RNA From Mathematical Models to Real Molecules

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2 R From Mathematical Models to Real Molecules 1. Sequences and Structures Peter Schuster Institut für Theoretische hemie und Molekulare Strukturbiologie der niversität Wien IMP enoma School Valdivia,

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4 Replication: D 2 D + Transcription: D R ucleotides mino cids Lipids arbohydrates Small Molecules Metabolism Food Waste Ribosom mr Protein Translation: R Protein sketch of cellular D metabolism

5 R as transmitter of genetic information R as adapter molecule R R as scaffold is the catalytic for supramolecular subunit in supramolecular complexes complexes D transcription... messenger-r translation genetic code leu protein R as working copy of genetic information ribosome????? R as catalyst R R is modified by epigenetic control R editing lternative splicing of messenger R ribozyme R as regulator of gene expression The R world as a precursor of the current D + protein biology R as carrier of genetic information R viruses and retroviruses R as information carrier in evolution in vitro and evolutionary biotechnology Functions of R molecules gene silencing by small interfering Rs

6 5' - end 2 1 5'-end 3 -end k =,,, P 2 2 a P 2 3 3'-end a 5 -end Definition of R structure P a a P 3' - end

7 T = 0 K, t T > 0 K, t T > 0 K, t finite Free Energy 3.30 S 10 S S 9 8 S 7 5 S 6 S 4 S S S 1 S 0 S 0 S 1 S 0 Minimum Free Energy Structure Suboptimal Structures Kinetic Structures Different notions of R structure including suboptimal conformations and folding kinetics

8 Stacking of free nucleobases or other planar heterocyclic compounds (6,9-dimethyl-adenine) The stacking interaction as driving force of structure formation in nucleic acids Stacking of nucleic acid single strands (poly-)

9 James D. Watson and Francis.. rick obel prize fifty years double helix Stacking of base pairs in nucleic acid double helices (B-D)

10 Å 54.4 = Å 56.2 Watson-rick type base pairs

11 = Deviation from Watson-rick geometry = Deviation from Watson-rick geometry Wobble base pairs

12 Sequence 5'-End DDMYMTP 3'-End 3'-End 5'-End 70 Secondary structure

13 Definition and physical relevance of R secondary structures R secondary structures are listings of Watson-rick and wobble base pairs, which are free of knots and pseudokots. D.Thirumalai,.Lee, S..Woodson, and D.K.Klimov. nnu.rev.phys.hem. 52: (2001): Secondary structures are folding intermediates in the formation of full three-dimensional structures.

14 Sequence 5'-End DDMYMTP 3'-End 3'-End 5'-End 70 Secondary structure Symbolic notation 5'-End 3'-End symbolic notation of R secondary structure that is equivalent to the conventional graphs

15 ircle representation of tr phe 5'-Ende 3'-Ende

16 '-Ende 3'-Ende Mountain representation of tr phe

17 Mountain representation used in structure prediction of medium size R molecules

18 Mountain representation used in structure prediction of large R molecules

19 Minimal hairpin loop size: n lp 3 Minimal stack length: n st 2 Recursion formula for the number of acceptable R secondary structures

20 omputed numbers of minimum free energy structures over different nucleotide alphabets P. Schuster, Molecular insights into evolution of phenotypes. In: J. rutchfield & P.Schuster, Evolutionary Dynamics. xford niversity Press, ew York 2003, pp

21 R sequence R folding: Structural biology, spectroscopy of biomolecules, understanding molecular function Biophysical chemistry: thermodynamics and kinetics Empirical parameters Inverse folding of R: Biotechnology, design of biomolecules with predefined structures and functions R structure Sequence, structure, and function

22 ow to compute R secondary structures Efficient algorithms based on dynamic programming are available for computation of minimum free energy and many suboptimal secondary structures for given sequences. M.Zuker and P.Stiegler. ucleic cids Res. 9: (1981) M.Zuker, Science 244: (1989) Equilibrium partition function and base pairing probabilities in Boltzmann ensembles of suboptimal structures. J.S.Mcaskill. Biopolymers 29: (1990) The Vienna R Package provides in addition: inverse folding (computing sequences for given secondary structures), computation of melting profiles from partition functions, all suboptimal structures within a given energy interval, barrier tress of suboptimal structures, kinetic folding of R sequences, R-hybridization and R/D-hybridization through cofolding of sequences, alignment, etc.. I.L.ofacker, W. Fontana, P.F.Stadler, L.S.Bonhoeffer, M.Tacker, and P. Schuster. Mh.hem. 125: (1994) S.Wuchty, W.Fontana, I.L.ofacker, and P.Schuster. Biopolymers 49: (1999).Flamm, W.Fontana, I.L.ofacker, and P.Schuster. R 6: (1999) Vienna R Package:

23 5 -end 3 -end Folding of R sequences into secondary structures of minimal free energy, 0 300

24 5 -end 3 -end 0 Free energy S 5 (h) S 6 (h) S 7 (h) S 4 (h) S 8 (h) (h) S 3 (h) S (h) 1 S 2 (h) S 9 Suboptimal conformations S 0 (h) Minimum of free energy The minimum free energy structures on a discrete space of conformations

25 5 -end 3 -end Edges: i j,k l S... base pairs (i) i i+1 S... backbone (ii) #base pairs per node = {0,1} (iii) if i j and l k S, then i<k<j i<l<j... pseudoknot exclusion i j k l Folding of R sequences into secondary structures of minimal free energy, 0 300

26 5 -end 3 -end free energy of stacking < 0 gij, kl + h( nl ) + b ( nb ) + i ( ni = ) stacks of base pairs hairpin loops bulges internal loops L Folding of R sequences into secondary structures of minimal free energy, 0 300

27 hairpin loop hairpin loop hairpin loop stack free end joint stack stack stack free end stack free end bulge internal loop hairpin loop stack hairpin loop multiloop hairpin loop stack stack stack Elements of R secondary structures as used in free energy calculations free end free end

28 Maximum matching n example of a dynamic programming computation of the maximum number of base pairs Back tracking yields the structure(s). [i,k-1] [ k+1,j ] i i+1 i+2 k j-1 j j+1 X i,k-1 X k+1,j X { X, max (( X + 1 X ρ )} i, j+ 1 = max i, j i k j 1 i, k 1 + k+ 1, j) k, j+ 1 Minimum free energy computations are based on empirical energies RStudio.lnk

29 Maximum matching j i n example of a dynamic programming computation of the maximum number of base pairs Back tracking yields the structure(s). [i,k-1] i i+1 i+2 k X X i,k-1 [ k+1,j ] j-1 j X k+1,j j+1 { X, max (( X + 1 X ρ )} i, j+ 1 = max i, j i k j 1 i, k 1 + k+ 1, j) k, j+ 1 1 * * * * * * * * * * * * * * * * * * * * * * * * * * 1 14 * * 15 * Minimum free energy computations are based on empirical energies RStudio.lnk

30 R sequence R folding: Structural biology, spectroscopy of biomolecules, understanding molecular function Biophysical chemistry: thermodynamics and kinetics Empirical parameters Inverse folding of R: Biotechnology, design of biomolecules with predefined structures and functions R structure Sequence, structure, and function

31 Minimum free energy criterion Inverse folding of R secondary structures The idea of inverse folding algorithm is to search for sequences that form a given R secondary structure under the minimum free energy criterion.

32 Structure

33 3 -end 5 -end Structure ompatible sequence

34 ompatible sequence Structure 5 -end 3 -end

35 ompatible sequence Structure 5 -end 3 -end Single nucleotides:,,, Single bases pairs are varied independently

36 ompatible sequence Structure 5 -end 3 -end Base pairs:,,, Base pairs are varied in strict correlation

37 ompatible sequences Structure 5 -end 5 -end 3 -end 3 -end

38 Structure Incompatible sequence 5 -end 3 -end

39 d =2 d = d = ity-block distance in sequence space 2D Sketch of sequence space Single point mutations as moves in sequence space

40 TTTTTTTTTTTTTTT... TTTTTTTTTTTTTT... amming distance d (I,I ) = (i) (ii) (iii) d (I,I) = d (I,I) = d (I,I) d (I,I) d (I,I) + d (I,I) The amming distance between sequences induces a metric in sequence space

41 Mutant class Binary sequences are encoded by their decimal equivalents: = 0 and = 1, for example, "0" =, "14" =, "29" =, etc. 31 ypercube of dimension n = 5 5 Decimal coding of binary sequences Sequence space of binary sequences of chain lenght n = 5

42 amming distance d (S,S ) = (i) (ii) (iii) d (S,S) = d (S,S ) = d (S,S ) d (S,S) d (S,S) + d (S,S) The amming distance between structures in parentheses notation forms a metric in structure space

43 Inverse folding algorithm I 0 I 1 I 2 I 3 I 4... I k I k+1... I t S 0 S 1 S 2 S 3 S 4... S k S k+1... S t I k+1 = M k (I k ) and d S (S k,s k+1 ) = d S (S k+1,s t ) - d S (S k,s t ) < 0 M... base or base pair mutation operator d S (S i,s j )... distance between the two structures S i and S j nsuccessful trial... termination after n steps

44 Initial trial sequences Stop sequence of an unsuccessful trial Target sequence Target structure S k Intermediate compatible sequences pproach to the target structure S k in the inverse folding algorithm

45 Minimum free energy criterion 1st 2nd 3rd trial 4th 5th Inverse folding of R secondary structures The inverse folding algorithm searches for sequences that form a given R secondary structure under the minimum free energy criterion.

