RNA Secondary Structure Alignment Based on Stem Representation

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1 RNA Seondry Struture Alignment Bsed on Stem Representtion Meng-Yi Wu, Chng-Biu Yng nd Kuo-Si Hung Deprtment of Computer Siene nd Engineering Ntionl Sun Yt-sen University, Kohsiung, Tiwn Abstrt The omprison methods for RNA or protein moleules re importnt nd bsi tools in moleulr biology. So fr, most omprison methods, suh s sequene lignment, re only pplible to the primry strutures of biomoleules. Indeed, the funtions of biomoleules hve lose reltionship in their strutures. The RNA seondry struture lignment problem is to lign two given RNA strutures to get the struture similrity. In ddition, it is lso helpful to predit the funtions of biomoleules nd to lssify them. In this pper, we design dynmi progrmming method for ligning two RNA seondry strutures tht do not ontin ny pseudoknot. The time omplexity of our lgorithm is O(N 4 ), where N is the number of bloks ontined in the given RNAs. We lso pply our lgorithm to rel biomoleules, the tr- NAs of Homo spiens mitohondrion, to evlute the prtibility of our lgorithm. We tke three trna genes, TRNG, TRNA nd TRNV, to test the performne of our lgorithm. From the viewpoint of humn eyes, the struture of TRNG is more similr to TRNA. Our lgorithm lso gets this result. 1 Introdution It is well-known tht RNAs ply n importnt role for the biohemil intertions, suh s RNA trnsription, reverse trnsription, protein trnsltion, nd mrna spliing. Some RNAs re lso ribozymes, nd some of them ontin geneti informtion of viruses, suh s HIV, SARS, nd flu. The funtions of RNAs re orresponding to their strutures nd RNA-protein inter- This reserh work ws prtilly supported by the Ntionl Siene Counil of the Republi of Chinnder ontrt NSC E tions. Atully, it is not esy to get the ext struture of RNA, beuse the struture my be different in vrious environments, suh s temperture nd pressure. Beuse of the importne of RNA struture, some methods hve been proposed to predit the RNA strutures [7, 10, 13, 17, 20]. The funtions of biomoleules hve lose reltionship in their strutures. Therefore, the struture omprison methods for RNA or protein moleules re importnt nd they re bsi tools in moleulr biology. So fr, most omprison methods re only pplible to the primry strutures of biomoleules. In generl, mny RNA sequenes hve very high similrity, but their seondry strutures or funtions my not be very similr. In other words, mny RNAs hve similr seondry strutures nd funtions, but their identities my not be high enough. Hene, this pper pys ttention to the RNA seondry struture lignment problem. Our gol is to lign two RNA moleules with given seondry strutures to get the struture similrity. For distinguishing different RNA strutures, our importnt job is to onstrut the struture similrity evlution method between RNAs. Sine there re mny sequene lignment methods, some reserhers try to inorporte the seondry struture informtion into primry struture [1, 3, 9]. Lin et l. [9] proposed two novel opertions, r-ltering nd r-breking. In n RNA struture lignment, n r-ltering opertion mens tht one nuleotide is deleted nd nother nuleotide is ligned to nuleotide of the other RNA, nd n r-breking opertion mens tht two nuleotides identilly re ligned to two nuleotides tht re not bse pir of the other RNA. The bove two opertions mke the problem hrder, but the biologil menings of these opertions my need more disussion. In nother wy, n RNA struture my be represented s tree. Severl lgorithms try to dopt 60

