Pseudoknots in RNA folding landscapes

Motivation: The function of an RNA molecule is not only linked to its native structure, which is usually taken to be the ground state of its folding landscape, but also in many cases crucially depends on the details of the folding pathways such as stable folding intermediates or the timing of the folding process itself. To model and understand these processes, it is necessary to go beyond ground state structures. The study of rugged RNA folding landscapes holds the key to answer these questions. Efficient coarse-graining methods are required to reduce the intractably vast energy landscapes into condensed representations such as barrier trees or basin hopping graphs (BHG) that convey an approximate but comprehensive picture of the folding kinetics. So far, exact and heuristic coarse-graining methods have been mostly restricted to the pseudoknot-free secondary structures. Pseudoknots, which are common motifs and have been repeatedly hypothesized to play an important role in guiding folding trajectories, were usually excluded. Results: We generalize the BHG framework to include pseudoknotted RNA structures and systematically study the differences in predicted folding behavior depending on whether pseudoknotted structures are allowed to occur as folding intermediates or not. We observe that RNAs with pseudoknotted ground state structures tend to have more pseudoknotted folding intermediates than RNAs with pseudoknot-free ground state structures. The occurrence and influence of pseudoknotted intermediates on the folding pathway, however, appear to depend very strongly on the individual RNAs so that no general rule can be inferred. Availability and implementation: The algorithms described here are implemented in C++ as standalone programs. Its source code and Supplemental material can be freely downloaded from http://www.tbi.univie.ac.at/bhg.html. Contact: qin@bioinf.uni-leipzig.de Supplementary information: Supplementary data are available at Bioinformatics online.


PART A: RNA FOLDING LANDSCAPES AND BASIN HOPPING GRAPH REVISITED
Given an RNA sequence σ, in this contribution, we consider the ensemble X = Xσ of secondary structures in which pseudoknots can be included. It has been proven that the cardinality |Xσ| grows exponentially with the length of σ (Akutsu (2000); Lyngso & Pedersen (2000) and the references therein) provided the stickiness of σ, i.e., the probability that two arbitrarily chosen nucleotides in σ can form a base pair, is relatively large. This is true for most biological RNA sequences, since the values of stickiness for RNAs are around 0.375 (Hofacker et al., 1994). This ensemble of RNA structures can be arranged as a graph, referred as RNA folding landscape, by defining a "move set", i.e. by specifying which pairs of secondary structures can be interconverted in a single step (Reidys & Stadler (2002) and the references therein). Each vertex of the RNA folding landscape, i.e., each RNA secondary structure x, is associated with an energy f (x). For the cases of pseudoknot-free structures, a well-established energy model allows us to explicitly compute f (x) for every structure s in terms of additive contributions for base pair stacking as well as hairpin loops, interior loops, bulges, and multiloops (Mathews et al., 1999). When pseudoknots appear, the evaluation of free energy gets more involved. The current energy models for pseudoknots are simple, heuristic extensions of the standard energy model that use "developer-defined" energy penalties to score pseudoknots. An alternative, rather general energy function for pseudoknotted structures has been derived from the "cross-linked gel model" (Isambert & Siggia, 2000), however it suffers from the same lack of experimental data. Furthermore, no open source implementation of this energy function is available.
A structure x ∈ X is a local minimum (LM) of the landscape if it does not have neighbors with lower energy. In particular, x is a global minimum or a minimum free energy structure (MFE) if its energy is minimal within X. For each LM x we define its gradient basin G(x) ⊂ X as the set of structures z ∈ X so that the unique gradient walk with starting point in z ends in x. We note for later reference that the gradient basins of all the LMs in the RNA folding landscape forms a partition of its configuration space X. This partitioning forms an intuitive coarse-grained model of the landscape. One key difference is the energetically favorable neighborhood relation between the basins, can be displayed in the basin hopping graph, but not in the barrier tree.
The cycle B h (x) of x at energy level h can be defined as a maximal connected subset of {z ∈ X|f (z) ≤ h} that contains x. In other worlds, B h (x) is the set of structures found by a flooding algorithm starting at x (Sibani et al., 1999;Flamm et al., , 2002. In particular, the basin B(s) = B f (s) (s) of s (Flamm et al., 2002) is the set of all points in X that can be reached from s by a path whose elevation never exceeds f (s).
A direct saddle between two LMs x and y is a structure s ∈ X with minimal energy so that both x and y are reachable from s by means of an adaptive walk. We call DS(x, y) = f (s) the direct saddle height between x and y. Not every pair of LMs is connected by direct saddles.
The saddle height S(x, y) between any two vertices x and y is the minimal value h for which y ∈ B h (x). In other words, S(x, y) is the level at which two cycles B h (x) and B h (y) "merge". If x and y are LMs connected by a direct saddle point then S(x, y) ≤ DS(x, y). A structure s ∈ X is called a saddle between x, y ∈ X if (i) f (s) = S(x, y) and (ii) there is a path P connecting x and y so that f (s) ≥ f (z) for all z ∈ P . A path P * connecting x and y in the landscape is energetically optimal if maxz∈P * f (z) = The y-axis denotes the (free) energies of the structures in the landscape. There are in total three gradient basins with local minima LM1, LM2 and LM3. Structures in the same gradient basin are labeled with the same color, except two saddles s1 and s2. In which, the structure s1 is a direct saddle and saddle between LM1 and LM2. An energetically optimal path between LM1 and LM2 pass through structures a, b, s1, c, d consecutively. Analogously, s2 is a direct saddle and saddle between LM2 and LM3 with an energetically optimal path passing through structures e, s2, f, and g. Note here, s1 is a saddle but not a direct saddle between LM1 and LM3. In fact, there does not exist any direct saddle between LM1 and LM3 since there does not exist any structure from which both LM1 and LM3 are reachable by adaptive walks. (Bottom) The resulting BHG of this landscape.
S(x, y). Energetically optimal paths are not necessarily unique. See Fig. 2 for an illustration of the concepts mentioned above. For RNA folding landscapes, the problems of computing saddle heights, saddle points and energetically optimal path are NP-hard (Maňuch et al., 2011). The basic idea of basin hopping graph (BHG) is to incorporate additional neighborhood information by considering LMs as adjacent if the transition between their corresponding basins are "energetically optimal". A schematic diagram of BHG for a toy landscape is illustrated in Fig. 1. In which, the transition from A to B on Fig. 1 is energetically optimal, since S(A, B) = DS(A, B) = 0, but the transition from A to D is not, since 1 = DS(A, D) > S(A, D) = 0.

