How RNA folds

Abstract

We describe the RNA folding problem and contrast it with the much more difficult protein folding problem. RNA has four similar monomer units, whereas proteins have 20 very different residues. The folding of RNA is hierarchical in that secondary structure is much more stable than tertiary folding. In RNA the two levels of folding (secondary and tertiary) can be experimentally separated by the presence or absence of Mg²⁺. Secondary structure can be predicted successfully from experimental thermodynamic data on secondary structure elements: helices, loops, and bulges. Tertiary interactions can then be added without much distortion of the secondary structure. These observations suggest a folding algorithm to predict the structure of an RNA from its sequence. However, to solve the RNA folding problem one needs thermodynamic data on tertiary structure interactions, and identification and characterization of metal-ion binding sites. These data, together with force versus extension measurements on single RNA molecules, should provide the information necessary to test and refine the proposed algorithm.

Introduction

The known biological functions of RNA continue to grow in number and to expand in scope. RNA has been transformed from a molecule with a minor role in protein synthesis, to an important player in all of molecular biology. Scarcely ten years ago, messenger RNA was thought to be a passive carrier of genetic information from DNA; its conformation random or irrelevant. Ribosomal RNA was believed to be a scaffold for the essential ribosomal proteins, and the transfer RNAs simply dull adapters that placed each amino acid in the location specified by the genetic code. Now we know that the ribosomal proteins mainly provide the scaffold that allows the ribosomal RNAs to catalyze peptide bond formation (Noller et al., 1992), and to perform their other functions; that transfer RNAs and their mimics are involved in replication, intron splicing, and translational regulation (Giegé et al., 1998); that the conformations of messenger RNAs, particularly at the 3′-and 5′-untranslated regions, determine the lifetimes of the RNAs and control the efficiency of translation (see, for example Spicher et al., 1998). Furthermore, it has been shown that stem-loops in the mRNA can bind to the protein product of the message to regulate its synthesis (McCarthy & Gualerzi, 1990), and that pseudoknots in retroviral mRNAs cause programmed frameshifts that produce the correct ratios of proteins required for viral propagation (Chamorro et al., 1992).

The importance of RNA has increased most rapidly since the discovery of catalytic RNA. The realization that RNA could be an enzyme has led to a multitude of studies of naturally occurring, and artificially evolved and selected ribozymes. It has become increasingly obvious that RNA-RNA, RNA-DNA and RNA-protein interaction are important in many different biological contexts. For example, there is recent evidence for the role of RNAs in memory at nerve synapses (for a review, see Lisman & Fallon, 1999).

A systematic search for non-coding RNAs in the human and other genomes is just beginning (Lowe & Eddy, 1999). There will certainly be many more unexpected RNA functions found.

The previous paragraphs have been written from an admittedly RNA-centric point of view, but they were intended to justify the effort to understand how RNA folds. We want to be able to interpret an RNA sequence in terms of its folded, three-dimensional functional form. Functional is emphasized because we do not necessarily need, or want, to know the coordinate of every atom in the structure to 1 Å resolution. The functional RNA may have regions where the conformations are not relevant. Some sequences may fold into a structure that is rigid and compact; other sequences may be designed to be flexible to maximize interaction with a particular ligand. We want to know the structure well enough to understand the function. This may mean that we need to know one part of the structure at atomic resolution, another part in terms of Watson-Crick base-pairs formed, and the rest of the molecule only approximately. One criterion for the importance of structure can be inferred from its constancy deduced from phylogenetic comparisons (Michel & Westhof, 1990). If base paired regions in an RNA molecule are conserved over many species, it is assumed that they are important in function. If only a single sequence is available for the RNA, secondary structure calculations can predict the base-paired regions. In viral RNAs the most stable calculated helices correlate with regions that are known to have important biological roles, such as internal ribosome entry sites (Palmenberg & Sgro, 1997).

Here we will try to systematize and generalize some of what is known about how RNA folds, and to suggest an algorithm to predict the three-dimensional structure of RNA from a knowledge of its sequence. For a review that describes the earlier work in this field, see Brion & Westhof (1997). Our goal is to encourage others to enter this relatively unpopulated field. If 10 % of protein fold researcher switched to RNA, the problem could be solved in one or two years. If the lack of competition in the field is not sufficiently enticing, we hope that by making sweeping statements about RNA folding we can at least goad people into proving us wrong.

