Pairwise RNA Edit Distance

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1 Pairwise RNA Edit Distance In the foowing: Sequences S 1 and S 2 associated structures P 1 and P 2 scoring of aignment: different edit operations arc atering arc removing 1) ACGUUGACUGACAACAC..(((...)))... 2) ACGAUCACGUACUAGCCUGAC...(((.((...)).)))...(((...)))..... A CGUUGACUGACAAC AC ACGAUCACGU ACUAGC CUGAC...(((.((...)).))). base deetion base match arc mismatch arc match Notation: S k [i]: position i in sequence k (for k = 1, 2). S k [i] is free if there is no arc incident in P k to i Jiang et a., 2002: above scoring scheme compexity of different probem casses agorithms

2 Edit Distance Scores base scoring: base mismatch w m, base inde w d. case 1: arc match and arc mismatch i 1 i 2 j 1 j 2 arc match (cost 0): S 1 [i 1 ] = S 2 [j 1 ] and S 1 [i 2 ] = S 2 [j 2 ] arc mismatch: S 1 [i 1 ] S 2 [j 1 ] or S 1 [i 2 ] S 2 [j 2 ] cost for mismatch: if both ends differ: w am if ony one differs: w am 2 in the foowing: different ways of deeting arcs cost: cost for deeting arc + cost for base operations case 2: arc breaking i 1 i 2 j 1 j 2 (i 1, i 2 ) in P 1, but (j 1, j 2 ) is not in P 2 cost: w b + possiby 2 w m.

3 Edit Distance Scores (Cont.) case 3: arc atering case 4: arc removing i 1 i 2 i 1 i 2 j 1 j 2 cost: w a + possiby w m. cost: w r remark: arc breaking/atering/remova can overap A U G G G A A G G G U U

4 Edit Distance Scores Summary operations on singe bases: base insertion/deetion (w d ) base mismatch (w m ) operations that act on both ends of an arc: 1. arc mismatch (w am ) 2. arc breaking (w b ) 3. arc atering (w a ) 4. arc removing (w r ) Exampe: (..)((.(.)))(..) CCGGAGGCCGCUCCCG CCG-ACCC-CGU-CC- (.).((...))...

5 Pan 1. Jiang agorithm soves the edit probem given the foowing restrictions: non-crossing (aka nested aka pseudoknot-free) input structures 1 pairwise aignment ony scoring restricted by w a = wr +w b show MAX-SNP-hardness without the restriction w a = wr +w b 2. 1 actuay, we wi see that crossing in at most one structure is OK

6 Restriction w a = w r +w b 2 Arc atering is at one end ike arc removing and at the other end arc breaking Restriction w a = wr +w b 2 captures that eft and right ends of arcs can be scored independenty if they are broken, deeted or atered. cost for arc end deetion wd end of w r,w b, and w a : w b = 2 w end b w r = 2 w end d w a = w r + w b 2 and breaking w end b = w end b + w end d instead i j end w b A k wd end j i end w d

7 cost for arc end deetion w end d Independent Arc Scoring and breaking w end b Hence: Cost arc breaking: w b = 2 w end b i 1 i 2 j 1 j 2 arc removing: w r = 2 w end d i 1 i 2 arc atering: w a = w end b + w end d i 1 i 2 j 1 j 2 of breaking or removing one end of the arc is independent of whether the other end is broken/removed or not. Ony the cost of matching one end of an arc is dependent on whether the other end is matched, too.

8 Exampe cost for arc end deetion wd end and breaking wb end arc breaking: w b = 2 wb end arc removing: w r = 2 wd end arc atering: w a = wb end + wd end (..)((.(.)))(..) CCGGAGGCCGCUCCCG CCG-ACCC-CGU-CC- (.).((...))...

9 How to make a DP agorithm for aignment? dynamic programming compute optima aignment recursivey from optima aignments of fragments questions to answer: what kind of fragments do we consider? ( semantics of a matrix entry) how to compute the soutions for a these fragments? ( recursion equation) compexity detais (evauation order, impementation detais,...)

10 Semantics of DP entry D(i, i, j, j ) D(i, i, j, j ) is the minimum cost of aigning the fragment [i, i ] of the first sequence to the fragment [j, j ] of the second sequence given that no arcs are matched that have one end inside these fragments and one end outside. Remarks The additiona restriction makes the aignment of the fragments independent of the aignment of the remaining parts. We wi see ater, why it is not sufficient to ook at (aignments of) prefixes, as done for pain sequence aignment.

