Relation Between Protein Structure, Sequence Homology and Composition of Amino Acids
From: ISMB-95 Proceedings. Copyright © 1995, AAAI (www.aaai.org). All rights reserved.
Relation between Protein Structure, Sequence Homology and Composition of Amino Acids
Eddy Mayoraz Inna Dubchak Ilya Muchnik RUTCOl~--Rutgers University’s Department of Chemistry, RUTCOl~--ttutgers University’s Center for Operations Research, University of California Center for Operations Research, P.O. Box 5062, New Brunswick, at Berkeley, Berkeley, P.O. Box 5062, New Brunswick, NJ 08903-5062, CA 94720, NJ 08903-5062, [email protected] dubchak%lcbvax.hepnet @lbl.gov [email protected]
factor are identical in the topology of the chain fold and similar in overall structure, yet the two proteins Abstract are dissimilar in sequence with less than 20%identical A methodof quantitative comparisonof two classifica- residues (Sander & Schneider 1991). Opposite example tions rules applied to protein folding problemis pre- sented. Classification of proteins based on sequence (Kabsch & Sander 1984): octapeptides from subtilisin homologyand based on amino acid composition were (2SBT) and imInunoglobulin (3FAB) are dissimilar compared and analyzed according to this approach. structure yet 75%identical in sequences. The main ob- The coefficient of correlation betweenthese classifica- stacle to developmentof strict criteria for calculation tion methods and the procedure of estimation of ro- of homologythreshold is the limited size of empirical bustness of the coefficient are discussed. database. Our study is based on the classification scheme 1 Introduction 3D_ALIof Pascarella and Argos (Pascarella & Argos One of the most powerful methods of protein structure 1992) that merges protein structures and sequence in- prediction is the model building by homology(Hilbert formation. It classifies the majority of the knownX-ray et al. 1993). Chothia and Lesk (Chothia & Lesk 1986) three-dimensional (3D) structures (254 proteins suggested that if two sequences can be aligned with protein domains) into 83 folding classes, 38 of them 50%or greater residue identity they have a similar fold. having two or more representatives, and the other 45 This threshold of 50%is usually used as a "safe defini- containing only a single protein example. This group- tion of sequence homology" (Pascarella & Argos 1992) ing is based on a structural superposition amongpro- and in conventional opinion grants a reasonable confi- tein structures with a similar main-chain fold, either dence that a protein sequence has chain conformation performed by the authors or collected from the litera- of the template excluding less conserved regions. Note ture. that in biology, homology implies an evolutionary re- Sequences from protein primary structure data bank lationship which is not measurable, but in this paper, (SWISSPROT)are associated with all 254 3D struc- according to Pascarella & Argos 1992, we use this term tures providing that they have not less than 50%se- to denote some measure of sequence similarity. quence homology and at least 50% of 3D structure But it was shown that structure information can residues were alignable (Sander & Schneider 1991). be transferred to homologous proteins only when se- Each of them is labeled by the number of one of 83 quence similarity is high and aligned fragments are long classes that includes a protein with maximumhomol- (Sander & Schneider 1991). It is known that homolo- ogy. This labeling can be considered as a classification gous proteins can have completely different 3D struc- procedure based on sequence homology. tures. For example, ras p21 protein and elongation Another simple measure of protein similarity often 1First author gratefully acknowledgesthe support of used is the distance function in the 20-dimensional vec- RUTCORand DIMACS. tor space of amino acid