Cost Partitioning Techniques for Multiple Sequence Alignment. Mirko Riesterer,

Transcription

1 Cost Partitioning Techniques for Multiple Sequence Alignment Mirko Riesterer,

2 Agenda. 1 Introduction 2 Formal Definition 3 Solving MSA 4 Combining Multiple Pattern Databases 5 Cost Partitioning 6 Experiments 7 Conclusion Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 2

3 Introduction Multiple Sequence Alignment Biological sequences mutate during evolution Insertion, deletion, substitution Some mutations are more likely (A G / C T) Observe phylogenetic relationships Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 3

4 Introduction Multiple Sequence Alignment Insert gaps within sequences Maximize correspondence between letters in columns Sequences ACGTG ACTAG CGTAG Alignment ACGT-G AC-TAG -CGTAG Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 4

5 Introduction Judging the alignment quality Count matches/mismatches Score matrix Point accepted mutation (PAM n ) matrix (Dayhoff et al., 1978) Blocks substitution matrix (BLOSUM) (Henikoff and Henikoff, 1992) Score matrix: A C T G A C T G Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 5

6 Agenda. 1 Introduction 2 Formal Definition 3 Solving MSA 4 Combining Multiple Pattern Databases 5 Cost Partitioning 6 Experiments 7 Conclusion Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 6

7 Formal Definition Family of sequences S = {s 1,, s n } over alphabet Σ and Σ = Σ Alignment Matrix A n m = a ij, where a ij Σ a i without is exactly s i No column contains only Score matrix: A C T G A C T G Sequences: ACT CTG Alignment A: A C T C T G C A = =7 Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 7

8 Formal Definition Score matrix can be viewed as function sub Σ Σ N Given alignment A and score matrix sub. Pair score m C A ij = k=1 sub(a ik, a jk ) Sum of pairs score C A = 1 i<j n A C ij Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 8

9 Formal Definition Shortest Path Problem Directed acyclic graph G = V, E V = x 1,, x n x i = 0,, l i } E = e 0,1 n v, v + e v, v + e V, e 0}. Figure: 2D graph alignment Figure: 3D edge structure ( Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 9

10 Agenda. 1 Introduction 2 Formal Definition 3 Solving MSA 4 Combining Multiple Pattern Databases 5 Cost Partitioning 6 Experiments 7 Conclusion Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 10

11 Solving MSA Needleman-Wunsch algorithm Dynamic programming approach Generates zero-based index table with optimal scores Dim n, lengths l: Complexity O l n Figure 2: Needleman-Wunsch score table using a score matrix Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 11

12 Solving MSA Pattern databases Family of sequences S = {s 1,, s n }. A pattern is a subset P S, P 2. A pattern database (PDB) is the perfect heuristic h for the subproblem induced by pattern P. Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 12

13 Solving MSA Heuristic search estimators Family of sequences S = {s 1,, s n }. h pair (Ikeda and Imai, 1994): h pair v = 1 i<j n h ij (v) Uses the information of every 2-dimensional PDB Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 13

14 Solving MSA Heuristic search estimators Family of sequences S = {s 1,, s n }. h all,k (Kobayashi and Imai, 1998): h all,k v = 1 n 2 k 2 1 x 1 < <x k n h x 1,,x k v Uses the information of every 3-dimensional PDB Every pair of sequences appears n 2 times normalize k 2 If k = 3, lenghts ~ 500, each PDB contains 10 8 vertices! Branching factor 2 n 1 Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 14

15 Solving MSA Heuristic search estimators Family of sequences S = {s 1,, s n }. h one,k (Kobayashi and Imai, 1998): k n h one,k v = h x 1,,x k v + h x k+1,,x n v + h x i,x j (v) i=1 j=k+1 1 or 2 higher-dimensional PDBs + remaining 2-dimensional PDBs Avoids normalization by choosing PDBs carefully h pair h one,k h all,k Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 15

16 Agenda. 1 Introduction 2 Formal Definition 3 Solving MSA 4 Combining Multiple Pattern Databases 5 Cost Partitioning 6 Experiments 7 Conclusion Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 16

17 Combining Multiple Pattern Databases Additivity A pattern collection of S = s 1,, s n is a collection P = P 1,, P m, P i S. P is non-conflicting, if no pair of elements of P conflict. Then the sum of PDBs is additive Pattern collection heuristic Admissible, if P is non-conflicting m h P v = h P i(v) i=1 Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 17

18 Combining Multiple Pattern Databases Conflicting pattern collections may violate admissibility Parts may still be useful? Canonical PDB heuristic (Haslum et al., 2007) h CAN v = max h P (v) s MNS P S Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 18

19 Combining Multiple Pattern Databases Post-hoc optimization (Pommerening et al., 2013) Use linear programming to solve constrained problem Pattern collection is strictly conflicting if m i=0 P i > 1 Let w 1,, w m be the solution to the linear program that maximizes m h PHO v = w i h P i(v) s. t. w i 1 for all strictly conflicting pattern collections S P i:p i S s. t. 0 w i 1 for all P i i Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 19

20 Combining Multiple Pattern Databases Post-hoc optimization (Pommerening et al., 2013) h all,k equals h PHO if we choose the same patterns Proof sketch: Four sequences S = s 1, s 2, s 3, s 4 of length 1 s 1 = A, s 2 = C, s 3 = T, s 4 = G Score matrix: A C T G A C T G P = P 1 = s 1, s 2, s 3, P 2 = s 1, s 2, s 4,, P 3 = s 1, s 3, s 4,, P 4 = s 2, s 3, s 4 h P 1 s = 3 h P 2 s = h P 3 s = h P 4 s = 1 h PHO s = = 3 = = h all,3 Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 20

