RNA Secondary Structure Prediction

Transcription

1 RN Secondary Structure Prediction Perry Hooker S 531: dvanced lgorithms Prof. Mike Rosulek University of Montana December 10, 2010 Introduction Ribonucleic acid (RN) is a macromolecule that is essential to living biological systems. In addition to its role in the central dogma of molecular biology as an agent of information transfer (Fig. 1), certain specific RN molecules are not translated into proteins and instead operate functionally in a number of different capacities. In fact, recent evidence suggests that most of the genomes of mammals and other complex organisms is transcribed into noncoding RN (ncrn) [8]. Examples of ncrn molecules include ribosomal RN, a core component of the cellular protein manufacturing mechanism, small nuclear RN, which is involved in RN splicing and regulation of transcription factors, and micro RN, which is responsible for translational gene repression. The function of a given ncrn depends on the molecule s structure in Transcription Translation DN RN Protein Reverse Transcription Figure 1: The central dogma of molecular biology, showing information flow. 1

2 3-dimensional space, or tertiary structure. 1 In RN, tertiary structure is derived from secondary structure, which is defined as the set of all base-pair interactions among an RN s component nucleotides (Figs. 2, 3). 2 Primary structure refers to the sequential ordering of an RN s nucleotides. urrently, the most well-established experimental method for determining RN secondary structure involves working backward from a tertiary structure obtained via X-ray crystallography. However, experimental methods for determining RN structure are generally expensive and time-consuming. Since all of the information in a strand of RN is captured by the molecule s primary structure, the ability to determine the function of an RN molecule given only the sequence of its nucleotides is theoretically possible. Free Energy Minimization Modern techniques for RN secondary structure prediction are based on the concept of free-energy minimization. These algorithms depend on two principles: First, that the structure adopted by an RN molecule in solution is that which minimizes the total Gibbs free energy G of the solution [16], and second, that the free energy for a given secondary structure can be estimated for any sequence [7, 15]. Gibbs Free Energy Gibbs free energy is a measure of the process-initiating work obtainable from a thermodynamic system at constant temperature and pressure, and is typically expressed in terms of enthalpy H, entropy S, and absolute temperature (Eq. 1). G = H T S (1) Reactions which increase the total entropy of the universe occur spontaneously and are characterized by negative free energy values. The tendency of systems (e.g., strands of RN in solution) to adopt states which minimize G is a consequence of the Second Law of Thermodynamics. 1 The tertiary structures of DN and RN are typically quite different: while DN molecules usually exist as fully base-paired double helices, most RN is single-stranded and hence adopts highly complex 3D forms. The genesis of these complex structures is further aided by the increased ability of RN to form hydrogen bonds due to the extra hydroxyl group in the RN s ribose sugar backbone. 2 These interactions include the canonical Watson-rick base pairs & G U, as well as alternate bonding patterns like wobble bases or Hoogsteen base pairs. 2

3 Estimating G for RN free energy estimate for a given secondary structure depends on the number and type of nucleotides involved in base-pairing interactions, as well as the resultant arrangements of those nucleotides which are not base-paired. These arrangements or motifs fall into two broad categories: nested loops (Fig. 2) and pseudoknots (Fig. 3). Formally, given an RN sequence S and a set of base pairs S which define a secondary structure, nested loops satisfy i j, i j S, either i < i < j < j or i < j < i < j. By contrast, i j and i j form a pseudoknot if i < i < j < j. Base pairs reduce the total free energy of a secondary structure, and regions of contiguous base pairs (or stacked bases) further reduce the structure s total free energy. Loop regions, on the other hand, increase the total free energy of a secondary structure; different loop motifs make different contributions to the final free energy estimate (Table 1) for a given secondary structure. G estimates for various motifs are based on optical melting experiments [7]. lgorithmic Techniques Nussinov [9] and Zuker [19] produced the first polynomial-time dynamic programming algorithms for RN secondary structure prediction in the late 1970s. Today, most RN structure prediction algorithms still use some form of dynamic programming, though other approaches employing genetic algorithms have been implemented with some success [5]. Minimum free-energy secondary structures which do not include pseudoknots can be found using dynamic programming in O(N 3 ) time, where N is the number of nucleotides in the sequence [9, 19, 10, 18]. By contrast, the task of determining RN secondary structures which include arbitrary pseudoknots has been shown to be NP-complete [6, 1], though there have been successful attempts to predict pseudoknotted structures in polynomial time using a limited set of all possible pseudoknot types [14]. Dynamic Programming Without Pseudoknots The first dynamic programming algorithm for predicting RN secondary structure sought only to maximize the number of nested base-pair interactions in a sequence S composed of nucleotides S 1... S n. This maximization problem can be approached by examining subsequences S ij of length p, and 3