2 the methods for solving tree problems to reonstrut the lignment nd to evlute the similrity between two RNAs [4 6, 11, 14, 15, 18]. Some methods [5, 6, 14, 15] do not use RNA nuleotide informtion, beuse it is diffiult to represent the stems nd loops in tree. Zhng [18] proposed method to solve WLCS (weighted lrgest ommon sub-struture) problem on two RNA trees by pplying the ordered tree edit distne lgorithm [19]. Lu et l. proposed nother mesurement of edit distne between lbeled trees [11], whih n be pplied on the omprison of RNA seondry strutures. If RNA seondry strutures n be represented s lbeled trees, one my lulte the edit distnes between these lbeled trees for ompring nd lssifying RNA seondry strutures. Their method is n lterntive mesurement of the similrity between RNA seondry strutures. Form the viewpoint of RNA strutures with psudoknots, some reserhers lso studied the RNA struture lignment problem [2,8,12,16]. For two given RNAs, whih one struture is known nd the other struture is unknown, Lenhof et l. [8] proposed polyhedrl pproh to lign the two RNAs. The pproh n be pplied to two RNA strutures whih hve tertiry intertions or pseudoknot. If the strutures of two RNAs re known, there re mny methods to lulte the similrity or to lign these two RNAs [2, 12, 16]. M et l. [12] proved tht lulting edit distne of two RNAs is n NP-hrd problem, where the strutures of RNAs ontin psudoknots. From the moleulr biologil or biomedil viewpoint, the stems nd loops in n RNA struture my possess some funtions or indue the intertions with other moleules, proteins nd omplexes. The stems nd loops should be importnt hrteristis for RNA strutures. If we regrd stems or loops s bloks, n RNA sequene n be represented s sequene with stem bloks nd loop bloks. The blok-representing RNA sequenes led to novel RNA struture similrity nd lignment problem. In this pper, we propose dynmi progrmming lgorithm for lulting the similrity nd lignment of two blokrepresenting RNAs with stems nd loops. The rest of this pper is orgnized s follows. In Setion 2, we introdue the RNA strutures nd relted definitions. Our RNA seondry struture lignment lgorithm bsed on stem representtion is desribed in Setion 3. We improve our lgorithm to redue the number of required indies in Setion 4. Setion 5 shows our experimentl results nd we onlude this pper in Setion 6. 2 Preliminries RNA is single strnd, omposed of nuleotides (bses) denine (A), gunine (G), ytosine (C) nd uril (U). The sequene onsisting of the nuleotides A, G, C nd U is lled the primry struture of RNA. The seondry struture of n RNA is formed by its primry struture whih is folded bk on itself. In seondry struture of n RNA, there re three possible types of bse pirs: G C, A=U, nd G U with triple, double nd single hydrogen bond, respetively. An RNA sequene is represented s string of N R elements R = r 1 r 2 r i r NR, where r i {A, G, C, U}, 1 i N R. (r i, r j ) denotes bse pir of seondry struture of R, where 1 i < j N R. For onveniene, given two nuleotides x nd y in n RNA sequene, if x is loser to 5 nd y is loser to 3, we ll x is in the left position of y nd y is in the right position of x. We n ut n RNA seondry struture into liner struture tht ontins the informtion of the seondry struture. In the liner struture, there re mny bloks ording to the informtion of the seondry struture. A blok is defined to be ontiguous sequene (or string) tht is omposed of severl nuleotides. Hene, the blok string of N B elements is denoted s B = b 1 b 2 b i b NB, where b i is blok, 1 i N B. We define stem s group of stked pirs, nd the stem in n RNA seondry struture is ontiguous mong these stked pirs. In stem, the stked pirs n be divided into two stem bloks ording to their indies. The left stem blok is loser to 5, nd the right stem blok is referred to s the blok loser to 3. We use < b i, b j > to denote the two stem bloks in the sme stem, where b i nd b j re the left nd right stem bloks, respetively. Another group of ontiguous unpired nuleotides is lled loop blok. The blok string B of n RNA seondry struture onsists of loop bloks nd stem bloks. The following exmple illustrtes n RNA struture. R = A 1 G 2 G 3 C 4 C 5 A 6 A 7 U 8 U 9 G 10 C 11 U 12 G 13 C 14 A 15 A 16 G 17 G 18 C 19 C 20 U 21 G 22 C 23 C 24 C 25 A 26 C 27 G 28 As shown in Figure 1, (G 2, U 9 ) nd (G 3, U 8 ) re bse pirs in R, where bse pirs G 2 nd G 3 re ontiguous, U 8 nd U 9 re lso ontiguous. The set of stked pirs (G 2, U 9 ) nd (G 3, U 8 ) form one stem. The left stem blok is omposed of G 2 nd 61

