Computational and physical models of RNA structure

Transcription

1 omputational and physical models of RN structure Ralf Bundschuh Ohio State niversity July 25, 2007 Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

2 Outline of all lectures Outline Part I : Statistical physics of RN Part II : Quantitative modeling of force-extension experiments Part III : RN folding kinetics Part IV : microrn target prediction Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

3 Outline of part I 1 Boltzmann partition function 2 Molten RN 3 Molten-native transition 4 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

4 Outline of part I Boltzmann partition function 1 Boltzmann partition function Secondary structure Partition function Recursion equation 2 Molten RN 3 Molten-native transition 4 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

5 Boltzmann partition function Secondary structure Definition of RN secondary structure Definition n RN secondary structure on an RN sequence of length N is a set S of base pairs (i, j) with 1 i < j N that fulfill the following conditions: Each base is involved in at most one base pair No pseudoknots, i.e., if (i, j) and (k, l) are base pairs with i < k, either i < k < l < j or i < j < k < l Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

6 Boltzmann partition function Diagrammatic representation Secondary structure n RN secondary structure can be represented by an arch diagram Every base is end point to at most one arch Two arches never cross One to one correspondance between arch diagrams that fulfil the above conditions and secondary structures Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

7 Energy models Boltzmann partition function Partition function Each structure S has a certain (free) energy E[S] Different models possible ll base pairs are equal E[S] = ε 0 S (ε 0 < 0) Energy associated with base pairs E[S] = (i,j) S e(i, j) Turner parameters (nearest neighbor model): energy associated with loops Base pair stacks Hairpin loops Interior loops Bulges Multi-loops multiloop bulge hairpin loop stacking loop interior loop Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

8 Partition function Boltzmann partition function Partition function Definition The partition function of an RN molecule with energy function E[S] is given by Z S e E[S] RT R = cal mol K Temperature T in Kelvin RT 0.6 kcal mol Ensemble free energy F = RT ln Z Thermodynamics completely specified if partition function is known Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

9 Boltzmann partition function alculating partition functions Recursion equation Definition Let Z i,j be the partition function for the RN molecule starting at base i and ending at base j. Denote this quantity by i j. Observation For the base pairing energy model the Z i,j obey the recursion equation = + i j i j 1 j Σ k i k 1 k k+1 j 1 j j 1 Z i,j = Z i,j 1 + Z i,k 1 e e(k,j) RT Zk+1,j 1 k=i alculate from shortest to longest substrands O(N 3 ) algorithm for arbitrary sequence Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

10 Outline of part I Molten RN 1 Boltzmann partition function 2 Molten RN Energy model z transform Properties 3 Molten-native transition 4 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

11 Molten RN Energetics in molten phase Energy model Definition In the molten phase of RN every base can pair with every other base equally well, i.e., e(i, j) = ε 0. Properties pplies to repetitive sequences:,, pplies to arbitrary sequences at temperatures close to denaturation on a coarse-grained level Minimum energy is always N 2 ε 0 Structural entropy plays a major role Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

12 Molten RN Molten phase partition function Energy model Reminder = + i j i j 1 j Σ k i k 1 k k+1 j 1 j Simplification j 1 Z i,j = Z i,j 1 + Z i,k 1 e e(k,j) RT Zk+1,j 1 k=i In the molten phase the partition function depends only on the length j i of the substrand, not on i and j individually: Z i,j (j i + 2) onsequence N 1 (N + 1) = (N) + q (k)(n k) with q e ε 0 RT k=1 Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

13 z transform Molten RN z transform Definition For any series Q(N) the z transform Q is defined as Q(z) Q(N)z N N=1 Properties Function of the complex variable z nalytic outside a radius of convergence Discrete version of Fourier transform Back transform: Q(N) = 1 Q(z)z N 1 dz onvolution property: 2πi N 1 k=1 Q(k)W (N k) = Q(z) Ŵ (z) Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

