Estimating statistical significance of sequence alignments
MICHAEL WATERMAN Departments of Mathematics and Molecular Biology, University of Southern California, Los Angeles, California 90089-1113, U.S.A.
SUMMARY Algorithms that compare two proteins or DNA sequences and produce an alignment of the best matching segments are widely used in molecular biology. These algorithms produce scores that when comparing random sequences of length n grow proportional to n or to log(n) depending on the algorithm parameters. The Azuma-Hoeffding inequality gives an upper bound on the probability of large f deviations of the score from its mean in the linear case. Poisson approximation can be applied in the i logarithmic case.
algorithm parameters determine this behaviour. In 1. INTRODUCTION the linear growth region, the Azuma-Hoeffding Sequence comparison algorithms are widely applied to inequality gives an upper bound for P(S - lE(S) produce aligned amino acid and nucleotide sequences. > 7.). In the logarithmic region, Poisson approxima- The DNA databases, DDBJ, EMBL and GenBank, contain tion can be applied to give good estimates for the about 180 x lo6 basepairs as of Spring 1994 and they probability of large scores. Numerical studies are double in size every two years. New DNA sequences are performed for both these approximations. compared to the DNA databases and translations of DNA sequences into amino acid sequences are com- pared to the protein databases. These database searches 2. ALGORITHM find relationships of newly determined sequences to Our sequences will be x = x1x2 . . . xn and known sequences, providing hypotheses as to the y = y1y2.. . y, for deterministic letters xi and yj and evolution and function of the new sequences. (Barker X = XlX2.. . X, and Y = Y1 Y2.. . Y, for iid letters Xi & Dayhoff 1982; Doolittle et al. 1983). This is one of the and 5. For ease of exposition we take s(x, y) to be the ways that computation is essential to the practice of score of aligned letters and g(k) = k6 to be the penalty modern biology. of a k letter indel. This makes a penalty of 6 per The sequence comparison algorithms produce deleted letter. The first algorithm is for global scores representing the similarity of the molecules. If alignment, where all of x must be aligned with all of x and y are two aligned letters, s(x, y) is the associated y. The algorithm is an application of dynamic similarity score. A gap of k letters receives a score programming which solves the alignment problem - -g(k). Thus with s(x, y) = I(x = y) and g(k) = Sk, the by building up solutions to subproblems. Set alignpent A Si,j = S(xlx2.. .xi, yly2.. yj). An alignment achiev- b . -1 ing score S,,j must end in one of these ways goo d ga -d
has score 1 + 0 - 6 + 1 = 2 - 6 = S(A). Note that there is a deletion of the second ‘0’ in good (or an because 1, aligning deletions, is not valid in insertion of ‘0’between a and d in gad). The problem alignment. Optimality requires the alignment preced- of sequence alignment is to find the highest scoring ing the final aligned letters to be optimal if the overall alignments. Insertions and deletions (indels) make alignment is. Therefore alignment a hard computational problem. As there are thousands of sequences in a database, it Sij = max{Si-l,j - 6,Si-l,j-l + s(xi, yj), S,,j-l - 6). is not possible to look at each comparison. Instead the To begin the recursion set Soj = -6j and Si,o = -6i. scores should be screened by estimates of statistical This algorithm takes O(nrn) time. significance so that the scientist only examines the Alignments are determined by tracing back from most statistically significant alignments. the optimal score S(x, y) = S,,, to determine the steps In the next section a commonly used alignment from (O,O) to (n,rn). algorithm is presented. When applied the random Sequences that are known to be related by descent sequences of length n, the scores grow with n either from a common ancestor should be aligned by global proportional to n or proportional to log(n). The alignment. Many sequences have one or more
Phil. Trans. R. Soc. Lond. B (1994) 344, 383-390 0 1994 The Royal Society Printed in Great Britain 383 .~
