Stability of the branching structure of an RNA molecule is an important
condition for its function. In this letter we show that the melting
thermodynamics of RNA molecules is very sensitive to their branching geometry
for the case of a molecule whose groundstate has the branching geometry of a
Cayley Tree and whose pairing interactions are described by the Go model.
Whereas RNA molecules with a linear geometry melt via a conventional continuous
phase transition with classical exponents, molecules with a Cayley Tree
geometry are found to have a free energy that seems smooth, at least within our
precision. Yet, we show analytically that this free energy in fact has a
mathematical singularity at the stability limit of the ordered structure. The
correlation length appears to diverge on the high-temperature side of this
singularity.Comment: 4 pages, 3 figure

B. Alberts

P. B. Moore

Ralf Bundschuh

Robijn Bruinsma

English

arXiv

In this Letter we show that the melting thermodynamics of RNA molecules is very sensitive to the branching geometry. We find that, when pairing interactions are described by a Gō model, unbranched RNA molecules with a linear geometry melt via a conventional continuous phase transition with classical exponents while RNA molecules with the branching geometry of a Cayley tree, with coordination number three, have a free energy that shows no thermodynamic singularity within numerical precision. Nevertheless, we provide an analytical proof that the free energy does have a mathematical singularity at the stability limit of the ordered structure. The correlation length appears to diverge but only on the high-temperature side of this singularity

