Prediction and statistics of pseudoknots in RNA structures using exactly clustered stochastic simulations A. Xayaphoummine , T. Bucher , F. Thalmann , and H. Isambert * Laboratoire de Dynamique des Fluides Complexes , Centre National de la Recherche Scientifique Universite Louis Pasteur , Institut de Physique , 3 Rue de l'Universite , 67000 Strasbourg , France Communicated by M. Gromov , Institut des Hautes Etudes Scientifiques , Bures-sur-Yvette , France , October 6 , 2003 -LRB- received for review March 28 , 2003 -RRB- Ab initio RNA secondary structure predictions have long dismissed helices interior to loops , so-called pseudoknots , despite their structural importance . Here we report that many pseudoknots can be predicted through long-time-scale RNA-folding simulations , which follow the stochastic closing and opening of individual RNA helices . The numerical efficacy of these stochastic simulations relies on an -LRB- n2 -RRB- clustering algorithm that computes time averages over a continuously updated set of n reference structures . Applying this exact stochastic clustering approach , we typically obtain a 5 - to 100-fold simulation speed-up for RNA sequences up to 400 bases , while the effective acceleration can be as high as 105-fold for short , multistable molecules -LRB- < 150 bases -RRB- . We performed extensive folding statistics on random and natural RNA sequences and found that pseudoknots are distributed unevenly among RNA structures and account for up to 30 % of base pairs in G C-rich RNA sequences -LRB- online RNA-folding kinetics server including pseudoknots : http : kinefold.u-strasbg . fr -RRB- . T he folding of RNA transcripts is driven by intramolecular GC AU GU base-pair stacking interactions . This primarily leads to the formation of short , double-stranded RNA helices connected by unpaired regions . Ab initio RNA-folding prediction restricted to tree-like secondary structures is now well established -LRB- refs . 1 7 , ref . 8 and references therein , www.bioinfo.rpi.edu applications mfold , and www.tbi . univie.ac.at -RRB- and has become an important tool to study and design RNA structures , which remain by and large refractory to many crystallization techniques . Yet , the accuracy of these predictions is difficult to assess , despite the precision of stacking interaction tables -LRB- 7 -RRB- , due to their a priori dismissal of pseudoknot helices -LRB- Fig. 1A -RRB- . Pseudoknots are regular double-stranded helices that provide specific structural rigidity to the RNA molecule by connecting different `` branches '' of its otherwise more flexible , tree-like secondary structure -LRB- Fig. 1 A and B -RRB- . Many ribozymes , which require a well defined 3D enzymatic shape , have pseudoknots -LRB- 9 17 -RRB- . Pseudoknots are also involved in mRNA ribosome interactions during translation initiation and frameshift regulation -LRB- 18 -RRB- . Still , the overall prevalence of pseudoknots has proved difficult to ascertain from the limited number of RNA structures known to date . This recently has motivated several attempts to include pseudoknots in RNA secondary structure predictions -LRB- 19 21 -RRB- . There are two main obstacles to include pseudoknots in RNA structures : a structural modeling problem and a computational efficiency issue . In the absence of databases for pseudoknot energy parameters , their structural features have been modeled at various descriptive levels by using polymer theory -LRB- 19 , 21 , 22 -RRB- . From a computational perspective , pseudoknots have proved not easily amenable to classical polynomial minimization algorithms -LRB- 20 -RRB- because of their intrinsic nonnested nature . Instead , simulating RNA-folding dynamics has provided an alternative avenue to predict pseudoknots -LRB- 21 , 22 -RRB- in addition to bringing PNAS December 23 , 2003 vol . 100 no. 26 some unique insight into the kinetic aspects of RNA folding -LRB- 8 , 21 -RRB- . Yet , stochastic RNA-folding simulations can become relatively inefficient due to the occurrence of short cycles among closely related configurations -LRB- 22 -RRB- , which typically differ by a few helices only . Not surprisingly , similar numerical pitfalls have been recurrent in stochastic simulations of other trapped dynamical systems -LRB- ref . 23 and references therein and refs . 