composition. Several groups 2Thesecond author is workingat the Structural Biology (Nakashima e~ al. 1986; Chou 1989; Dubchak et al. Division of LawrenceBerkeley Laboratory (Division Direc- 1993) have shown that the amino acid composition tor Prof. Sung-HouKim), supported by US Department of of a protein provides significant correlation with its Energy. SThe third author gratefully acknowledges the sup- structural class, and a numberof studies were devoted port of DIMACS(grant NSF-CCR-91-1999) and RUTCOR to a quantitative description of this relationship. In (grant F49620-93-1-0041). all these studies, a protein is characterized as a vec-
240 ISMB-95 tor of 20-dimensional space where each of its compo- Thesecond dataset, used as ~es~ingset~ contains 2338 nent is defined by the composition of one of 20 amino proteinsequences. This set, disjoint from the training acids. Recognition schemes based on percent composi- set,is the subsetof all the sequencesof the SWIS- tion are fairly effective for simple classifications where SPROT database,whose homology with at leastone proteins are described in terms of the following struc- of the254 sequences of thetraining set is greaterthan tural classes: all a, allfl, c~+~, a/fl, and irregular. It is or equalto 50%.For eachof these2338 sequences, the obvious that the difficulty of recognition grows rapidly followingitems are available: with the number of classes. Even in the distinction between a +/3 and a/fl classes, serious problems exist . completesequence of aminoacids; because parameter vectors of these types of structure ¯ to the the trainingset are located too close in the parameter space. That is a pointer sequence in which hasthe largesthomology with this sequence; why amino acid composition is considered a crude rep- resentation of a protein sequence and not a tool for ¯ the valueof the homologywith thismost similar protein structure prediction in a context where more sequencein thetraining set. than four structural classes are to be distinguished. In this study, we investigate the correlation between Homology,denoted dh in this work,is a real num- two classifications of protein sequences with unknown ber between0 and 1 showinga proportionof homol- structure, which are obtained by the same simplest ogousresidues in a sequence.Besides homology, we nearest neighbor association procedure (Duda & Hart willstudy simpler measures of similaritybetween se- 1973). These two classifications differ only in the mea- quences,based only on the rate of each ~mlnoacid sure of similarity of proteins: one is based on homology, in thesequence, and independent of therelative posi- while the other uses a distance in the space of amino tionof theseamino acids. The compositionof a protein acid composition. Wedefine and calculate correlation sequenceis definedas a 20-dimensionalvector of coeffi- coefficient between them and propose a technique to cientsin [0,1] indicatingthe rate (number of instances estimate the robustness of this coefficient of correla- of theacid divided by thetotal length of thesequence) tion. High correlation between these two classifications of eachamino acid in the sequence.Very natural dis- would show a hidden power of amino acid composition tancemeasures on the compositionspace [0, i]2° are for folding class prediction and at the same time would providedby the normsLp and willbe denotedalp: cast some doubt on the application of homologyto this prediction without a careful examination. 20 The remaining part of this paper is divided into - - 3 sections. Section 2 describes the methods and data i----I used in this study. The results of our analysis are re- In thisstudy, we willretain only dl andd½, since the ported in section 3. The last section contains a brief formeris easyto interpret,and the latter provides the discussion. highestcorrelation with dh, as we observedempirically.
