BIOINF 4120 Bioinforma2cs 2 - Structures and Systems -

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1 BIIF 4120 Bioinforma2cs 2 - Structures and Systems - liver Kohlbacher SS RA Structure Part I verview RA Types of RA and their biological func@on Two- dimensional structure Three- dimensional structure RA structure predic@on Formal defini@on of secondary structure Combinatorics of structures ussinov folding algorithm Basics of RA RA (Riboucleic Acid) is a biopolymer consis@ng of A sugar- phosphate backbone ne of four bases bound to the sugar (A, C, G, U) The RA World hypothesis suggests that RA was the primary carrier of life Proteins and DA evolved later to take over specific func@ons from RA (biocatalysis, more reliable informa@on storage) RA can have numerous biological func@on, the most well- known is its role as mra in transcrip@on/transla@on 1

RA Func2on RA has many biological func@ons and not all of them are yet fully understood: Informa@on transfer (mra) Amino acid transport (tra) Cataly@c func@on (ribozymes) Gene regula@on (sira) Genome

jpg The RA Zoo Transfer RA (tra) Transports amino acids to the ribosome Messenger RA (mra) Blue print for protein biosynthesis, maturing/splicing of mra creates eucaryo<c intron/exon structure

2 RA Func2on RA has many biological and not all of them are yet fully understood: transfer (mra) Amino acid transport (tra) (ribozymes) Gene (sira) Genome (RA viruses) hdp://images2.clinicaltools.com/images/gene/mrna.jpg The RA Zoo Transfer RA (tra) Transports amino acids to the ribosome Messenger RA (mra) Blue print for protein biosynthesis, maturing/splicing of mra creates eucaryo<c intron/exon structure Ribosomal RA (rra) Central part of the ribosome, responsible for tra recogni<on and pep<dyl transferase ac<vity Small interfering RA (sira) and micro RA (mira) Involved in gene regula<on and silencing. RA Func2on Small RAs Just when scien2sts thought they had deciphered the roles played by the cell's leading actors, a familiar performer has turned up in a stunning variety of guises. RA, long upstaged by its more glamorous sibling, DA, is turning out to have star quali2es of its own. (Science, 298 (2002), 2296) 2

3 RA Func2on hxp:// RAi hdp:// RA Chemistry Four bases Adenine (A) Cytosine (C) Guanine (G) Uracil (U) A/G: purine bases C/U: pyrimidine bases 2 Adenine 2 Guanine 2 Cytosine Uracil U in RA replaces thymine (T) used in DA 7- purine pyrimidine 3

4 RA Chemistry Sugar backbone contains ribose instead of deoxyribose (in DA) Sequence of bases read from 5 to 3 end gives primary structure Due to the reac@ve hydroxyl group, ribose is much less stable than deoxyribose Sugars are connected by phosphate groups Phosphate bears a nega@ve charge RA P B - B P - B 2 -hydroxy DA P B - B P - B 2 -deoxy RA Chemistry Base + sugar: nucleoside ucleoside + P i : nucleo4de C- G and A- U are complementary Called Watson- Crick pairs Stabilized by three/two hydrogen bonds C- G base pair P ucleoside ucleo2de P A- U base pair (full nucleo2des) Wobble Pairs Wobble pairs (G- U) occur as well Stability of different base pairs varies C- G > A- U > G- U As with DA, the three hydrogen bonds of C- G pairs contribute most to the overall energy G- U wobble pair 4

RA Chemistry RA usually occurs single- stranded (ssra), although double- stranded RA occurs as well (dsra) Preferences are different for DA (which preferen@ally occurs as dsda) RA

usually does RA Secondary Structures D. Mount, Bioinforma@cs, p.