3 loop blok stem blok loop blok A GGCCAAUUGCUGCAAGGCCUGCCCACG A C C G G stem blok A A U U loop blok G G A A stem blok (b) C C loop blok stem blok loop blok left stem blok right stem blok left stem blok right stem blok C U G G C G C C C A C G Figure 1: The seondry struture nd the bloks of RNA sequene R = AGGCCAAUUGCUGCAAGGCCUGCCCACG. () The seondry struture of R. (b) All loop bloks nd ll stem bloks of R, where stem hs left stem blok nd right stem blok. G 3. The right stem blok is omposed U 8 nd U 9. Similrly, the bse pirs (G 13, C 23 ), (C 14, G 22 ) nd (A 15, U 21 ) form nother stem, where the left stem blok is omposed of G 13, C 14 nd A 15, nd the right stem blok is omposed U 21, G 22 nd C 23. In ddition, C 4, C 5, A 6 nd A 7 re ontiguous unpired nuleotides. In ft, C 4 C 5 A 6 A 7 is one of loop bloks in R. The RNA seondry struture hs two types nmed non-pseudoknot nd pseudoknot. The nonpseudoknot type hs no ross for eh stem of n RNA seondry struture, the pseudoknot type my hve rossing stems in n RNA seondry struture. For exmple, onsider string blok B = b 1 b 2 b 3 b 4 b 5 b 6, where b 2 nd b 4 re loop bloks nd other bloks re stem bloks. If < b 1, b 6 > nd < b 3, b 5 > re two stems, we sy B belongs to the non-pseudoknot type. If < b 1, b 5 > nd < b 3, b 6 > re two stems, then they form ross nd the RNA B hs pseudoknot. Note tht our lgorithm only onsiders the RNA strutures without ny pseudoknot. 3 RNA Seondry Struture Alignment Algorithm In this setion, we shll propose new dynmi progrmming method for performing n lignment on two RNA seondry strutures. For onveniene, we regrd the two RNA sequenes s the mster sequene nd the slve sequene. () The mster sequene is of length N R, denoted s R = r 1 r 2 r NR, nd the slve sequene is of length N R, denoted s R = r1r 2 rn. If R there re N B bloks ontined in the seondry struture of R, the N B bloks form blok string B = b 1 b 2 b NB. Similrly, B = b 1b 2 b N, B where R ontins N B bloks. An RNA seondry struture ontins set of stems, eh onsisting of two stem bloks. Suppose there re N S nd N S stem bloks in the seondry strutures of R nd R, respetively. Let S denote stem blok string nd S = s 1 s 2 s i s NS, where s i is stem blok, 1 i N S. Let S denote nother stem blok string nd S = s 1s 2 s j s N, where S s j is stem blok, 1 j N S. Let p(s i ) denote the stem prtner of stem blok s i. Tht is, < s i, p(s i ) > nd < p(s i ), s i > re stems. We define segment to be group of ontiguous nuleotides whih pper within blok or ross severl bloks. The symbol [x s, x e ] represents the segment strting t nuleotide x s nd ending t nuleotide x e. Similrly, the symbols with subsripts s nd e represent the strting nd ending positions, respetively. We define funtion ρ(s i, s j ) to deide whether two stem bloks s i nd s j form stem or not. If two stem bloks s i nd s j form stem, then ρ(s i, s j ) = 1; otherwise, ρ(s i, s j ) = 0. In ddition, we denote the omplement of ρ(s i, s j ) s ρ(s i, s j ). For onveniene, we regrd ρ(s i, s j ) nd ρ(s i, s j ) s Boolen funtions in our formuls. Let stemlign(< s i, s j >, < s k, s l >) denote the lignment sore funtion between stem < s i, s j > of R nd stem < s k, s l > of R, where 1 i < j N S nd 1 k < l N S. Let segmentlign(t, z) denote the segment lignment sore funtion, where t is segment of R nd z is segment of R. The opertor represents to find the next stem blok nd represents to find the previous stem blok. Finlly, let f(x, y, u, v, α, β, γ, δ) define the lignment sore funtion in our lgorithm, where x nd y re the stem bloks of R, u nd v re the stem bloks of R, α nd β re segments of R, nd γ nd δ re segments of R. Therefore, the initil indies of the lignment sore funtion f(x, y, u, v, α, β, γ, δ) of our lgorithm re expressed in Figure 2. Hene, the initiliztion of our lgorithm is summrized s follows. 1. x = s 1 nd y = s NS in R. 2. u = s 1 nd v = s N S in R. 62