2 Material and methods 2.2 Scheme of analysis Thissection describes a methodfor the evaluationof In a first stage,20-dimensional vectors of composition correlationsbetween two measuresof proteinsequence are constructedfor each sequence of trainingand test- similarityas wellas a techniqueto measuretheir ro- ingsets. For any distance measure d, theclosest neigh- bustness,by experimentsbased on two setsof data. bor(according to d) in tr~inlng set is identified for all sequences in the testing set. This operation induces 2.1 Material a new classification of the testing set, where each se- quence is associated with the class of its closest neigh- As it was mentionedin the introduction,our first bor. Note that this procedure, executed here for dl dataset,from now on referredto as ~rainingse~, consistsof 254 proteinsequences with known 3D- and d½, is identical to that performed by Pascarella structure.For eachof the proteinsequences in the and Argos (PascareUa & Argos 1992) for dh. trainingset we have: Next step is the comparison of two different classi- fications based on dh, dl and d~. These comparisons * completesequence of aminoacids; are carried out by the computation of the coe~Ncien~of agreemen~between two classifications, i.e. the number ¯ a labelin {1,...,83}that denotes the classrepre- of times when two classifications agree. More detailed sentingits 3D structure. comparisons containing a table indicating the number
Mayoraz 241 of times two classifications agree, versus the intensity other uses, it aims at the extraction of simple logical of each of distance measures, was also performed. patterns able to explain the repartition of the samples The coefficients of agreement between different clas- of a training set between their various classes. Since sifications obtained this way are random variables de- the methodessentially deals with logical variables, the pending on the testing set. In order to make any valid first stage of the whole analysis is a binarization of conclusion, it is essential to estimate the robustness of given data. these coefficients. To do so, we suggest to reduce the In manypractical problems, a large set of attributes amount of information available in training and test- is available for each dataset, some of them play a cru- ing sets from whole space of amino acid composition to ciai role in the studied classification, while others are more and more simple spaces of this space. To be valid, completely irrelevant. Therefore, the binarization de- each of these simplifications of the composition space scribed here tends to extract the minimal amount of must maintain the cons/s~enev of the training set, i.e. information, while keeping consistency of the training any two sequences of the training set belonging to two set. different classes must’be different in the simpler space. For easy interpretation of the encoding, each binary At every step of this simplification process, new dis- attribute corresponds to one particular threshold and tances are defined in the simpler space, the process this attribute is true for a particular data when the of classifying by a new distance measure is executed, corresponding original attribute has a value equal or and the coefficient of agreement between new classifi- greater than this threshold. The binarization problem cation and the one induced by dh is calculated. The consists in finding a small numberof such thresholds, ratio between each of these new coefficients of agree- so that the resulting encoding maintains the consis- ment and the original one (agreement between dh and tence of the training set. dx or dx) is interpreted as a measure of the robust- Moreformally, let say that there are n original at- hess of t~e original coefficient of agreement. Indeed, if tributes consisting all of continuous values, and the a strong simplification of the composition data deteri- training set S E ~n is partitioned into c classes orates only partially the agreement between the clas- S = S1offTo...uSc. An encodinge of this space sification induced by homology and the one based on ~" will be consistent with S if and only if it