14 Molten RN z transform z transform for molten RN Ĝ(z) can be calculated: N 1 (N + 1) = (N) + q (k)(n k) k=1 N 1 (N + 1)z N = (N)z N + qz N (N + 1)z N = Ĝ(z) + qĝ 2 (z) N=1 zĝ(z) (1) = Ĝ(z) + qĝ 2 (z) zĝ(z) 1 = Ĝ(z) + qĝ 2 (z) k=1 Ĝ(z) = 1 [ ] z 1 (z 1) 2q 2 4q (k)(n k) Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

15 Back transform I Molten RN z transform Integral expression for (N) (N) = 1 Ĝ(z)z N 1 dz 2πi [ 1 = z 1 4πqi = 1 4πqi (z 1) 2 4q (z 1) 2 4q z N 1 dz ] z N 1 dz Singularity structure Branch cut from 1 2 q to z q Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

16 Back transform II Molten RN z transform Reminder (N) = 1 4πqi (z 1) 2 4q z N 1 dz Leading behavior For large N only the singularity with largest real part contributes (N) z0 N = (1 + 2 q) N Prefactor µc (N) N 3/2 (1 + 2 q) N µ 0 (µ c µ) α e µn dµ Γ(1 + α)n (1+α) e µcn Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

17 Molten RN Properties Properties of molten RN ( ) q 1/2 (N) N 3/2 (1 + 2 q) N 4πq 3/2 N 3/2 characteristic behavior due to entropy an be observed in pairing probability: (k)(n k) 3/2 (N k) 3/2 Pr{1 and k paired} = q k (N + 1) N 3/2 k 3/2 Free energy is F = RT ln (N) RTN ln(1 + 2 q) For q 1: F RTN ln(2 q) = RT 2 N ln(q) = N ε 0 2 For q = 1: (N) =number of secondary structures N 3/2 3 N. Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

18 Outline of part I Molten-native transition 1 Boltzmann partition function 2 Molten RN 3 Molten-native transition Model Solution Results 4 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

19 Molten-native transition Model Model for molten-native transition Observation Structural RNs have to fold into a specific native structure there must be something in the sequence that prefers this structure Model se perfect hairpin as native structure 1 N/2 1 1 N N ssign binding energy ε 1 to all native base pairs ssign binding energy ε 0 to all other base pairs N Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

20 Molten-native transition Solution Partition function Definition Let Z(N; q, q) be the partition function of the ō model for the molten-native transition with 2N 2 bases, q = e ε 0/RT, and q = e ε 1/RT Definition Let W (N; q) Z(N; q, q = 0) be the partition function of the ō model for 2N 2 bases in which native contacts are disallowed. Observation Z(N; q, q) =(p.f. with 0 native contacts) + (p.f. with 1 native contacts) + (p.f. with 2 native contacts) +... = Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

21 Molten-native transition Solution Individual terms 0 native contacts (p.f. with 0 native contacts) = = W (N; q) z-transform: Ŵ (z) 1 native contact (p.f. with 1 native contacts) = z-transform: qŵ 2 (z) = q N 1 k=1 W (k; q)w (N k; q) 2 native contacts (p.f. with 2 native contacts) = = q 2 1 k 1 <k 2 <N W (k 1; q)w (k 2 k 1 ; q)w (N k 2 ; q) z-transform: q 2 Ŵ 3 (z) Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

22 Putting it all together Molten-native transition Solution Summing up Ẑ(z; q, q) = Ŵ (z) + qŵ 2 (z) + q 2 Ŵ 3 (z) +... = Ŵ (z) 1 qŵ (z) What is Ŵ (z)? For q = q we have Z(N; q, q = q) = (2N 1; q) Ẑ(z; q, q) = Ê(z; q) N=1 (2N 1; q)z N Ŵ (z) = Ê(z) 1 + qê(z) Ẑ(z; q, q) = Ê(z) 1 ( q q)ê(z) Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