The protein folding problem, the search for rules to predict the three-dimensional structures of proteins from their primary sequences, has attracted scientists for almost five decades. Structural prediction in protein folding is difficult for two main reasons. First, because it takes 20 different amino acid residues to build a protein molecule. Thus, the number of distinct interactions among these residues is large, depending not only on the nature of the residues (hydrophobic, hydrophilic, polar, etc.), but also on each residue’s detailed structure. Second, because the existence of the various secondary structural elements (α-helices, parallel and antiparallel β-sheets, β-turns, random coils, etc.) is contextual, i.e. these elements form and are stable in the context of the rest of the protein, but may not form when they are isolated in solution. The energies that stabilize secondary structural elements in proteins are comparable to the energies involved in tertiary interactions. Thus, the formation of secondary structure depends on the nature of the tertiary folding contacts, and vice versa. This implies that the contributions of secondary and tertiary interactions to the energetic stability of a protein are, in principle, not separable and, therefore, no simple rules may exist to predict the three-dimensional structure of the protein from its sequence.

By comparison, RNA folding is simpler. Only four nucleotides, each made of a base, a ribose, and a phosphate, are used as building blocks of the structure. The four bases are very much alike; there are two purines and two pyrimidines that differ only in the placement of carbonyl and amino groups and their interactions are either through hydrogen bonding or base stacking. Each ribose is a five-membered ring with only two main conformations. The phosphate groups with their long-range electrostatic interactions might be considered as providing theoretical difficulties not present in protein folding, but they actually simplify the RNA folding problem both theoretically and experimentally. Electrostatic interactions are well understood theoretically, and experimentally the effective charge on the phosphate groups can be controlled by means of the ionic strength of the solvent.

There are only four basic secondary structure elements in RNA (helices, loops, bulges, and junctions). The helices are A-form Watson-Crick duplexes; the loops, bulges and junctions are all non-Watson-Crick regions terminated by one or more helices. Finally, because the energies involved in the formation of secondary structure are larger than those involved in tertiary interactions, secondary structural elements can exist and be stable by themselves. Thus, the energetic contributions of the secondary and tertiary structural elements are separable, and it is possible to treat the energy of tertiary interactions as a perturbation on the energy stabilizing the secondary structure.

The different challenges posed by the protein and RNA folding problems can be traced to the distinct ways in which the information contained in the sequence of these two molecular species controls the secondary and tertiary structures. We can think of the information in the amino acid sequence of a protein as branching into secondary and tertiary structural elements and also flowing between these elements in both directions. In an RNA, in contrast, the information in the sequence flows linearly, and largely in one direction, first to the secondary and then to the tertiary structure. An RNA molecule can thus be thought of as possessing a hierarchical structure in which the primary sequence determines the secondary structure which, in turn, determines its tertiary folding, whose formation alters only minimally the secondary structure. A folding algorithm then is simply a description of the information flow from the primary to the tertiary structure.

The postulated hierarchical nature of the RNA structure is undoubtedly an oversimplification, but it furnishes the basis on which a predictive folding algorithm can be generated. First, the primary structure can be used to establish the secondary structure through the use of simple, robust rules of secondary folding. Next, the secondary structure can be used to predict the tertiary contacts in the structure, following again simple tertiary folding rules.

A useful algorithm to predict RNA folding need not tell us anything about the folding pathway of an RNA molecule; the hierarchical nature of the algorithm depends on the relative stabilities of the various structures, not the kinetics of folding. However, we think that the thermodynamics and kinetics of folding are directly linked in RNA. Consider an unfolded single strand of RNA at a temperature and ionic strength in which no base-pairs are stable. The single-stranded RNA is a flexible polynucleotide that has many different conformations. The temperature is now lowered slightly, or the ionic strength is raised slightly, so that base-pair formation is barely possible. Loops and bulges decrease the entropy of the single strand, so they form only if the free energy decrease of base pair formation more than balances the cost of loop closure. We thus expect to find some small amount of secondary structure involving the most stable G+C-rich helices. One or more hairpin loops have formed with the smallest, least destabilizing loops (tetraloops) favored. Larger hairpin loops can form if the loss of configurational entropy of closing the loop is compensated by increasing numbers of successive base-pairs. As the temperature is further lowered, or the ionic strength is raised further, increasing amounts of secondary structure form, including the entire range of loops, bulges and junctions. The structures formed at the higher temperatures do not necessarily survive at lower temperatures and some rearrangements can occur. Eventually, at low temperature or high ionic strength, the least stable structures involving tertiary interactions form.