5 RA Chemistry RA usually occurs single- stranded (ssra), although double- stranded RA occurs as well (dsra) Preferences are different for DA (which occurs as dsda) RA prefers to form intramolecular base pairs instead, whereas DA prefers to form intermolecular base pairs (double helices) RA molecules thus fold into more complex shapes than DA usually does RA Secondary Structures D. Mount, p. 209 Secondary Structure Representa2on Secondary structure is usually depicted as a graph Besides the structural drawings seen on the last slide, linear or circular representa@ons are common Dot plots can be used as well Structure: hdp:// 5

Secondary Structure Representa2on Structure: hdp://www.rnaso`.

molecule_id=spr_00016 molecule_id=spr_00016 RA Secondary

Secondary structure is essen@al for func@on! hdp://rna.ucsc.

6 Secondary Structure Representa2on Structure: hdp:// Secondary Structure Representa2on Structure: hdp:// RA Secondary Structure RA secondary structure can become quite complex Secondary structure is for hdp://rna.ucsc.edu/rnacenter/images/figs/thermus_16s_2ndry.jpg hdp://www3.interscience.wiley.com:8100/legacy/college/boyer/ /structure/tra/trna_diagram_small.gif 6

RA Ter2ary Structure Three- dimensional structure can be determined with x- ray crystallography (details on that later when we

) More complex than 2D structure, relevant to explain func@on Base pairs in 2D structure and 3D structure are of course the

7 RA Ter2ary Structure Three- dimensional structure can be determined with x- ray crystallography (details on that later when we talk about proteins!) More complex than 2D structure, relevant to explain Base pairs in 2D structure and 3D structure are of course the same! Example: Phe- tra from yeast GCGGAUUUAGCUCAGUUGGGAGAGCGCCA GACUGAAGAUCUGGAGGUCCUGUGUUCGA UCCACAGAAUUCGCACCA tra Ter2ary Structure PDB: 1EVV Formaliza2on Def. 1: An RA sequence of length n is a string s = (s 1, s 2, s 3,, s n ) with s i 2 RA = {A, C, G, U} 8 i. Def. 2: An RA secondary structure for a sequence s is a set P of ordered base pairs (i, j) with 1 i < j n such that j - i > 3 (bases are not too close to each other bend radius!) {i, j} Å {i, j } = (base pairs do not conflict) 7

bend radius!) {i, j} Å {i, j } = (base pairs do not conflict) estedness Def.

8 Formaliza2on - Example Def. 2: An RA secondary structure for a sequence s is a set P of ordered base pairs (i, j) with 1 i < j n such that j - i > 3 (bases are not too close to each other bend radius!) {i, j} Å {i, j } = (base pairs do not conflict) estedness Def. 3: A secondary structure is called nested, if for any two base pairs (i, j) and (i, j ), w.l.o.g. i < i, we have either 1. i < j < i < j, i.e. (i, j) precedes (i, j ), or 2. i < i < j < j, i.e. (i, j) includes (i, j ) Crossing, not nested estedness and Pseudo Knots Pseudo knots arise from non- nested secondary structures Base pairs from different stems cross each other In this example the loop region of one hairpin is also part of another stem In the arc diagram, the base pairs are thus crossing, not nested We will only consider nested secondary structures, as the methods presented here become intractable for pseudo- knoxed structures hdp://en.wikipedia.org/wiki/image:pseudoknot.svg 8

Pseudo Knots Pseudo knots are not uncommon and occur in numerous RA structures Some of them are even func@onally relevant and highly conserved A prominent example is telomerase Legassie & Jarsqer,

9 Pseudo Knots Pseudo knots are not uncommon and occur in numerous RA structures Some of them are even relevant and highly conserved A prominent example is telomerase Legassie & Jarsqer, Structure, 14 (2006), 1603 Secondary Structure Predic2on Predic@ng the secondary structure of RA is a hard problem Formal problem deﬁni2on Given an RA sequence Iden@fy the correct secondary structure Ques2ons ow many possible secondary structures? ow to dis@nguish the correct structure from the others? Combinatorics of Sec. Structures ow many possible (diﬀerent) secondary structures are theore2cally possible for a given sequence? For simplicity, we will exclude pseudo knots and allow all possible base pairings for now Thus, coun@ng secondary structures comes down to coun@ng diﬀerent base pairings for a sequence of a given length We will denote a sequence of length n as [1, n] ow many secondary structures are there for n = 1 and n = 2? 9