4 x s... x e p(x) p(y) y s... y e β s β e x s... x e p(x) p(y) y s... y e α s α e β s β e Stem4 α e α s u s... u e v s... v e u s... u e v s... v e γ s γ e δ s δ e γ s γ e δ e δ s Figure 4: Illustrtion of Cse 2. Figure 2: The illustrtion of initil indies of our lgorithm. α s α e x s... x e v s... v e u s... u e γ s γ e δ s δ e Figure 3: Illustrtion of Cse 1. y s... y e β s 3. In R, if α is not null (empty), then α = b 1 ording to B, α s = (b 1 ) s nd α e = (b 1 ) e, i.e., [α s, α e ] = [(b 1 ) s, (b 1 ) e ]. Otherwise, α is null, α s = x s nd α e = x s 1, i.e., [α s, α e ] = [x s, x s 1]. It is similr for γ in R. If γ is not null, then [γ s, γ e ] = [(b 1) s, (b 1) e ]. Otherwise, γ is null, [γ s, γ e ] = [u s, u s 1]. 4. In R, if β is not null, then β = b NB ording to B, β s = (b NB ) s nd β e = (b NB ) e, i.e., [β s, β e ] = [(b NB ) s, (b NB ) e ]. Otherwise, β is null, β s = y e + 1 nd β e = y e, i.e., [β s, β e ] = [y e + 1, y e ]. It is similr for δ in R. If δ is not null, then [δ s, δ e ] = [(b N B ) s, (b N B ) e]. Otherwise, δ is null, [δ s, δ e ] = [v e + 1, v e ]. In our lgorithm, the input dt re two RNA seondry strutures without ny pseudoknot. In other words, their seondry strutures look like ( (( ( ))) ), where () represents stem nd represents loop blok. We divide our lignment lgorithm into four ses ording to the situtions of stem bloks s follows. Cse 1: ρ(x, y) = 1 nd ρ(u, v) = 1. Figure 3 illustrtes Cse 1. We n further divide the se into three subses. Cse 1.1: Stem < x, y > ligns with stem < u, v >. We must find solution for β e stem bloks x 1, y 1, u 1 nd v 1. Thus, f(x, y, u, v, α, β, γ, δ) = f(x 1, y 1, u 1, v 1, [x e + 1, (x 1) s 1], [(y 1) e + 1, y s 1], [u e + 1, (u 1) s 1], [(v 1) e + 1, v s 1]) + stemlign(< x, y >, < u, v >)+segmentlign(α, γ)+ segmentlign(β, δ). Cse 1.2: < x, y > is not ligned with < u, v >, nd < x, y > of R beomes prt of segment. We must find solution for stem bloks x 1, y 1, u nd v. Thus, f(x, y, u, v, α, β, γ, δ) = f(x 1, y 1, u, v, [α s, (x 1) s 1], [(y 1) e + 1, β e ], γ, δ). Cse 1.3: < x, y > is not ligned with < u, v >, nd < u, v > of R beomes prt of segment. We must find solution for stem bloks x, y, u 1 nd v 1. Thus, f(x, y, u, v, α, β, γ, δ) = f(x, y, u 1, v 1, α, β, [γ s, (u 1) s 1], [(v 1) e + 1, δ e ]). Cse 2: ρ(x, y) = 0 nd ρ(u, v) = 1. Figure 4 illustrtes Cse 2. This se n be further divided into three subses s follows. Cse 2.1: Find the prtner p(y) of stem blok y. We must find solution for stem bloks x, p(y) 1, u nd v. Thus, f(x, y, u, v, α, β, γ, δ) = f(x, p(y) 1, u, v, α, [(p(y) 1) e + 1, β e ], γ, δ). Cse 2.2: Find p(x) of x. We must find solution for stem bloks p(x) 1, y, u nd v. Thus, f(x, y, u, v, α, β, γ, δ) = f(p(x) 1, y, u, v, [α s, (p(x) 1) s 1], β, γ, δ). Cse 2.3: Stem < u, v > of R is prt of segment. We must find solution for stem bloks x, y, u 1 nd v 1. Thus, f(x, y, u, v, α, β, γ, δ) = f(x, y, u 1, v 1, α, β, [γ s, (u 1) s 1], [(v 1) e +1, δ e ]). 63

5 α s α e x s... x e y s... y e u s... u e p(u) p(v) v s... v e γ s γ e δ s δ e Stem4 α s α e Figure 5: Illustrtion of Cse 3. x s... x e p(x) p(y) y s... y e γ s γ e u s... u e p(u) p(v) Stem4 Figure 6: Illustrtion of Cse 4. β s Stem4 β e v s... v e Cse 3: ρ(x, y) = 1 nd ρ(u, v) = 0. Figure 5 illustrtes Cse 3. We n further divide the se into three subses. Cse 3.1: Find p(v) of v. We must find solution for stem bloks x, y, u nd p(v) 1. Thus, f(x, y, u, v, α, β, γ, δ) = f(x, y, u, p(v) 1, α, β, γ, [(p(v) 1) e + 1, δ e ]). Cse 3.2: Find p(u) of u. We must find solution for stem bloks x, y, p(u) 1 nd v. Thus, f(x, y, u, v, α, β, γ, δ) = f(x, y, p(u) 1, v, α, β, [γ s, (p(u) 1) s 1], δ). Cse 3.3: Stem < x, y > of R is prt of segment. We must find solution for stem bloks x 1, y 1, u nd v. Thus, f(x, y, u, v, α, β, γ, δ) = f(x 1, y 1, u, v, [α s, (x 1) s 1], [(y 1) e + 1, β e ], γ, δ). Cse 4: ρ(x, y) = 0 nd ρ(u, v) = 0. Figure 6 illustrtes Cse 4. We n further divide the se into six subses s follows. Cse 4.1: Find p(y) of y. We must find solution for stem bloks x, p(y) 1, u nd v. Thus, f(x, y, u, v, α, β, γ, δ) = f(x, p(y) 1, u, v, α, [(p(y) 1) e + 1, β e ], γ, δ). β s δ s β e δ e Cse 4.2: Find