fulfills composition, we can conclude that our evaluation of the following property: the original agreement (with the complete information on composition) is robust. Vk, le {1,...,c},k#l VaeS~ Vbe~ e(a)#e(b). It should be mentioned that the simpler spaces used (1) in our approach are binary, and the distance used is If the binary encoding e :/Rn --* {0, 1}m is obtained by the Hammingdistance. Thus, it happens very often m threshold values placed on some of the n continuous that the nearest neighbor is not unique, and even that variables, then e(a) e(b) if andonlyif th ereis one several of them belong to different classes. If this is threshold t E ~ along one variable i E {1,..., n} such a case, the classification follows a simple voting rule that associating the class with the largest numberof nearest neighbors, and in case of ties, the new sequence is put either ai>_tandbi
242 ISMB--95 to each threshold (~, i) E Sincea reductionfrom the continuousspace ~20 to thediscrete space {0, 1} 12 is substantial,we are go- ingto exploreintermediate reductions, obtained by in- creasingthe right-hand-side of the constraints in (3), 8.t¯ ~(t,i)6T 8tiabzti >_ 1 VG ¯ S k, b ¯ SI, ~ # I fromi to 2, 3, up to 10.Note that in a solutionof (3) withthat right-hand-side set to k, everypairs of se- ¯ . ¯ {o, 1} v(t, i) ¯ quencesa and b fromdifferent classes can be distin- (3) guishedby at leastk differentthresholds of thesolu- where s~b = 1 if t, i, a and b satisfy relation (2), and tion.The increase observed on thesize of thesolution s~b = 0 otherwise. A subset of candidate thresholds was almostlinear: around five additional thresholds corresponds to each binary vector m ¯ {0, 1}[TI, and werenecessary for each next value of k. clearly, such a subset fulfill relation (1) if and only if all the constraints of the linear program (3) are satisfied. 3 Results A simple interpretation of this information reduc- It is known that in SWISSPROTdatabase, i.e. among tion procedure by binarization is the following. Some 35t000 proteins, nearly 90% have a homology less hyperplanes (thresholds) are placed in the original con- tinuous space (here ~20). Each of these hyperplanes than 50%with any of proteins with known structure, normal to a vector of the basis of the continuous space. and no conventional technique is knownto predict their These hyperplanes are chosen in such a way that each 3D-structure from their sequence. of the hyperboxes they delimit contains only sequences In thispaper we analyzeonly ~ 7% (2338exactly) belonging to the same class. The smallest number of of all knownprotein sequences having relatively high hyperplanes fulfilling this property is desired. homologyto classified3D-structures. The integer linear program (3), known as the 8el Table1(a) (resp. (b)) shows the distribution of cover{rig problem, has been widely studied. It is well 2338sequences according to theirdistances to their known to be hard to solve exactly (NP-Complete). nearestneighbor with respectto homologydh along the rowsand withrespect to dl alongthe col-runsin However, a large number of heuristics are available in the literature, and provide solutions close to min- (a)(resp. d½ in (b)).The wholefield of correlations imal for most of the real life problems (see for example betweenthese two distributions is displayed in eachof Beasley 1990 for a good algorithm and for a survey)¯ thesetwo 10 by 10 tables,and highcorrelations can be Moreover, in our application it is not essential to get observedat leastin casesof highsimilarity in protein the minimal subset of thresholds fulfilling (1), and sequences. reasonably small subset is sufficient. It turned out that On the right-hand-side of each of these tables, a 2 by 2 the simplest greedy heuristic described in figure 1 is table reports this same field of correlations where the giving somesatisfactory results. bipartitions are based on threshold values (0.85 for dh, 22 for dl, and 17 x 100 for d~,) chosen to highlight the ¯ 9 ~ = (0,..., 0), is thematr ix of constraints in ( 3) fact that most of the data hes on the diagonal of these main loop: wh£1e (A contains at least one row) loop 2 by 2 tables. j: index of the column of A with the more ls Zj~I The main result, presented in table 2, shows the per- suppress from A every row i with alj = 1 centage of identical classifications between the proce- end loop po st op’c iL4.za¢ion (op¢ioned.): dure based on homology (using dh) and the one based find the smallest vector ~_ m still solution of (3) on composition (using dl or d½). These percentages are detailed for each of 69 non empty classes among83 (a Figure 1: Simplest greedy algorithm for the set cover- classification based on homologyassociates none of the ing problem. 