23 Molten-native transition Solution Behavior of Ê(z) Expression for Ê(z) Ê(z) = N=1 (2N 1; q)z N can be calculated similarly to Ĝ(z). Ê(z) = 1 2q 1 [ ] (z 1) 4qz 2 4q + (z + 1) 2 4q. Properties Vanishes as z Square root branch cut at z 0 = q Finite limit Ê(1 + 2 q) at branch cut E(z ) 0 E(z) Other branch cuts have smaller real part z 0 z Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

24 Molten-native transition Solution Singularity structure of Ẑ(z) andidate singularities Ẑ(z; q, q) = Ê(z) 1 ( q q)ê(z) E(z ) 0 E(z) 0.1 Square root branch cut at z 0 = q 0.05 ~ 1 q q Pole at z 1 ( q) given by Ê(z 1) = 1/( q q) z Dominant singularity Square root branch cut at z 0 if 1/( q q) > Ê(z 0) Pole at z 1 ( q) if 1/( q q) < Ê(z 0) Phase transition z 0 z (q) ~ 1 Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

25 Molten-native transition Results Properties of molten-native transition haracterization of phase transition ritical bias q c = q + 1/Ê(z 0) If q < q c regular molten behavior Z N 3/2 (1 + 2 q) N native base pairs do not play any role If q > q c native behavior with Z N 0 z 1 ( q) N finite fraction of native base pairs but still many bubbles For q q c we get Z N 0 q N only native base pairs onclusion It takes a finite amount of sequence bias to enforce a native structure. Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

26 Outline of part I Summary 1 Boltzmann partition function 2 Molten RN 3 Molten-native transition 4 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

27 Summary Summary of part I The partition function of RN can be calculated in polynomial time symptotic behavior for homogeneous RN can be calculated by analytical methods The partition function of molten RN has a characteristic N 3/2 behavior It takes a finite amount of sequence bias to enforce a native structure. Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

28 Outline of part II 5 Force-extension experiments 6 Quantitative modeling 7 Results for simple force-extension experiments 8 Nanopores 9 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

29 Outline of part II Force-extension experiments 5 Force-extension experiments Motivation Experimental setup 6 Quantitative modeling 7 Results for simple force-extension experiments 8 Nanopores 9 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

30 Force-extension experiments Motivation Experimental methods to determine RN structure Experimental methods X-ray crystallography Nuclear Magnetic Resonance Biochemical evidence: protection essays orrelated mutation analysis Problem RN is very floppy two RN molecules rarely have the same three-dimensional structure Idea Look at one RN molecule at a time. Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

31 Experimental setup Force-extension experiments Experimental setup ttach ends of RN molecule to two beads Keep beads at fixed distance R with optical tweezers Measure force f on beads as function of distance R r Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

32 Measurement Force-extension experiments Experimental setup Subject P5ab hairpin of Tetrahymena thermophila group I intron Data (Liphardt, Onoa, Smith, Tinoco, and Bustamante, Science, 2001) Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

33 Outline of part II Quantitative modeling 5 Force-extension experiments 6 Quantitative modeling Force from free energy Secondary structures Backbone Putting it back together 7 Results for simple force-extension experiments 8 Nanopores 9 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

34 alculating the force Quantitative modeling Force from free energy r Basic physics energy = force distance Force from free energy Need free energy F (r) at fixed end to end distance r force f = F (r) r Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

35 Partition function I Quantitative modeling Force from free energy r Free energy from partition function F (r) = RT ln Z(r) need partition function Z(r) at fixed distance r Partition function Z(r) = secondary structures S polymer configurations P with distance r and structure S e E[S]+E[P] RT Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

36 Partition function II Quantitative modeling Force from free energy r Z(r) = = = secondary structures S secondary structures S N m=0 secondary structures S with m exterior bases polymer configurations P with distance r and structure S polymer configurations P of the exterior bases with distance r and structure S polymer configurations P of the exterior bases with distance r and structure S e E[S]+E[P] RT e E[S]+E[P] RT e E[S]+E[P] RT Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