p(v) of v. We must find solution for stem bloks x, y, u nd p(v) 1. Thus, f(x, y, u, v, α, β, γ, δ) = f(x, y, u, p(v) 1, α, β, γ, [(p(v) 1) e + 1, δ e ]). Cse 4.3: Find p(y) of y nd p(v) of v. Thus, we divide this subse into two mpping groups. The first group is for stem bloks x, p(y) 1, u nd p(v) 1. The seond group is for stem bloks p(y), y, p(v) nd v. We must find the solution for both groups. Thus, f(x, y, u, v, α, β, γ, δ) = f(x, p(y) 1, u, p(v) 1, α, [(p(y) 1) e + 1, p(y) s 1], γ, [(p(v) 1) e + 1, p(v) s 1]) + f(p(y), y, p(v), v, [p(y) s, p(y) s 1], β, [p(v) s, p(v) s 1], δ). Cse 4.4: Find p(x) of x. We must find solution for stem bloks p(x) 1, y, u nd v. Thus, f(x, y, u, v, α, β, γ, δ) = f(p(x) 1, y, u, v, [α s, (p(x) 1) s 1], β, γ, δ). Cse 4.5: Find p(u) of u. We must find solution for stem bloks x, y, p(u) 1 nd v. Thus, f(x, y, u, v, α, β, γ, δ) = f(x, y, p(u) 1, v, α, β, [γ s, (p(u) 1) s 1], δ). Cse 4.6: Find p(x) of x nd p(u) of u. Thus, we divide the subse into two mpping groups. The first group is for stem bloks x, p(x), u nd p(u). The seond group is for stem bloks p(x) 1, y, p(u) 1 nd v. We must find solution for the two groups. Thus, f(x, y, u, v, α, β, γ, δ) = f(x, p(x), u, p(u), α, [p(x) e + 1, p(x) e ], γ, [p(u) e +1, p(u) e ])+f(p(x) 1, y, p(u) 1, v, [p(x) e +1, (p(x) 1) s 1], β, [p(u) e + 1, (p(u) 1) s 1], δ). 4 Improvement on Spe The funtion f(x, y, u, v, α, β, γ, δ) hs n 8- dimensionl rgument, nd the mximum rnge of eh rgument in one dimension is the mximum number of bloks in the blok strings (i.e., mx(n B, N B )). One n esily observe tht both time omplexity nd spe omplexity of our lgorithm re O((mx(N B, N B )) 8 ). For prtil implementtion in omputers, the formul is infesible when the number of RNA bloks is lrge. Aordingly, we hve to try to redue the required memory spe. Tht is, we 64

6 will try to derese the dimensions (i.e., the number of indies) of the dynmi progrmming formul f(x, y, u, v, α, β, γ, δ). It will led to the design of new dynmi progrmming formul f(i, j, k, l) whose funtion is the sme s tht of f(x, y, u, v, α, β, γ, δ). Before introduing formul f(i, j, k, l), we need modify the symbols, terms nd funtions defined in formul f(x, y, u, v, α, β, γ, δ), nd define some more. Let I(x, S, B) denote the blok in the blok string B whih is orresponding to the stem blok x of stem blok string S, where 1 x N S nd 1 I(x, S, B) N B. In other words, if s x = b i, then I(x, S, B) = i. Let I(x, B, S) denote the stem blok of S orresponding to blok x of B, but I(x, B, S) = null if we n not find the orresponding stem blok. Let L(x, i, j) denote the mximl blok between bloks i nd x, where I(L(x, i, j), B, S) null nd i p(l(x, i, j)) j. In other words, L(x, i, j) = mx i y x {y y is blok of B, I(y, B, S) null nd i p(y) j}. Similrly, R(x, i, j) = min x y j {y y is blok of B, I(y, B, S) null nd i p(y) j }. In ddition, we define LN(x, i, j) = mx i y x {y y is blok of B, I(y, B, S) null nd i p(y) j}. Similrly, R(x, i, j) = min x y j {y y is blok of B, I(y, B, S) null nd i p(y) j }. LN(x, i, j) nd RN(x, i, j) re used for finding the next stems. The bove funtions for S nd B re defined similrly. In the new dynmi progrmming lgorithm, the initil vlues of the vribles in f(i, j, k, l) re set tht i nd j re the blok indies in B, nd i = 1 nd j = N B in R; k nd l re the blok indies in B, nd k = 1 nd l = NB in R. Then formul f(i, j, k, l) is given in Figure 7. segmentlign(i, j) = min(n R (i), n R (j)) φ, where φ is the sore for ligning two nuleotides nd it is onstnt, nd n R (i) nd n R (j) re the lengths of segments i nd j, respetively. (8) For exmple, onsider two segments i = CGAAGUC nd j = AAUGAGCUG, where segments i nd j hve 7 nd 9 nuleotides, respetively. If φ = 1 in Eqution 8, then segmentlign(i, j) = min(7, 9) 1 = 7. When φ = 0, our lgorithm only ligns the stem prt of RNA seondry strutures. Here, we will set φ to 1 when disussing the performne of our lgorithm. In the stem lignment problem, we onsider only bse pirs here. Tht is, we only lign the bse pirs in both