2338 with some 14 classes). The total percentages are then reported, and finally, the percentages of identical In our application IS[ : 254 and the number of classification are computedfor three groups of classes: classes is c = 83. By applying the simple rule de- in eachclass of thefirst group, at least31 sequences scribed above, the number of candidate thresholds is areassociated according to thehomology classification; ITI = 1608 and the set covering problem has originally eachclass of the secondgroup contains between 6 and 31~058 constraints. However, the size of the solution 30 sequences;and classes of thethird group contain not produced by the greedy algorithm was 12. This means morethan 5 sequences,always according to the clas- in particular that the matrix of constraints has a high sificationbased on homology.Throughout the whole density making the set covering problem easy, and that table,each of the valuesare givenonce counting any thesize of 12 is probablyvery close to theoptimal. sequences,once counting only the sequenceswith ho-
Mayoraz 243 (°) d~_>55 >50 _>45 _>40 _>35 >3O _>~5 _>20 _>15 <15 ~1 >22 <22 lOOdh E 50-54 29 18 33 34 36 45 34 21 11 10 271 55-59 19 12 25 44 54 54 50 20 13 3 294 60-64 21 14 21 22 42 35 70 12 8 3 248 65-69 16 15 12 26 35 63 60 27 19 8 281 1462 236 70-74 12 16 22 31 36 27 30 18 10 6 2O8 75-79 10 12 18 15 23 23 28 26 9 2 166 8O-84 7 28 19 34 22 15 41 46 9 9 23O 85-89 4 5 14 22 10 15 24 57 33 54 238 90-94 1 9 13 20 7 14 24 62 45 37 232 213 427 95-99 1 0 2 2 0 0 5 22 33 105 170 all 120 129 179 250 265 291 366 311 190 237 2338
(b) ~1/~ _~45 ~40 ~35 ~30 >-25 >-20 >15 >-10 >5 <5 all >17 <17 lOOd~ E 50-54 98 37 54 29 17 9 12 5 6 4 271 55-59 70 40 70 45 32 17 14 3 1 2 294 60--64 53 23 43 47 39 24 11 5 2 1 248 65-69 46 26 41 57 58 34 13 6 0 0 281 1542 156 70-74 47 11 25 41 33 26 14 8 1 2 2O8 75-79 24 8 10 17 32 38 23 10 1 3 166 80--84 28 6 9 16 23 40 49 37 19 3 230 85-89 15 i 2 7 10 20 59 58 52 14 238 90-94 23 3 1 4 7 9 23 53 85 24 232 150 490 95-99 3 O 1 0 2 1 5 5 38 115 170 all 407 155 256 263 253 218 223 190 205 168 2338
Table h Distribution of the distances to the closest neighbors according to dl and d~. versus the distances to the 2 closest neighbors according to dh.
mology to their nearest neighbor between 0.5 and 0.7, coefficient of similarity is muchsmaller for the small and once counting only the sequences with homology classes (77%for the classes with at most 5 elements, for to their nearest neighbor greater than 0.7. d½), than for the bigger classes (at least 87%). More- The results are almost similar between the classifi- over, for the 20 largest classes (more than 30 elements), cation based on dl and the one based on dx. The first somesingularities should be observed: the coefficient 3 one coincides 86%of the time with the classification of similarity is incredibly low for one class (13% for based on homology, while the second one coincides for class 35); intermediate for three classes (from 69% 87%of the sequences. We carried out similar experi- 79%for classes 13, 29 and 38); above 90%for the 16 ments for many other distances dp. For the Euclidian other big classes; and even 100%for few of them of normd2, the global percentage of identical classifica- cardinality 40 and more (classes 31, 58, 59 and 72). tion is 83%, and it drops downto 71%for the distance Obviously, this coefficient of similarity depends d~. Among6 different values of p < 1, p = 0.51 gave strongly on the homology, as it is illustrated in ta- the highest correlation (87.37%) with