11 Recursion for D(i, i, j, j ) D(i, i, j, j ) = D(i, i 1, j, j ) + w d + ψ 1 (i )(wd end w d ) D(i, i, j, j 1) + w d + ψ 2 (j )(wd end w d ) D(i, i 1, j, j 1) + χ(i, j )w m + (ψ 1 (i ) + ψ 2 (j ))wb end min if (, ) = ((i 1, i ), (j 1, j )) P 1 P 2 for some i 1, j 1 D(i, i 1 1, j, j 1 1) + D(i 1 + 1, i 1, j 1 + 1, j 1) +(χ(i 1, j 1 ) + χ(i, j )) wam 2 Notation ψ 1 (i) = 1 if i is paired in structure 1, 0 otherwise. (ψ 2 (i) anaogous) χ(i, j) = 1 if S 1 [i] S 2 [j], 0 otherwise.

12 An optimized version: Jiang Agorithm D(i, i, j, j ) aignment of subsequences in principe: a regions [i..i ] and [j..j ]. O(n 2 m 2 ) space But: not a entries are considered a j i Hence: O(nm)-matrices M for each pair of arcs,. Each matrix: O(nm) entries M (i, j)

13 Jiang Recursion reformuated recursion: M (i 1, j) + w d +ψ 1 (i)(w end d w d ) a 2 broken bond i i 1 j aigned to gap M (i, j) = min M (i, j 1) + w d +ψ 2 (j)(w end d w d ) M (i 1, j 1) + χ(i, j)w m +(ψ 1 (i) + ψ 2 (j))wb end a 1 a 2 i broken bond j 1 i 1 broken bond j 1 j i j aigned to gap M (i 1, j 1) +M a 1 (i 1, j 1) +(χ(i, j ) + χ(i, j)) wam 2 i j a 1 j a 2 i

14 Compexity time compexity: O(nm) arc pairs O(nm) aignment beow arcs = O(n 2 m 2 ) time remaining question: space compexity: each entry of some M ony depends on other entries of the same matrix Ma a1 2 and fina entries of arc pairs of smaer arcs: r 1 r 1 r r store fina vaues in separate O(nm) matrix F (in recursion, repace ookup M a 1(i 1, j 1) by F (a 1, a 2 )) it suffices to keep ony F and one M in memory simutaneousy. compute a M ordered (increasing) according to size of and

15 Compexity Matrix F : O(nm) space ony one Matrix M at a time: O(nm) space argument: for computing one entry M (i, j), recurse ony to F (, a 2 ) for smaer a 1, a 2 or entries of the same matrix M consequence: reuse space for M TOTAL: O(nm + nm) = O(nm) space drawback: traceback requires recomputation but ony O(min(n, m)) many matrices M need to be recomputed.

16 What about Pseudoknots? Why doesn t the agorithm work for pseudoknots? ast recursion case does not cover cases where matched arcs cross (compare Nussinov) ony matching of crossing arcs is a probem pseudoknots in ony one of the structures are OK.

17 The aignment hierarchy Aignment approaches have different imitations concerning the two input structures the common superstructure (e.g. for tree aignment nested) the set of edit operations aignment hierarchy cassifies aignment probems as input1 input2 superstructure with input1,input2,superstructure being one of pain: ony pain sequence (no basepairs at a) nest: ony nested structures (no pseudoknots) cross: crossing structures (pseudoknots) unim: unimited, aso severa base pairs per base possibe. Exampes: cross nest unim: Jiang agorithm nest nest nest: tree aignment

18 The aignment hierarchy besides the imitations of input and superstructure, the scoring scheme (set of edit operations) is an important difference between the various aignment probems / agorithms. Overview: aignment hierarchy (Bin&Touzet, SPIRE 2006) structures scoring schemes no atering+removing no arc atering a operations nest nest nest O(n 4 ) O(n 4 ) O(n 4 ) nest nest cross O(n 3 og(n)) NP-compete nest nest unim O(n 3 og(n)) NP-compete NP-compete cross nest cross O(n 3 og(n)) NP-compete cross nest unim O(n 3 og(n)) NP-compete Max SNP-hard cross cross cross NP-compete NP-compete cross cross unim NP-compete NP-compete Max SNP-hard unim nest unim O(n 3 og(n)) NP-compete Max SNP-hard unim cross unim NP-compete NP-compete Max SNP-hard unim unim unim NP-compete NP-compete Max SNP-hard O(n 3 og(n)): P.Kein, ESA 1998 O(n 3 ): E.Demaine et a., ICALP 2007

MONTE CARLO SIMULATIONS

MONTE CARLO SIMULATIONS Current physics research 1) Theoretica 2) Experimenta 3) Computationa Monte Caro (MC) Method (1953) used to study 1) Discrete spin systems 2) Fuids 3) Poymers, membranes, soft matter