stems. Let x be one stem of the seondry struture of R nd n bp (x) be the number of bse pirs in x. Similrly, let y be one stem of the seondry struture of R nd n bp (y) be the number of bse pirs in y. Hene, stemlign(x, y) is given s follows. stemlign(x, y) = (min(n bp (x), n bp (y))) λ τ, where λ nd τ re onstnts. (9) For exmple, suppose two stems x = AAG CUU nd y = AGGCC GGCCU, stem x hs 3 bse pirs nd stem y hs 5 bse pirs. If λ = 2 nd τ = 4 in Eqution 9, then stemlign(x, y) = (min(3, 5)) 2 4 = 36. We n see tht there re four indies in the improved formul f(i, j, k, l), nd the number of ll ses re fixed. Therefore, it is trivil tht the time omplexity of our improved dynmi progrmming lgorithm is O(N 4 ), where N is the number of bloks ontined in the given RNAs (i.e. N = mx(n B, N B )). segmentlign(i, j) represents the segment lignment sore funtion, where i is segment of R nd j is segment of R. The lultion of segmentlign(i, j) is to find the miniml length between segments i nd j. Indeed, one n esily modify the funtion segmentlign(i, j) to lign segments i nd j. We do not onsider whether the ligned nuleotides re the sme or not, beuse the RNA seondry struture lignment ppreites the lignment of the struturl shpe. We shll onsider only the segment length here. Hene, segmentlign(i, j) is given s follows. 5 Experimentl Results In this setion, we will show our experimentl results nd nlyze the performne of our lgorithm. Our lgorithm is implemented by C++ using Borlnd C++ Compiler 5.5 on PC with AMD Duron T M proessor 800 MHz nd 256 MB RAM. Our testing dt re obtined from NCBI ( There re 22 trnas in Homo spiens mitohondrion. In our experiments, we use the method in [7] to predit the seondry strutures of trnas. 65

7 f 1 (i, j, k, l) = mx RN(i, i, j) x LN(j, i, j) RN(k, k, l) y LN(l, k, l) { f(i, x 1, k, y 1) + f(x, p(x), y, p(y)) + f(p(x) + 1, j, p(y) + 1, l) } (1) f 2 (i, j, k, l) = segmentlign([i s, (R(i, i, j) 1) e], [k s, (R(k, k, l) 1) e]) +segmentlign([(l(j, i, j) + 1) s, j e ], [(L(l, k, l) + 1) s, l e ]) +stemlign(< R(i, i, j), L(j, i, j) >, < R(k, k, l), L(l, k, l) >) + f(r(i, i, j) + 1, L(j, i, j) 1, R(k, k, l) + 1, L(l, k, l) 1) (2) f 3 (i, j, k, l) = mx f 4 (i, j, k, l) = mx f(i, R(i, i, j) 1, k, R(k, k, l) 1) + f(r(i, i, j), LN(p(L(j, i, j)), i, j), R(k, k, l), L(l, k, l)) + f(ln(p(l(j, i, j)), i, j) + 1, j, L(l, k, l) + 1, l) f(i, RN(p(R(i, i, j)), i, j) 1, k, R(k, k, l) 1) + f(rn(p(r(i, i, j)), i, j), L(j, i, j), R(k, k, l), L(l, k, l)) + f(l(j, i, j) + 1, j, L(l, k, l) + 1, l) f(i, R(i, i, j) 1, k, RN(R(k, k, l), k, l) 1) + f(r(i, i, j), L(j, i, j), RN(R(k, k, l), k, l), LN(L(l, k, l), k, l)) + f(l(j, i, j) + 1, j, LN(L(l, k, l), k, l) + 1, l) (3) f(i, R(i, i, j) 1, k, R(k, k, l) 1) + f(r(i, i, j), L(j, i, j), R(k, k, l), LN(p(L(l, k, l)), k, l)) + f(l(j, i, j) + 1, j, LN(p(L(l, k, l)), k, l) + 1, l) f(i, R(i, i, j) 1, k, RN(p(R(k, k, l)), k, l) 1) + f(r(i, i, j), L(j, i, j), RN(p(R(k, k, l)), k, l), L(l, k, l)) + f(l(j, i, j) + 1, j, L(l, k, l) + 1, l) f(i, RN(R(i, i, j), i, j) 1, k, R(k, k, l) 1) + f(rn(r(i, i, j), i, j), LN(L(j, i, j), i, j), R(k, k, l), L(l, k, l)) + f(ln(l(j, i, j), i, j) + 1, j, L(l, k, l) + 1, l) (4) f 5(i, j, k, l) = mx f(i, R(i, i, j) 1, k, R(k, k, l) 1) + f(r(i, i, j), LN(p(L(j, i, j)), i, j), R(k, k, l), L(l, k, l)) + f(ln(p(l(j, i, j)), i, j) + 1, j, L(l, k, l) + 1, l) f(i, R(i, i, j) 1, k, R(k, k, l) 1) + f(r(i, i, j), L(j, i, j), R(k, k, l), LN(p(L(l, k, l)), k, l)) + f(l(j, i, j) + 1, j, LN(p(L(l, k, l)), k, l) + 1, l) f(i, p(l(j, i, j)) 1, k, p(l(l, k, l)) 1) + f(p(l(j, i, j)), j, p(l(l, k, l)), l) f(i, RN(p(R(i, i, j)), i, j) 1, k, R(k, k, l) 1) + f(rn(p(r(i, i, j)), i, j), L(j, i, j), R(k, k, l), L(l, k, l)) + f(l(j, i, j) + 1, j, L(l, k, l) + 1, l) f(i, R(i, i, j) 1, k, RN(p(R(k, k, l)), k, l) 1) + f(r(i, i, j), L(j, i, j), RN(p(R(k, k, l)), k, l), L(l, k, l)) + f(l(j, i, j) + 1, j, L(l, k, l) + 1, l) f(i, p(r(i, i, j)), k, p(r(k, k, l))) + f(p(r(i, i, j)) + 1, j, p(r(k, k, l)) + 1, l) (5) f 6(i, j, k, l) = { segmentlign([is, j