We have described the hierarchical folding of the RNA as the conditions for folding are improved. Now let us consider the kinetics of folding when a single-stranded RNA is placed in a folding environment. Complementary base-pairs will collide randomly, but a single base-pair is never stable in aqueous solution. Hydrogen bonding provides at most 1 kcal mol⁻¹ of stabilization; this is not enough to counteract the entropy loss of even a hairpin loop of four nucleotides (ΔG₃₇°=+4.5 kcal mol⁻¹). This means that consecutive base-pairs must form to close a hairpin loop; this is the first step in folding. The rate of hairpin loop formation thus depends on the frequency of collision between complementary bases that initiate a sequence of base-pairs. This rate depends on the effective concentration of one base relative to another, i.e. on the effective relative volume that is accessible to the two bases. The effective relative volume depends, in turn, on the number of nucleotides separating the two bases, so small hairpin loops will form at the shortest times. Later the stems will grow forming internal loops and bulges, and eventually forming junctions of several stems. Tertiary interactions can occur among two, or more, secondary structural elements, but will almost always follow the formation of these elements. The explanation is that the kinetics of folding favors formation of small hairpin loops for the same reasons that make these loops thermodynamically stable. Small loops have their ends close together; this means that the ends collide more often and do not lose much entropy on closing. The main conclusion is that the folding of RNA is sequential as well as hierarchical. We use sequential here to describe the kinetics of formation of secondary structure before tertiary structure; we are not referring to the folding of RNA during its biological synthesis.

We recognize that folding does not always occur on the path towards the lowest free energy structure. Clearly, during folding, secondary contacts may transiently form that must be undone before other interactions occur that lead the molecule in the direction of the final global free energy minimum. Of course, some species can remain kinetically trapped in non-equilibrium states. The enthalpy necessary to break the non-equilibrium base-pairs determines the activation energy of the kinetic trap.

There is still substantial disagreement in the literature as to what constitutes secondary and what constitutes tertiary structures in RNA Chastain and Tinoco 1991, Moore 1999. The definitions we use here, and the logic for their choice, will become clear from the discussion.

Let us look in detail at the folding of a model RNA to form a hairpin tetraloop, the most abundant of the hairpin loops found in RNA (Woese et al., 1990). As an example, we show the formation of a tetraloop as a function of temperature and Na⁺ concentration in Figure 1. The melting temperature was measured Antao et al 1991, Antao and Tinoco 1992 at pH 7, 0.1 mM EDTA in 1 M NaCl, 10 mM phosphate buffer (T_m=67.7 °C), and in 10 mM phosphate buffer (T_m=60.1 °C), and extrapolated to the other salt concentrations. The broken lines indicate the region where the hairpin is partially formed (between 10 % and 90 %). The Figure illustrates several facts about RNA folding. The melting temperature is expected to vary logarithmically with salt concentration. The slope depends theoretically on the change in the number of ions bound on melting the structure (Cantor & Schimmel, 1980). Here we find a change in melting temperature of about 3.8 °C per tenfold change in Na⁺ concentration. For long RNA strands, and depending on the base composition of the duplex, values of from 8 to 20 °C per tenfold change in Na⁺ have been found (Steger et al., 1980). The transition region is broad, here about 20 °C; the width depends on the enthalpy of melting, the more stable the stem the sharper the melting. The ion concentration is specified in terms of a dialyzed concentration to emphasize the fact that the total ion concentration depends on the RNA concentration. Clearly, an RNA strand of ten nucleotides at 1.0 mM concentration will have a total concentration of 10 mM monovalent positive ions in the absence of any added salt.