24 27 -RRB- . To address this computational efficiency issue and capture the slow folding dynamics of RNA molecules , we developed a generic algorithm that greatly accelerates RNA-folding stochastic simulations by exactly clustering the main short cycles along the explored folding paths . The general approach , which may prove useful to simulate other trapped dynamical systems , is discussed in Theory and Methods . In Results , the efficacy of these exactly clustered stochastic -LRB- ECS -RRB- simulations is first compared with nonclustered RNA-folding simulations , before being used to predict the prevalence of pseudoknots in RNA structures on the basis of the structural model introduced in ref . 21 , and reviewed briefly hereafter . Theory and Methods Modeling and Visualizing Pseudoknots in RNA Structures . We model the 3D constraints associated with pseudoknots using polymer theory . The entropy costs of pseudoknots and internal , bulge , and hairpin loops are evaluated on the same basis by modeling the secondary structure -LRB- including pseudoknots -RRB- as an assembly of stiff rods -LRB- representing the helices -RRB- connected by polymer springs -LRB- corresponding to the unpaired regions -RRB- -LRB- Fig. 1C -RRB- . In practice , free-energy computations involve the labeling of RNA structures into constitutive `` nets '' -LRB- shown as colored circuits in Fig. 1C -RRB- to account for the stretching of the unpaired regions linking the extremities of pseudoknot helices -LRB- see ref . 21 for details -RRB- . In addition , free-energy contributions from base-pair stackings , terminal mismatches , and coaxial stackings are taken from the thermodynamic tables measured by the Turner Laboratory -LRB- 7 -RRB- . The main limitation of this structural model is the absence of hardcore interactions , which could stereochemically prohibit certain RNA structures with either long pseudoknots -LRB- e.g. , 11 bp , one helix turn -RRB- or a large proportion of pseudoknots -LRB- e.g. , 30 % of formed base pairs -RRB- . However , we found that such stereochemically improbable structures account for 1 10 % of all predicted structures depending on G C content -LRB- see Results -RRB- . Hence , in practice , neglecting hardcore interactions is rarely a stringent limitation except for a few , somewhatpathological cases . Abbreviation : ECS , exactly clustered stochastic . * To whom correspondence should be sent at the present address : Institut Curie , Section de Recherche , Centre National de la Recherche Scientifique Unite Mixte de Recherche 168 , 11 Rue Pierre et Marie Curie , 75005 Paris , France . E-mail : herve.isambert@curie.fr . 2003 by The National Academy of Sciences of the USA 15310 15315 www.pnas.org cgi doi 10.1073 pnas .2536430100 Fig. 1 . -LRB- A -RRB- An RNA secondary structure with pseudoknots . -LRB- B -RRB- Minimum set of helices defined as `` pseudoknots '' and visualized for convenience by colored , single-stranded regions connected by two straight lines . -LRB- C -RRB- The entropic cost of the actual 3D structural constraints is evaluated by modeling RNA helices as stiff rods -LRB- black -RRB- and single-stranded regions as ideal polymer springs . Colored , single-stranded circuits define quasiindependent structural domains referred to as `` nets '' in ref . 21 . stochastic openings and closings of individual RNA helices -LRB- 28 -RRB- . The time-limiting step to transit between two structures sharing essentially all but one helix can be assigned Arrhenius-like rates , k exp -LRB- G kT -RRB- , where kT is the thermal energy . k , k which reflects only local stacking processes within a transient nucleation core , has been estimated from experiments on isolated stem-loops -LRB- 28 -RRB- -LRB- k 108 s 1 -RRB- , whereas the free-energy differences G between the transition states and the current configurations -LRB- Fig. 2 -RRB- can be evaluated by combining the Modeling RNA-Folding Dynamics and Straightforward Stochastic Algorithm . RNA-folding kinetics is known to proceed through rare Fig. 2 . Stochastic transitions over a thermodynamic barrier G to close and open an individual helix between two neighbor RNA structures , i and j. Nucleation of the new helix usually involves some local unzipping of nearby helices at the barrier and further base-pair rearrangements to reach equilibrium in the new structure j -LRB- 21 -RRB- . ECS Simulations . As in the case of RNA-folding dynamics , the simulation of other trapped dynamical systems generally presents a computational efficiency issue . In particular , powerful numerical schemes have been developed to compute the elementary escape times from traps for a variety of simulation techniques -LRB- see ref . 23 and references therein and refs . 