37 Partition function III Quantitative modeling Force from free energy Z(r) = = = N m=0 secondary structures S with m exterior bases N m=0 secondary structures S with m exterior bases N m=0 secondary structures S with m exterior bases r polymer configurations P of the exterior bases with distance r and structure S e E[S] RT e E[S] RT polymer configurations P of m exterior bases with distance r e E[S]+E[P] RT e E[P] RT polymer configurations P of ssrn with m bases with distance r e E[P] RT Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

38 Partition function IV Quantitative modeling Force from free energy N Z(r) = Q(m)W (m, r) m=0 with and Q(m) W (m, r) secondary structures S with m exterior bases e E[S] RT polymer configurations P of ssrn with m bases with distance r e E[P] RT Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

39 Recursion equation Quantitative modeling Secondary structures Reminder i j = i j 1 j + k i k 1 k k+1 j 1 j j 1 Z i,j = Z i,j 1 + k=i Σ Z i,k 1 e e(k,j) RT Zk+1,j 1 Definition Let Q j (m) be the partition function for the first j bases with exactly m exterior bases ( j). eneralization = Σ j 1 j 1 j k 1 k 1 k k+1 j 1 j j 1 Q j (m) = Q j 1 (m 1) + k=1 Q k 1 (m)e e(k,j) RT Zk+1,j 1 Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

40 Quantitative modeling Properties of recursion equation Secondary structures Properties: = Σ j 1 j 1 j k 1 k 1 k k+1 j 1 j j 1 Q j (m) = Q j 1 (m 1) + O(N 3 ) complexity Q(m) = Q N (m) k=1 an be easily generalized to Turner parameters Q k 1 (m)e e(k,j) RT Zk+1,j 1 an be calculated by modifying the Vienna package (Hofacker, Fontana, Stadler, Bonhoeffer, Tacker, and Schuster, Monatshefte f. hemie, 1994) Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

41 Quantitative modeling Backbone Polymer physics Need W (m, r) = polymer configurations P of ssrn with m bases with distance r e E[P] RT Model Elastic freely jointed chain Persistence length 1.9nm/base distance 0.7nm Partition function alculation of W (m, r) is standard polymer physics. Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

42 RNpull Quantitative modeling Putting it back together Putting it together Q(m), W (m, r) N Z(r) = Q(m)W (m, r) m=0 F (r) = RT ln Z(r) F (r) f (r) = r Web server Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

43 Results for simple force-extension experiments Outline of part II 5 Force-extension experiments 6 Quantitative modeling 7 Results for simple force-extension experiments Hairpin real molecule Why the force-extension curve is smooth? 8 Nanopores 9 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

44 Hairpin Results for simple force-extension experiments Hairpin pply to P5ab hairpin of Tetrahymena thermophila group I intron Quantitative agreement! Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

45 Results for simple force-extension experiments The full group I intron real molecule The group I intron of Tetrahymena thermophila roup I intron contains pseudo-knot! Quantitative modeling ignores pseudo-knot known inactive conformation (Pan and Woodson, J. Mol. Biol., 1998) P5c P5a P5b P6b P5 P4 P6a P1 5 P2.1 P2 P9.1a P9 P9.1 P7 lt P3 3 P8 P9.2 omputational result No sign of secondary structure! <F> [pn] <R> [nm] Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

46 Results for simple force-extension experiments What s happening? Why the force-extension curve is smooth? Intermediate structure Look at intermediate structure (here r = 100nm) Like Socks on the @ Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

47 Results for simple force-extension experiments What s happening? Why the force-extension curve is smooth? ompensation effect Extension r is increased One of the socks disappears The other socks take up the slag No rapid change in force as sock disappears smooth force-extension curve Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

48 Outline of part II Nanopores 5 Force-extension experiments 6 Quantitative modeling 7 Results for simple force-extension experiments 8 Nanopores Introduction Force-extension experiments 9 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

