miller, 1998) and then by structure (Gerstein & Levitt, 1988){{{,}}} <<< --- >>> and calculated scores for sequence and structural similarity.

Web-accessible database
The results of all the pairwise comparisons are available via a searchable database on the world wide web at http://bioinfo.mbb.yale.edu/align/scop  The query engine allows searches of individual SCOP pairs, all pairs which include a given SCOP domain, or all pairs containing any SCOP domain contained in a given PDB entry.
Traditional scores: RMS and percent identity
The sequence-structure relation<<<,>>> as expressed by the root mean square (RMS) of the aligned Ca distances and percent sequence identity<<<,>>> has been previously characterized as an exponential function (see equation in Figure 3 caption) (Chothia & Lesk 1986; Flores et al. 1993; Russell & Barton, 1994; Russell et al. 1997).  As Figure 3 illustrates, our data displays a similar trend.  However, we have one thousand times as many data points (30,000 as opposed to 30).
The main difference between our results and the data from Chothia & Lesks’s 1986 study is a result of differences in RMS trimming methods.  The dependence of the RMS separation on trimming method restricts its usefulness in comparing data.  By trimming we refer to the process of removing the worst fitting aligned atoms from the RMS calculation.  One does this because the small distances between well-matched alpha-carbons have much less of an effect on the root mean square than do very large distances between poor matches.  The untrimmed score of divergent protein domains is then concerned primarily with the poorly matched residues instead of the conserved core.  Trimming alleviates this effect by restricting the RMS calculation to {{{only include}}} those residues believed to be in the conserved core.  However, the degree to which one trims is to some extent arbitrary, and this choice affects the baseline of the reported RMS scores.  In this investigation we considered only the better half (50%) of matched residues in a given pair of protein domains.  Chothia & Lesk chose a somewhat different threshold.  Figure<<<s>>> 3C and 3D demonstrate the effect of trimming.
Analogous alignment similarity scores: Smith-Waterman Score and Structural Comparison Score
For sequence alignments, the sum of the similarity matrix values for the alignment, less the total gap penalty, often performs better than percent identity for expressing sequence similarity (Abagyan & Batalyov, 1997).  (Sometimes this is called the Smith-Waterman score (Smith & Waterman, 1981.))  An analogous alignment score for structure is the Structural Comparison Score, described by Levitt and Gerstein (1998).  The two similarity scores, referred to as Sseq and Sstr, respectively, increase for more similar pairs, whereas RMS increases for more divergent pairs.  Specifically, Sstr is the score maximized by the structural alignment program we used (Gerstein and Levitt, 1998).  It can be calculated from any pair of aligned structures according to the function:

where M, d0, and Ngap are constants, usually set to 10, 5, and 10; and where di is the distance between each aligned pair of Ca atoms and the sum is carried over all aligned pairs.
The main advantage of Sstr over RMS separation in describing structural similarity is that the Ca to Ca distance, di, appears in the denominator of the calculation.  This means that the smallest distances, corresponding to the best matches in the conserved core, are most significant in determining the score.  Hence, the need for trimming is eliminated.  Sstr is also advantageous because it takes gaps into account and, when used in conjunction with Sseq, because of their fundamental analogy. <<<[Preceding sentence not clear. How does clause beginning with “because” relate to rest of sentence???]>>>
Figure 4A displays the relation between structural and sequence similarity as expressed by Sstr and Sseq.  Figure<<<s>>> 4C and 4D show calibration curves relating each of these scores back to approximate RMS separation and percent identity, respectively.  This helps one get an intuitive feel for the degree of relationship in terms of the more traditional scores.  Figure 4B adds a third axis, alignment length, and demonstrates that Sstr depends greatly on this quantity.  Although Sstr and Sseq are “better” scores than RMS and percent sequence identity, the heavy dependence of both of these on length limits their usefulness in many situations.  For instance, two pairs of similar domains with equal percent sequence identities but different lengths can have drastically different Sseq scores.
Probabilistic scores: P-values expressing sequence and structure similarity significance
Probabilistic scores can to a great degree overcome the length-dependence problems associated with the alignment scores.  Probabilistic measures are advantageous because they express similarity not by an arbitrary “score” but by a statistical significance: the likelihood that such a similarity could be achieved by chance.  This likelihood is also called the P-value.  We used calculations (described in detail in the Figure 5 caption) determined by Levitt & Gerstein (1998) to obtain P-values based directly on Sstr and Sseq. We refer to the P-values as Pstr and Pseq.  For Pseq we could equally well have used the numbers from one of the popular sequence similarity programs (i.e. BLAST or FASTA) as all these values have been shown to be perfectly proportional to each other (Levitt & Gerstein, 1998; Brenner et al. 1998).
Pseq and Pstr can be used to express the relation between structure and sequence similarity on a more fundamental level.  Figure 5A shows a log-log (base 10) plot of Pstr against Pseq.  Because it is log-log the trends can be visualized as straight lines.  Two straight lines are necessary to fit the points well, with the discontinuous boundary between the lines located at the beginning of the twilight zone.  The different slope of the line at low sequence similarity reveals that in the twilight zone there is a different relation between the significance of structural similarity and that of sequence similarity.  For domain pairs in the twilight zone (according to the percent identity to Pseq calibration in 5B), a given structural similarity is more significant (smaller P-value) than the always insignificant sequence similarity.  In contrast, for pairs with more than about 30% identity the structural similarity is less significant than the sequence similarity.  One possible interpretation of why the relative statistical significances reverse is the following:  Structure is more highly conserved than sequence, so structural similarity is not as significant as sequence similarity, provided that significant sequence similarity exists.  In the twilight zone sequence similarity is by definition not significant, but because all pairs in our comparison by definition share at least the same fold<<<,>>> there is still a significant amount of structural similarity.
Class differences
The division of SCOP into classes based on secondary structural composition allows easy investigation of whether there are any deviations from the common similarity relationships on account of secondary structure characteristics.  Figure 6A reveals that secondary structural composition does not markedly affect the trends in sequence and structure similarities.  However<<<,>>> the larger average length of alpha/beta domains compared to domains in the other classes results in a deviation in the length-dependent Sstr (6B).  The consistency among length-independent scores applies for certain individual folds as well.  The immunoglobulin fold makes up an appreciable fraction of all the b-pairs (Figure 1C)<<<,>>> yet the results are not affected if these pairs are left out.
Linking sequence and structure to function
Difficulties of functional comparison
Indeed there is a clear, well-characterized relationship between sequence and structure similarity, which can be used to precisely transfer structural annotation based on the degree of sequence homology.  In genome analysis, however, one is usually more interested in finding a functional annotation for an ORF based on similarity to well-known proteins; yet the sequence-function and structure-function relationships have not been as explicitly characterized.  The fundamental obstacle to extending this and similar investigations to deal with function is the absence of a clear measure of functional similarity.  Although we were able to present three different quantitative measures of structural relatedness, an analogous situation for function does not exist.  How can one express quantitatively the degree of similarity between a triosephosphate isomerase and a glucose-6-phosphate isomerase?  How do they compare to a trp repressor?  
It is especially difficult to define functional similarity because the definition of function itself is often vague.  It is difficult to describe the function of any protein in a few words, not to mention in a standardized framework.  Consequently, the precise hierarchical fold classifications for domains in the PDB do not have functional equivalents.
