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«Abstract The rugged nature of the RNA folding landscape is determined by a number of conflicting interactions like repulsive electrostatic potential ...»

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Chapter 2

Theory of RNA Folding: From Hairpins

to Ribozymes

D. Thirumalai(*) and Changbong Hyeon


The rugged nature of the RNA folding landscape is determined by a number

of conflicting interactions like repulsive electrostatic potential between the charges on

the phosphate groups, constraints due to loop entropy, base stacking, and hydrogen

bonding that operate on various length scales. As a result the kinetics of self-assembly

of RNA is complex, but can be easily modulated by varying the concentrations, sizes, and shapes of the counterions. Here, we provide a theoretical description of RNA folding that is rooted in the energy landscape perspective and polyelectrolyte theory.

A consequence of the rugged folding landscape is that, self-assembly of RNA into compact three-dimensional structures occurs by parallel routes, and is best described by the kinetic partitioning mechanism (KPM). According to KPM one fraction of mol- ecules (Φ) folds rapidly while the remaining gets trapped in one of several competing basins of attraction. The partition factor Φ can be altered by point mutations as well as by changing the initial conditions such as ion concentration, size and valence of ions.

We show that even hairpin formation, either by temperature or force quench, captures much of the features of folding of large RNA molecules. Despite the complexity of the folding process, we show that the KPM concepts from polyelectrolyte theory, and charge density of ions can be used to explain the stability, pathways and their diversity, and the plasticity of the transition state ensemble of RNA self-assembly.

2.1 Introduction The landmark discovery that RNA molecules are ribozymes (RNA enzymes) (Guerriertakada et al. 1983; Kruger et al. 1982) has triggered an intense effort to decipher their folding mechanisms. In the intervening years an increasing repertoire of cellular functions has been associated with RNA (Doudna and Cech 2002). These D. Thirumalai Department of Chemistry and Biochemistry, University of Maryland, College Park, College Park, MD 20742, USA e-mail: thirum@umd.edu N.G. Walter et al. (eds.) Non-Protein Coding RNAs 27 doi: 10.1007/978-3-540-70840-7_2, © Springer-Verlag Berlin Heidelberg 2009 28 D. Thirumalai, C. Hyeon include their role in replication, translational regulation, viral propagation etc.

Moreover, interactions of RNA with each other and with DNA and proteins are vital in many biological processes. Even, the central chemical activity of ribosomes, namely, the formation of the peptide bond in the biosynthesis of polypeptide chains by ribosomes near the peptidyl transfer center, involves only RNA, leading many to suggest that ribosomes are ribozymes (Nissen et al. 2000; Yusupov et al. 2001). The appreciation that RNA molecules play a major role in a number of cellular functions has made it important to establish the structure – function relationship. Thus, the need to understand, at the molecular level the ribozyme activity, inevitably leads to the question: How do RNA molecules fold?

In little over a decade great success has been achieved in an attempt to answer this question because of progress on a number of fronts. The number of experimentally determined high resolution RNA structures (Ban et al. 2000; Cate et al. 1996; Nissen et al. 2000; Yusupov et al. 2001) continues to increase which has enabled us to understand the interactions that stabilize the folded states. Single molecule (Ma et al. 2006;

Onoa et al. 2003; Russell et al. 2002b; Woodside et al. 2006; Zhuang et al. 2000) and ensemble experiments (Zarrinkar and Williamson 1994; Koculi et al. 2006; Pan et al.

1999) using a variety of biophysical methods combined with theoretical techniques (Thirumalai and Woodson 1996; Thirumalai and Hyeon 2005) have led to a conceptual framework for predicting various processes by which RNA molecules fold.

There are two aspects to RNA folding. The first is the prediction of the folded structures from sequence (Hofacker 2003; Zuker and Stiegler 1981). The second problem concerns the mechanisms by which assembly of the three dimensional functionally competent structure forms, start from the unfolded conformations. In this chapter we describe the folding mechanisms from the energy landscape perspective with focus on the polyelectrolyte aspects of RNA.

At a first glance it might appear that the RNA folding problem should be simple at least in comparison to the better investigated problem of protein folding (Tinoco and Bustamante 1999). However, there are several reasons why RNA folding is a difficult problem.