4 Hairpin loop G U G 30 G Bulge loop G 20 G G U G U G Interior loops U 40 G Interior loop G } U G Stacked bases 10 G U 3 5 U U G G G U Multiloop G G G G G G Exterior loop G G G U U Hairpin loop U 60 Figure 2: Secondary structure of trn-phe from Haloarcula marismortui, a single-celled archaeon found in the Dead Sea, showing the different types of nested loops and stacked base pairs. Table 1 shows the G values for the various motifs. Figure generated with mfold [20]. 4

5 Structural element δg Information External loop ss bases & 1 closing helices. Stack External closing pair is G 1-73 Stack External closing pair is 2 -G 72 Stack External closing pair is 3 -G 71 Stack External closing pair is G 4-70 Stack External closing pair is 5 -G 69 Helix base pairs. Multi-loop 1.90 External closing pair is 6 -G 68 4 ss bases & 3 closing helices. Stack External closing pair is 50 -G 66 Stack External closing pair is 51 -G 65 Stack External closing pair is 52 -G 64 Stack External closing pair is G Helix base pairs. Hairpin loop 4.40 losing pair is G Stack External closing pair is U 8 -G 47 Stack External closing pair is 9 -U 46 Stack External closing pair is G Stack External closing pair is 11 -G 44 Stack External closing pair is U Helix base pairs. Bulge loop 1.40 External closing pair is 13 -G 42 Stack External closing pair is G Helix base pairs. Bulge loop 1.60 External closing pair is 16 -U 40 Stack External closing pair is 17 -G 38 Helix base pairs. Bulge loop 3.80 External closing pair is U Stack External closing pair is G 19 -U 34 Helix base pairs. Interior loop External closing pair is G Stack External closing pair is G Helix base pairs. Hairpin loop 4.40 losing pair is 24 -U 29 Table 1: Free energy values for the structure in Figure 2. The total G for the structure is From mfold [20]. 5

6 U U U G G G U Figure 3: simple pseudoknot. incrementally increasing p at each step. t each iteration, the maximization algorithm tests the ability of each nucleotide S k : i k < j to pair with S j, and then recursively calculates the best score in the subsequences to either side of S k. The key observation is that at each successive step of the algorithm, the values of the subsequences S i(k 1) and S (k+1)j have already been computed in an earlier iteration. The general recursive formula for computing the maximum number of nested, paired bases in a sequence from i to j is given in Equation 2 [9]. M(i, j) = max i k<j L min { M(i, k 1) + M(k + 1, j 1) + 1 M(i, j 1) Here, L min is the minimum number of bases that must separate two nucleotides involved in a base-pairing interaction. 3 Refinements to this process use more detailed free-energy approximations [19]. During execution, these refined algorithms maintain two matrices of free-energy scores for a subsequence S ij - the matrix V (i, j) (Eq. 3) represents the best possible score with i and j paired to one another, and W (i, j) (Eq. 4) is the best score regardless of whether i and j are paired. The explicit calculation of energy scores on substructures in which i and j are paired enables these algorithms to incorporate energy scores for whole motifs as well as individual base-pairs. sample of the thermodynamic stacking and destabilizing energies used to calculate a free energy estimate for a given secondary structure can be found in Table 1. The terms, eh, es, and el found in Equation 3 are the energy functions for hairpin loops, stacked base pairs, and internal loops & bulges, respectively. 3 Steric constraints limit the minimum loop size to three nucleotides, and most loops include 4-8 bases. (2) 6