24 27 -RRB- . Still , a pervasive problem usually remains for most applications due to the occurrence of short cycles among trapped states , and heuristic clustering approaches have been proposed to overcome these `` numerical traps '' -LRB- 29 -RRB- . To capture the slow folding dynamics of RNA molecules , we developed an exact stochastic algorithm that accelerates the simulation by numerically integrating the main short cycles among trapped states . This approach being quite general , it could prove useful to simulate other small , trapped dynamical systems with coarse-grained degrees of freedom . In a nutshell , the ECS algorithm aims at overcoming the numerical pitfalls of kinetic traps by `` clustering '' some recently explored configurations into a single , yet continuously updated cluster A of n reference states . These clustered configurations are then collectively revisited in the subsequent stochastic exploration of states . Although stochasticity is `` lost '' for the individual clustered states , its statistical properties , however , are exactly transposed at the scale of the set A of the n reference states . This is achieved as follows . For each pathway CA on A , a m A CA m p statistical weight WCm lk is defined , where k and l run over all consecutive states along CA from its `` starting '' state i to its m `` exiting '' state j on A . The n n probability matrix PA that sums A the statistical weights WCm over all pathways CA on A between m any two states i and j of A is then introduced , PNAS December 23 , 2003 vol . 100 no. 26 15311 Xayaphoummine et al. . MATHEMATICS BIOPHYSICS Although the presence of pseudoknots in an RNA structure is not associated with a unique set of helices , it is convenient for visualization and statistics purposes to define the set of pseudoknots as the minimum set of helices that should be imagined broken to obtain a tree-like secondary structure -LRB- Fig. 1B -RRB- . Finding such a minimum set -LRB- with respect to the number of base pairs or their free energy -RRB- amounts to finding the maximum tree-like set among the formed helices and can be done in polynomial time by using a classical `` dynamic programming '' algorithm . stacking energy contributions and the global coarse-grained structural model described above -LRB- Fig. 1C -RRB- . Simulating a stochastic RNA-folding pathway amounts to following one particular stochastic trajectory within the large combinatorial space of mutually compatible helices -LRB- 22 -RRB- . Each transition in this discrete space of RNA structures corresponds to the opening or closing of a single helix , possibly followed by additional helix elongation and shrinkage rearrangements to reach the new structure 's equilibrium compatible with a minimum size constraint for each formed helix -LRB- 21 -RRB- -LRB- base-pair zipping unzipping kinetics occurs on much shorter time scales than helix nucleation dissociation -RRB- . For a given RNA sequence , the total number of possible helices -LRB- which roughly scales as L2 , where L is the sequence length -RRB- sets the local connectivity of the discrete structure space and therefore the number of possible transitions from each particular structure . Formally , we consider the following generic model . Each structure or `` state '' i is connected to a finite , yet possibly state-to-state varying number of neighboring configurations j via transition rates kji -LRB- the right-to-left matrix ordering of indices is adopted hereafter -RRB- . Because kji is the average number of transitions from state i to state j per unit time , the lifetime ti of configuration i corresponds to the average time before any transition toward a neighboring state j occurs , i.e. , ti 1 j kji , and the transition probability from state i to state j is pji kjiti , 1 , as expected , for all states i. with j pji Hence , in the straightforward stochastic algorithm -LRB- 21 , 22 -RRB- , each new transition is picked at random with probability pji while the effective time is incremented with the lifetime ti of the current configuration i. -LSB- In principle , the approach can be adapted to stochastically drawn lifetimes from known distributions Pi -LRB- t -RRB- with mean lifetime ti . This effectively yields a -LRB- n3 -RRB- ECS algorithm in this case . -RSB- However , as mentioned in the Introduction , the efficiency of this approach is often severely impeded by the existence of kinetic traps consisting of rapidly exchanging states . CA CA PA ji m : j4i W Cm m : j4i j4i A CA m p lk , -LSB- 1 -RSB- and the exit probability to make a transition outside A from the state j is noted : p eA 1 A pk j. Hence , starting from state i , j k A A the probability to exit the set A at state j is p eAP ji , with Ap AP ji j j j 1 , for all i of A. Thus , in the ECS algorithm , one first chooses