Figure 1 shows the folding behavior of a particularly simple RNA molecule, but it illustrates the more general case. As the temperature of an RNA is decreased slowly, the single strand begins to form stem loops. At the higher temperatures (and depending on the salt concentration) there will be multiple species: unfolded, partly folded, and fully folded. Some stems will grow longer, but they are eventually interrupted by internal loops and bulges. There is competition among the growing stems; a kinetically favored loop may be replaced by a thermodynamically more favored one. Of course conformations formed first need not be replaced by more stable ones, the less stable form can be kinetically trapped if the cooling is too fast. Even if the solution is at thermodynamic equilibrium, the region between the broken lines in Figure 1 reminds us that more than one species may be present.

Figure 2(a) shows the P5abc region of the Tetrahymena thermophila group I self-splicing intron. The base-pairing (secondary structure) shown is based on NMR studies of the A-rich bulge (Luebke et al., 1997), the P5b stem loop (Kieft & Tinoco, 1997), and a truncated version of P5abc (Wu & Tinoco, 1998). NMR is particularly appropriate for learning secondary structure because the imino proton resonances are well separated from the rest of the spectrum, and one peak is seen for each A·U and each G·C pair. Each G·U pair shows two imino resonances. We expect that the P5abc RNA starts folding at the two GNRA (N is any nucleotide; R is a purine) tetraloops near nucleotides 25 and 45 in the Figure. These hairpin loops should form as the temperature is lowered to allow the first base-pairs to be stable; they should also form first kinetically when the temperature is dropped below the melting range. The tetraloop stems then grow and form a three-stem junction (nucleotides 10-56 in Figure 2(a)). The 5′ and 3′ ends have now been brought close together; they form the closing stem that produces the A-rich bulge and a single U-base bulge. A bulge is distinguished from an internal loop by continuous pairing of the bases flanking the bulge. It is possible, but very unlikely, that the 5′ and 3′ ends would pair first as the temperature is lowered. In the single strand the ends are far apart (which slows the kinetics of loop formation), and formation of the base-pair would greatly decrease the entropy of the RNA. Formation of the base-pair between the ends is thus kinetically and thermodynamically unfavored until most of the other pairs have already formed.

It may seem uneccessary to discuss the kinetics of folding if we are trying to predict only the final folded state. If the folded RNA is at equilibrium with respect to its conformation the folding path is irrelevant. However, knowledge of the folding path will make it easier to understand, or to predict, the final result. If the folded RNA is not at equilibrium, then the actual conformation can depend crucially on the path. We expect that most RNAs exist naturally in their thermodynamically most stable conformations; the free energy is a minimum. We can prepare RNA in their most stable conformation by slow cooling. Of course, we can also trap RNA in unstable forms by quick cooling, or by changing solvent at low temperature. Interactions with proteins, other RNAs, or any ligand can also change the conformation by stabilizing a higher free energy species, or catalyzing the refolding of a kinetically trapped species.

The program mfold (Zuker, 1989) caluclates secondary structures for RNAs in order of increasing free energy; the calculations take a few minutes for RNAs of a few hundred nucleotides. This program can be accessed through the internet at http://mfold1.wustl.edu/∼mfold/rna/form1.cgi. The free energies of stems and loops used by the program come from experimentally measured values on oligonucleotides obtained by Turner and co-workers (reviewed by Burkard et al., 1999). The main assumption is that the free energies of stems depend on nearest-neighbor sequences of base-pairs only, and that free energies of stems and loops are additive. The minimum free energy secondary structure for the P5abc sequence in Figure 2(a) was obtained in seconds by mfold. The calculated structure at 37 °C, 1 M NaCl is identical with the NMR-deduced one; it has a calculated free energy, ΔG°, of −27.7 kcal mol⁻¹. The RNA studied by NMR (56 nt) was a truncated version of P5abc with a shorter P5b stem and a shorter P5a terminal stem. However, the measured and calculated structures both have the same two tetraloops and the same five base-pair P5c stem. The A-rich bulge with its flanking stems is the same in both structures. The junctions are the same in both structures, except that the imino proton of A13·U53 is not seen in the NMR. This is not unusual for fraying base-pairs at the ends of stems; the terminal 5′G·C3′ pair is rarely seen in an NMR imino spectrum. Good agreement is usually found between thermodynamically based calculated and experimental secondary structures (Mathews et al., 1999).