1. The building blocks of RNA are the four nucleotides each with a base, ribose, and phosphate groups. The bases (two purines and two pyrimidines), that are chemically similar, interact with each other either through hydrogen bonding or base stacking. The secondary structural elements (helices, loops, bulges) are independently stable which gives the impression that the three dimensional assembly is built much the same way as complicated architecture using prefabricated building blocks. However, the difficulty arises not only because of the chemical similarity of the nucleotides but also due to the polyelectrolyte nature arising from the charged phosphate groups.

2. The bases, their ability to form hydrogen bonds through Watson–Crick (WC) pairing withstanding, are all hydrophobic. The uniformity of the hydrophilic backbone along with lack of diversity in the bases make RNA closer to a “homopolymer” than polypeptide chains (Thirumalai and Hyeon 2005). The “homopolymer” nature of nucleic acids results in RNA structures being able to adopt alternate structures i.e., the stability gap between the folded and the other 2 Theory of RNA Folding: From Hairpins to Ribozymes 29 Fig. 2.1 View of the states of RNA as a free energy spectrum. The conformations in the NBA are separated from those in the competing basins of attraction (CBA) by the stability gap Δ. The structures in the CBA, while misfolded, can have many native-like features. Rapid folding without long pauses in the CBAs is likely if Δ/kBT 1. Figure adapted from (Guo et al. 1992) misfolded structures is not large (Fig. 2.1). As a result, the energy landscape of RNA, even at the secondary structural level, is rugged containing many metastable conformations that serve as kinetic traps.

3. At some level, WC base pairing does simplify the prediction of RNA secondary structures. However, not all nucleotides are engaged in WC base pairing.

Analysis of RNA secondary structures shows that the number of base-pairs (NBP) varies with sequence length N as NBP = 0.27 × N. The linear growth of NBP with N with slope 0.5 is expected if all the nucleotides are engaged in Watson–Crick base pairings. However, the slope is only 0.27 (Dima et al. 2005). This shows that 46% of the sequence, which is computed using NBP/N ≈ (1 − x)/2, constitute non-pairing regions such as bulges, loops, dangling ends, and other motifs. The bulges and loops are important structural elements that glue the independent helices together to make the RNA structures compact.

4. Finally, the folding mechanisms can be greatly altered by changing the nature of counterions which makes it necessary to consider explicitly the polyelectrolyte nature of RNA. In particular, the important role of valence, shape and size of the counterions (Koculi et al. 2004, 2006, 2007) in modulating the secondary structures and possibly altering them during the course of tertiary structure formation, are difficult to predict (Chauhan and Woodson 2008; Thirumalai 1998; Wu and Tinoco 1998). The varying flexibilities of different regions of RNA, the homopolymer character of the building blocks, the key role of counterions in the folding process, and the presence of alternate structures render RNA folding a challenging problem.

2.2 Structural Characteristics of RNA

–  –  –

the RNA native structures available in the Protein Data Bank (PDB) can be used to infer the general characteristics of the shapes and flexibility of folded RNA.

Native Structures are Compact: If RNA structures are compact then their volumes are expected to scale as V ∼ RG3 ∼ a3N, where RG is the radius of gyration, a is an effective monomer length. More generally, Flory showed that RG ∼ aN v where the Flory exponent v = 1/3 for maximally compact structures, v = 1/2 for polymers in Θcondition, and v = 3/5 for flexible polymers in good solvents. As RNA is a polyelectrolyte valence, shape, and concentration (C) of counterions can alter solvent quality, and hence RG. At low C, RNA is expanded and the transition to a compact structure occurs only when C exceeds the midpoint of the unfolded to folded transition.

Computation of the sizes of RNA structures using the PDB coordinates reveals that RG, follows the Flory scaling law, namely, RG = aNN1/3 Å (Hyeon et al. 2006).

The pre-factor, aN = 5.5 Å, corresponds approximately to the average distance between the phosphate groups (≈5.8 Å) along the ribose-phosphate backbone. For a given N, the approximate volume of RNA is larger than that of proteins whose RG scales as RG = 3.1 N1/3 Å (Dima and Thirumalai 2004; Hyeon et al. 2006). In other words, RNA molecules are more loosely packed than proteins, which are probably linked to their folding being dependent on accommodation of counterions to form compact structures. The difference is due to the larger size of the nucleotides compared to amino acids and the nature of interactions that stabilize the folded states of RNA and proteins.