homology. The ble 2. It should be mentioned that we also carried out fact that distances dp, p < 1 give a better result has some experiments to evaluate this coefficient of simi- an easy explanation. With a distance dl, the gaps larity against the length of the protein sequences and between any of the coefficients of two vectors are con- our results, not reported in this short paper, agreed sidered with the same strength, while in dp, p < 1, with Sander and Schneider’s work (Sander & Schneider more importance is accorded to the small gaps than to 1991) that higher similarities between these two clas- the big ones. The higher similarity with homologyob- sification procedures are found for longer sequences. tained for p < 1 reflects the fact that in the homology To conclude this section, let us consider the results procedure, a higher attention is accorded to the amino obtained after reduction of the information contained acids whose cardinalities are about the same in the two in the composition space, according to our procedure sequences. described in section 2.3. Without giving details on The coefficient of similarity of classification rules ob- which thresholds occur in the solution of the inte- served in table 2 is a statistical measure depending on ger linear program(3) for each right-hand-side /~, the dataset, and thus, its stability depends on the num- is worth mentioning that the amino acids F, I, T and ber of samples. We found interesting to observe these Wwere used most often (if the number of thresholds coefficients along a partition of the classes according to placed along each of these amino acids is summedup their cardinalities. As expected, it turned out that the for 8 different solutions of (3), we got a total higher
244 ISMB-95 class # of proteins % dh = dl %da =d~ label total da < 0.7 > 0.7 total dh =< 0.7 > 0.1’ total da _< 0.7 > 0.7 1 1 0 1 100 100 1aa 1aa 2 3O 2O 10 90 85 100 90 85 100 3 14 7 7 93 86 100 93 86 100 4 4 1 3 100 100 100 100 100 100 5 14 9 5 100 100 100 100 100 100 6 70 42 28 90 83 100 94 90 100 7 84 24 60 87 63 97 94 83 98 8 98 74 24 94 92 100 93 91 100 9 2 1 1 100 100 100 100 100 100 10 7 3 4 86 67 100 86 67 100 11 2 1 1 50 0 100 50 0 100 12 3 2 1 67 50 100 67 50 100 13 69 34 35 62 29 94 70 44 94 14 41 34 7 85 82 100 98 97 100 15 534 182 352 90 70 100 92 75 100 16 280 221 59 95 93 100 96 95 100 17 12 6 6 83 67 100 78 67 83 18 141 15 126 90 27 98 95 60 99 19 39 14 25 87 71 96 97 93 100 20 76 50 26 92 94 88 91 92 88 21 5 5 0 40 40 40 40 22 71 60 11 99 98 100 90 88 100 23 1 1 0 0 0 0 0 24 36 19 17 89 79 100 100 100 100 25 10 7 3 90 86 100 90 86 100 27 2 0 2 100 100 100 100 28 7 2 5 100 100 100 100 100 100 29 67 52 15 72 65 93 69 60 100 30 60 46 14 97 96 100 93 91 100 31 41 20 12 100 100 100 100 100 100 32 8 0 8 100 100 100 100 33 2 1 1 100 100 100 100 100 100 34 9 1 8 89 O 100 89 0 100 35 122 14 108 10 7 10 13 7 14 36 4 3 1 100 100 100 100 100 100 37 18 3 15 94 67 100 94 67 100 38 47 26 21 72 50 100 79 62 100 40 1 1 0 100 100 100 100 43 6 3 3 100 100 100 100 100 100 44 1 0 1 100 100 100 100 45 2 2 0 100 100 100 100 46 2 1 1 50 0 100 50 0 I00 48 2 2 0 50 50 0 0 49 12 9 3 100 100 100 100 100 100 50 3 3 0 100 100 67 67 52 18 3 15 83 67 87 89 67 93 54 9 7 2 89 86 100 89 86 100 56 1 1 0 0 0 0 0 57 2 1 1 50 0 100 100 100 100 58 41 9 32 100 100 100 100 100 100 59 65 27 38 100 100 100 100 100 100 60 2 2 0 0 0 50 50 62 3 0 3 100 100 100 100 63 4 0 4 100 100 100 100 64 11 10 1 91 90 100 73 70 100 65 2 2 0 0 0 0 0 67 3 3 0 100 100 100 100 68 3 3 0 33 33 33 33 70 5 3 2 100 100 100 100 100 100 71 2 1 1 50 O 100 100 100 100 72 40 6 34 100 100 100 100 100 100 73 9 8 1 78 75 100 78 75 100 74 11 9 2 82 78 100 91 89 100 76 30 21 9 97 95 100 97 95 100 77 1 1 0 100 100 100 100 78 4 1 3 100 100 100 100 100 100 80 7 0 7 100 100 100 100 81 2 1 1 50 0 100 50 0 100 82 3 2 1 67 50 100 100 100 100 all classes 2338 1151 1187 86 80 91 87 83 91 card< 5 74 45 29 74 58 100 77 62 100 5 < card ~ 30 242 128 114 92 86 98 91 85 98 30 < card 2022 978 1044 85 81 90 87 84 90
Table 2: Percentage of similar classification between the rule resulting from dh and the ones obtained by dl and d½.