at random with A probability p eAP ji the reference state j of A from which a new j transition toward a state k outside A will then be chosen stochastically with probability pkj p eA . Meanwhile , the physical j A quantities of interest , such as the cumulative time lapse t ji to exit the set A from j starting at i , are exactly averaged over all -LRB- future -RRB- pathways from i to j within A , as explained in the next subsection . Finally , the new state k is added to the reference set A while another reference state is removed , so as to update A , as discussed in The -LRB- n2 -RRB- Algorithm . with the path average time lapse to exit the set A. Let us ~ introduce the time-lapse transform of PA : -LSB- PA -RSB- -LCB- t -RCB- PA -LCB- t -RCB- , A A which sums the weighted cumulative lifetimes -LRB- Cmth -RRB- Cmplk over all pathways C A on A between any two states i and j of A , m CA are not connected -RRB- . We will assume that A has n states and B has m states and that their probability and path average matrices PA , ~ ~ PA , PB , and PB are known . Starting at state i of A , we find that the probability to exit on j of B after crossing once and only once from A to B is p eB -LRB- PBTBAPA -RRB- ji , where we have used matrix j notations . Let us consider a particular path from i in A to j in B crossing once and only once from A to B , with statistical weight -LRB- Bplk -RRB- pba -LRB- Apl k -RRB- . Its contribution to the average time to exit somewhere from the union of A and B is B A B A th j4b B a4i th j4b B p lk p ba a4i A pl k th j4b B j4b p lk p ba a4i A pl k A Exact Averaging over All Future Pathways . We start the discussion p lk p ba j4b a4i th a4i pl k -LSB- 4 -RSB- or in matrix form for any `` direct '' pathway from A to B P BT BAP A P B -RSB- T BAP A P BT BA PA , -LSB- 5 -RSB- PA ji t ~ ji PA t m : j4i CA m CA m th j4i j4i p lk , -LSB- 2 -RSB- where the th values are summed over all consecutive states h -LRB- from i to j included -RRB- along each pathway C A . Hence , the mean m time t iA to exit A from any state j of A starting from configuration A A eA ~ A i is ti j pj Pji -LCB- t -RCB- . However , in the context of the ECS algorithm , A the time lapse of interest is t ji , the mean time to exit A from a A A A ~ A ~ A particular state j , tji peAPji -LCB- t -RCB- peAPji Pji -LCB- t -RCB- Pji . j j The average of any path cumulative quantity of interest xi can ~ be similarly obtained by introducing the appropriate PA -LCB- x -RCB- matrix . In particular , the instantaneous efficiency of the algoA rithm is well reflected by the average pathway length ji between any two states of A , A ji A A which implies that applying the usual differentiation rules to any combination of probability matrices yields the correct combined path average matrices -LRB- defining -LSB- TBA -RSB- ij 0 for all i and j -RRB- . Note that this out-of-equilibrium calculation of path average quantities is reminiscent of the usual equilibrium calculation of thermal averages through differentiation of an appropriate partition function . Indeed , the probability matrices introduced here are `` partition functions '' over all pathways within a set of reference states . to the calculation of the probability and path average matrices PC ~ and PC for the union C of two disjoint sets A and B. Defining PAb PATAB and PBa PBTBA , we readily obtain the probability matrix PC as an infinite summation over all possible pathway loops between the sets A and B -LRB- I is the identity matrix -RRB- , PC QAA QBA QBB QAB Q AA Q AB , with Q BA Q BB I PAbPBa PBaPAb PAbPBa PBaPAb 2 2 The -LRB- n2 -RRB- Algorithm . With this result in mind , we can now return ~ ji PA A / P A , ji A -LSB- 3 -RSB- C ~ A where P ji -LCB- -RCB- m : j4i Cm 1 Cm plk , with Cm1 corresponding to the length of the pathway C A -LRB- 1 is added at each state along m A each pathway C A -RRB- . Hence , starting from state i , ji corresponds m to the average number of transitions that would have to be performed by the straightforward algorithm before exiting the A set A at state j . As expected , ji can be very large for a trapped dynamical system , which accounts for the efficiency of the present algorithm . Because the approach is exact , there is , however , no a priori requirement on the trapping condition of the states of A , and the algorithm can be used continuously . Similarly , the time average of any physical quantity yi -LRB- such as the pseudoknot proportion of an RNA molecule -RRB- can be calculated by introducing the appropriate time-weighted matrix A ~ PA -LCB- yt -RCB- . For instance , the time average energy E ji over all