384 M. Waterman Statistical signijcance of sequence alignments segments or interva that align well but can be Table 1. (a) Best .,ea1 aligament; (6) declumping for the otherwise unrelated. In this case local alignment second best alignment algorithms are recommended. The following local (4 alignment algorithm of Smith & Waterman (1981) is - CGTCCAATAOCCAAT a modification of the global alignment algorithm. TOOlOOOOlOOOOOO 1 Define cloo2looooollooo TOOlllOOlOOOOOOl H(x, y) = max(0; S(xk... xi,yl.. . yj): G010000000100000 1 Hi,j = max{Hi-l,j - 6, Hi-1,j-l s(xi,yj), Hij-1 - 6,0}, E + . CGTCCAATAGCCAAT with Hij = 0 if either i or j are 0. The score TOOlOOOOlOOOOOOl H(x,y) = max{Hij : 1 < i < n, 1 Phil. Trans. R. SOC.Lond. B (1994) Statistical signijcance of sequence alignments M. Waterman 385 When a(p,S) < 0 the situation is very different. then Positive scores of local alignments are rare events. It can be proved that for a certain constant 6, for all lE(epG> < exp [/P/(z 2 ci)]. E > 0, i=l An outline of the proof is given in Williams (1991). < (2 + r)b) + 1, To apply this to sequence alignment let s* = max{s(x,y)}, s, = min{s(x, y)}, and indel penalty and it is conjectured that limH,/logn -+ 26. A g(k) = a + Pk. Then set heuristic for this result goes as follows, Let c = max{min{2s* + 4g(l),2s* - 2s,},O}. s(x,y) = -00 when x # y and 6 = 00. H, is then the longest exactly matching region. Set p = P(X = Y). We can obtain with some work (Arratia & Waterman Neglecting end effects there are n2 places to start an 1994): alignment of length m so the expected number is about n2pm. Solving 1 = n2pm yields m = 2 logl/p(n). So we have learned that H, has linear growth for This bound will be examined with data below, but it { (p,6): a(p, 6) > 0) while H, has logarithmic growth gives exponential decay of deviations from E(&). In for {(p,8): a(p, 6) < 0). The graph of a(p,6) = 0 addition equation (2) can be extended to H, in the appears in figure 1. It is the location of a phase linear region. transition between linear and logarithmic growth of Steele (1986) has a result on the variance for non- score as sequence length n + 00. These results appear symmetric statistics that has a similar sytle upper in Arratia & Waterman ( 1994). bound. Applied to alignment we obtain The linear and logarithmic growth regions have Var(S,) < n( 1 - P)C, (3) quite distinct statistical behaviour. In the next two sections we will give some theory and numerical where p = P(Xl = Y1). simulations to illustrate this. Equations (2) and (3) give us bounds on important quantities for alignments that are generally interesting in the linear region and for global alignments. We are 4. THE LINEAR REGION interested in several questions. Does Azuma-Hoeffding provide useful bounds on the tail probabilities? Does For (p,6) such that a(p,S) > 0, we have Steele’s result provide a useful bound on the variance? Hn sn lirn- - = lim - = a(p, 6). How do these results compare between global align- n-+m n n-tm n ments and local alignments in the linear region? Therefore, divided by n,H, and S, behave the same To explore this we first look at the LCS problem way. This holds because the average score per pair of (s(x, y) = I(z = y) and 6 = 0) for P(X = A) = letters is positive and it is always advantageous to 1 - P(X = B) E (0,l). In figure 2 we give histograms extend to an essentially global alignment. of 1000 scores for n = 250,500,750,1000 along with Now some results are given for alignments with graphs of the corresponding estimates P(Sn- lES 2 yn) a(p,6) > 0. First we give a lemma that deserves to be versus y (dotted line) and the bound e-?n/(24 (solid well known. line) where c + 2. Clearly these bounds are not useful. Corresponding graphs appear in figure 3 where Lemma 1 (Azuma-Hoeffding) Let 20 = 0, Z1, Z2, global alignment with the Dayhoff PAM250 matrix . . . be a martingale relative to {F,} so that Zn-l = with g(k) = 5 + k. Here c = 58. The decay of the tail lE(WIFn-l),n > 1. rf there is a sequence of positive probabilities is slower but Azuma-Hoeffding is not constants c, such that useful. Repeating this analysis for local alignment in figure 4 does not change the curves very much. 12, - zn-1l< c, for n 2 1, These linear region local alignments are truly global alignments. For each of these three situations (LCS, global PAM250 and local PAM250), the mean of is . S, 0.8 known to grow like a . n and Var(S,) < n( 1 -p)c. The means and variances are shown in figure 5. Certainly Var(S,) looks linear in n for all three cases. 0.6 . 