21 Combining Multiple Pattern Databases A factored representation of MSA with operators i,j O = o x,y x,y 1 i < j n, 0 x l i, 0 y l j } i,j An operator o x,y x,y affects heuristic h P if s i, s j P Example: e.g. edge 3,3,5 4,3,6 is factored into 3 operators: 1,2 {o 3,3 4,3 1,3, o 3,5 4,6 2,3, o 3,5 3,6 } Basic factors for opreators in higher dimensions Less operators than defining all operators Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 21

22 Agenda. 1 Introduction 2 Formal Definition 3 Solving MSA 4 Combining Multiple Pattern Databases 5 Cost Partitioning 6 Experiments 7 Conclusion Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 22

23 Cost Partitioning Better estimates than using max. among available? Patterns only consider parts of the problem Combine multiple heuristic values Distribute operator costs among them Formal (Seipp et al., 2017) Given a pattern collection of size m Cost partitioning is a tuple C = c 1,, c m CP heuristic is h C v σ m i=1 h P i,c i (v) s. t. σ m i=1 c i o c o Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 23

24 Cost Partitioning Greedy zero-one cost partitioning (Haslum et al. 2005; Edelkamp, 2006) Assign full costs to at most one PDB Multiple PDBs affected? Greedily chose from ordering Assign full costs to c i o if o aff(h P i) and o i 1 j=1 aff(h P j) Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 24

25 ҧ ҧ ҧ Cost Partitioning Saturated cost partitioning (Seipp and Helmert, 2014) Assign exploitable parts of the costs to components Remainder can contribute to other components Saturated cost function Assigns the least possible cost without changing outcome Formal: saturate(h P, c) is the minimal cost function c c with h P,c v = h P,c (v) Saturated cost partitioning C = c 1,, c m remaining cost functions c 0,, c m ҧ c 0 ҧ = c c i = saturate h P i, ҧ c i = c i 1 c i c i 1 Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 25

26 Cost Partitioning Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 26

27 Agenda. 1 Introduction 2 Formal Definition 3 Solving MSA 4 Combining Multiple Pattern Databases 5 Cost Partitioning 6 Experiments 7 Conclusion Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 27

28 Experiments Using MSASolver Java program by Matthew Hatem BAliBASE Benchmark Reference Set 1 (Thompson et al., 1999) Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 28

29 Experiments Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 29

30 Agenda. 1 Introduction 2 Formal Definition 3 Solving MSA 4 Combining Multiple Pattern Databases 5 Cost Partitioning 6 Experiments 7 Conclusion Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 30

31 Conclusion GZOCP No benefit over existing heuristics PHO LP Solver in every search step Expensively computed PDB may be left unused Future Work Implement other cost partitioning techniques Generate PDBs automatically e.g. like Haslum et al. (2007) M&S heuristics (Dräger et al., 2009; Helmert et al., 2014) Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 31

32 Thank you for your attention.

33 Bibliography MO Dayhoff, RM Schwartz, and BC Orcutt. A model of evolutionary change in proteins. In Atlas of protein sequence and structure, volume 5, pages National Biomedical Research Foundation Silver Spring, MD, Steven Henikoff and Jorja G Henikoff. Amino acid substitution matrices from protein blocks. Proceedings of the National Academy of Sciences, 89: , Saul B Needleman and Christian D Wunsch. A general method applicable to the search for similarities in the amino acid sequence of two proteins. Journal of molecular biology, 48 (3): , Takahiro Ikeda and Hiroshi Imai. Fast a algorithms for multiple sequence alignment. Genome Informatics, 5:90 99, Hirotada Kobayashi and Hiroshi Imai. Improvement of the A* algorithm for multiple sequence alignment. Genome Informatics, 9: , Patrik Haslum, Adi Botea, Malte Helmert, Blai Bonet, Sven Koenig, et al. Domainindependent construction of pattern database heuristics for cost-optimal planning. In AAAI, volume 7, pages , Florian Pommerening, Gabriele Röger, and Malte Helmert. Getting the most out of pattern databases for classical planning. In Francesca Rossi, editor, Proceedings of the 23rd International Joint Conference on Artificial Intelligence (IJCAI 2013), pages , Jendrik Seipp, Thomas Keller, and Malte Helmert. A comparison of cost partitioning algorithms for optimal classical planning. In Proceedings of the Twenty-Seventh International Conference on Automated Planning and Scheduling (ICAPS 2017). AAAI Press, Patrik Haslum, Blai Bonet, Héctor Geffner, et al. New admissible heuristics for domainindependent planning. In AAAI, volume 5, pages 9 13, Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 33

34 Bibliography Stefan Edelkamp. Automated creation of pattern database search heuristics. In International Workshop on Model Checking and Artificial Intelligence, pages Springer, Jendrik Seipp and Malte Helmert. Diverse and additive cartesian abstraction heuristics. In Proceedings of the Twenty-Fourth International Conference on Automated Planning and Scheduling (ICAPS 2014). AAAI Press, pages , Julie D. Thompson, Frédéric Plewniak, and Olivier Poch. Balibase: a benchmark alignment database for the evaluation of multiple alignment programs. Bioinformatics (Oxford, England), 15:87 88, Klaus Dräger, Bernd Finkbeiner, and Andreas Podelski. Directed model checking with distance-preserving abstractions. International Journal on Software Tools for Technology Transfer, 11(1):27 37, Malte Helmert, Patrik Haslum, Jörg Hoffmann, and Raz Nissim. Merge-and-shrink abstraction: A method for generating lower bounds in factored state spaces. Journal of the ACM (JACM), 61(3):16, Cost Partitioning Techniques for Multiple Sequence Alignment, Mirko Riesterer, Universität Basel 34