Pseudoknot energy model of 1-structures in gfold
In this section, we give a brief review of the energy model for evaluating 1-structures introduced in gfold ( Reidys et al., 2011). A full-fledged version is available in the supplementary material of their original paper.
In the pseudoknot energy model of gfold, except pseudoknotted loops, all other types of loops are evaluated according to the standard Turner 1999 energy model (Mathews et al., 1999). The energy contributions of pseudoknotted loops are evaluated as an extended version of multiloops.
More precisely, the energy of an external pseudoknot (a pseudoknot not covered by any base pair) is evaluated as (1) In which, the parameter βT ype is is the penalty of forming a pseudoknot of Type H, K, L, or M, B is the number of base pairs forming the pseudoknot, and U is number of unpaired nucleotides inside the loop. Since the number of crossing base pairs is always at least two, a multiloop is formed whenever a pseudoknot is nested in a base pair. In these cases, the penalty parameter βT ype is replaced by β mul T ype . Otherwise, if a pseudoknot is nested in another pseudoknot, then βT ype is replaced by β pseudo T ype . The energy parameters for pseudoknots used in the gfold are listed in Table 1.
There is a heavy penalty for forming a pseudoknot inside another pseudoknot or multiloop, which may be due to a lack of experimental evidence of such complicate pseudoknotted structures. As a result of the relatively heavy penalties to form a pseudoknot, a gradient walk starting from a pseudoknot-free structure can not end in a pseudoknotted LM.

Adaptations in gfold program
We have made some necessary adaptions in gfold (Bon et al., 2008;Reidys et al., 2011) to implement the adaptive sampling schedule used in RNAlocmin. First, an additional option for the ξ-scaling procedure required in RNAlocmin is implemented. Secondly, the output format of gfold is tailored for its usages in the ξ-scaling procedure including an option to vary the sample sizes.
The original output file of gfold is still kept as an output option in the modified version, which is available on the webpage https://github.com/marcelTBI/gfold.