Calculated structures with higher free energies are also available from the program mfold. They would correspond to less stable secondary structures if the thermodynamic prediction was correct. One can also think of them as alternate secondary structures that are just as likely to be correct as the minimum free energy one, because of errors in the parameters and assumptions in the calculation. The fifth most stable calculated structure (ΔG°=−26.4 kcal mol⁻¹) has another tetraloop instead of the A-rich bulge, and the three-stem junction has been changed to a four-stem junction (see Figure 2(b)). The calculated free energy is within 5 % of the minimum, well within errors in the calculation. The 19th most stable calculated structure (ΔG°=−22.0 kcal mol⁻¹) has the P5c stem loop changed. The tetraloop has been changed to a loop of five nucleotides, and the stem has only four base-pairs with a one-base bulge. The calculated free energy differs by 25 % from the minimum. If this secondary structure for P5abc is correct, the thermodynamic calculation is incorrect; K_eq for the less stable form relative to the most stable is 10⁻⁴. The 20 calculated lowest free energy conformations are minor variations of the three shown. The P5a and P5b stems are stable and do not vary, the less stable A-rich bulge and P5c region can vary significantly.

Many other RNA secondary structures have been determined (for a review, see Conn & Draper, 1998). The main structural aspects of the secondary folding not included in Figure 2 are the wide variety of non-Watson-Crick base-pairs (base-base mismatches) that have been found. Each base can form hydrogen bonds with every other base in many different orientations (Gesteland et al., 1999); many have been seen. Only the very common sheared G·A base-pairs are shown in Figure 2.

When P5abc is part of the entire ribozyme, or part of the P4-P6 domain of the ribozyme, the secondary structure seen is closest to the one with the least stable free energy (ΔG°=−22.0 kcal mol⁻¹) shown in Figure 2(c). The reason for this change is the formation of tertiary structure. The structure of P4-P6 was determined by X-ray diffraction in a crystal (Cate et al., 1996a); a schematic of the P5abc domain is shown in Figure 3. The five nucleotide A-rich bulge has been changed to two bulges, a four nucleotide bulge and a single guanine bulge. The guanine base is linked to the P5c stem by a bound Mg²⁺. The P5c stem loop is changed from the tetraloop to a pentaloop closed by only three base-pairs. Two new G·A pairs bound to Mg²⁺ are formed. Essentially the same base-pairing scheme seen in Figure 3 was found by phylogenetic analysis (Michel & Westhof, 1990) of many group I introns. Covariation in sequences with the same function, of bases that are Watson Crick complements indicates base-pairing. The secondary structure thus found corresponds to that present in the functional form i.e. the secondary structure in the presence of tertiary structure.

Comparison of Figure 2, Figure 3 shows one example of how the formation of tertiary structure can change the secondary structure. For P5abc the presence of Mg²⁺ and the presence of the rest of the ribozyme causes some of the less stable parts of the secondary structure to rearrange. Two adenine bases in the A-rich bulge interact with a loop in P4; the P5b tetraloop docks with its receptor in J4/J5; the P5c stem loop is connected to the surrounding RNA by Mg²⁺. These tertiary interactions not only change the three-dimensional arrangement of the RNA; they also change the base-pairing. There are many other examples where the base-pairing is not changed. In tRNA the base-pairing in the cloverleaf is not changed when the tRNA is in its folded, functional form. Tertiary folding in tRNA involves formation of base triples and loop-loop base-pairing, but the stems and loops in the secondary structure are not changed.

Positively charged ions have a large general effect on RNA folding because of the negative charge on each phosphate group; their effect is like that of ionic strength on polyelectrolytes. However, divalent ions affect the tertiary folding much more than the secondary structure; specific metal ion binding occurs. (For recent reviews with references to the earlier literature, see Draper and Misra 1998, Misra and Draper 1999.)

Divalent ion binding sites in RNA secondary structures seem to be pre-formed sites. This means that the site exists in the absence of the metal ion, and the structure of the site does not change significantly on binding the metal. Examples of these include G·U base-pairs in stem regions. A G·U base-pair has only hydrogen bond acceptors in the major groove: N7 and the carbonyl oxygen of G, and the 4-carbonyl oxygen of U. The A·U and G·C base-pairs have amino hydrogen bond donors from either cytosine or adenine bases in the major groove. Therefore G·U base-pairs, in particular runs of G·U pairs, are expected to be sites for binding solvated magnesium ions, Mg(H₂O)₆²⁺.