Several functional classifications have been developed to date.  One is the ENZYME system developed by the Enzyme commission (EC) developed to classify enzymes by reaction type (Webb, 1992).  [You say by reaction mechanism but then later that it doesn’t consider mechanism so I changed it to reaction type.]  This classification has several drawbacks.  First of all, it does not consider catalytic reaction mechanisms (Riley, 1998), often ignoring obvious similarities.  Secondly, it presumes a 1:1:1 relationship between gene, protein, and reaction, although this is often not the case.  (An enzyme can have two functions<<<,>>> or two polypeptides from two different genes can oligomerize to perform a single function.)
An alternative to the EC approach to functional assignment is organism{{{al}}} classification of protein functions.  Several such schemes exist; for instance, EcoCyc for E. coli (Karp et al., 1998), MIPS for yeast (Mewes et al., 1998), the FLYBASE functional classification for Drosophila (Ashburner, 1994), and EGAD for human (write to them ????). These classifications possess some advantages. They have additional levels of hierarchy (cellular and organism{{{al}}}) that help{{{s}}} present a more comprehensive picture of genotype-phenotype relationships.  On the other hand<<<,>>> these classifications have much room for improvement.  For example, there is no standardized vocabulary to allow key word searches among multiple databases and across organisms, and there are also inconsistencies in category numbering style.
Our functional classification of SCOP domains
Given the discussed limitations, we constructed a simple functional classification for the SCOP domains included in our comparison{{{,}}} <<<; our classification is >>> based on {{{merging}}} <<<a merger of>>>  many of the existing functional annotations.  First we used pairwise comparison to cross-reference the PDB domains against the Swissprot database (Bairoch{{{.}}} & Apweiler, 1998), as done in Hegyi & Gerstein (1999).  We chose to assign protein functions according to Swissprot because {{{it}}} <<<Swissprot>>>  provides more comprehensive functional annotations than SCOP.  We were initially able to divide all entries into enzymes and nonenzymes, {{{which represent}}} <<<a division that represents>>> the highest level of functional difference in our classification scheme (Figure 2).
For the enzyme category<<<,>>> we transferred Enzyme Commission (Webb, 1992) numbers to those SCOP domains with a one-to-one match to a Swissprot enzyme.  Only one-to-one matching entries could be considered because Swissprot assigns Enzyme numbers to entire proteins<<<,>>> whereas SCOP is a domain-based classification; {{{therefore}}} we could {{{only}}} be confident about the classification of <<<only>>>those domains which mapped to an entire Swissprot entry.
In the absence of an EC-type classification for nonenzymes we assigned functions to nonenzymatic SCOP domains according to the FLYBASE functional classification of Ashburner [need reference].  The hierarchical structure of this classification makes it well suited for classifying nonenzymatic SCOP entries in a manner comparable to the ENZYME assignments for the enzymes.  Another advantage of this classification is that it is more compatible with the makeup of our data set than the Escherichia coli and Saccharomyces cerevisiae classifications, as Drosophila is a multicellular organism and many of the SCOP entries come from animals.  We were able to use the FLYBASE classification as a framework<<<,>>> to which we added functional categories and individual proteins{{{,}}} <<< --- >>> such as “Hemoglobin{{{”,}}} <<<,”>>> which was added to the “Physiological Processes – Respiration” category.  Another example <<<of an>>> addition is <<<the>>>  “Physiological processes - Immunity” category (Figure 2B), to which we added immune system proteins.  We also modified the numbering scheme slightly in order to assign a unique number to each protein domain (Figure 2B).
As discussed earlier, the universal functional classification of proteins is very difficult, and {{{maybe not}}} <<<may not be >>> possible with the current level of knowledge about genes, proteins and genomes. Consequently, our classification of SCOP proteins based on merging pre-existing schemes is somewhat incomplete and inconsistent, and retains many of the limitations of its components (Hegyi & Gerstein, 1999; Riley, 1998).  It is not yet broad enough to include many plant, virus, and bacterial proteins.  Nevertheless, as we were able to classify a very large number of the total 30,000 pairs, it was sufficient for our analysis.
Determining functional similarity
Using our functional classification <<<,>>> we were able to assign a level of functional similarity to each domain pair.  According to our scheme a pair can have {{{either}}} no functional similarity, <<<it can be>>> an enzyme paired with a nonenzyme, or <<<it can have>>> one of three levels of similarity:
(i)	General similarity. Both domains are enzymes or both nonenzymes.
(ii)	Same functional class.  Both domains share the first component of their ENZYME or FLYBASE numbers, e.g. 1.1.1.1 Alcohol dehydrogenase and 1.3.1.1 Cortisone beta-reductase or (for nonenzymes) 3.3.2.1.2 Calcicyclin and 3.6.3.2.1 Calmodulin.
(iii)	Same precise function.  Both domains share three components of their ENZYME or FLYBASE number -- e.g. 1.1.1.1 Alcohol dehydrogenase and 1.1.1.3 Homoserine dehydrogenase or (for non-enzymes) 1.2.9.1.1.1 Arc repressor and 1.2.9.1.1.1 C-jun (both transcription factors).  A pair which shares precise function must also, by definition, share functional class and general similarity.
Based on those assignments we calculated the percentage of total pairs at a given level of sequence or structural similarity {{{that possess}}} <<<possessing>>>  each level of functional similarity. The results appear in Figure 5.
Sequence and function
The relation between sequence similarity and functional similarity behaves as one might expect, with sigmoidal curves that drop off sharply at particular conservation thresholds, and with the three levels of functional similarity (precise function, functional class, and general similarity) having progressively lower thresholds.  Figure 7A shows that precise function is not conserved below 30-35% sequence identity, whereas functional class is conserved for sequence identities as low as 20-25%.  Below 20% the general similarity is no longer conserved; among pairs of approximately 7% sequence identity <<<,>>> about 40% are enzymes paired with nonenzymes.  It is important to note that in all the pairs considered here the domains share the same fold.  Functional similarity at low percent identities (e.g. 7%) would be much less for all possible pairs of domains rather than <<<for???>>> just those with the same folds. <<<[Preceding sentence not clear.]>>>