7 Figure 4: graphical representation of the terms used in V (i, j). Dashed lines represent a region of arbitrary secondary structure. Dotted lines separate distinct components of each term, and roughly correspond to the + operator in each expression. () The structure represented by the first term, which handles hairpin loops. (B) The relationship between stacked base pairs. () Represents internal loops; the condition i i + j j > 2 ensures that there is at least one unpaired base between either i and i or j and j. (D) generalized multiloop structure. eh(i, j) es(i, j, i + 1, j 1) + V (i + 1, j 1) V (i, j) = min min {el(i, j, i, j ) + V (i, j )} i<i <j <j i i+j j >2 min {W (i + 1, i+1<i <j i 1) + W (i, j 1)} V (i, j), W (i + 1, j), W (i, j) = min W (i, j 1), min {W (i, i i <j i ) + W (i + 1, j)} (3) (4) Figure 4 is a graphical representation of the terms used to compute V (i, j). Figures 5 & 6 show graphical representations of the recursions used to form V (i, j) and W (i, j). The algorithms described here use only a single RN sequence as input. State-of-the-art systems for RN secondary structure prediction incorporate information derived from multiple sequences. These methods use the assumption that similar RN sequences generally have similar functional roles [4], and will therefore adopt similar secondary structures even if they differ slightly in primary structure and/or are found in different organisms. Such functionally similar sequences are deemed homologous. Homologous 7

8 sequences can be identified using covariance models or other expert techniques; structure predictions which make use of this additional information are generally termed consensus models. Figure 5: Graphical representation of the general recursion for V (i, j). wavy line between bases i and j indicates i is base-paired to j [14]. Figure 6: Graphical representation of the recursion for W (i, j). dotted line from i to j means the relation between bases is unknown [14]. Dynamic Programming With Pseudoknots The dynamic programming algorithms for RN secondary structure prediction reviewed thus far depend on the notion that the free energy of an RN sequence can be expressed as an optimal combination of the free energies of motifs formed from non-overlapping subsequences of the full sequence (Fig. 7). These algorithms are efficient because at each iteration they re-use previously calculated energies of smaller motifs formed from shorter contiguous subsequences. These standard dynamic-programming algorithms thus fail to predict pseudoknots because pseudoknotted motifs are structures formed from overlapping sequences of RN (Fig. 8). During the execution of such a standard algorithm, at any given stage a minimum free energy pseudoknotted structure may exist which incorporates base-pairing interactions with nucleotides that have already been implicitly examined by earlier iterations of the algorithm. This suggests that the general class of pseudoknotted structures does not exhibit the optimal substructure property required for a successful application of dynamic programming. However, RN pseudoknots occur relatively rarely in vivo [17]. Furthermore, the set of pseudoknot configurations that have actually been observed in nature is limited. Rivas and Eddy [14] developed a dynamic programming algorithm capable of predicting all known types of RN pseudoknots in O(N 6 ) time and O(N 4 ) space; their algorithm recursively calculates pseudoknotted configurations by explicitly considering certain classes of pseudoknots along with the general class of nested loops. Theoretical bounds 8

9 Figure 7: linear representation of the trn secondary structure shown in Figure 2. rcs represent base-pair interactions. Note that every arc connecting base i to base j defines a contiguous sequence of bases Sij such that k : i < k < j, if k is paired to l then i < l < j. Generated with VRN [3]. Figure 8: linear representation of the pseudoknot in Figure 3, showing the non-nested base-pair interactions characteristic of pseudoknots. 9

10 have since been improved to O(N 4 ) time and O(N 2 ) space [12]. Heuristic approaches offer further improvement on the accuracy and efficiency of pseudoknot prediction [13], as do updates to the free energy models used for determining structure energetics [2]. Pseudoknots & omputational omplexity The problem of RN secondary structure prediction with generalized pesudoknots has been shown to be NP-complete under simplified energy models via reductions from longest common subsequence [1] and a version of 3ST where each literal occurs at most twice (2LIT3ST) [11] [6]. Both reductions use large, carefully-constructed RN sequences and energy functions. In Lyngsø and Pedersen [6], an instance φ of 2LIT3ST with c clauses and v variables is encoded as a strand s φ of infinite-alphabet 4 RN, and they show that φ has a satisfying assignment if and only if the energy E of s φ s optimal secondary structure under a specified model is strictly greater than a certain value. Their proof is outlined below: Translate each clause i = (l 1 l 2 l 3 ) into a clause substring: i = 1,1 (l 1 ) 1 1,1 1,2 (l 2 ) 1 1,1 1,2 (l 3 ) 1 1,2 Translate each variable x i which occurs twice positively and twice negatively in φ into a variable substring: V i = V 1 (x 1 ) 2 (x 1 ) 1 V 1 ( x 1 ) 2 ( x 1 ) 1 V 1 reate the sequence s φ = c V 1 V 2... V v ompute the energy E of the optimal secondary structure under the energy model E(S) = E(i j,i + 1, j 1) i j S where 1 if X and Y are complementary and do not form a pesudoknot E(X i Y j, V i+1, W j 1 ) =. with a neighboring base 0 else 4 The standard RN alphabed has four bases: Guanine, denine, ytosine, & Uracil. 10