10 Combinatorics of Sec. Structures Theorem: Let S(n) be the number of secondary structures for the sequences of length n. Then S(0) = 0, and S(1) = 1, and for n 2, Example: S(2) = 1, S(3) = 2, S(4) = 4, S(5) = 8, S(6) = 17, S(7) = 37, Combinatorics of Sec. Structures All 17 secondary structures for n = 6: Waterman, p. 329 Combinatorics of Sec. Structures Proof: bvious for n = 1 and n = 2, therefore, S(1) = S(2) = 1. Assume S(k) is known for 1 k n. Consider the sequence [1, n + 1]. Either n + 1 is not paired or n + 1 is paired with j for 1 j n 1. Case 1: n + 1 is not paired There are S(n) structures, since that is the number of secondary structures [1, n] can form. 2 n+1 1 n Waterman, p

11 Combinatorics of Sec. Structures Case 2: n + 1 is paired with j [1,j- 1] and [j+1,n] can form S(j- 1) and S(n- j) secondary structures independently. Therefore, j- 1 j j+1 n+1 n Waterman, p. 329 Combinatorics of Sec. Structures Recursively defined S(n) has a closed- form approxima@on (without deriva@on; see Waterman, p. 330): The number of possible secondary structures is thus exponen@al in sequence length Example: tra, n = 76, S(76) ~ Secondary Structure Predic2on As usual in nature, RA also tries to assume a state of minimal energy Since hydrogen bond forma@on yields energy, RA will try to maximize the number of base pairs A simple secondary structure predic@on algorithm could thus iden@fy the secondary structure with the maximum number of base pairs among all structures Complica2on: exponen@al number of possible secondary structures! 11

ussinov Folding Algorithm Similar to pairwise sequence alignment, dynamic programming can help to solve this op@miza@on problem Ruth ussinov and coworkers proposed such an algorithm in 1978 The

12 ussinov Folding Algorithm Similar to pairwise sequence alignment, dynamic programming can help to solve this problem Ruth ussinov and coworkers proposed such an algorithm in 1978 The algorithm explores all best secondary structures recursively using a similar argument we used in the last proof Ruth ussinov Dynamic programming allows to construct the best solu@ons efficiently based on the best solu@ons for smaller subsequences hdp://ccr.cancer.gov/staff/staff.asp?profileid=6892 R. ussinov, G. Pieczenik, J.G. Griggs, D.J. Kleitman. Algorithms for Loop Matchin. SIAM J. Appl. Math. 35 (1978), ussinov Folding Algorithm Consider all possible ways to build up an op@mal secondary structure for sequence [i,j] from smaller structures Case 1: i remains unpaired and [i+1,j] is op@mal Case 2: j remains unpaired and [i,j- 1] is op@mal i+1 i j i j- 1 j ussinov Folding Algorithm Case 3: i and j are paired and [i+1, j- 1] is op@mal Case 4: [i,j] consists of op@mal substructures [i,k] and [k+1,j] i+1 j- 1 i j i k k+1 j 12

13 ussinov Folding Algorithm ussinov s algorithm has two stages, similar to the Smith- Waterman algorithm Fill stage fill the dynamic programming matrix (DP matrix) Traceback stage traceback through the DP matrix to obtain one of the op@mal secondary structures Given RA sequence s = s 1 s 2 s n utput maximal number (1,n) of base pairs for s ussinov Folding Algorithm Ini2aliza2on of the DP matrix (i, i- 1) = 0 8 i, 2 i k (i,i) = 0 8 i, 1 i k Filling the matrix For k = 2 to n: For j = k to n: i = j k + 1 (with ±(i,j) := 1, if i,j complementary, 0 otherwise) ussinov Folding Algorithm Ini2aliza2on of the DP matrix (i, i- 1) = 0 8 i, 2 i k (i,i) = 0 8 i, 1 i k Example: S = GGGAAACCU G G G A A A C C U G 0 G 0 0 G 0 0 C 0 0 C 0 0 U