Energy parameters in BHG ψ and BHG •
For comparison purpose, in this publication we often consider, for an RNA whose ground state is pseudoknot-free the full BHG ψ including pseudoknotted LMs and a pruned BHG • from which first all pseudoknotted LMs are removed and then the BHG-adjacency is recomputed using only pseudoknot-free structures along the paths.
To make sure that the energy parameters are coherent, in both BHG ψ and BHG • , we are obliged to use the standard Turner energy model (Mathews et al., 1999) without considering dangle energies as implemented in the ViennaRNA Package with options -d0 -P rna turner99.par . This is because in the energy model used in gfold, the penaltyparameters for pseudoknots are only trained under the standard Turner energy model (Mathews et al., 1999) at 37 • C without taking the dangle energies into consideration.

Determine valid base pairs to add into a secondary structure
Given a 1-structure S, we first need to construct the conflict graph of S. The vertices of conflict graph are constructed based on the relations between any two helices a = (la, ra; da) which is a set of base pairs {(la, ra), (la + 1, ra − 1), . . . , (la + da, ra − da)} and b = (l b , r b ; d b ) of S: 1. crossing, denoted by a⊥b if la < l b < ra < r b or its symmetric case is true; 2. nesting, denoted by a b if la < l b < r b < ra or its symmetric case is true; These two relations give rise to a partition of the helices of S into gap-sets via requiring that two helices a and b belong to the same gap-set if a b and they cross with the same set of helices in S. For example, in Fig. 3 (A), there are in total 6 helices {a1, a2, . . . , a6} in a pseudoknotted structure and 5 gap-sets {{a1, a6}, {a2}, {a3}, {a4}, {a5}}. Each gap-set is represented as a vertex in its conflict graph shown in Fig. 3 (B). Furthermore, we draw an edge in the conflict graph between two vertices, if their corresponding helices cross with each other. In Fig. 3 (B), two gapsets {a4} and {a1, a6} are adjacent in the conflict graph given that a1⊥a4 and a6⊥a4.
Adding a base pair a in S therefore, in the "worst" case, is equivalent to add a vertex (and potential incident edges) into the conflict-graph of S accordingly, see Fig. 3 (D) for an example. Thus all we need is to test whether the components of the resulting conflict graph has some component other than the 5 valid types shown in Fig. 3 (C). In particular, we only need to consider the components which contain base pairs crossing with a. gives rise to an invalid structure since the component including a7 in the resulting conflict graph is not one of the five valid types shown in (C). Adding a base pair (b7, b9) gives rise to a valid structure since the according conflict graph stays the same except the helix a4 thickens by the base pair (b7, b9).

Methods
From a microscopic point of view, the dynamics on an RNA folding landscape can be described by a continuous-time Markov process with infinitesimal generator R = (ryx) . The transition rate ryx from a secondary structure x to y is non-zero only if x and y are adjacent, i.e., if they differ by adding/removing a single base pair. Typically the Metropolis rule, the following formula is used to assign microscopic rates Here, f evaluates the (free) energy of x, R is the universal gas constant, T is the absolute ambient temperature and r0 is a parameter used to gauge the time axis from experimental data. Here we simply use r0 = 1, implicitly defining our time unit. Denote the probability that an RNA molecule has the secondary structure x at time t by Px,t, the dynamics is governed by the master equation dPx,t/dt = y rxyPy,t with rxx = − y =x ryx. This linear system of differential equations can be exactly solved by explicitly computing P(t) = exp(tR) · P(0) for short RNA molecules ∼ 30nt, where P(t) is the vector of Px,t for all possible structures x. The program treekin (Wolfinger et al., 2004) provides an implementation of this method.
Even for RNA molecules of moderate size, direct computation of the matrix exponential becomes impossible due to the exponential growth of the underlying state space. An alternative is to perform stochastic simulations as is done in the kinfold program , however this turns out to be rather time consuming for large RNA molecules. Wolfinger et al. (2004) used barrier trees (Flamm et al., 2002) to assign a macro state to each local minimum and recalculate rates between these. This approximation has shown excellent agreement to the full-process computed from Eqn. 1 with all possible structures, but its exhaustive nature limits its applicability to molecules up to ∼ 80nt.
We observed that the computation of matrix exponentials in treekin becomes numerically unstable when some transition rates are very small. We therefore use a Padé approximation and the scaling and squaring method described in (Al-Mohy & Higham, 2009) and implemented in the function f01ecc of the NAG library Mark 9 with time complexity of O(N 3 ) (N is the dimension of the matrix and thus the number of the LMs in our case).