Bundschuh, Ralf

Bruinsma, Robijn

KnowledgeBank at OSU

Melting of Branched RNA MoleculesRalf Bundschuh1 and Robijn Bruinsma21Department of Physics, Ohio State University, Columbus, Ohio 43210-1117, USA2Department of Physics and Astronomy, The University of California at Los Angeles, Los Angeles, California 90049, USA(Received 1 August 2007; published 7 April 2008)In this Letter we show that the melting thermodynamics of RNA molecules is very sensitive to thebranching geometry. We find that, when pairing interactions are described by a Go¯ model, unbranchedRNA molecules with a linear geometry melt via a conventional continuous phase transition with classicalexponents while RNA molecules with the branching geometry of a Cayley tree, with coordination numberthree, have a free energy that shows no thermodynamic singularity within numerical precision.Nevertheless, we provide an analytical proof that the free energy does have a mathematical singularityat the stability limit of the ordered structure. The correlation length appears to diverge but only on thehigh-temperature side of this singularity.DOI: 10.1103/PhysRevLett.100.148101 PACS numbers: 87.15.A, 64.60.F, 87.14.gn, 87.15.bdA fundamental principle of statistical mechanics statesthat phase transitions are not possible for one-dimensionalsystems unless long-range interactions are present. It thuscame as a surprise when Poland and Scheraga (PS) showed[1] that an infinite, linear molecule composed of twoflexible polymer strands bound together by a local attrac-tive interaction does undergo a true phase transition whenthe two strands separate. The required long-range correla-tions are due to the fact that the partition function of astrand separation ‘‘bubble’’ has a power-law dependenceon size [2]. This PS mechanism is encountered in thecontext of the denaturation of RNA molecules [3]. RNAmolecules usually operate in a single-stranded mode.Pairing between complementary bases of the strand pro-duces the ‘‘secondary structure’’ [4]: a treelike graph ofunpaired ‘‘bubbles’’ linked by paired double-helical seg-ments. The minimum-energy secondary structure can bepredicted from the primary sequence of nucleotides [5].Melting of the secondary structure of an RNA moleculeproduces a ‘‘molten-globule’’ state with the molecule fluc-tuating over a range of different secondary structures [6].In his pioneering paper of 1968 [7], de Gennes showed thatthe partition function GL of a large RNA moleculefluctuating over all possible secondary structures, withidentical pairing energies, has a power-law dependenceon size of the form zL0=L with   3=2. Subsequently,Bundschuh and Hwa [8] showed that if the ground statesecondary structure of an RNA molecule is a long, uniformhairpin, then the molecule undergoes a continuous phasetransition to the molten-globule state with the same melt-ing thermodynamics as that of the PS model.Actual RNA secondary structures have a heterogeneous,branched, treelike form. In this Letter we will discuss theeffects of branching on the melting thermodynamics. Theground state secondary structure will be assumed to be aCayley tree (see Fig. 1) with coordination number three.The RNA strand traces out the perimeter of the tree, start-ing and ending at the root of the tree, with each branch ofthe tree occupied by a complementary base pair. The totalsize of the molecule is indexed by the level k of the tree,which is related to the total sequence length Nk byNk  2k2  2 bases (a k  1 tree is here a three-armedstar with one base pair per arm). If one numbers the basesof the strand then the ‘‘designed’’ ground state will bedenoted by the set S  fi1; j1; i2; j2; . . . ; iM; jMg ofcomplementary pairs.The folding energy of the molecule will be described bya ‘‘Go¯ model’’ [9] where one assigns a binding energy ~"to a pair belonging to the set S and a weaker binding energy" if the pair does not belong to S (the ‘‘nonspecific’’binding energy). We will allow in general any secondarystructure provided it does not have circuits (or ‘‘pseudo-knots’’). In the Go¯ model, the finite temperature partitionFIG. 1 (color online). Single-stranded RNA molecule having abranched secondary structure that follows the outline of a Cayleytree with coordination number three. Nucleotides are schemati-cally indicated by circles, bonds between nucleotides and com-plementary pairing by solid lines around the perimeter or acrossthe structure, respectively. (a) Ground state structure with pairingrestricted to a complementary ‘‘native’’ pair for each branch ofthe Cayley tree. (b) In a molten-globule bubble (hatched) allpossible pairing interactions are permitted. (c) For a fixed set S0of specific base pairs (solid lines) the set CS0 is defined as allpossible sets of base pairs compatible with the base pairs in set S0(dashed lines are one example of such a set). Note that theadditional base pairs are constrained to the loops of the structureS0 so the sum over all possible base-pairing configurationsfactorizes into a factor from each such loop.PRL 100, 148101 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending11 APRIL 20080031-9007=08=100(14)=148101(4) 148101-1 © 2008 The American Physical Societyfunction is given by Zq; ~q;S  XS0S~qjS0jXS002CNSS0qjS00j; (1)where q  exp and ~q  exp~ are the Boltzmannweights of nonspecific and specific pairs, respectively. Thefirst sum in Eq. (1) is over all subsets S0 of S. Once such asubset S0 is fixed, CNSS0 denotes the collection of all setsS00 of nonspecifically paired bases such that the union S0 [S00 defines an acceptable, circuit-free secondary structurefor the molecule as a whole [see Fig. 1(c)]. The difficulty inevaluating the partition function resides in the fact that inthe second sum we cannot allow pairing between specificpairs that do not belong to S0. In order to simplify thesecond sum, we apply a variant of the binomial theorem tothe first sum over S0 by replacing it with a double sum overall possible ways to divide S0 into two parts S1 and S2whose union S1 [ S2 equals S0: XS0S~qjS0jfS0  XS1SXS2Sx1~q qjS1jqjS2jfS1 [ S2:(2)Here, Sx1 denotes the complement of S1, and fT is anyfunction defined on the set T. Replacing the sum over S0 