14 ussinov Folding Algorithm For k = 2 to n: For j = k to n: i = j k + 1 ±(i,j) = 1 for C- G, A- U i: ver@cal, j: horizontal 1 st step: k = 2, j = 2, i = 1 ±(i,j) = 0 G G G A A A C C U G 0 0 G 0 0 G 0 0 C 0 0 C 0 0 U 0 0 ussinov Folding Algorithm For k = 2 to n: For j = k to n: i = j k + 1 ±(i,j) = 1 for C- G, A- U i: ver@cal, j: horizontal k = 2, j = 3, i = 2 ±(i,j) = 0 G G G A A A C C U G 0 0 G G 0 0 C 0 0 C 0 0 U 0 0 ussinov Folding Algorithm For k = 2 to n: For j = k to n: i = j k + 1 ±(i,j) = 1 for C- G, A- U i: ver@cal, j: horizontal k = 4, j = 9, i = 6 ±(i,j) = 1 (A- U) G G G A A A C C U G G G C C U

15 ussinov Folding Algorithm traceback(i,j): if i < j: if (i,j) = (i+1,j): // case 1 traceback(i+1,j) elif (i,j) = (i+1,j): // case 2 traceback(i,j- 1) elif (i,j) = (i+1,j- 1) + ±(i,j): // case 3 print (i,j) traceback(i+1,j- 1) else for k = i+1 to j- 1: // case 4 if (i,j) = (i,k)+ (k+1,j): traceback(i,k) traceback(k+1,j) break ussinov Folding Algorithm For k = 2 to n: For j = k to n: i = j k + 1 ±(i,j) = 1 for C- G, A- U i: ver@cal, j: horizontal last step: k = 9, j = 9, i = 1 ±(i,j) = 0 G G G A A A C C U G G G C C U 0 0 ussinov Folding Algorithm G G G A A A C C U G G G C C U 0 0 Pairs: (2,8) (3,7) 15

ussinov Folding Algorithm Missing bits and pieces in the algorithm Bend radius of loops For the sequence GGGAAAUCC the algorithm pairs neighboring bases (A- U)! ow to fix this? Assignment!

16 ussinov Folding Algorithm Missing bits and pieces in the algorithm Bend radius of loops For the sequence GGGAAAUCC the algorithm pairs neighboring bases (A- U)! ow to fix this? Assignment! ot all base pairs are equal The algorithm does not account for the different stability of different base pairs 1 G 0 2 G 8 3 G 7 4 A 0 5 A 0 6 A 0 7 C 3 8 C 2 9 U 0 bpseq format utput in a format that can be read by other tools bpseq is a simple format that describes base pairings Assignment! So`ware Tools Jviz RA structure visualiza@on tool Resources References Kay ieselt, Lecture RA Secondary Structure from Algorithms in Bioinforma@cs M. S. Waterman. Introduc@on to Computa@onal Biology Maps, sequences and genomes. Chapman & all, Boca Raton, 1995 J. Setubal, J. Meidanis. Introduc@on to computa@onal molecular biology, PWS, Boston, MA, 1997 (Chapter 8.1, pp. 246) D.W. Mount. Bioinforma@cs. Sequences and genome analysis, 2001 M. Zvelebil, J.. Baum. Understanding Bioinforma@cs, Garland Science, ew York, 2008 (Chapter 11.9, pp. 455) R. ussinov, G. Pieczenik, J.G. Griggs, D.J. Kleitman. Algorithms for Loop Matchin. SIAM J. Appl. Math. 35 (1978), So`ware Jviz RA structure visualiza@on tool hxp://jviz.cs.sfu.ca/ Websites FLI Jena RA World hxp:// leibniz.de/ra.html 16

proteins are the basic building blocks and active players in the cell, and

proteins are the basic building blocks and active players in the cell, and 12 RN Secondary Structure Sources for this lecture: R. Durbin, S. Eddy,. Krogh und. Mitchison, Biological sequence analysis, ambridge, 1998 J. Setubal & J. Meidanis, Introduction to computational molecular