e ], [k s, l e ]) if i = j nd k l or i j nd k = l 0 if i > j or k > l f 1 (i, j, k, l) (6) (ρ(i, j) ρ(k, l) + ρ(r(i, i, j), L(j, i, j)) ρ(r(k, k, l), L(l, k, l))) f 2 (i, j, k, l) f(i, j, k, l) = mx (ρ(i, j) ρ(k, l) + ρ(r(i, i, j), L(j, i, j)) ρ(r(k, k, l), L(l, k, l))) f 3 (i, j, k, l) (ρ(i, j) ρ(k, l) + ρ(r(i, i, j), L(j, i, j)) ρ(r(k, k, l), L(l, k, l))) f 4(i, j, k, l) if i < j nd k < l (7) (ρ(i, j) ρ(k, l) + ρ(r(i, i, j), L(j, i, j)) ρ(r(k, k, l), L(l, k, l))) f 5 (i, j, k, l) segmentlign([i s, j e ], [k s, l e ]) f 6(i, j, k, l) otherwise Figure 7: The new dynmi progrmming formul f(i, j, k, l). 66

8 Some prmeters re used to mesure the LCS (longest ommon subsequene) identity of the two sequenes. M in(l) denotes the lignment length of the miniml LCS lignment. The LCS identity is defined s follows. LCS identity = LCS length Min(L) 100% Then, some prmeters re used to mesure the struture similrity of the seondry strutures of R nd R. Sore(R, R ) denotes the sore of the seondry struture lignment of R nd R by using our lgorithm. Mx(R, R ) denotes the mximl sore between the two sores whih R nd R re ligned with themselves, respetively. Tht is, Mx(R, R ) = Mx{Sore(R, R), Sore(R, R )}. Then the struture similrity is defined s follows. struture similrity = Sore(R, R ) Mx(R, R ) 100% We shll hnge vlues of some prmeters in Eqution 9. λ dvnes (mkes) tht the lignment sore of bse pirs between two stems is nonliner enhnement nd τ denotes the stem sore ontributed by ligning two bse pirs in two stems. We test struture similrities of eh pir of 22 trnas by using our lgorithm, where λ = 1, 1.5 nd 2, nd τ = 2, 3, 4, 5 nd 6. Consequently, we hve ome to the onlusion tht λ n be set to 1.5 nd τ n be set to 4. Though mny RNA sequenes shre very high identity, their seondry strutures or funtions my not be very similr. Mentime, the seondry strutures nd funtions of mny RNA sequenes re very similr, but they my not shre high sequene identity. Hene, we will disuss the performne of our lgorithm ording to the similrities of sequenes nd strutures. Tble 1 shows the result bsed on λ = 1.5 nd τ = 4, where the input dt is TRNG nd TRNA for Cse A, nd TRNG nd TRNV for Cse B. They re trnas of Homo spiens mitohondrion. In Tble 1, we n see tht LCS identity is % nd % of Cse A nd Cse B, respetively. Tht is, the LCS identity of Cse B is higher thn Cse A. The RNA seondry strutures of Cse A re predited by using dynmi progrmming method [7]. Figure 8 nd Figure 9 show the seondry strutures of Cse A nd Cse B, respetively. In Figure 8 nd Figure 9, we n see tht the two seondry strutures of Cse A re more similr thn those of Cse B. u g g u u u u g u u g u u u u g u u u u g u g () g g u g u g u g u g u u g g u u g u g g g u g g g g u u u u g g g u g g Figure 8: The optiml lignment for Cse A by using our lgorithm nd λ = 1.5 nd τ = 4. () The seondry struture of TRNG. (b) The seondry struture of TRNA. Note: The seondry strutures re predited by using dynmi progrmming method [7]. In Tble 1, we n see tht the struture similrity of Cse A is % nd struture similrity of Cse B is %. In other words, the struture similrity of Cse A is higher thn Cse B by pplying our lgorithm, but the LCS identity of Cse B is higher thn Cse A. Hene, our lgorithm is good pproh for RNA seondry struture lignment. 6 Conlusion In this pper, we propose n RNA seondry struture lignment lgorithm to evlute the seondry struture similrity of two RNAs bsed on bloks for deling with stems nd loops. For the prtil implementtion in omputers, the eight indies of our originl formul mke the progrm infesible when the number of RNA bloks is lrge. We improve the formul into four-index one. We n see tht there re four indies in the improved formul, both time omplexity nd spe omplexity of our improved dynmi progrmming lgorithm re redued to O(N 4 ), where N is the number of bloks ontined in the given RNAs. For the experiment of the biologil dt, we (b) 67