Cobalt hexammine and osmium hexammine ions in the crystal structure of the P4-P6 domain of the T. thermophila group I intron Cate and Doudna 1996, Cate et al 1997 first identified major groove binding sites for fully hydrated Mg²⁺ at tandem G·U base-pairs. Ion binding was studied in solution using Mn(H₂O)₆²⁺ (Allain & Varani, 1995), and Co(NH₃)₆³⁺ Colmenarejo and Tinoco 1999, Kieft and Tinoco 1997. The Mn²⁺ is paramagnetic and broadens the NMR resonances of nearby nuclei. The cobalt hexammine has 18 protons to provide intermolecular nuclear Overhauser effect (NOE) cross-peaks to precisely locate the RNA binding site. Addition of Mg²⁺ to the RNA causes changes in chemical shifts of the guanine and uracil imino protons at the binding site. These shifts not only confirm the location of the ion binding site, but they can be used to measure an equilibrium constant for the Mg²⁺ binding. Values of dissociation constants, K_d, in the millimolar range have been found Colmenarejo and Tinoco 1999, Kieft 1997 for tandem G·U base-pairs.

Guanine·uracil binding sites have been characterized in solution for the P1 helix (Allain & Varani, 1995), the P5 helix (Colmenarejo & Tinoco, 1999), and the P5b stem loop (Kieft & Tinoco, 1997) of the Tetrahymena group I intron ribozyme. These sites are pre-formed in the double helices of the secondary structure, but the sites are all near tertiary interactions in the ribozyme. They are secondary structure ion binding sites that facilitate tertiary structure formation. The G·U pair in the P1 helix is right next to the splice site; it is at the center of the catalytic reaction. The tandem G·U pairs in the P5 helix are part of the docking site for the P9 loop (Golden et al., 1998), and the tandem G·U pairs in the P5b stem loop are near the GAAA tetraloop that docks with its receptor in J6a/6b Cate et al., 1996b). Presumably the bound metal ions reduce the electrostatic repulsion of the phosphate groups from the different secondary structural elements that must come together in the tertiary structure. In fact, the adenine platform in the J6a/6b tetraloop receptor (a secondary structural element) has a specific binding site for a potassium ion (Basu et al., 1998) that further reduces the net negative potential at the tertiary interaction.

A crystal structure of a 5 S ribosomal RNA domain (Correll et al., 1997) found magnesium ions bound in the major groove of an internal loop with three non-Watson-Crick base-pairs. Hexahydrated Mg(H₂O)₆²⁺ was found, but pentahydrated magnesium ions were also seen, with the remaining metal coordination to phosphate oxygen atoms, or N7, or to carbonyl oxygen atoms.

We conclude that after the secondary structure forms, metal ions bind to the pre-formed sites; the metal ion is not required for the formation of the site. However, the presence of the ion in the site favors the formation of tertiary structure by reducing electrostatic repulsion of the phosphate ions.

Some metal ion binding sites are formed only after, or during, the formation of tertiary structure. For example, when a single strand at the end of a stem loop folds back to form a pseudoknot, a metal ion binding site can form. Such a site has been seen in the mouse mammary tumor virus pseudoknot that promotes the programmed frameshift vital to the propagation of this retrovirus (Gonzalez & Tinoco, 1999). The metal ion, either Mg(H₂O)₆²⁺ or Co(NH₃)₆³⁺, binds in a pocket between the major groove of the stem and the short loop that crosses the stem. The binding site is formed as the tertiary structure, the pseudoknot, is formed. Divalent ions are known to stabilize pseudoknots relative to their constituent hairpins Qiu et al 1996, Wyatt et al 1990. This does not mean that a pseudoknot forms only in the presence of divalent ions; high concentrations of univalent ions can replace the divalent ions. An analogous binding site can form when two hairpin loops pair to form kissing hairpins. The ion binding site has not been characterized yet, but Mg²⁺ is bound when the kissing complex is formed (Gregorian & Crothers, 1995). Complete base-pairing of the two loops requires tight turns in the loops that brings several phosphates very close together (Chang & Tinoco, 1997). They may produce the preferential divalent ion binding sites.