Comparison to Wolfinger et al. (2004)'s folding dynamic approximation
To demonstrate the quality of the BHG approximation, we present the comparison to the barrier tree based coarse graining of folding kinetics for several examples. We show that our approximation reflects a qualitatively correct description of the process, as well as important quantitative details, such as, the ordering of the top frequent structures and the time needed to converge to the thermodynamic equilibrium distribution. The time for an RNA to reach the equilibrium is evaluated as the first time t, when the Euclidean distance between computed distribution P(t) and the Boltzmann equilibrium distributions is less than a threshold 10 −5 . The examples include the following: the Pyaiella Littoralis Group II Intron (PDB 01042, 34nt, Fig. 4), the pseudoknot domain of tmRNA from E. coli (PKB49,30nt,Fig. 5) and Legionella pneumophila (PKB67, 30nt, Fig. 6), a synthetic tetraloop-receptor (PDB 00924, 86nt, Fig. 7), and a Hammerhead ribozyme (type III) (RFA 00398, 54nt, Fig. 8). The LMs that appear in both kinetics plots are marked with same color, otherwise with black.
For longer RNAs, the exponential growth of LMs in the BHG poses computational difficulties in our continuous time Markov chain based folding simulations, since the number of LMs considered is exactly the dimension of the infinitesimal generator R. The number of LMs on BHG can be easily beyond 10 5 for an RNA of length ∼ 100nt, even with additional restriction on their energy range. Furthermore, our observations show that only a small portion of the whole set of LMs on BHG play important roles in the kinetic simulations, most of the LMs only contribute in fast fluctuations and the resulting computational cost. For example, the folding kinetics of the Pyaiella Littoralis Group II Intron shown in   ((((....))))))))))))((.(((....))).)) ((((((((..(....)..))))))))......... which is an LM of energy -37.80 kcal/mol. This kinetic process was not started in the open chain structure given that there are more than 10000 LMs in between the open structure and the ground state structure which is beyond the feasible range of the Wolfinger et al.
's folding dynamic approximation. The process was run until convergence to the thermodynamic equilibrium distribution. The x-axes and y-axes indicate the time and population probabilities, respectively.

PART D: QUASI-STEADY-STATE REDUCTION
We first partitioned all the LMs found into two categories: (G) important LMs (with high degree in our case) which are the "good" ones to keep and (B) intermediate LMs which are the "bad" ones to be neglected. Next, we re-arrange the ordering of the LMs based on their categories so that the rate matrix R and population vector P(t) can be rewritten into the following format In which, PG(t) and PB(t) denotes the population subvectors of the good and bad states respectively. Submatrix GB contains the transition rates from good states to bad states. The remaining three sub-matrices GG, BG and BB are defined analogously. Accordingly, dP(t) dt = P(t)R can be written as and furthermore PG(t) = PG(0) · e GG−GB·BB −1 ·BG .
In which, the Schur complement GG − GB · BB −1 · BG can be computed efficiently given that the rate matrix R is sparse. Due to properties of Schur complement, it can be computed iteratively -reducing a single LM at each step (the matrix BB is a scalar). Then the time complexity of such a single step is O(c 2 ), where c is the number of neighbors of this LM. Assign b = dim(BB) and assume that degree of all reduced LM is small and bounded by some cmax (cmax b), then the whole time complexity is O(bc 2 max ). However, if the matrix is dense (cmax ∼ b) this reduction is equally time consuming as naive computation of BB −1 and thus unfeasible for our purposes.
In practice, this heuristic works reasonably well and has been implemented as part of the BHGbuilder program.