49 Nanopores Nanopores Introduction What is a nanopore? nanopore is a little hole is so small that only single-stranded but no double-stranded RN can pass through it can be placed between two chambers such that there is only one nanopore connecting the chambers Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

50 Nanopores Nanopores Introduction Types of nanopores natural ion channel (α-hemolysin) (Meller, J. Phys. ond. Mat., 2003) solid state pore (Storm et al., Nature Mat., 2003) Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

51 Nanopores Force-extension experiments ombining nanopores with force-extension experiments Suggestion ombine a nanopore with a force-extension setup "cis" "trans" n nucleotides f r Prediction The force will rise at every structural element The structural elements will break in their order along the sequence Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

52 Nanopores Force-extension experiments Simulation approach "cis" "trans" n nucleotides f Simulation Fix r(t) to be linear (set by experiment) Degree of freedom: number n of base in the pore t each time n can increase: calculable gain in mechanical energy on the right potentially loss in binding energy calculable from E[S] decrease: calculable loss in mechanical energy on the right Monte arlo simulation (see also part III) r Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

53 Simulation result Nanopores Force-extension experiments pply to group I intron of Tetrahymena thermophila 50 <f> [pn] n= P9.1 P9 P7 lt P3 P9.2 P8 P6 P4-P5a P5bc <R> [nm] P2.1 P2 P1 P5 P5a P1 P4 P5c 5 P6a P5b P6b P2.1 P2 P9.1a P9.1 P9 P9.2 3 P7 lt P3 P8 Signature of every structural element an extract sequence position of stalling sites Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

54 Structure reconstruction I Nanopores Force-extension experiments Repeat pulling in opposite direction <f> [pn] <f> [pn] n= n= <R> [nm]...<...>...<...<...>...<... 5 (((((((((((...))...)))))))))...((((((((((...)))))))))).(((((((( ccccccccc...ccccccccc...aaaaaaaaaa...aaaaaaaaaa.gggggg.....>...>...<..<...<...<... (((.((...)).))))).))))))...((((((...((((((...(((.(((((...gggggg...jjjj...kkkkkk...iiii......>...<...>...>...>..>...>. (((..(((((((((...)))))))))..(((...)))...)))...).)))))))...))))))....llll...llll...iiii...kkkkkk....>..>.<...<...<...>...>...<.....)).))))((...((((...((((((((...))))))))..))))...))...(.(((((..((((....jjjj...dddddddd...dddddddd...hhhhh......<..<...>...>.<...<...>...<...<....(((((((...)))))))...))))))))))((((.(((((...)))))(((((((...(((...eeeeeee...eeeeeee...hhhhh...ffff...ffffbbbbbbb......>...>...>...>... 3 (...))))...))))))).((((((((.((((((...)))))).))))))))..))...))....bbbbbbb... Match stalling sites by sequence comparison reconstruct structure Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

55 Nanopores Force-extension experiments Structure reconstruction II 3 P9 P9.1 P9.2 P9.0 P9.1a P8 P2 P2.1 5 P1 P7 lt P3 P5b P5a P5 P5c P6b P6a P4 3 P9 P9.1 P9.2 P9.0 P9.1a P8 P2 P2.1 5 P1 P7 P3 P5b P5a P5 P5c P6b P6a P4 Overall structure reconstructed correctly an distinguish different structures on the same sequence Even pseudoknot reconstructed Just needs to be implemented experimentally... Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

56 Outline of part II Summary 5 Force-extension experiments 6 Quantitative modeling 7 Results for simple force-extension experiments 8 Nanopores 9 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

57 Summary of part II Summary Quantitative description of single-molecule experiments possible Force-extension curves do not reveal secondary structure information due to compensation effects Single-molecule experiments reveal structure information with the help of a nanopore Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

58 Outline of part III 10 Motivation 11 Molecular Dynamics 12 Monte arlo simulation 13 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

59 Outline of part III Motivation 10 Motivation What is kinetics? Why care about kinetics? 11 Molecular Dynamics 12 Monte arlo simulation 13 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