5. THE LOGARITHMIC REGION 6 0.4. When we move to the case where sn 0.2 . lim - = a(p,8) < 0, n+m n the positive scoring local alignments are rare events 0.0 1 and the statistical behaviour of local alignment scores 0.0 0.2 0.4 0.6 0.8 H, = H(Xl . . . X,, Y1... Y,) is very different from that cc obtained with parameters’ in the linear region. In Figure 1. Phase transition boundary. recent years Poisson approximations have been given Phil. Trans. R. SOC.Lond. B (1994) 386 M. Waterman Statistical signijcance of sequence alignments 120 A 8 80 $ 40 0 190 195 200 205 1.o 150 0.8 h g 100 0.6 1 0.4 g 50 0.2 & \ 0 0.0 385 390 395 400 405 410 415 150 1.o x 0.8 8 100 0.6 J 0.4 50 % 0.2 \ 0 0.0 590 600 610 620 150 x g 100 0.6 J 0.4 Lt" 50 0.2 790 800 810 820 0.0 0.2 0.4 0.6 0.8 1.o score Y Figure 2. Histograms and tail probabilities for LCS: (a) n = 250; (b) n = 500; (c) n = 750; and (d) n = 1000. for special cases of Hn and related statistics. For the tion is that the number of clumps exceeding the centre longest alignment with up to a given fraction of by c, with t = logl/*(rnn) + c, is Poisson with mean mismatches (Arratia et al. 1990) or a given fraction of A ymnf' = yf. indels (Neuhauser 1994) the Chen-Stein method E (Arratia et al. 1989) has been applied to give Poisson A Poisson with mean yrnnk' has no scores as large as t approximations. Our problem requires handling with probability e-rmnf. -i scoring and weighted indels. To put this style of Poisson approximation in Let s(x,y) be a scoring function so that context we refer to the Poisson clumping heuristic max,,ys(s,y) > 0 and lE(s(X,Y)) < 0, and let (Aldous 1989). In this model clumps are located ? g(k) = 00 for all k 2 1 so there are no indels. Define according to a Poisson process, and them clump sizes 5 E (0, 1) to be the largest root of 1 - E(A-s((X)Y))= 0. are assigned independently to the clumps. For our Then sequence comparison problem, the number of (alignment) clumps with score exceeding a test value n+wlim (Hn/logl/t(n2)) = 1, (4) t = centre + c has an approximate Poisson distri- in probability. When we compare sequences of length bution with mean A. The probability that at least n and m the divisor becomes logl/t(nm), which we call one score exceeds t is 1 - B(no score exceeds the centre of the distribution of H,,. For random t) = 1 -e-'. This model has only been rigorously sequences there is a constant y that can be determined established for the case described in the preceding numerically (by solving an equation) such that paragraph. None the less we provide numerical evidence that Poisson clumping model holds in the P{H(X,Y) > t = logl/t(mn) + c) - 1 - e-ymnt'. (5) entire logarithmic region. Alignment clumps are marked by the end (i,j) of optimal local alignments The first result equation (4) obtaining the centre and the score HtJ is the clump size. logllt(nm) for scoring was given in Arratia et al. There is an obvious way to estimate 6 and y by (1988). Later Karlin & Altschul (1990) extended the using equation (5). Simulate N scores H(X,Y) for result to the more general scoring schemes described sequences of length n and m. The distribution function above and presented equation (5) which is a Poisson FH(t) is e-ynm*'from the Poisson assumption. Taking a approximation. The idea of the Poisson approxima- log-log transformation of the empirical distribution Statistical signijcance of sequence alignments M. Waterman 387 120 80' 40 0.4 8 0.2 9 100 8 50 & l.O 0.8 r- 0.2 h 0.8 2 100 0.6 a 0.4 g- 50 cr 0.2 700 800 900 0.0 0.2 0.4 0.6 0.8 1.0 score Y Figure 3. Histograms and tail probabilities for global alignment using PAM250; (a) n = 250; (b) n = 500; (c) n = 750; and (d) n = 1000. function, we expect log(y) + log(nm) + t log([) and by sequence with a database sequence of length ni. The fitting this curve y and 6 can be estimated. As we use model is a straightforward sample of N comparisons, we call P(H; t) = e-ymnlt'. this method direct estimation. A drawback is the time < (6) required to do N = 1000 (say) comparisons with To obtain iid random variables, the random variables n = m = 900. u.= e-ymn,fHi A more efficient method of estimation that tests the Poisson clumping model is as follows. Let H(jl denote are all uniform (0,l). While the N sequences in the size of the i-th largest clump. Using our Swisprot are not independent, we proceed as if they computational algorithm to declump we produce are. Ordering U(l)2 U(2)2 . . . 