There are differences between the functional conservation thresholds of enzymes and nonenzymes, with enzymes appearing to more highly conserve precise function than nonenzymes <<<,>>> but nonenzymes conserving functional class more highly than enzymes.  This may reflect that the nonenzyme functional classes are broader (<<<with a higher tendency towards conservation>>> {{{hence more conservation}}}) than those of the enzymes<<<,>>> while the nonenzymatic precise functions are more specific.
When Pseq is used as the measure of sequence similarity (Figure 7B) the results look somewhat different—it appears that functional class is conserved for the entire range of sequence similarities.  In this case percent sequence identity is more discriminating than Pseq because functional class diverges only at sequence similarities that are low enough that they have little or no statistical significance<<<.>>>  {{{—that}}} <<<That>>> data is compressed near the y-axis of the graph.
Structure and function
The relation between structural similarity and functional similarity is somewhat less straightforward than that between similarity in sequence and function.  The relation between RMS and functional similarity shown in Figure 7C appears broadly similar to that for percent identity and functional similarity; however the thresholds for conservation of the various levels of functional similarity are less sharp.  Sstr and Pstr are less revealing with respect to functional similarity.  The reason for this is: for RMS very structurally similar pairs are all clustered between 0 and 0.5 angstroms.  Pstr does not have a set limit for maximum similarity, and Sstr has a large range of scores for similar pairs due to the length dependency; so the broad sigmoids are blurred further <<<so>>> {{{such}}} that they are no longer apparent.
Conclusion
In this paper we considered functional and structural annotation transfer by analyzing the relationships between similarity in sequence, structure, and function.  The ~30,000 protein domain pairs of varying levels of similarity (at least the same fold) <<<that>>> we constructed out of the SCOP classification show a sequence-structure relationship consistent with previous research.  The relationship is consistent among the secondary-structural classes and holds <<<a potential ???>>>  for newer probabilistic scoring methods.  The sequence-function and structure-function relationships have not been studied as precisely {{{in}}} <<<due to the >>> lack of a robust functional classification and measure of functional similarity.  To overcome this we constructed our own classification by merging and extending the ENZYME and FLYBASE schemes and assigning levels of functional similarity.  Our measures of frequency of functional similarity over the domain of sequence and structural similarities provide curves and thresholds relating function to sequence and structure.  Percent sequence identity was the most discerning measure for comparison to functional similarity {{{as it provides}}} <<<, providing>>> higher resolution in the regions of low sequence similarity where functional class diverges.
Implications
The sequence-structure and sequence-function relationships provide useful information for fold and function annotation.  Table 1 summarizes the relative advantages of the different scoring methods we used to characterize the sequence-structure relationship.  Using the trends in sequence and structure similarity, one can assess the degree to which structural and functional annotation can be “transferred” between similar sequences given the level of sequence similarity.  The sequence and function similarity thresholds establish minimum requirements of sequence similarity for reliable function prediction.  Note that because the protein domain pairs considered here all share the same fold, the numbers for all possible pairs will differ in the region of very little sequence identity, in which the sequence similarity is not enough to indicate the same fold. 
Practically, then, when one searches an uncharacterized ORF against known structures, if the ORF matches a structure with a good e-value or percent identity <<<,>>> then the curves presented here can be used to check how the functional and detailed structure annotation will transfer.  For example, if an unknown ORF matches a PDB structure with an e-value of 0.001 and a percent identity of 30%, then one can be assured that it has the same fold (Brenner et al. 1998) and according to our analysis it has a 67% chance of having the same exact function.  Furthermore it has a 99% chance of having the same functional class and its structure probably diverges from the known structure by a trimmed RMS of 0.65Å.
Future Directions
There are a number of directions in which we could extend this analysis.  With respect to the sequence-structure relation, we can reduce the overrepresentation of the immunoglobulins and improve the calculation of Pstr (redo the fit to the extreme value distribution) to eliminate the residual length-dependency.
In the functional realm, we can investigate if and how the sequence-function and structure-function relationships vary for different categories of proteins.  For example, although we found consistency of the sequence-structure relationship among secondary structural classes, Hegyi & Gerstein (1999) found that the distribution of enzymes and nonenzymes varies with secondary structural class.  A related issue is that of conformational changes.  It is conceivable that among domains with very similar sequences but structures that differ by a conformational change, function is less conserved than it is among similar sequences with more similar structures.
Perhaps the most important direction in which to further this work is the augmentation of the functional classification.  With the growing amount of fully sequenced genomes there is a need for the development of a comprehensive system of functional classification of proteins.  We have begun preliminary work on developing a complete functional classification for the entire fold library.  It is a difficult process, as many existing organism-specific classifications will have to be merged, but the end result will have the advantage of not being centered on any one organism.  Such a “universal” classification would allow much better transfer of functional annotation.