11 If E > (3c + v) then φ is not satisfiable. The proof then demonstrates the equivalence of an infinite RN alphabet and the standard four-letter RN alphabet. Though the energy functions on which these reductions are based do not perfectly emulate real-world conditions, complexity results should remain valid as long as the natural (complex) model can be reduced to the simple model by fixing parameters [6]. References [1] Tatsuya kutsu. Dynamic programming algorithms for RN secondary structure prediction with pseudoknots. Discrete pplied Mathematics, 104(1-3):45 62, [2] Mirela S. ndronescu, ristina Pop, and nne E. ondon. Improved free energy parameters for rna pseudoknotted secondary structure prediction. RN, 16(1):26 42, [3] Kevin Darty, lain Denise, and Yann Ponty. Varna: Interactive drawing and editing of the RN secondary structure., [4] S R Eddy and R Durbin. RN sequence analysis using covariance models. Nucleic cids Research, 22(11): , Jun [5] P Gultyaev, F H van Batenburg, and W Pleij. The computer simulation of RN folding pathways using a genetic algorithm. J Mol Biol, 250(1):37 51, Jun [6] Rune B. Lyngsø and hristian N. S. Pedersen. RN pseudoknot prediction in energy based models. Journal of omputational Biology, 7:2000, [7] David H. Mathews and Michael Zuker. RN secondary structure prediction. John Wiley & Sons, Ltd, [8] John S. Mattick and Igor V. Makunin. Non-coding rna. Human Molecular Genetics, 15(suppl 1):R17 R29, 15 pril [9] R Nussinov and B Jacobson. Fast algorithm for predicting the secondary structure of single-stranded RN. Proceedings of the National cademy of Sciences of the US, 77(11): , Nov

12 [10] Ruth Nussinov, George Pieczenik, Jerrold R. Griggs, and Daniel J. Kleitman. lgorithms for loop matchings. SIM Journal on pplied Mathematics, 35(1):68 82, [11] hristos H. Papadimitriou. omputational complexity. [12] Jens Reeder and Robert Giegerich. Design, implementation and evaluation of a practical pseudoknot folding algorithm based on thermodynamics, [13] Jihong Ren, Baharak Rastegari, nne ondon, and Holger H Hoos. Hotknots: heuristic prediction of RN secondary structures including pseudoknots. RN, 11(10): , Oct [14] E Rivas and S R Eddy. dynamic programming algorithm for RN structure prediction including pseudoknots. Journal of Molecular Biology, 285(5): , Feb [15] I Jr Tinoco, P N Borer, B Dengler, M D Levin, O Uhlenbeck, D M rothers, and J Bralla. Improved estimation of secondary structure in ribonucleic acids. Nature: New Biology, 246(150):40 41, Nov [16] I Jr Tinoco, O Uhlenbeck, and M D Levine. Estimation of secondary structure in ribonucleic acids. Nature, 230(5293): , pr [17]. Xayaphoummine, T. Bucher, F. Thalmann, and H. Isambert. Prediction and statistics of pseudoknots in rna structures using exactly clustered stochastic simulations. Proceedings of the National cademy of Sciences of the United States of merica, 100(26): , Dec. 23, [18] M. Zuker, D.H. Mathews, and D.H. Turner. lgorithms and Thermodynamics for RN Secondary Structure Prediction: Practical Guide. NTO SI Series. Kluwer cademic Publishers, [19] M Zuker and P Stiegler. Optimal computer folding of large RN sequences using thermodynamics and auxiliary information. Nucleic cids Research, 9(1): , Jan [20] Michael Zuker. Mfold web server for nucleic acid folding and hybridization prediction. Nucleic cids Research, 31(13): ,