2 U(N),recall that scores H(1) 2 Hp) 2 . . . H(N, for the first N clumps. lE(U(,))= i/(N + 1). Letting the i-th score in this list Scores 2 t should, by the clumping heuristic, be H;, constitute a random sample of H(X,Y) that exceed t. Using one comparison, then, we can declump and obtain a sample to estimate 5 and y as above. This and procedure is called declumping estimation. Now that we have described two methods to -log -log (N- 1)) + log(mni) ZZ estimate [ and y that are accomplished by simula- ( tion. Earlier we have shown that the distribution fits - logy - H,* logp. (7) independently simulated data very well even when mn is changed (Waterman & Vingron 1994). Here we A comparison of human a hemoglobin with the will explore the fit to the Swisprot database with Swisprot database was performed. Using both direct N = 14642 protein sequences. and declumping estimates of y and 5, equation (7) The test of fit is done as follows. A query sequence was used in the following .way to obtain figure 6a of length m is compared with N database sequences. (direct) and figure 66 (declumped). We simulated Score Hi comes from comparison of the query 1000 pairs sequences of length 900, iid with letter Phil. Tram. R. Sac. Lond. B (1994) 388 M. Waterman Statistical signijcance of sequence alignments 150 x 8 100 $ 50 0 h c0 s 100, bH 0. 0.0 300 350 400 450 500 J \GI 150 1 0.0 300 550 600 650 700 -750 150 x 8 100 $ 50. 42 0.2 700 750 800 850 900 950 0.0 0.2 0.4 0.6 0.8 1 .o score Y Figure 4. Histograms and tail probabilities for local alignment using PAM250; (a) n = 250; (b) n = 500; (c) n = 750; and (d) n = 1000. frequencies of hemoglobin and of the database. If the mates give about the same fit, giving a good test of the fit were perfect the data would lie on the solid line at Poisson clumping heuristic. 45". Interestingly both direct and declumped esti- Because sequences are known to have dependent 2500 200 500 200 400 600 800 loo0 200 ' 400 600 800 loo0 n n Figure 5. Means and variances as a function of n for: (a) LCS; (b) global PAM250; and (c) local PAM250. Phil. Trans. R. SOC.Lond. B (1994) Statistical SigniJicance of sequence alignments M. Waterman 389 18 vi -3 + 14 3 .& v." 3 10 v7 8 T 6 .' 5 10 15 -log(Ytmog@) Figure 8. Direct estimates with m = 142 and n = 350. likelihood model. Our approach is closely related to that of Mott (1992). He understood the implications of the phase transition results of Arratia & Waterman, and used the extreme value form of the distribution function, which is another face of Poisson approxima- tion. Mott used a four-parameter model. Using .- equation (6), the density function for H is f(t)= ymn log( l/t).$e-Tmnf'. 6 10 14 18 The likelihood of Hl, H2,.. . is then -lOg(ytmog@) Figure 6. (u) Direct estimates; (b) declumping estimates. letters, the lack of fit may simply be due to database and sequences not having iid letters. In figure 7 we show the fit from simulating sequences by a Markov chain (with estimated transition probabilities from Swis- prot). The small improvement is consistent with early work of Smith et al. (1985). In figure 8 direct estimation is used with m = 142 The equations (the length of a hemoglobin) and n = 350 (approxi- dloaIL - mately the median database sequence length). The fit -- - -u is better than that of figure 6, almost entirely due to at an improved estimate of y, which scales the 'clump and volume' ymn. d log IL -- -0 Notice that we have been fitting the distribution of dY - scores from a database comparison by estimating parameters by simulation, not by using the data to become estimate parameters. We now develop a maximum N N N ym niHitH1= 0, i=l and N ym i=l nitH' - N = 0, (9) the maximum likelihood equations. There is a quick application. In figure 6a 5 appears to be quite good. Using that value .f+ we use equation (9) to re-estimate y: N The fit shown in figure 9 is not as good as the earlier 5 10 15 fits. Finally we solve the maximum likelihood -log(yt~og@) equations (8) and (9) to obtain the fit in figure 10, Figure 7. Estimates from Markov sequences. unfortunately of about the same quality as figure 9. Phil. Trans. R. SOC.Lond. B (1994) 390 M. Waterman Statistical SigniJicance of sequence alignments Chvital, V. & Sankoff, D. 1975 Longest common I subsequences of two random sequences. J. appl. Probab. 