Summary of LMs and gfold-sampling structures in the lower parts of RNA molecules' landscapes
We analyze the composition of the LMs in the "lower" part of the energy landscapes of various RNA molecules listed in Table 2. In which, "lower" part means that we only consider LMs with negative free energies and within 10 kcal/mol above the minimum free energy of the whole folding landscape. We contrast RNAs with pseudoknots in their ground state structures selected from Pseudobase++ (PKB ID), (Han et al., 2002;Taufer et al., 2009) and pseudoknot-free structures from the RNA STRAND database (Andronescu et al., 2008) (RNA STRAND ID). As well the transcriptional Bacillus subtilis riboswitch (Bsu) with an H-type pseudoknotted structure or a pseudoknot-free structure as its ground state depending on the presence of preQ1 (Suddala et al., 2013). Note here, we select the molecules such that their ground state structures predicted by gfold have both sensitivity and PPV beyond 80%, so that the effects caused by the prediction software can be limited.
The mean and standard deviation (STDEV) for the number and proportion of LMs of each type (pseudoknotted (H, K, L) or pseudoknot-free (N)) are given in Table 3 based on 10 independent samples. Furthermore, in Table 4 we report the analogous information of the gfold-sampling structures starting from which these LMs are derived by simulating gradient walks. In both tables, the numbers regarding to the M-type LMs or structures are omitted, since structures of such type were not observed in any of the experiments. Comparison of 10 independent samples shows that, despite that sometimes the sample sets of structures have relatively large deviations, the derived LM sets vary only slightly, confirming that the sampling is sufficient for the purpose of detecting LMs.

Robustness of BHG-approach in estimating saddle heights
As shown in Table 3, the LM sets are fairly stable when sufficiently large sets are sampled. Of course, the LM sets obtained from independent samplings are usually not identical since the high energy LMs grow exponentially in number and thus cannot be exhaustively collected in practise. The BHGs constructed based on these LM sets therefore will differ in vertex sets, edge sets and the weights (saddle heights) on edges. We therefore show that these BHGs nevertheless agree to high accuracy on the low-energy LMs, and the edges between them. As a consequence, the estimations of saddle heights between them are also robust.
To this end, we first compute BHGs based on 10 independent samples for a given RNA sequence, collect the set of the common LMs in these BHGs and then evaluate the standard deviations of saddle heights between all these pairs of common LMs. The average standard deviations are reported in Table 5. Note that the number of common LMs is different from the number of LMs generated with RNAlocmin in Table 3. This is because the heuristic algorithm constructing the BHGs first selects the non-shallow LMs from the initial LM set generated from RNAlocmin and then iteratively expands this set of non-shallow LMs by adding intermediate LMs detected in the path searching procedure. For three examples PDB 00542, PDB 00702, and SRP 00005, the evaluation described above is computationally infeasible due to the large numbers (more than 13 thousands) of the common LMs in their BHGs. Given a set of K LMs, evaluating saddle heights between all pairs of these LMs requires O(K 3 ) time using a variant of Dijkstra's algorithm to detect the corresponding shortest min-max paths. Instead of the entire set, we therefore evaluate only the lowest 1000 common LMs.
Given that the averaged deviations are less than 0.26 kcal/mol, we conclude that our method in estimating saddle heights is robust.