60 Kinetics introduction Motivation What is kinetics? Thermodynamics What structure(s) does an RN molecule take on? Kinetics How long does it take an RN molecule to reach a certain structure? long which pathway does an RN molecule get to a certain structure? Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

61 Motivation Why is kinetics important? Why care about kinetics? Why is kinetics a problem? Take, say, a short RN with 50 bases Has roughly = structures Let s be generous and say that it can explore a structure per picosecond it takes 500,000,000s 18yr to explore all structures! Processes that need to happen in time: rrn have to fold riboswitches have to get from one configuration to another terminator hairpins have to form in time for termination RN viruses have to be folded for packaging splicing might depend on secondary structure Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

62 Outline of part III Molecular Dynamics 10 Motivation 11 Molecular Dynamics eneral principle Pros and cons 12 Monte arlo simulation 13 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

63 Molecular Dynamics What is Molecular Dynamics? eneral principle Model an RN molecule atom by atom dd water and salt atom by atom or as an effective medium hoose a force field Integrate Newton s equations Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

64 Molecular Dynamics Pros and cons dvantages and disadvantages of Molecular Dynamics dvantages First principle based modeling Full three dimensional structure included Disadvantages Need many atoms ( 30 atoms per base) an only simulate very short sequences (10 base pairs) an only simulate very short times (at most µs) Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

65 Molecular Dynamics What can be done with MD? Pros and cons Structural dynamics of very small building blocks: Sorin et al., RN simulations: probing hairpin unfolding and the dynamics of a NR tetraloop, J. Mol. Biol Sarzynska et al., Effects of base substitutions in an RN hairpin from molecular dynamics and free energy simulations, Biophys J Hart et al., Molecular dynamics simulations and free energy calculations of base flipping in dsrn, RN Mazier et al., Molecular dynamics simulation for probing the flexibility of the 35 nucleotide SL1 sequence kissing complex from HIV-1Lai genomic RN, J Biomol Struct Dyn Nystrom et al.,molecular dynamics study of intrinsic stability in six RN terminal loop motifs, J Biomol Struct Dyn Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

66 Outline of part III Monte arlo simulation 10 Motivation 11 Molecular Dynamics 12 Monte arlo simulation Method Base pair level approach Helix level approach 13 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

67 Monte arlo simulation What is Monte arlo dynamics? Method Description of a system State space Energy for each state llowed transitions from state to state Procedure hoose random initial state S 0 Repeat as often as necessary for i = 1, 2, 3,...: Enumerate all states into which state S i 1 is allowed to transition. Pick one of these states T randomly alculate E E[S] E[T ] If E 0 let S i T If E < 0 pick a random number r If r < e E RT let S i T, otherwise let S i S i 1 Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

68 Monte arlo simulation Method Properties of Monte arlo dynamics Boltzmann distribution Transitions have to be chosen such that every state can be reached from every other state The sequence S 0, S 1,... visits each state with a frequency proportional to its Boltzmann probability p = e E[S] RT /Z It avoids calculating a partition function Monte arlo and real time In general, Monte arlo dynamics has nothing to do with real dynamics of system Monte arlo dynamics is a good representation of real dynamics if: The allowed transitions correspond to true elementary steps of real dynamics There is a good physical reason for all downhill transitions occuring all with the same rate k 0 In this case, the real time is t = i/k 0 for the ith step Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

69 Numerical tricks Monte arlo simulation Method illespie sampling Instead of accepting or rejecting a move, choose a move in every step but increase time by a larger chunk if moves are unfavorable. (illespie, J. Phys. hem. 1977) State clustering If algorithm revisits the same state in a local minimum over and over again, precalculate the distribution and residence time in local valley by diagonalizing a reasonably sized matrix and directly choose a move that leads out of the valley. Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