inEq. (1) by a double sum over S1 and S2 gives Zq; ~q;S  XS1S~q qjS1j XS32CS1qjS3j; (3)where CS1 is defined in the same way as CNSS0, exceptthat the restriction excluding specific pairing has beenlifted. The specific pairs were included through the sumover S2 in Eq. (2). The statistical weightPS32CS1qjS3j nowcan be written as the product of the statistical weights forthe individual loops LS1 linking the clusters of specificpairs belonging to S1: Zq; ~q;S  XS1S~q qjS1j YfLS1gGLS1: (4)Here, GL / z0qL=L is the familiar ‘‘molten-globule’’partition function for a strand of length L, which dependsonly on q. The partition function can be viewed as a sumover all possible bubble configurations, with GL thepartition function of the bubble [10].Equation (4) has a form for which we can constructexplicit recursion relations. The first step is to removefrom the sum the open loop located at the root of the tree(see Fig. 1): Zq; ~q;S  XNk=2n0G2nWk; n; (5)with n the number of base pairs in the root bubble. Here,Wk; n is a restricted partition function, i.e., the partitionfunction of a molecule with n accessible bases in the openbubble at the root, but not including the configurations ofthe open bubble. We cut the tree into two equal sizedsubtrees with level index k 1. The number of accessiblebase pairs of the two subtrees together must add to n 1,as we removed one pair by the cutting operation. Becausewe permit no circuits, the restricted partition function of alevel k tree for n > 0 can be expressed in terms of a productof the restricted partition functions of two k 1 levelsubtrees: Wk; n  Xn1m0Wk 1; mWk 1; n 1m; (6)with Wk 1; m  0 if m> 2k  1. The n  0 case, i.e.,a tree with no bubble at the root, must be treated separately.Take the first complementary pair at the root of the tree outof the partition function, and then sum over all possiblesizes for the bubble that immediately follows this pair(including bubbles of zero size) and treat those bubblesas the bubble at the root of a new tree that can again be cutinto two equal parts in the same way as before. UsingEq. (4), we obtain a second recursion relation: Wk; 0  ~q q X2k1n10X2k1n20Wk 1; n1Wk 1; n2G2n1  n2	: (7)Equations (6) and (7) together constitute a complete setof recursion relations from which Wk; n can be obtainedby iterative solution. The initial conditions for the recur-sion relations are W1; 0  ~q q1 4q q2  2~q~q2	, W1; 1  ~q q2, W1; 2  2~q q, andW1; 3  1, as follows by inspection. We carried outthis iteration procedure numerically, up to level k  19,for different values of ~q  exp~ and for fixed q  4. InFig. 2 we show the second derivative of the free energy persite with respect to ~q, which effectively correspond to theheat capacity. As one increases the value of k, a maximumdevelops near ~q  80. However, within the numericalprecision, the free energy per site does not develop athermodynamic singularity in the large N limit. Thismust be contrasted with the Bundschuh and Hwa case,where the molecule had a uniform, linear ground state, inwhich case the heat capacity very clearly develops such asingularity already for much smaller system sizes (left-hand inset of Fig. 2).In order to examine subleading contributions to the freeenergy, i.e., terms that are small compared to the leadingterm proportional to N, we also computed the ‘‘pinchingfree energy’’ Fk=kBT ’ lnZk 1  2 lnZk: (8)For example, in a molten-globule phase the partition func-tion should have the asymptotic scaling form azN0 =N3=2for large N. The pinching free energy Fk=kBT ’ 32 k 2 ln2 lna then should have a linear dependencePRL 100, 148101 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending11 APRIL 2008148101-2on k, with slope 3=2. In an ordered phase, the partitionfunction should scale as azN0 for large N, in which caseFk should be a constant independent of k. The right-hand inset of Fig. 2 shows that, for ~q values up to 80, Fkindeed has a linear dependence on k, for large k, with aslope close to 3=2 ln2. This indicates that, for ~q valuesbelow 80, the tree is in the molten-globule phase. Since forthe corresponding case of a linear ground state the meltingpoint is as low as ~qc  18:4 for q  4:0, we are forced toconclude that branching has a powerful destabilizing effecton the ordered state.For smaller k values, the pinching free energy is aconstant, which indicates that the ordered ground statedominates over shorter length scales. The crossover pointseparating the two regimes can be interpreted as a corre-lation length  whose physical meaning would be that ofthe typical size of smaller ordered Cayley tree-type struc-tures imbedded in a larger molten-globule state. The valueof  increases with ~q and beyond ~q  80 it exceeds ourmaximum system size (N  106). A fit to a power law  /~qc  ~q produces a correlation length exponent  ’ 2:1and a critical ~qc  80.Can we really be sure that the ordered phase is thermo-dynamically stable at any finite temperature? The re-stricted partition function is expected to have the scalingform WN; n  wnzn0 in the ordered phase, with wnthe fraction of configurations that have an open bubble atthe root of size n. If we insert this ansatz into the recursionrelation Eq. (6), we obtain, for positive n, the followingfixed-point condition: wn  Xn1m0wmwn 1m: (9)This equation can be solved by applying the discreteLaplace transform w^z  P1m0wmzm. The solutionw^z  z2 z24  zw0qhas a branch cut starting at z 4w0, with w0 an undetermined constant. After applyingan inverse Laplace transform, one finds that wn actuallyhas the same scaling form as the partition function of amolten globule: wn / expfn ln1=4w0	gn3=2: (10)The mathematical origin of the n3=2 factor is here acombinatorial factor that reflects the different ways onecan partition the open bubble between the two subtrees. Wemay interpret  1= ln1=4w0	 as the characteristic sizeof a molten-globule bubble at the root of the tree in theordered phase. Numerical iteration of the recursion rela-tions for Wk; n for ~q  150 and q  4 were found to beconsistent with Eq. (10). If we now use WN; n  wnzN0in the remaining recursion relation Eq. (7), together withEq. (9), then one obtains the following self-consistencyrelation