PART F: SADDLE HEIGHT CHANGES
BETWEEN BHG ψ AND BHG • 6.1 Histograms of saddle height changes between BHG ψ and BHG • In the following, we only consider for RNAs whose ground states are pseudoknot-free. For each of such RNAs, the full BHG ψ including pseudoknotted LMs and a pruned BHG • from which first all pseudoknotted LMs are removed and then the BHG-adjacency is recomputed using only pseudoknot-free structures along the paths. The re-evaluation may result in the removal of adjacencies from BHG ψ . Five examples (PDB 00542, PDB 01042, RFA 00398, SRP 00194 and SRP 00284) were not shown given that the differences between BHG ψ and BHG • are relatively small.
Histograms of saddle height changes between BHG ψ and BHG • for 8 RNAs are shown in Fig. 9-15. In each example, the x-axes of the top and bottom histograms denote the exact changes (in kcal/mol) and relative changes (in %), respectively. The y-axes denote the corresponding numbers of pseudoknot-free LMs pairs with such saddle changes. The colors of the histograms indicate the types of pseudoknotted structures appear in the energetically optimal paths between LM pairs. For example, the pink color (Type HK) indicates that the energetically optimal paths contain structures 6.2 Using --depth parameter in findpath to improve negative saddle height differences between BHG ψ and BHG • In general, saddle heights between the LMs in BHG • will increase compared to BHG ψ . In practice, however, the inclusion of additional LMs during the recomputation of the adjacencies can in rare cases lead to an apparent decrease in saddle heights and furthermore negative saddle heights differences between BHG ψ and BHG • . For example, see Fig. 16. In such cases, the saddle heights in BHG ψ are overestimated due to the heuristic nature of the findpath program  used to estimate saddle heights. Findpath performes a bounded breadth-first search algorithm that at each depth only keeps the m most promising candidates. The option "-depth" with default value 10 is used to specify m and therefore balances speed versus accuracy. As we show in Fig. 17, once we increase the candidate number to 100, all negative saddle height differences are eliminated, see Fig. 16. Fig. 9. Histograms of saddle height changes between BHG ψ and BHG • of the transcriptional preQ1 riboswitch of Bacillus subtilis (Bsu, Suddala et al. (2013)). The relevant saddle heights are generated with findpath using default parameter --depth=10.  2004)). The relevant saddle heights are generated with findpath using default parameter --depth=10. Fig. 11. Histograms of saddle height changes between BHG ψ and BHG • of a signal recognition particle of M. Jannaschiii (PDB 00879, Hainzl et al. (2005)). The relevant saddle heights are generated with findpath using default parameter --depth=10.  Davis et al. (2005)). The relevant saddle heights are generated with findpath using default parameter --depth=10. Fig. 13. Histograms of saddle height changes between BHG ψ and BHG • of a signal recognition particle RNA provided in the SRPDB database (SRP 00005, Rosenblad et al. (2003)). The relevant saddle heights are generated with findpath using default parameter --depth=10. Histograms of saddle height changes between BHG ψ and BHG • of a signal recognition particle RNA provided in the SRPDB database (SRP 00094, Rosenblad et al. (2003)). The relevant saddle heights are generated with findpath using default parameter --depth=10. Fig. 15. Histograms of saddle height changes between BHG ψ and BHG • of a tmRNA provided in the tmRDB database (TMR 00272, Knudsen et al. (2001)). The relevant saddle heights are generated with findpath using default parameter --depth=10.   Merianos et al. (2004)). The relevant saddle heights are generated with findpath using parameter --depth=100.

PART G: FOLDING KINETICS OF RNAS
In the following we only consider for RNAs whose ground states are pseudoknot-free. For each of such RNAs, the full BHG ψ including pseudoknotted LMs and a pruned BHG • from which first all pseudoknotted LMs are removed and then the BHG-adjacency is recomputed using only pseudoknot-free structures along the paths. The reevaluation may result in the removal of adjacencies from BHG ψ .
Seven examples (PDB 00542, PDB 00924, PDB 01042, RFA 00398, SRP 00194, SRP 00284 and TMR 00272) were not shown given that the differences in kinetics between BHG ψ and BHG • are relatively small. The kinetics on BHG • and BHG ψ of three RNAs are shown in the top and bottom of Fig. 12, 13, and 14, respectively. The process was started in the open chain structure and run until convergence to the thermodynamic equilibrium distribution. Dotted vertical line indicates when the simulation reaches its equilibrium. The LMs that appear in both kinetics plots are marked with the same color, otherwise pseudoknot-free and pseudoknotted LMs are marked with black and red, respectively. The sums of the population probabilities of pseudoknot-free and pseudoknotted LMs on BHG ψ are shown with blue and red broken lines, respectively.

PART H: MAXIMUM LIKELIHOOD CRITERION WITHIN UPPER TIME LIMIT T
Given a trajectory U = (s0, t0, s1, t1, . . . , s k−1 , t k−1 , s k ) that the molecule started in s0, where it stayed for time t0, then transitioned to s1, where it stayed for time t1, and so on until eventually it reached s k , where it remained until time T . The likelihood of such a trajectory U is when ti ≤ T and 0 otherwise. In our cases, we have λs i = s j rs i s j and Ps i ,s i+1 = rs i s i+1 λs i . Therefore, equation reduces to Π k−1 i=0 rs i s i+1 · e −( k−1 i=0 λs i t i +λs k (T − t i )) .
We compute the optimal paths for two cases T = 0 and T = 10 11 for the SV11 sequence shown in Fig. 21. Notice here when T = 0, all ti has to be 0 as well. Eqn. 2 furthermore reduces to Π k−1 i=0 rs i s i+1 Therefore the Criterion C can be seen as a special case of the Criterion B when T = 0.  Fig. 19. Folding kinetics of a signal recognition particle RNA of M. Jannaschiii (PDB 00879, Hainzl et al. (2005)). Fig. 20. Folding kinetics of a signal recognition particle RNA provided in the SRPDB database (SRP 00005, Rosenblad et al. (2003)).