Mayoraz 245 class # of proteins % d~ = dl %dh =d½ label total d~ < 0.7 > 0.7 total dh ~_ 0.7 > 0.7 total dh _< 0.7 > 0.7 1 1 0 1 i00 100 100 100 2 30 20 10 90 85 i00 90 85 100 3 14 7 7 93 86 I00 93 86 100 4 4 1 3 I00 I0¢ I00 100 100 100 5 14 9 5 I0~ I00 i00 100 100 100 6 70 42 28 90 83 I00 94 90 100 7 84 24 60 87 63 97 94 83. 98 8 98 74 24 94 92 100 93 91 100 9 2 1 1 100 100 100 100 100 100 10 7 3 4 86 67 100 86 67 100 11 2 1 1 50 0 100 50 0 100 12 3 2 1 67 50 I00 67 50 100 13 69 34 35 62 29 94 70 44 94 14 41 34 7 85 82 100 98 97 100 15 534 182 352 90 70 100 92 75 100 16 280 221 59 95 93 100 96 95 100 17 12 6 6 83 67 100 75 67 83 18 141 15 126 90 27 98 95 60 99 19 39 14 25 87 71 96 97 93 100 20 76 50 26 92 94 88 91 92 88 21 5 5 0 40 40 40 40 22 71 60 11 99 98 100 90 88 100 23 1 1 0 0 0 0 0 24 36 19 17 89 79 i00 100 i00 lOO 25 10 7 3 90 86 100 90 86 I00 27 2 0 2 100 100 100 100 28 7 2 5 100 100 100 100 100 100 29 67 52 15 72 65 93 69 60 100 30 60 46 14 97 96 100 93 91 100 31 41 29 12 100 100 100 100 100 100 32 8 0 8 100 100 100 100 33 2 1 1 100 100 100 100 100 100 34 9 1 8 89 0 100 89 0 100 35 122 14 108 10 7 10 13 7 14 36 4 3 1 100 100 100 100 100 100 37 18 3 15 9,1 67 100 94 67 100 38 47 26 21 72 50 100 79 62 100 40 1 1 0 100 100 100 100 43 6 3 3 100 lOG 100 100 100 100 44 1 0 1 IO0 100 100 100 45 2 2 0 100 100 100 100 46 2 1 1 50 0 i00 50 0 100 48 2 2 0 50 50 0 0 49 12 9 3 100 100 100 100 100 100 50 3 3 0 100 100 67 67 52 18 3 15 83 67 87 89 67 93 54 9 7 2 89 86 100 89 86 100 56 1 1 0 0 0 0 0 57 2 1 1 50 0 100 100 100 100 58 41 9 32 100 100 100 100 100 100 59 65 27 38 100 100 100 100 100 100 6O 2 2 0 0 0 50 50 62 3 0 3 100 100 I00 I00 63 4 0 4 100 100 100 100 64 11 10 1 91 90 100 73 70 I00 65 2 2 0 0 0 0 0 67 3 3 0 100 100 100 100 68 3 3 0 33 33 33 33 70 5 3 2 100 100 100 100 100 100 71 2 1 1 50 0 100 100 100 100 72 40 6 34 100 100 100 i0¢ 100 i0~ 73 9 8 1 78 75 100 78 75 100 74 11 9 2 82 78 100 91 89 100 76 30 21 9 97 95 100 97 95 100 77 1 1 0 100 100 100 IO0 78 4 1 3 100 100 100 100 100 100 8O 7 0 7 100 100 100 100 81 2 1 1 50 0 100 50 0 100 82 3 2 1 67 50 100 100 I00 I00 all classes 2338 1151 1187 86 8O 91 87 83 91 card< 5 74 45 29 74 58 100 77 82 100 5 < card < 30 242 128 114 92 86 98 91 85 98 30 < card 2022 978 1044 85 81 90 87 84 90
Table2: Percentageof similarclassification between the rule resulting from dh and the onesobtained by dl and d~. 3
246 ISMB.-95 than 12 for eac~h of these four amino acids), while can recognize if a given sequence is a T-cell receptor or and Y were almost never used (only once each among not. 10 different solutions). This information should be con- As third example, one of the interesting problem in sidered with a lot of care, since each of these vectors molecular biology is analysis of interactions between solution of (3) are obtained by a greedy procedure, and different proteins and peptides and one of the applica- it is concivable that a different procedure would make tion problems is to classify sequence database on the some different uses to various amino acids. However,a ability of proteins to interact with somegiven proteins. more robust evaluation of the frequency of each amino Weexpect that manymore classification of this kind acids could yield some very usefull information for the will appear in near future and it is necessary to develop design of new distance measures in the composition an approach that allows to compare various methods space. For example, one could weight each amino acid of classification and to showtheir relationship. This by its frequency. approach should compare methods, enhance their im- As it was already mentioned, the