A ~ A pathways between any two states i and j of A is E ji P ji -LCB- Et -RCB- A A CA CA CA ~ ~ mp m : j 4 i mEhth P ji -LCB- t -RCB- , where P ji -LCB- Et -RCB- lk . The actual calculation of the probability and path average ~ matrices PC and PC over a set C of N states will be performed recursively in the next subsection . As an intermediate step , we first consider hereafter the unidirectional connection between two disjoint sets A and B. Let us hence introduce the transfer matrix TBA from set A to BA set B defined as T ji pji , where pji is the probability to make BA a transition from state i of A to state j of B -LRB- T ji 0 if i and j PA PB LA PA -LSB- 6 -RSB- LB PB PBaLAPA I PAbLBPB , -LSB- I PAbPBa -RSB- 1 and LB -LSB- I PBaPAb -RSB- 1 . where LA ~ ~ ~ ~ Defining also PAb PATAB and PBa PBTBA , we finally obtain ~ the path average matrix PC from simple `` differentiation '' of the `` partition function '' PC -LRB- Eq . 6 -RRB- ~ PC ~ QAA ~ QBA ~ QBB ~ QAB ~ and LB ~ ~ Q AA Q AB , with ~ ~ Q BA Q BB ~ LA PA ~ LB PB ~ LA PA ~ PBaLAPA ~ PAbLBPB ~ PBaLAPA -LSB- 7 -RSB- ~ PAbLBPB ~ LB PB ~ PAbPBa LA ~ PBa LA PA ~ PAb LB PB ~ LB PBaPAb ~ where , LA ~ LA PAbPBa ~ PBaPAb LB . Xayaphoummine et al. . 15312 www.pnas.org cgi doi 10.1073 pnas .2536430100 Q AA ~ Q AA L AP A , ~ LA PA I T AB Q BA ~ P B AB Q Q BA , Q BB -LSB- 8 -RSB- Results Performance of the ECS Algorithm . Before applying the ECS algorithm to investigate the prevalence of pseudoknots in RNA structures , we first focus on the efficacy of the approach by studying the net speed-up of the ECS algorithm with respect to the straightforward algorithm . As illustrated on Fig. 3 for a few natural and artificial sequences , there is an actual 101 - to 105-fold increase of the ratio `` simulated time over CPU time '' between ECS and straightforward algorithms -LRB- Fig. 3 , black lines -RRB- for RNA shorter than 150 nt . This improvement runs parallel to the A expected speed-up -LRB- Fig. 3 , gray lines -RRB- as predicted by ji -LRB- Eq . 3 -RRB- , as long as the number n of reference states is not too large -LRB- typically n 50 here -RRB- , so that the -LRB- n2 -RRB- update routines do not significantly increase the operation count as compared with the straightforward algorithm . Hence , the ECS algorithm is most efficient for small trapped systems -LRB- when the dynamics can be appropriately coarsegrained -RRB- , although a several-fold speed-up can still be expected with somewhat larger systems such as the 394-nt-long group I intron pictured in Fig. 4A . Alternatively , using this exact approach may also provide a controlled scheme to obtain approximate coarse-grained dynamics for larger systems . The C routines of the ECS algorithm are freely available on request . PNAS December 23 , 2003 vol . 100 no. 26 15313 ~ which eventually provides PA and PA by using the Sherman Morrison formula -LRB- 30 -RRB- to invert I TAB R QBA , PA LA 1 Q AA I Q AB T BA AA Q , Q BB -LSB- 9 -RSB- ~ B ~ PA I I Q AB T Q BB BA ~ Q AA P Q AB Q BA Q BB -LSB- 10 -RSB- T AB Q BA . 1 T AB Q BA Hence , the single state B can be removed from the set of reference C in -LRB- n2 -RRB- operations to yield the updated probability ~ and path average matrices PA and PA. . Note , however , that this continuous updating procedure , using alternatively Eqs . 6 and 7 and Eqs . 9 and 10 in succession , is Xayaphoummine et al. . MATHEMATICS Eqs . 6 and 7 are valid for any sizes n and m of A and B. Hence ~ PC and PC can be calculated recursively starting from N isolated ~ states and 2N 1 1 matrices Pi -LSB- 1 -RSB- and Pi -LCB- x -RCB- -LSB- xi -RSB- , with i 1 , N , where xi is the value of the feature of interest in state i. Clustering those states 2 by 2 , then 4 by 4 , etc. , by using Eqs . 6 ~ and 7 finally yields PC and PC in -LRB- N3 -RRB- operations -LRB- i.e. , by matrix inversions and multiplications -RRB- . However , instead of recalculating everything back recursively from scratch each time the set of reference states is modified , it turns out to be much more efficient to update it continuously each time a single state is added . Indeed , Eqs . 