<<<Note: The asterisked entries vary from the usual style in that each word in a title is capped (as opposed to first-word-only), or in that they use quotes around the title.>>>
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Tables
Table 1: Summary of scoring methods
The table lists the schemes presented here for characterizing the sequence-structure relationship, along with their relative advantages and disadvantages.

Sequence Similarity
	Structural Similarity	Features	Limitations	
Percent sequence identity	RMS Ca separation	Well understood, in use	RMS depends most highly on worst matches, requiring arbitrary trimming
	
Sseq	Sstr	Analogous similarity scores, Sstr depends most highly on best matches
	Dependence on alignment length	
Pseq	Pstr	Statistical significance, unified framework for different comparisons	Not as familiar as RMS and % identity, some residual length-dependency

Figures
Figure 1: Overview
This figure schematically depicts certain aspects of our comparison methodology.
Part A illustrates the paradigm relating sequence to structure to function.  There has not been as much assessment of functional annotation transfer based on structure as there has been with sequence-based structural and functional annotation transfer.
Part B shows how we conceptualized our analysis in terms of pairs.  A few example SCOP domains (identified on the left and bottom) are included from our comparison.  In the figure the shape represents fold and the pattern represents function.  We have highlighted some example categories of pairs:  a pair that shares fold and function, a pair that shares fold but not function, and a pair that shares neither fold nor function.  The latter category of pairs is not considered in our investigation; we only looked at paired domains with the same fold.  The SCOP domain d4tima_ is not paired with anything in the figure, because it is represented by d5tima_, which is the same species and protein.  For each level of pairs (fold, superfamily, family) cluster representatives were chosen for the level below:
(i)	For family pairs, one representative was selected from each species/protein, the level below, and then paired with all the other representatives within its family. <<<Preceding not clear--how to “, the level below,” relate to the rest??>>>
(ii)	For superfamily pairs, one representative was chosen from each family, unless there were domains in the family that shared less than 40% sequence identity, in which case additional representatives were included, each not more than 40% identical to the other representatives from the family.  (This occurs, for instance, for the globins.).
(iii)	Likewise for fold pairs, one representative was chosen from each superfamily, more if there were domains with less than 40% sequence identity.
<<<???Is the following a part of subsection (iii)???>>> For example, one representative from each species/protein was selected to construct pairs within a family.  In the illustration these representatives are designated by *.
Part C subdivides the pairs into the four SCOP classes from which they were composed: (i) all-a, domains consisting of a-helices; (ii) all-b, domains consisting of b-sheets; (iii) a/b, domains with integrated a-helices and b-strands; and (iv) a+b, domains with segregated a-helices and b-strands.  We initially set apart the immunoglobulins from the rest of the all-b pairs because we realize that their large number biases our data.  However, we compared the results for the immunoglobulin pairs to all other pairs and found that they <<<generally>>> exhibit the same behavior {{{of}}} <<<as>>>  the other pairs {{{in general}}}; the trends we observe are very general.  Therefore we decided to leave them in the comparison.