12, 306-315. Doolittle, R.F. 1981 Similar amino acid sequences: chance or common ancestry? Science, Wash. 214, 149-159. Doolittle, R.F. et al. 1983 Simian sarcome virus onc gene, v-sir, is derived from the gene (or genes) encoding a platelet- derived growth factor. Science, Wash. 221, 275. Karlin, S. & Altschul, S.F. 1990 Methods for assessing the statistical significance of molecular sequence features by using general scoring schemes. Proc. natn. Acad. Sci. U.S.A. 87, 2264-2268. I, Mott, R. 1992 Maximum-likelihood estimation of the 6 10 14 18 statistical distribution of Smith-Waterman local sequence -log(y)-~og@) similarity scores. Bull. Math. Biol. 54, 59-76. Neuhauser, C. 1994 A Poisson approximation for sequence Figure 9. Maximum likelihood y. comparisons with insertions and deletions. Ann. Stat. (In the press.) r’ Smith, T.F. & Waterman, M.S. 1981 Identification of Common Molecular Subsequences. J. Mol. Biol. 147, 195- 197. . Smith, T.F., Burks, C. & Waterman, M.S. 1985 The statistical distribution of nucleic acid similarities. Nuclear Acid Res. 13, 645-656. Steele, J.M. 1986 An Efron-Stein inequality for nonsym- metric statistics. Ann. Stat. 14, 753-758. Waterman, M.S. & Eggert, M. 1987 A new algorithm for best subsequence alignments with application to tRNA- rRNA comparisons. J. molec. Biol. 197, 723-728. Waterman, M.S. & Vingron, M. 1994 Sequence com- parison significance and Poisson approximation. Stat. Sci. I .* (In the press.) 6 10 14 18 Williams, D. 1991 Probability with martingales. Cambridge -my)-Hog@) University Press. Figure 10. Maximum likelihood estimates. Discussion In Waterman & Vingron (1993) we applied these ideas to Newat, a database assembled by R. Doolittle R. A. ELTON (Medical Statistics Unit, University of (1981) that removes closely related sequences. In that Edinburgh, U.K.). Are there examples where com- database the sequences are more likely to be parisons of biologically unrelated sequences using independent. Our maximum likelihood fit of Newat Professor Waterman’s method give quite high prob- scores is better than we achieve here with Swisprot. It abilities? This would provide reassuring negative remains to be seen whether with a ‘single copy’ evidence for the validity of the theory. Also, how version of Swisprot will give improved fits. does the fact that real DNA is not composed of random sequences of equally common bases affect the 1.. Software is available by anonymous ftp from hto-e.usc.edu. distribution of his statistics? This research was supported by grants from the National Institutes of Health and the National Science Foundation. - M. WATERMAN.The primary issue is to be able to detect weak alignments that are biologically signifi- REFERENCES cant from alignments that score high simply due to Aldous, D. 1989 Probability approximations via the Poisson the large number of sequences and alignments. There clumping heuristic. New-York: Springer-Verlag. are at least two difficulties with running human ~1 Arratia, R., Goldstein, L. & Gordon, L. 1989 Two hemoglobin versus the Swisprot database. The first moments suffice for Poisson approximations: The Chen- comes from the non-independence of the sequences in Stein method. Ann. Probab. 17, 9-25. the database. As mentioned in the paper, another test Arratia, R., Gordon, L. & Waterman, M.S. 1990 The in Vingron & Waterman (1994) applies the theory to Erdos-Rtnyi Law in distribution, for coin tossing and Newat, a ‘single copy’ database. The fit is very good sequence matching. Ann. Statist. 18, 539-570. there with the right number of non-homologous Arratia, R., Morris, P. & Waterman, M.S. 1988 Stochastic sequences with high scores. The second problem Scrabble: a law of large numbers for sequence matching comes from the fact that real protein sequences do with scores. J. appl. Prob. 25, 106-1 19. Arratia, R. & Waterman, M.S. 1994 A phase transition for not have iid letters. As noted in Smith et al. (1985) the score in matching random sequences allowing there is a lack of sensitivity of score distribution on the deletions. Ann. appl. Prob. 4, 200-225. dependencies in biological sequences. In any case Barker, W.C. & Dayhoff, M.O. 1982 Viralsrcgeneproduct are dependencies that can be modeled can easily be related to the catalytic chain of mammalian CAMP-depen- included in our simulations. We have not found this to dent protein kinase. Proc. natn. Acad. Sci. U.S.A. 79, 2836. be necessary in practice. Phil. Trans. R. SOC.Lond. €3 (1994)