70 The Vienna approach Monte arlo simulation Base pair level approach Specification States: all secondary structures Energies: calculated by Turner model Transitions: Opening of a base pair losing of a base pair Repairing of a base Flamm et al., RN folding at elementary step resolution, RN 2000 kinfold program in Vienna package Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

71 Properties Monte arlo simulation Base pair level approach Validity Every state can be reached from every other Transitions are reasonable elementary steps losing of a base is an activated process itself with a roughly constant free energy barrier Time scales Time for closing a base pair experimentally determined to be 1µs Monte arlo time is real time in µs Time for closing a hairpin of length 4: e E[S] RT µs e µs 1ms Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

72 Monte arlo simulation Isambert and Siggia approach Helix level approach Specification Precompute all possible helices of minimum length 3 States: all possible combinations of non-overlapping helices (Note: this includes pseudo-knots!) Energy: Turner stacking energies for stacks stick and string model for loop entropies Transitions: formation and removal of a complete helix Isambert and Siggia, Modeling RN folding paths with pseudoknots: pplication to hepatitis delta virus ribozyme, PNS 2000 Xayaphoummine et al., Kinefold web server for RN/DN folding path and structure prediction including pseudoknots and knots, Nucleic cids Research 2005 Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

73 Results I Monte arlo simulation Helix level approach Pseudoknots Test on 7 relatively short RN molecules with experimentally known pseudoknots For 5 out of the 7 correct pseudo-knotted is found For 1 the pseudo-knotted structure is only marginally higher than minimum energy structure For 1 specific Mg binding is suspected Isambert and Siggia, PNS 2000 Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

74 Results II Monte arlo simulation Helix level approach Kinetics Folding of HDV ribozyme 1/3 of the time folds within a fraction of a second 2/3 of the time trapped for up to a minute Molecule has to be functional after 4s to self-cleave! If folding co-transcriptional only 10% of molecules get trapped Isambert and Siggia, PNS 2000 Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

75 Outline of part III Summary 10 Motivation 11 Molecular Dynamics 12 Monte arlo simulation 13 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

76 Summary of part III Summary Dynamics of RN folding is important for many biological processes Molecular dynamics is useful to study fast structural changes of small substructures Monte arlo simulations can describe folding of realistically sized molecules Monte arlo simulations can describe folding including pseudo-knots The elementary time scale for closing of a base pair on top of an existing step is on the order of 1µs Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

77 Outline of part IV 14 Introduction to microrns 15 First generation target prediction 16 urrent target prediction methods 17 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

78 Outline of part IV Introduction to microrns 14 Introduction to microrns What are microrns Biogenesis Functions omputational problems 15 First generation target prediction 16 urrent target prediction methods 17 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

79 Introduction to microrns What is a micro RN? What are microrns Definition microrn is an approximately 22 nucleotide long RN that regulates expression of other genes Properties microrn are post-transcriptional regulators microrn themselves are temporally and spatially regulated There are most likely hundreds of microrn in higher eukaryotes Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

80 Introduction to microrns How are microrn made? Biogenesis microrn is made in four steps: 1 Transcription of the primary RN (pri-mirn) by RN polymerase (in the nucleus) 2 Processing into hairpin-like RNs called pre-mirn (in the nucleus) 3 leavage into the mature microrn by Dicer (in the cytoplasm) 4 Incorporation into the RN-induced silencing complex (RIS) Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

81 Introduction to microrns What do microrn do? Functions In plants microrns in RIS complexes bind to exactly complementary regions of the mrns of target genes The mrn of the target gene is cut and degraded In animals microrns in RIS complexes bind to partially complementary regions of the mrns of target genes The mrn of the target gene is prevented from translation Processes microrns are known to be involved in Development Immune system ancer Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

82 Introduction to microrns omputational problems omputational problems ene finding problem iven a genome sequence, predict the microrn sequences of the organism (in plants and animals). Target prediction problem iven a microrn sequence, predict the genes that are regulated by that microrn (in animals). Here, only talk about the target prediction problem Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