Figure 2 Functional Classification of enzymes and nonenzymes
Part A divides the pairs by general function.  There are three categories of pairs: (i) enzymes paired with nonenzymes (no general functional similarity), labeled “ENZ/~ENZ”; (ii) enzymes paired with enzymes (same general function), labeled “ENZ/ENZ”; and (iii) nonenzymes paired with nonenzymes (same general function)<<<???? Should there be a “labeled as”  for (iii) too???>>> .  Pairs for which one or both domains could not be identified as enzyme or nonenzyme are not included in this chart.
Enzymes are classified according to the EC system (Webb, 1992).  The first component of the number represents the nature of reaction and is called class.  There are six classes: oxidoreductases, transferases, hydrolases, lyases, isomerases, and ligases.  The next level is subclass.  It refers to the chemical groups on which the enzyme acts.  For example, the first class, oxidoreductases, {{{have}}} <<<has>>> 19 subclasses that are arranged according to the donor group that undergoes oxidation (CH-OH, aldehyde or oxo group, CH-CH group, etc).  For another group of enzymes (hydrolases) subclass is determined by the nature of the bond:  ester bond, peptide bond, etc.  The next level is sub-subclass.  For oxidoreductases this indicates the acceptor group: NAD(+) and NADP(+), or cytochrome; for hydrolases the sub-subclass represents the nature of substrate (carboxylic ester hydrolases, thiolester hydrolases, etc.).  The fourth level represents a unique number for each individual enzyme, for example, 1.1.1.1: Alcohol dehydrogenase. 

Part B shows how we adapted the functional classification of Drosophila gene products developed by M. Ashburner.  This classification is part of FLYBASE (Ashburner & Drysdale, 1994).  In Ashburner’s classification all members of the highest level are labeled 0, representatives of the next level are labeled 1, and all lower levels are labeled {{{in a similar manner}}} two through nine.  We changed the numbering scheme so that it will reflect the hierarchical nature of the classification.  This figure illustrates sections of the original and modified classification.  
<<<In the following paragraph, please put all periods and commas inside the end-quotes, to follow non-British style.>>> The top level in the FLYBASE classification scheme is called “Function primitive” (level 0) and includes 5 classes: “Metabolism”, “Intracellular protein traffic”, “Cell structure”, “Developmental process”, “Physiological process”, and “Behavior”.  The next level after “Function primitive” is “Process” or “Molecule” (level 1 in Ashburner’s classification).  For “Function primitive – Metabolism” the processes are “Carbohydrate metabolism”, “Nucleotides and nucleic acids metabolism”, etc.  For “Function primitive – Cell Structure” the “Process” can be “Nucleus”, “Mitochondrion”, “Membrane”, etc.  The next level is “Pathway” or “Macromolecule”(level 2 in the original classification).  “Pathway” can include “Metabolic pathway”, “Signaling pathway”, or “Developmental pathway”.  The “Macromolecule” category includes “Protein” and “Nucleic Acid”.  We added categories to the original classification in order to classify some mammalian proteins that are widely represented in SCOP but are absent from Ashburner’s Drosophila classification.  These categories include immune system proteins (labeled “new” in Part B) and respiratory proteins such as hemoglobin and myoglobin that we added to “Function primitive – Physiological process – Respiration”. 

Part C shows the overall hierarchy of our final scheme and identifies the different levels of similarity.  If two proteins are both enzymes or both nonenzymes, then they possess general functional similarity.  If they share the first component of their classification numbers, then they are in the same functional class.  If they share the first three components of their enzyme numbers (or the equivalent for nonenzyme numbers, depending on category) then they have the same precise function.
A significant difference between the two main branches of the hierarchy is that the levels of the ENZYME classification do not correspond exactly to those in the FLYBASE system because the fly classification is more extensive than the enzyme classification.  For instance, the FLYBASE classification takes into account cellular function (cytoskeleton, metabolic pathways, etc.) and phenotypic function (morphology, physiology, behavior) that are absent from the ENZYME scheme.  This makes our classification of SCOP proteins somewhat unbalanced, as nonenzymes have much broader and more loosely defined functional classes.  As a consequence, while each enzyme is assigned a four-component number, the length of a nonenzyme number varies<<<,>>>  depending on the functional category to which it belongs.  For example, myosin is assigned a number which happens to have the same length as EC numbers: 3.12.1.1.  However, transcription factors are numbered 1.12.9.1.1.1.  We took into account this varying hierarchy depth in deciding how many components are necessary to identify precise function<<<s???>>> in each category.  Note that what we mean by domains having the same precise function is not the same as the domains coming from the same essential protein.