6 and 7 can be calculated in -LRB- n2 -RRB- operations only , when m 1 and n N 1 , as we will show below . Naturally , a complete update also requires the removal of one `` old '' reference state each time a `` new '' one is added so as to keep a stationary number n of reference configurations . As we will see , this removal step can also be calculated in -LRB- n2 -RRB- operations only . The -LRB- n2 -RRB- - operation update of the reference set , which we now ~ outline , relies on the fact that TAB , PAb , and PAb are n 1 ~ matrices and that TBA , PBa , and PBa are 1 n matrices when m 1 and n N 1 -LRB- PB and LB are simple 1 1 matrices for a single state B -RRB- . Because we operate on vectors , the Sherman Morrison formula -LRB- 30 -RRB- can then be used to calculate the n n matrix LA -LSB- I PAb R PBa -RSB- 1 -LSB- I PAb R PBa -LRB- 1 PAb PBa -RRB- -RSB- . Hence , not only LA but also any matrix product LAM , where M is an n n matrix , can be evaluated in -LRB- n2 -RRB- operations -LSB- by first calculating PBaM followed by PAb R -LRB- PBaM -RRB- -RSB- . Noticing that the same ~ ~ reasoning applies for the n n matrices PAb R PBa and PAb R PBa provides a simple scheme to add a single reference state to A and ~ obtain matrices PC and PC in -LRB- n2 -RRB- operations by using Eqs . 6 and 7 . To achieve the reverse modification consisting in removing one state B from the reference set C , it is useful to first imagine ~ that the original PC and PC were obtained by the addition of the single state B to the n-configuration set A , as given by Eqs . 6 and 7 . Identifying row QBA , column QAB , and their intersection QBB corresponding to the single state B readily yields the vectors PAb QAB QBB , PBa TBA -LRB- as PB -LSB- 1 -RSB- -RRB- and , hence , the n n matrix A -RSB- 1 -LSB- L I PAb R PBa I -LRB- QAB R TBA -RRB- QBB . This gives the following relations between the known LA , TAB , TBA , QAA , QBB , ~ ~ ~ QBA , QAB , PB , and QAA and the unknown PA and PA , Fig. 3 . -LRB- A -RRB- Expected -LRB- gray lines -RRB- and actual -LRB- black lines -RRB- speed-up of the approach with respect to the straightforward algorithm -LRB- see main text -RRB- . i , Bistable molecule in Fig. 4C -LRB- with a combinatorial structure space of 37 possible helices -RRB- ; ii , 67-nt-long molecule with reverse sequence of the bistable molecule in Fig. 4C -LRB- 38 possible helices -RRB- . The -LRB- n2 -RRB- algorithm becomes unstable above 40 reference states in this case -LRB- see main text -RRB- . iii , Hepatitis delta virus ribozyme -LRB- Fig. 4B -RRB- -LRB- 84 possible helices -RRB- ; iv , average speed-up for random 100-nt-long RNA sequences with 50 % G C content ; v , group I intron ribozyme -LRB- Fig. 4A -RRB- -LRB- 894 possible helices -RRB- . -LRB- B -RRB- Net speed-up distribution among random 100-nt-long RNA sequences with 50 % G C content -LRB- iv in A -RRB- for a cluster of 40 reference states . expected to become numerically unstable after too many updates of the reference set . For 1 n 300 , we have usually found that A the small numerical drifts -LSB- as measured , e.g. , by iA -LRB- Ap eAP ji j j 1 -RRB- 2 0 -RSB- can simply be reset every nth update by recalculating ~ matrices PA and PA recursively from n isolated states in -LRB- n3 -RRB- operations so as to keep the overall -LRB- n2 -RRB- - operation count per update of the reference set . Another important issue is the choice of the state to be removed from the updated reference set . Although this choice , in principle , is arbitrary , the benefit of the algorithm strongly hinges on it -LRB- for instance , removing one of the most statistically visited reference states usually ruins the efficiency of the method -RRB- . We have found that a `` good choice '' is often the state j with the lowest `` exit frequency '' from the current state i -LSB- i.e. , A A 1 tj i minA -LRB- 1 t ji -RRB- -RSB- , but other choices may sometimes prove j more appropriate . BIOPHYSICS Fig. 4 . RNA structure prediction with the ECS algorithm . Structures were drawn by using RNAMOVIES software -LRB- 31 -RRB- adapted to visualize predicted pseudoknots . -LRB- A -RRB- The 394-base-long Tetrahymena group I intron . The lowest free-energy structure found shares 80 % base-pair identity with the known 3D structure , including the two main pseudoknots , P3 and P13 -LRB- 11 , 12 , 14 17 -RRB- . -LRB- B -RRB- The 88-base-long hepatitis delta virus ribozyme . Predicted structure shares 93 % base-pair identity with the known 3D structure , including the main pseudoknot , P2 -LRB- 21 -RRB- -LSB- but not the 2-bp-long P1 .1 -LRB- 13 -RRB- -RSB- . -LRB- C -RRB- The two structures of a bistable , 67-nt-long artificial RNA molecule . context of RNA-folding dynamics , the present approach can be used to evaluate time averages for a variety of physical features of interest , such as the free energy along the folding paths , the fraction of time particular helices are formed , the extension of an RNA molecule unfolding under mechanical force -LRB- 32 -RRB- , the end-to-end distance of a nascent RNA molecule during transcription , etc. . Here we report results on the prediction