for the remaining unknown constant w0: w0  ~q q2iI 1zG^zw^1=z2dz; (11)where G^z is the discrete Laplace transform of GL [11].G^z has a branch cut that terminates at z  1 2 qp 2.The integration contour in Eq. (11) must run inside anannulus in the complex plane that surrounds the originpassing the real axis outside the branch cut of G^z butinside the branch cut of w^1=z that starts at z  1=4w0.That means that the contour integral only can be carried outas long as w0  1=41 2 qp 2. The partition functiondevelops a mathematical singularity when the two branchcuts merge. At that point, the partition function wn /1=4w0	n=n3=2 of the root bubble has the same form asthe partition function Gn / 1 2 qp 2	n=n3=2 for amolten globule of the same size. We thus identify w0 1=41 2 qp 2 as the stability limit of a finite temperatureordered phase. Note that the (low-temperature) correlationlength  1= ln1=4w0	 cannot diverge at the stabilitylimit. The critical value ~qc for ~q at the stability limit is noweasily obtained by noting that w0 is small compared to 1.Expanding the argument of the contour integral in powersof w0 leads toFIG. 2 (color online). Second derivative of the free energywith respect to the Boltzmann weight ~q of specifically pairedbases plotted as a function of ~q for different values of the level kof the Cayley tree ground state. The free energy was computednumerically from the recursion relations Eqs. (6) and (7) andexpressed in units of NkBT with N the sequence length of theRNA strand. The arrow denotes the location of the mathematicalsingularity associated with melting of the root of the Cayley tree.Left-hand inset: Same except that the ground state is a linearhairpin. A thermodynamic singularity develops near~qc  18:4 with mean-field critical exponents. Right-handinset: Numerically computed ‘‘pinching’’ free energy Fk[see Eq. (8)] versus the level k of the Cayley tree. For ~q largerthen 90, Fk=kBT is independent of k, consistent with theordered ground state. For ~q less than 20, Fk=kBT can be fittedby the scaling relation Fk=kBT ’ 32 k 2 ln2 lna asso-ciated with the molten-globule state. The crossover point be-tween these two regimes for intermediate values of ~q marks thesize of the ordered, correlated regions in the molten-globulestate. For ~q above 80, the size of these correlated regions exceedsthe system size.PRL 100, 148101 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending11 APRIL 2008148101-3 ~q q1  w0  21 qw02 51 6q 2q2w03     : (12)If Eq. (12) is combined with w0  1=41 2 qp 2 onefinds that for q  4 the singularity is located at ~qc  92:6.The numerically computed free energy per site exhibits nosingular dependence on ~q in that range (see Fig. 2). This isnot inconsistent because wn only contributes a sublead-ing term to the total free energy. On the other hand, thecorrelation length obtained from the pinching free energyappears to diverge near ~qc. We encountered, however,strong finite-size effects in the numerical solution of therecursion relations for ~q values in the range between 80 and90, which make it difficult to numerically explore thecritical properties in more detail.In summary, a branched RNA molecule in the form of aCayley tree does have a mathematical singularity in thefree energy at a temperature where the ordered ground statebecomes unstable. This singularity does not correspond toa conventional phase transition, however, as the numeri-cally computed specific heat does not exhibit an anomaly atthe singularity. On the high-temperature side, the numeri-cally computed correlation length appears to diverge yetthe mathematical singularity itself is not associated with adivergence of the correlation length. We conclude thatbranching (i) ‘‘smears out’’ the melting transition and(ii) destabilizes the ordered phase but without suppressingit altogether.Experimental studies comparing the melting character-istics of large, branched RNA molecules with that of linear,unbranched molecules that could probe this exotic form ofmelting would be very interesting. An important questionin this respect would be the role of excluded-volumeinteractions and of ‘‘tertiary’’ pairing interactions, i.e.,pairing interactions that introduce, for example, pseudo-knots. Excluded-volume interactions in general tend tosuppress thermal fluctuations and possibly could restorethe thermodynamic singularity in the free energy per sitethat was encountered for linear molecules. Tertiary inter-action could have the effect of turning a branched, second-ary template into a three-dimensional gel-like structure, inwhich case the transition to the molten-globule state couldresemble the melting transition of a bulk solid material.We would like to thank the Aspen Center for TheoreticalPhysics for its hospitality. R. B. would like to acknowledgesupport by the NSF under DMR Grant No. 0404507.[1] D. Poland and H. A. Scheraga, J. Chem. Phys. 45, 1464(1966).[2] The order of the phase transition depends delicately on therole of excluded-volume interactions.[3] See, e.g., B. Alberts et al., Molecular Biology of the Cell(Garland, New York, 1994); P. B. Moore in The RNAWorld, edited by T. T. Cech, J. F. Atkins, and R. F.Gesteland (Cold Spring Harbor Laboratory, Cold SpringHarbor, NY, 2005).[4] For a review, see P. G. Higgs, Q. Rev. Biophys. 33, 199(2000).[5] M. Zuker and P. Stiegler, Nucleic Acids Res. 9, 133(1981).[6] P. G. Higgs, J. Phys. I (France) 3, 43 (1993).[7] P. G. de Gennes, Biopolymers 6, 715 (1968).[8] R. Bundschuh and T. Hwa, Phys. Rev. Lett. 83, 1479(1999).[9] N. Go¯, J. Stat. Phys. 30, 413 (1983).[10] For an expanded version of the proof of Eq. (4), see R.Bundschuh, ‘‘Unified Approach to Partition Functions ofRNA Secondary Structures,’’ in Lecture Notes Series,Institute for Mathematical Sciences, National Universityof Singapore (Singapore University Press and WorldScientific, Singapore, to be published).[11] The exact form is G^z  z4q zp4q  zp  12  4qq  zp  12  4qq 	.PRL 100, 148101 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending11 APRIL 2008148101-4

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Melting of Branched RNA Molecules

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