Figure 3: RMS as a function of percent identity
Part A shows a simple scatter plot of our pairs, relating RMS separation to percent sequence identity.  This is similar to the presentation in Chothia & Lesk (1986)<<<,>>> but in this survey we look at 30,000 pairs—1,000 times the number they compared.  Outliers (pairs with RMS scores further than two standard deviations from the mean for their percent identity) are excluded from this graph; they represent domains which are very closely related with the exception of a conformational change.  We simplify our graph in Part B, which shows a number of fits to the data.  For each percent identity bin we show the median RMS value, indicated by , and the top and bottom quartile RMS values, indicated by the bars.  Two fits are drawn through the median RMS values.  The thin line, labeled “SINGLE”, <<<comma inside end-quote>>>is a simple exponential fit through the medians.  It has the form:
R = 0.21e0.0132(100 – I)
where R is the RMS separation and I is the percent sequence identity.
The thick line, labeled “MULTI”, <<<comma inside end-quote>>> is a multigraph fit<<<,>>> which is described in the Figure 3 caption.  The relation between RMS and percent identity according to this fit is expressed by the equation:
R = 0.18e0.0187(100 – I)
The twilight zone of sequence identity and below is labeled “TZ”. <<<period inside end-quote>>> In this region, sequence similarity is not as significant and is not as reliable for predicting structural similarity.  This is why the median values in this area of the graph deviate significantly from the fits, which only consider data above 20% sequence identity.
For reference we include the original data points from Chothia and Lesk’s 1986 paper (Lesk, personal communication), indicated by X.  Their data follows the form:
R = 0.40e0.0187(100 – I)
The difference between the Chothia and Lesk trend and our relation is due to the different trimming methods used in calculating the RMS score.  Chothia and Lesk imposed a 3Å cutoff in determining the conserved core residues; we defined the core as <<<query: “the better matching”>>>  the better matching (in terms of Ca distances) half (50%) of the residue pairs.  Parts C and D demonstrate the effect our trimming has on median RMS values.  The RMS values in Part C are calculated from all the matched residues in each pair; the values in Part D are calculated from <<<query: “the better matching”>>>  the better matching 50% of the residues.

Figure 4: Similarity Scores: Structural Comparison Score as a function of Smith-Waterman Score
Alignment similarity scores Sstr and Sseq have certain advantages over RMS and percent identity scores for expressing the sequence-structure relation.  Sstr is calculated according to

(where M and d0 are constants, di is the Ca distance, and Ngap is the number of gaps)
(Gerstein & Levitt, 1998; Levitt & Gerstein, 1998).  Sseq is calculated using the BLOSUM50 matrix (Henikoff & Henikoff, 1992) with gap opening<<<s?>>> and extension penalties of –12 and –2, respectively.
Part A of this figure is analogous to part B of Figure 2.  From the original 30,000 pairs we show the median Sstr value for each Sseq bin, along with quartile bars above and below.  Again the twilight zone and below is labeled “TZ”.  <<<punctuation inside end-quote>>>The thin line, marked “SINGLE”, <<<punctuation inside end-quote>>>is a simple fit to the median Sstr values in this graph; it has the form:
Sstr = 2144 – 1106exp(-0.00544Sseq)
The thick fit, marked “MULTI”, <<<punctuation inside end-quote>>>is the multigraph fit, explained below.  It follows the equation:
Sstr = 2157 – 787exp(-0.0028Sseq)
The equations presented here provide a simple approximation of the observed trends; as Part B illustrates, they are nothing more than simple approximations.  The main disadvantage of Sstr as a measure of structural similarity is its heavy length dependency for pairs of structurally similar protein domains.  Part B is a surface plot of the median Sstr as a function of Sseq and alignment length (the number of matched residue pairs).  It is clear that the size of the aligned domains plays a major role in the resulting Sstr, even though our fits do not take length into account.
Parts C and D relate Sseq and Sstr to the more familiar percent identity and RMS measures.  The fits were used to convert between scoring schemes in constructing the multigraph fit.
We derived the multigraph fit in order to create one set of equations and parameters that would relate sequence and structural similarity using either the percent identity and RMS scheme or the Sseq and Sstr scheme, and allow translation between them.  We simultaneously performed least-squares-fits to the median values in four graphs:  Figure 2B, Figure 3A, and the calibrations of Sseq to percent identity and Sstr to RMS, Figure 3C and 3D, respectively.  In all cases, we ignored data in and below the sequence identity twilight zone (labeled “TZ”).  The parameters in 3A are dependent on the parameters in 2B via the mentioned calibrations.