9 Tble 1: The experimentl results of Cse A nd Cse B bsed on λ = 1.5 nd τ = 4. Cse A: TRNG nd TRNA. Cse B: TRNG nd TRNV. λ τ Cse LCS length Min(L) LCS identity Sore(R, R ) Mx(R, R ) Struture similrity A % % B % % u g g u u u u g u u g u u u u g u u u u g u g u u g g g g g u u g u u g g u g u g g g g u Referenes tke three of trnas, tht re TRNG, TRNA, nd TRNV, s the input dt of our lgorithm. In the result, we see tht the struture similrity of TRNG nd TRNA is higher, but their LCS identity is lower thn TRNG nd TRNV. Hene, our blok onept nd our lgorithm should be good pproh nd method for RNA seondry struture lignment. In ddition, it is lso helpful to predit the funtions of biomoleules nd to lssify them. In the future, we will test more rel dt to verify the prtility of our lgorithm. The psudoknots re not onsidered in our lgorithm, it is worth wy to develop new RNA struture lignment method to del with psudoknots. [1] V. Bfn, S. Muthukrishnn, nd R. Rvi, Computing similrity between RNA strings, Proeedings of the 6th Annul Symposium on Combintoril Pttern Mthing, pp. 1 16, [2] G. D. Collins, S. Le, nd K. Zhng, A new method for omputing similrity between RNA strutures, Informtion Sienes, Vol. 139, pp , [3] F. Corpet nd B. Mihot., RNAlign progrm: Alignment of RNA sequenes using both primry nd seondry strutures, Computer Applitions in the Biosienes, Vol. 10, No. 4, pp , () Figure 9: The optiml lignment for Cse B by using our lgorithm nd λ = 1.5 nd τ = 4. () The seondry struture of TRNG. (b) The seondry struture of TRNV. Note: The seondry strutures re predited by using dynmi progrmming method [7]. (b) [4] T. Jing, L. Wng, nd K. Zhng, Alignment of trees - n lterntive to tree edit, Theoretil Computer Siene, Vol. 143, No. 1, pp , [5] S. Y. Le, R. Nussinov, nd J. V. Mizel, Tree grphs of RNA seondry strutures nd their omprisons, Computers nd Biomedil Reserh, Vol. 22, pp , [6] S. Y. Le, J. Owens, R. Nussinov, J. H. Chen, B. A. Shpiro, nd J. V. Mizel, RNA seondry strutures: Comprison nd determintion of frequently reurring substrutures by onsensus, Computer Applitions in the Biosienes, Vol. 5, No. 3, pp , [7] R. C. T. Lee, Computtionl biology. Deprtment of Computer Siene nd Informtion Engineering, Ntionl Chi-Nn University, [8] H. P. Lenhof, K. Reinert, nd M. Vingron, A polyhedrl pproh to RNA sequene struture lignment, Journl of Computtionl Biology, Vol. 5, No. 3, pp ,

10 [9] G. H. Lin, B. M, nd K. Zhng, Edit distne between two RNA strutures, Proeedings of the fifth Annul Interntionl Conferene on Computtionl Biology, pp , [10] M. C. Lin, C. B. Yng, nd K. S. Hung, Predition of RNA seondry strutures by geneti lgorithms, Proeedings of the 6th World Multionferene on Systemis, Cybernetis nd Informtis, SCI 2002, Vol. 12, pp , [19] K. Zhng nd D. Shsh, Simple fst lgorithms for the editing distne between trees nd relted problems, SIAM Journl on Computing, Vol. 18, No. 6, pp , [20] M. Zuker nd D. Snkoff, RNA seondry strutures nd their predition, Mthemtil Biosiene, Vol. 46, pp , [11] C. L. Lu, Z. Y. Su, nd C. Y. Tng, A new mesure of edit distne between lbeled trees, Proeedings of the 7th Annul Interntionl Computing nd Combintoris Conferene (COCOON 2001), Vol. 2108, pp , [12] B. M, L. Wng, nd K. Zhng, Computing similrity between RNA strutures, Theoretil Computer Siene, Vol. 276, pp , [13] C. Notredme, E. O Brien, nd D. G. Higgins, RAGA: RNA sequene lignment by geneti lgorithm, Nulei Aids Reserh, Vol. 25, No. 22, pp , [14] B. A. Shpiro, An lgorithm for ompring multiple RNA seondry strutures, Computer Applitions in the Biosienes, Vol. 4, No. 3, pp , [15] B. A. Shpiro nd K. Zhng, Compring multiple RNA seondry strutures using tree omprisons, Computer Applitions in the Biosienes, Vol. 6, No. 4, pp , [16] Z. Wng nd K. Zhng, Alignment between two RNA strutures, Proeedings of the 26th Interntionl Symposium on Mthemtil Foundtions of Computer Siene, pp , [17] M. S. Wtermn nd T. F. Smith, RNA seondry struture: A omplete mthemtil nlysis, Mthemtil Biosiene, Vol. 42, pp , [18] K. Zhng, Computing similrity between RNA seondry strutures, In Proeedings of IEEE Interntionl Joint Symposi on Intelligene nd Systems, pp ,