of pseudoknot prevalence in RNA structures . They have been obtained by performing several thousands of stochastic RNAfolding simulations including pseudoknots . As explained in Theory and Methods , the structural constraints between pseudoknot helices and unpaired connecting regions are modeled by using elementary polymer theory -LRB- Fig. 1C -RRB- -LRB- 21 -RRB- and added to the traditional base-pair stacking interactions and simple loops ' contributions -LRB- 7 -RRB- . We found that many pseudoknots can be predicted effectively with such a coarse-grained kinetic approach probing seconds to minutes folding time scales . No optimum `` final '' structure is actually predicted , as such , in this folding kinetic approach . Instead , low free-energy structures are visited repeatedly as helices stochastically form and break . Fig. 4A represents the lowest free-energy secondary structure found for the 394-nt-long Tetrahymena group I intron , which shows 80 % base-pair identity with the known 3D structure , including the two main pseudoknots , P3 and P13 -LRB- 11 , 12 , 14 17 -RRB- . A number of smaller known structures with pseudoknots are also compared with the 15314 www.pnas.org cgi doi 10.1073 pnas .2536430100 Pseudoknot Prediction and Prevalence in RNA Structures . In the Fig. 5 . Distribution of pseudoknot proportion among formed base pairs for 50-nt-long -LRB- A -RRB- , 100-nt-long -LRB- B -RRB- , and 150-nt-long -LRB- C -RRB- random sequences of increasing G C content . Projected lines correspond to the average pseudoknot proportion in 50-nt-long -LRB- blue -RRB- , 100-nt-long -LRB- red -RRB- , and 150-ntlong -LRB- green -RRB- random sequences . All three average curves are displayed -LRB- B Inset -RRB- . Open -LRB- and filled -RRB- symbols in C correspond to known -LRB- and predicted -RRB- pseudoknot proportions for the Tetrahymena group I intron -LSB- Fig. 4A -LRB- triangles -RRB- -RSB- and hepatitis delta virus ribozyme -LSB- Fig. 4B -LRB- 13 , 21 -RRB- -LRB- circles -RRB- -RSB- . lowest free-energy structures found with similar stochastic RNAfolding simulations in ref . 21 . In addition , to facilitate the study of folding dynamics for specific RNA sequences , we have set up an online RNA-folding server including pseudoknots -LRB- http : kinefold.u-strasbg . fr -RRB- . Beyond specific sequence predictions , we also investigated the general prevalence of pseudoknots by studying the `` typical '' proportion of pseudoknots in both random RNA sequences of increasing G C content -LRB- Fig. 5 -RRB- and in 150-nt-long mRNA fragments of the Escherichia coli and Saccharomyces cerevisiae genomes . The statistical analysis was done as follows : for each random and genomic sequence set , 100 1,000 sequences were sampled , and three independent folding trajectories were simulated for each of them by using the ECS algorithm . A minimum duration for each trajectory was determined so that 80 90 % of sequences visit the same free-energy minimum structures along their three independent trajectories . The time average proportion of pseudoknots was then evaluated , considering this fraction Xayaphoummine et al. . of sequences had likely reached equilibrium -LRB- including the 10 20 % of still unrelaxed sequences does not significantly affect global statistics -RRB- . In practice , slow folding relaxation limits extensive folding statistics to sequences up to 150 bases and 75 % G C content , although individual folding pathways can still be studied for molecules up to 250 400 bases depending on their specific G C contents . The results for 50-nt-long -LRB- Fig. 5A -RRB- , 100-nt-long -LRB- Fig. 5B -RRB- , and 150-nt-long -LRB- Fig. 5C -RRB- random sequences show , first , a broad distribution in pseudoknot proportion from a few percent of base pairs to 30 % for some G C-rich random sequences . Such a range is compatible with the various pseudoknot contents observed in different known structures -LRB- e.g. , see triangles and circles in Fig. 5C -RRB- . Second , the average proportion of pseudoknots -LRB- Fig. 5B Inset -RRB- slowly increases with G C content , because stronger -LRB- G C-rich -RRB- helices are more likely to compensate for the additional entropic cost of forming pseudoknots . Third , and perhaps more surprisingly , this average proportion of pseudoknots seems roughly independent of sequence length except for very short sequences with low G C content -LRB- Fig. 5B Inset -RRB- , in contradiction to a naive combinatorial argument . Fourth , we found that the cooperativity of secondary structure rearrangements amplifies the structural consequences of pseudoknot formation ; typically , a structure with 10 helices including 1 pseudoknot conserves not 9 but only 7 8 of its initial helices -LRB- whereas 2 3 new nested helices concomitantly form -RRB- if the single pseudoknot is excluded from the structure prediction . 