Figure 5: Probabilistic scores: p-values
Pseq and Pstr are p-values calculated from Sseq and Sstr according to the following calculations (Levitt & Gerstein, 1998):

Pseq = 1 – exp(-exp(-Sseq/a + ln(nm) + b/a))
where a = 5.84, b = -26.3, and n and m are the lengths of the sequences

Pstr = 1 – exp(-exp(-Z)) where Z is a function of Sstr and N, the number of matched residues:
Z = (Sstr – (c ln(N)2 + d ln(N) + e))/(f ln(N) + g), N < 120
Z = (Sstr – (a ln(N) + b))/(f ln(120) + g), N  120
At N = 120: a ln(N) + b = c ln(N)2 + d ln(N) + e and a = 2c ln(N) + d
a = 171.8, b = -419.4, c = 18.4, d = -4.50, e = 2.64, f = 21.4, g = -37.5

Part A of this figure is analogous to Figure 3A and Figure 2B, with the exception of the fits.  It is a log-log (base 10) plot relating Pseq and Pstr.  We show the median log(Pstr) value for each log(Pseq) bin, along with quartile bars above and below.  We have added approximate percent identity and RMS values to the x and y axes to aid interpretation of the graph in terms of more familiar scores.  The values were calculated using the calibration curves in Parts B and C.
The straight-line nature of the log-log plot reveals distinct relations inside and outside the twilight zone, labeled “TZ”. <<<punctuation inside end-quote>>> (The area of percent identity below the twilight zone does not appear in Pseq graphs—there is no significance for such low sequence similarity; thus all data points in that zone appear at Pseq = 1 or log[Pseq] = 0.)  The thick line in the figure is fit to the median Pstr values for Pseq values outside the twilight zone; its equation is
Pstr=10-10Pseq.05
The thin line is fit to the data inside the twilight zone; it follows the relation:
Pstr=10-6Pseq.274
For reference we include the dotted line, representing the function Pstr = Pseq, where sequence and structural similarity are equally significant.  See text for a discussion of how the two trends might be interpreted with respect to this line.

Figure 6: SCOP class differences
Previously it has been observed that secondary structural composition does not cause deviations from the trends in structure and sequence similarity (Flores et al. 1993).  To test this <<<observation>>> we looked at the scores divided by SCOP class.  The following legend applies to the graphs:

Part A shows median RMS values for each percent identity bin.  The traditional scores reveal no dependency on class.  However, in Part B a/b pairs consistently score higher Sstr scores than pairs in other classes.  This is a consequence of the dependence of Sstr on length; domains in the alpha/beta class are longer on average than in the other classes.

Figure 7: Linking Sequence, Structure, and Function
We express functional similarity as the fractional percentage of pairs at a given level of sequence/structural similarity for which the paired domains share (according to our classification—see Figure 2), a precise function, functional class, or general similarity.  The following legend applies to all the graphs:

Part A relates functional similarity to sequence similarity in terms of percent identity.  The functional similarity appears as a sharp sigmoid, with distinct thresholds of divergence for precise function, functional class, and general similarity.  Enzymes are paired with nonenzymes only at very low percent identity, in and below the twilight zone (labeled “TZ”).  At slightly higher sequence identity pairs diverge with respect to functional class, and at around 30% identity with respect to precise function.  (Note that 50-100% identity is not shown because almost all domains that are that similar share function with their counterparts.)  
The same data is shown in Part B using Pseq as the measure of sequence similarity.  Only the divergence in precise function is visible because there is such little significance for the low sequence similarity at which functional class and general similarity diverge—all data points in that region appear at Pseq = 1 or log[Pseq] = 0 (the y-axis).
Part C illustrates that the structure-function relation is not as clearly defined as that for sequence and function.  Functional similarity expressed in terms of RMS separation appears as a broad sigmoid; there are thresholds of divergence for precise function, but the divergences in functional class and general similarity are more gradual. The thresholds are only apparent because RMS clusters the most structurally similar pairs between scores of 0 and 0.5.  For this reason, RMS is better at discerning functional similarity then Sstr and Pstr, which do not cluster the most similar pairs around a set limit.