83 Outline of part IV First generation target prediction 14 Introduction to microrns 15 First generation target prediction pproaches lgorithm Results 16 urrent target prediction methods 17 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

84 First generation target prediction pproaches for Drosophila pproaches Many algorithms proposed essentially simultaneously For Drosophila: Stark et al., Identification of Drosophila microrn targets, PLoS Biollogy (2003) Enright et al., MicroRN targets in Drosophila, enome Biology (2003) Rajewsky and Socci, omputational identification of microrn targets, Developmental Biology (2004) Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

85 First generation target prediction pproaches for mammals pproaches For mammals: Lewis et al., Prediction of mammalian microrn targets, ell (2003) John et al., Human MicroRN targets, PLoS Biology (2004) Kiriakidou et al., combined computational-experimental approach predicts human microrn targets, enes & Development (2004) ll approaches rather similar, but resulting predictions have only small overlap Here: Rajewsky and Socci Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

86 First generation target prediction lgorithm Features used in target prediction Observations: 1 MicroRN target sites are in 3 TRs. 2 lthough microrns are not exactly complementary to their targets they always contain a nucleus of length 6-8 bases with exact complementarity. 3 Even in the not exactly complementary regions, there is still a lot of stabilizing base pairing. 4 MicroRN target relationships tend to be conserved across species. Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

87 First generation target prediction Nucleus identification lgorithm sing observation 2: there is a nucleus ssign to every base pairing stretch a score by the formula score = w # base pairs + w # b.p. + w # b.p. se training sets of true targets and of true non-targets to choose w, w, and w such that they maximize the distance between expected scores of targets and non-targets w = 5, w = 2, w = 0 Fix a p-value threshold Score many random mrn sequences for a given microrn sequence and determine the score cutoff that belongs to the p-value cutoff Score all 3 TR regions of interest and keep only those for which the score exceeds the threshold. Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

88 sing other features First generation target prediction lgorithm sing observation 3: other bases pair as well ut 40 base window around identified nucleus se MFOLD to predict binding energy of microrn-mrn hybrid Keep only candidate targets with binding energy below 17.4kcal/mol sing observation 4: targets are conserved Keep only candidates for which a target is predicted in D. melanogaster as well as in D. pseudoobscura Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

89 First generation target prediction Results Results Test1 Search D. melanogaster hid transcript (not used in training) for bantam sites: 4 of the 5 experimentally known sites found, no false positive Test2 Prediction of target sites for 74 D. melanogaster microrns in 31 body patterning genes: found 39 putative target sites, some of them make biological sense Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

90 Outline of part IV urrent target prediction methods 14 Introduction to microrns 15 First generation target prediction 16 urrent target prediction methods dditional features Performance 17 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

91 urrent target prediction methods dditional features dditional features used for prediction Position of nucleus in microrn (5 end) onservation of nucleus in large multiple sequence alignments bsence of conservation outside nucleus n immediately upstream of the nucleus ombinatorics of target sites for multiple microrns Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

92 urrent target prediction methods Performance comparison Performance ssay Evaluate prediction method on 133 experimentally tested targets. (Stark, Brennecke, Bushati, Russell, ohen, ell 2005) Results Best methods: TargetScanS (Stark et al.) PicTar (rün et al.) Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

93 Outline of part IV Summary 14 Introduction to microrns 15 First generation target prediction 16 urrent target prediction methods 17 Summary Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98

94 Summary of part IV Summary microrns are a relatively new but important class of regulatory RNs There are many programs for microrn target prediction The main features used for target prediction are stability of a nucleus, stability of the full hybrid, and conservation between species Best current algorithms (TargetScanS and PicTar) have about 90% accuracy and 70% 80% sensitivity Reviews: Issac Bentwich, Prediction of microrns and their targets, FEBS Letters 2005 Nikolaus Rajewksy, microrn target predictions in animals, Nature enetics 2006 Ralf Bundschuh (Ohio State niversity) Modelling RN structure July 25, / 98