46 riterion of Minimum Free Energy Sequence Space Shape Space

47 R sequences as well as R secondary structures can be visualized as objects in metric spaces. t constant chain length the sequence space is a (generalized) hypercube. The mapping from R sequences into R secondary structures is many-to-one. ence, it is redundant and not invertible. R sequences, which are mapped onto the same R secondary structure, are neutral with respect to structure. The pre-images of structures in sequence space are neutral networks. They can be represented by graphs where the edges connect sequences of amming distance d = 1.

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49 S k = ψ( I. ) f k = fs ( k ) Sequence space Structure space Real numbers Mapping from sequence space into structure space and into function

50 S k = ψ( I. ) f k = fs ( k ) Sequence space Structure space Real numbers

51 S k = ψ( I. ) f k = fs ( k ) Sequence space Structure space Real numbers The pre-image of the structure S k in sequence space is the neutral network k

52 eutral networks are sets of sequences forming the same structure. k is the pre-image of the structure S k in sequence space: k = -1 (S k ) π { j (I j ) = S k } The set is converted into a graph by connecting all sequences of amming distance one. eutral networks of small R molecules can be computed by exhaustive folding of complete sequence spaces, i.e. all R sequences of a given chain length. This number, =4 n, becomes very large with increasing length, and is prohibitive for numerical computations. eutral networks can be modelled by random graphs in sequence space. In this approach, nodes are inserted randomly into sequence space until the size of the pre-image, i.e. the number of neutral sequences, matches the neutral network to be studied.

53 Step 00 Sketch of sequence space Random graph approach to neutral networks

54 Step 01 Sketch of sequence space Random graph approach to neutral networks

55 Step 02 Sketch of sequence space Random graph approach to neutral networks

56 Step 03 Sketch of sequence space Random graph approach to neutral networks

57 Step 04 Sketch of sequence space Random graph approach to neutral networks

58 Step 05 Sketch of sequence space Random graph approach to neutral networks

59 Step 10 Sketch of sequence space Random graph approach to neutral networks

60 Step 15 Sketch of sequence space Random graph approach to neutral networks

61 Step 25 Sketch of sequence space Random graph approach to neutral networks

62 Step 50 Sketch of sequence space Random graph approach to neutral networks

63 Step 75 Sketch of sequence space Random graph approach to neutral networks

64 Step 100 Sketch of sequence space Random graph approach to neutral networks

65 -1 k k j j k = ( S) I ( I) = S λ j = 12 / 27 = 0.444, λ k = (k) j k onnectivity threshold: / κ- λcr = 1 - κ -1 ( 1) λ λ lphabet size : = 4 > λ k cr.... < λ k cr.... network k is connected network k is not connected cr ,, Mean degree of neutrality and connectivity of neutral networks

66 connected neutral network

67 multi-component neutral network iant omponent

68 = = (=) = ( ) The six base pairing alphabets built from natural nucleotides,,, and

69 = = (=) = ( ) The six base pairing alphabets built from natural nucleotides,,, and

70 3'-End 3'-End 3'-End 3'-End 5'-End 5'-End 5'-End 5'-End lphabet Degree of neutrality Degree of neutrality of cloverleaf R secondary structures over different alphabets

71 omputed numbers of minimum free energy structures over different nucleotide alphabets P. Schuster, Molecular insights into evolution of phenotypes. In: J. rutchfield & P.Schuster, Evolutionary Dynamics. xford niversity Press, ew York 2003, pp

72 Reference for postulation and in silico verification of neutral networks

73 Structure S k eutral etwork k k k ompatible Set k The compatible set k of a structure S k consists of all sequences which form S k as its minimum free energy structure (the neutral network k ) or one of its suboptimal structures.

74 cknowledgement of support Fonds zur Förderung der wissenschaftlichen Forschung (FWF) Projects o , 10578, 11065, , and niversität Wien Jubiläumsfonds der Österreichischen ationalbank Project o. at-7813 European ommission: Project o. E Siemens, ustria The Santa Fe Institute and the niversität Wien The software for producing R movies was developed by Robert iegerich and coworkers at the niversität Bielefeld

75 oworkers Walter Fontana, Santa Fe Institute, M niversität Wien hristian Reidys, hristian Forst, Los lamos ational Laboratory, M Peter Stadler, Bärbel Stadler, niversität Leipzig, E Ivo L.ofacker, hristoph Flamm, niversität Wien, T ndreas Wernitznig, Michael Kospach, niversität Wien, T lrike Langhammer, lrike Mückstein, Stefanie Widder Jan upal, Kurt rünberger, ndreas Svrček-Seiler, Stefan Wuchty lrike öbel, Institut für Molekulare Biotechnologie, Jena, E Walter rüner, Stefan Kopp, Jaqueline Weber

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