Folded RNAs are Prolate Ellipsoids: Even though folded RNAs are compact, as assessed by RG, substantial deviations from sphericity have been found. When the shape of RNA molecules is characterized by the asphericity Δ and the shape parameters S that are computed using the eigenvalues of the moment of inertia tensor (Aronovitz and Nelson 1986; Hyeon et al. 2006), we find that a large fraction of folded RNA structures are aspherical and the distribution of S values shows that RNA molecules are prolate. The prolate ellipsoid shape of RNA renders their diffusion intrinsically anisotropic. The observed difference between shapes of RNAs and globular proteins is primarily due to the nature of interactions that stabilize the folded structures of RNA and proteins. Packing in RNA is not only determined by the favorable interactions between nucleotides but also by counter-ion mediated long-range interactions. The volume excluded by counterions affects packing, and consequently the shape of RNA structures.

Persistence Length of RNA shows Similarity to Polyelectrolytes. From the polymer perspective, flexibility of RNA is best assessed by its persistence length, lp, and its dependence on the changes in ionic strength. The overall compact RNA structure is formed by gluing together flexible (loops and bulges) and stiff helical regions.

Despite the potential variations in the flexibility it is useful to obtain estimates of the global lp. The total persistence length of RNA may be written as lp = lp0 + lpel where lp0 is the intrinsic persistence length and lpel is the electrostatic contribution. If RNA were a polyelectrolyte then lpel = lB /4κ2 A2 where the Bjerrum length lB = e2/4πεkBT (e is the unit of charge, ε is the dielectric constant, kB is the Boltzmann constant, and T is the temperature), for monovalent couterions κ 2 = 8πlBI (I is the ionic strength), and A is the average distance between the charges (Odijk 1977; Skolnick and Fixman 2 Theory of RNA Folding: From Hairpins to Ribozymes 31 1977). The lp values can be obtained from the distance distribution functions, which, for folded RNA molecules, can be directly computed using the PDB coordinates.

The persistence length of the folded RNA can be extracted by fitting, for r/RG 1, the distance distribution function P(r), which is computed using the coordinates of the folded RNA, to the wormlike chain model PWLC(r)∼exp{−1/(1−(lpr/RG2)2)} (Caliskan et al. 2005; Hyeon et al. 2006). The persistence length is scale-dependent and varies as lp = 1.5 N 0.33 Å (Hyeon et al. 2006). The dependence of lp on N implies that the average length of helices with stacks should increases as N grows.

In principle, as the counterion concentration decreases the changes in lp can be secured by obtaining P(r) using Small Angle X-ray Scattering (SAXS) experiments.

To date, SAXS data is available for only a few RNA molecules (Azoarcus ribozyme (Rangan et al. 2004), RNase P (Fang et al. 2002), and Tetrahymena ribozyme (Russell et al. 2002a) ). Surprisingly, analysis of P(r) for Azoarcus ribozyme and RNase P showed that the distance distribution function is well fit using PWLC(r) for the WLC model. As the concentration of Mg2+ and Na+ decreases lp increases (Caliskan et al. 2005) for Azoarcus ribozyme, lp ∼ 21 Å in the unfolded state, and lp ∼ 10 Å in the compact folded state. It is noteworthy that lp κ −2 which is predicted for polyelectrolytes (Odijk 1977; Skolnick and Fixman 1977) do not have globally compact folds like RNA molecules. Thus, not only does lp change dramatically as RNA folds, but it also exhibits the characteristics of polyelectrolytes especially at low ionic strength. Thus, how the polyelectrolyte problem is solved in RNA remains a key problem.

2.3 Rugged Folding Landscape and the Kinetic Partitioning Mechanism

The observed multiple folding routes and the associated heterogeneity of folding pathways can be anticipated from the energy landscape perspective (Thirumalai and Woodson 1996). The states for RNA (or for proteins for that matter) can be represented as a free energy spectrum (Guo et al. 1992). If the free energy gap (Δ in Fig. 2.1) is large, then trapping in one of the many Competing Basins of Attraction (CBAs) is not very probable. The presence of many alternate structures implies that the stability gap (especially when scaled by N) for RNA is not very large. As a result, RNA folding landscape is rugged (Fig. 2.2a), and is characterized by the presence of multiple minima that are separated by free energy barriers of varying heights.

The rugged nature of the energy landscape arises due to the presence of several competing interactions. Favorable hydrophobic stacking, and tertiary interactions favor chain compaction while the negatively charged interactions are better accommodated by extended structures. As a result RNA molecules are “frustrated” because not all interactions involving a given nucleotide can be simultaneously satisfied.

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