1 . Waterman , M. S. -LRB- 1978 -RRB- Stud . Found . Comb . Adv. Math . Suppl . Stud . 1 , 167 212 . 2 . Nussinov , R. , Pieczenik , G. , Griggs , J. R. & Kleitman , D. J. -LRB- 1978 -RRB- SIAM J. Appl . Math . 35 , 68 82 . 3 . Nussinov , R. & Jacobson , A. B. -LRB- 1980 -RRB- Proc . Natl. Acad . Sci . USA 77 , 7826 7830 . 4 . Zuker , M. & Stiegler , P. -LRB- 1981 -RRB- Nucleic Acids Res . 9 , 133 148 . 5 . McCaskill , J. S. -LRB- 1990 -RRB- Biopolymers 29 , 1105 1119 . 6 . Hofacker , I. L. , Fontana , W. , Stadler , P. F. , Bonhoeffer , L. S. , Tacker , M. & Schuster , P. -LRB- 1994 -RRB- Monatsh . Chem . 125 , 167 188 . 7 . Mathews , D. H. , Sabina , J. , Zuker , M. & Turner , D. H. -LRB- 1999 -RRB- J. Mol . Biol . 288 , 911 940 . 8 . Higgs , P. G. -LRB- 2000 -RRB- Q. Rev. Biophys . 33 , 199 253 . 9 . Pleij , C. W. A. , Rietveld , K. & Bosch , L. -LRB- 1985 -RRB- Nucleic Acids Res . 13 , 1717 1731 . 10 . Tinoco , I. , Jr. -LRB- 1997 -RRB- Nucleic Acids Symp . Ser . 36 , 49 51 . 11 . Lehnert , V. , Jaeger , L. , Michel , F. & Westhof , E. -LRB- 1996 -RRB- Chem . Biol . 3 , 993 1009 . 12 . Zarrinkar , P. P. & Williamson , J. R. -LRB- 1996 -RRB- Nat . Struct . Biol . 3 , 432 438 . 13 . Ferre-D'Amare , A. R. , Zhou , K. & Doudna , J. A. -LRB- 1998 -RRB- Nature 395 , 567 574 . 14 . Sclavi , B. , Sullivan , M. , Chance , M. R. , Brenowitz , M. & Woodson , S. A. -LRB- 1998 -RRB- Science 279 , 1940 1943 . 15 . Treiber , D. K. , Root , M. S. , Zarrinkar , P. P. & Williamson , J. R. -LRB- 1998 -RRB- Science 279 , 1943 1946 . Thus , neglecting pseudoknots usually induces extended structural modifications beyond the sole pseudoknots themselves . We compared these results with the folding of 150-nt-long sections of mRNAs from the genomes of E. coli -LRB- 50 % G C content -RRB- and S. cerevisiae -LRB- yeast , 40 % G C content -RRB- . These genomes exhibit similar broad distributions of pseudoknots despite small differences due to G C content inhomogeneity and codon bias usage -LSB- pseudoknot proportions -LRB- mean SD -RRB- : E. coli , 15.5 6.5 % -LRB- versus 16.5 7.9 % for 50 % G C-rich random sequences -RRB- ; yeast , 14 6.6 % -LRB- versus 15 7.3 % for 40 % G C-rich random sequences -RRB- -RSB- . Hence , genomic sequences seem to have maintained a large potential for modulating the presence or absence of pseudoknots in their 3D structures . Overall , these results suggest that neglecting pseudoknots in RNA structure predictions is probably a stronger impediment than the small intrinsic inaccuracy of stacking energy parameters . In practice , combining simple structural models -LRB- Fig. 1C -RRB- and ECS simulations provides an effective approach to predict pseudoknots in RNA structures . We thank J. Baschenagel , D. Evers , D. Gautheret , R. Giegerich , W. Krauth , M. Mezard , R. Penner , E. Siggia , N. Socci , and E. Westhof for discussions and suggestions . H.I. acknowledges a stimulating 2-month visit at the Institute for Theoretical Physics -LRB- University of California , Santa Barbara -RRB- , where the ideas for this work originated . This work was supported by Action Concertee Incitative Grants PC25-01 and 2029 from Ministere de la Recherche , France . ` 16 . Pan , J. & Woodson , S. A. -LRB- 1999 -RRB- J. Mol . Biol . 294 , 955 965 . 17 . Russell , R. , Millet , I. 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R. & Pande , V. S. -LRB- 2001 -RRB- Phys . Rev. Lett . 86 , 4983 4987 . 28 . Porschke , D. -LRB- 1974 -RRB- Biophys . Chem . 1 , 381 386 . 29 . Krauth , W. & Pluchery , O. -LRB- 1994 -RRB- J. Phys . A Math . Gen. 27 , L715 . 30 . Press , W. H. , Teukolsky , S. A. , Veterling , W. T. & Flannery , B. P. -LRB- 1992 -RRB- Numerical Recipes -LRB- Cambridge Univ. . Press , Cambridge , U.K. -RRB- , 2nd Ed . 31 . Evers , D. & Giegerich , R. -LRB- 1999 -RRB- Bioinformatics 15 , 32 37 . 32 . Harlepp , S. , Marchal , T. , Robert , J. , Leger , J.-F. , Xayaphoummine , A. , Isambert , H. & Chatenay , D. -LRB- 2003 -RRB- Eur . Phys . J. E , in press . Xayaphoummine et al. . PNAS December 23 , 2003 vol . 100 no. 26 15315 MATHEMATICS BIOPHYSICS

O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O B O O O O B O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O B O B O B O O 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O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

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