The crystal structure of the P4-P6 domain of the group I intron Cate and Doudna 1996, Cate et al 1997 have revealed a wide range of tertiary metal ion binding sites. Magnesium ions knit the tertiary structure of the P4-P6 domain together, and in so doing change the secondary structure. The A-rich bulge of P5abc is turned inside out when the tertiary structure forms. The isolated A-rich bulge (Luebke et al., 1997), and the A-rich bulge in P5abc without Mg²⁺ (Wu & Tinoco, 1998) has five nucleotides in the bulge (AAUAA) with the adenine bases stacked inside the bulge. In the crystal structure there are two Mg²⁺ bound to phosphate groups inside the bulge; the bases are on the outside with two adenine bases interacting with the P4 stem. The bulge now contains only four nucleotides (AAUA) because the terminal adenine base has replaced a G·U pair by an A·U and produced a single-base G-bulge. This guanine base is bonded by Mg²⁺ to the P5c stem. Two G·A base mismatches bound to Mg²⁺ occur in the three-stem junction (see Figure 3). The metal ions shown in the figure are called a magnesium ion core whose formation is necessary for the tertiary folding of the P4-P6 domain (Cate et al., 1997). Again, the binding sites only appear when the tertiary structure forms.

We thus distinguish two types of specifically-bound metal ions: (1) the metal ions that bind to pre-formed sites in secondary structure and do not change the structure of the site significantly; and (2) the metal ions that are part of the tertiary structure site; without the metal ion neither the site nor the tertiary structure exists.

The hierarchical and sequential folding of RNA greatly simplifies the prediction of the RNA folded state, or states, because it divides the problem in two. First, the possible base-pairing schemes of the molecule are calculated from its sequence to obtain the secondary structure. Next, the possible interactions between these base-paired structures are evaluated to obtain the functional, tertiary structure. This formal separation of the RNA folding problem into two parts has an experimental counterpart. Magnesium or other divalent ions preferentially stabilize tertiary structures; thus, the secondary structure of RNA can be characterized at low ionic strength in Na⁺, and then Mg²⁺ can be added to form the tertiary structure. The first part of the RNA folding problem is nearly solved; the second part is just beginning to be attacked.

As we have discussed in detail earlier, secondary structure can be predicted well using programs such as that by Zuker (1989). A range of secondary structures (helices, hairpin loops, internal loops, bulges, and junctions) is calculated as the first step of the folding algorithm. We favor the Zuker program ‘mfold’, but any method that gives many possible secondary structures consistent with the sequence is good. The results from the Zuker program depend on experimental thermodynamic data mainly from the Turner group (Mathews et al., 1999); as more data are obtained the predictions should improve. In whatever way we obtained possible secondary structures, there are extensive thermodynamic data that tell us how changing a secondary structure changes the free energy. We know how breaking base-pairs, or forming new ones, affects the thermodynamic stability of the molecule. We can thus assess which secondary structure elements can be changed without much cost in free energy. The kinetics of unfolding and refolding may also be important; the time scales can be estimated from the enthalpy changes involved.

In the second step of the algorithm, possible tertiary folds are formed from the various calculated secondary structures. We minimize the destabiliztion of secondary structure in forming the tertiary structure. Of course, we should also choose the most thermodynamically stable tertiary interactions, but thermodynamic data for these interactions are still very sparse. We will summarize the little that is known about tertiary interactions.