coefficient of sim- provementand create a classification of higher level. ilarity between the classification based on homology, In this paper we present a simple version of this ap- and the one based on composition is 87%(reap. 86%) proach. We focus on the problem of a comparison of with d~ (reap. dl) as distance measure on ~20, the two methods of protein folding classification with real composition space. With the classification based on istic numberof folding classes. Wedefine a coefficient Hammingdistance and vo~ing rule described in sec- of similarity among two given methods, and describe tion 2.2, this coefficient of similarity becomes 73%, a procedure to estimate a robustness of the coefficient 61%and 45%for solutions of (3) with right-hand-side for two sets of data. Weapplied these coefficient and /c = 10, 3 and 1 respectively (the respective size of bi- procedure to the study of the relation betweendifferent nary spaces are 59, 22 and 12). In other words, when measures of similarity for protein sequences. Weshow we extract 59 binary variables from 20 continuous ones that folding class prediction by sequence homologyand of the composition space, 86%of the similarity between amino acid composition are very close. the classification based on homologyand the one based On theone hand,this result emphasized restrictions on composition is preserved; 70% is preserved when forfolding class prediction by sequencehomology. On we express the composition space with nearly only one the otherhand, it showsa new opportunityfor us- binary variable for each amino acid; and the maxi- ing aminoacid composition in modelsof foldingclass mai compression of information, which reduces ~20 to prediction,since we foundthat this prediction can be {0, 1}i2, still maintains 52%of this similarity. In con- improved using different weights for different amino clusion, we claim that our estimation of this coefficient acids. of similarity is robust. Acknowledgments 4 Discussion Besidetheir support mentioned in the firstpage, the Molecularbiology is notedfor vastamount of empirical authorswould like to thankone anonymousreferee datawith "hidden" correct classification. A good ex- of ISMB’95for his appropriatecriticism and valuable ampleof thisis StructureAnalysis of Proteinsand Pep- comments. tides.There is a hugeamount of dataon theiramino acid sequences that need to be classified versus small References set of sequences with known three-dimensional (3D) structures. There are several classifications of proteins Beasley, J.E. 1990. A Lagrangian heuristic for set- with known3D structure, and the problem is to find a covering problems. Naval Research Logistics, 37: 151- good extension for sequences with unknownstructure. 164. A realistic3D classificationdivides sequences into a largenumber of classesand it providesan additional Crama,Y., Hammer,P.L. and Ibaraki,T. 1988. Cause- Effect 17~elatiouships and Partially Defined Boolean difficultiesforbuilding an extension. Functions. Annals of Opera~ion.s Research, 16: 299- Anotherexample. It is well-knownthat folding pat- 326. ternsof T-cellreceptors are verysimilar to thatof antibodies.In this case a classificationincludes only Chothia,C.,Lesk,A.M. 1986. The relationbetween two classes,where one of theseclasses is complemen- the divergenceof sequenceand structurein proteins. taxyto another.This complementary class usually has EMBOJournal, 5: 823-826. a verycomplicated description for its recognitionand the problemis to finda characteristicproperty that Chou,P.Y.1989. Predictionof proteinstructural
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248 ISMB-95