Figure 4 shows thermodynamic data for the formation of a secondary structure hairpin (the equilibrium involves both possible hairpins, but only one is shown in the Figure), followed by a tertiary structure pseudoknot in the presence and absence of Mg²⁺. In 200 mM NaCl, 10 mM Na phosphate, 100 μM EDTA (pH 6.4), the pseudoknot melts at 27.8 °C, and the hairpins melt at 63.2 °C (Gonzalez & Tinoco, 1999). Thus the pseudoknot is unstable at 37 °C, whereas the hairpins are still present above 60 °C. Adding micromolar amounts of Mg²⁺ stabilize both transitions, but the pseudoknot is preferentially stabilized, causing the two melting transitions to overlap. Even at a concentration of Mg²⁺ as low as 50 μM the hairpins already melt at over 90 °C so their thermodynamics cannot be analyzed, and the pseudoknot melts at 68 °C. At a Mg²⁺ concentration of 5 to 10 mM, typical of biochemical experiments, both transitions are at temperatures too high to measure. The figure gives the thermodynamic parameters for the transitions, and illustrates the small changes in ΔH° and ΔS° that can combine to produce changes in sign of ΔG°₃₇. Of course, to predict tertiary structure we need the thermodynamic data not for one pseudoknot, but for pseudoknot formation as a function of stem sequence and length, and as a function of loop lengths.

Kissing hairpins, or hairpin loop-hairpin loop complexes, were studied by Gregorian & Crothers (1995) in 50 mM Na⁺, 5 mM Mg²⁺. Four complexes with hairpin loops of complementary sequences of seven nucleotides were melted. Although the four complexes could all form three G·C base-pairs and four A·U base-pairs in the loop-loop helix, the melting temperatures varied from 32 °C to 59 °C. Kissing complexes with hairpin loops of five to seven nucleotides allow the maximum number of base pairs to form because of the 11 base-pairs per turn in an A-form helix. We thus need to characterize the sequence dependence of this interaction.

Non-Watson-Crick tertiary interactions include base-base mismatches, base-base triples formed in the major groove of the RNA, and base-nucleoside triples formed in the minor groove through hydrogen bonding to the 2′-hydroxyl of ribose-ribose zippers. Other loop-loop, loop-helix, and helix-helix interactions can occur. Many of these interactions have been seen in the group I intron ribozyme Cate et al 1996a, Golden et al 1998, but thermodynamic data that can be assigned to specific interactions is not extensive. Replacement of ribose with its 2′-hydroxyl group by deoxyribose indicates a ΔG°₃₇ of about −1 kcal mol⁻¹ per ribose-base interaction in the minor groove Bervilacqua and Turner 1991, Narlikar et al 1997. The knowledge about secondary structure thermodynamics and the lack of knowledge about tertiary structure is illustrated in Table 1.

The ability to predict tertiary folding awaits more thermodynamic data, but metal ion binding sites in a secondary structure can suggest the locations of tertiary interactions. Knowledge of the role of Mg²⁺ in tertiary structure formation will be very valuable.

At present, atomic coordinates can be obtained only from experimental measurements; we do not attempt to predict them. X-ray diffraction from crystals gives the highest accuracy, but the structure obtained is not always the desired structure (Holbrook & Kim, 1997). Tetraloops in solution tend to crystallize as double strands with internal loops Baeyens et al 1996, Holbrook et al 1991. A DNA enzyme with its RNA substrate crystallized as a four-stranded cruciform (Nowakowski et al., 1999). NMR provides atomic coordinates in solution for rigid parts of the RNA, and leads to ranges of coordinates for flexible parts. The uncertainty in coordinates may be closer to the truth in representing a flexible molecule in solution, than the coordinates obtained from a crystal. However, NMR is limited in the size of the RNA that can be analyzed at atomic resolution. The accessible size is continually increasing; at present the upper limit is about 100 nt. Bound metal ions and their hydration shells, as well as bound water molecules, can be seen in crystal structures. Ion binding sites can be identified in solution by NMR, and the thermodynamics of ion binding can be measured.

Prediction of correct atomic coordinates for RNA from sequence alone, i.e. without experimental data input, is very difficult (for a review, see Auffinger & Westhof, 1998). Moleculer mechanics or molecular dynamics calculations need experimental data to guide the calculation toward the the correct free energy minimum. Calculations that sort through possible local conformations (Gautheret et al., 1993) also need experimental guidance to select the right structure. Once an atomic resolution structure is obtained, electrostatic potentials can be calculated to suggest likely ion binding sites and to provide an estimate of the electrostatic contributions to the free energy as a function of conformation (Hermann & Westhof, 1998). The free energies required to dehydrate metal ions can be used to predict whether direct coordination, or outer shell coordination, of the metal ion will occur (Draper & Misra, 1998).