A New Computational Approach for Mechanical Folding Kinetics of RNA Hairpins

Based on an ensemble of kinetically accessible conformations, we propose a new analytical model for RNA folding kinetics. The model gives populational kinetics, kinetic rates, transition states, and pathways from the rate matrix. Applications of the new kinetic model to mechanical folding of RNA hairpins such as trans-activation-responsive RNA reveal distinct kinetic behaviors in different force regimes, from zero force to forces much stronger than the critical force for the folding-unfolding transition. In the absence of force or a low force, folding can be initiated (nucleated) at any position by forming the first base stack and there exist many pathways for the folding process. In contrast, for a higher force, the folding/unfolding would predominantly proceed along a single zipping/unzipping pathway. Studies for different hairpin-forming sequences indicate that depending on the nucleotide sequence, a kinetic intermediate can emerge in the low force regime but disappear in high force regime, and a new kinetic intermediate, which is absent in the low and high force regimes, can emerge in the medium force range. Variations of the force lead to changes in folding cooperativity and rate-limiting steps. The predicted network of pathways for trans-activation-responsive RNA suggests two parallel dominant pathways. The rate-limiting folding steps (at f = 8 pN) are the formation of specific basepairs that are 2-4 basepairs away from the loop. At a higher force (f = 11 pN), the folding rate is controlled by the formation of the bulge loop. The predicted rates and transition states are in good agreement with the experimental data for a broad force regime.     

Determination of thermodynamics and kinetics of RNA reactions by force

Single-molecule methods have made it possible to apply force to an individual RNA molecule. Two beads are attached to the RNA; one is on a micropipette, the other is in a laser trap. The force on the RNA and the distance between the beads are measured. Force can change the equilibrium and the rate of any reaction in which the product has a different extension from the reactant. This review describes use of laser tweezers to measure thermodynamics and kinetics of unfolding/refolding RNA. For a reversible reaction the work directly provides the free energy; for irreversible reactions the free energy is obtained from the distribution of work values. The rate constants for the folding and unfolding reactions can be measured by several methods. The effect of pulling rate on the distribution of force-unfolding values leads to rate constants for unfolding. Hopping of the RNA between folded and unfolded states at constant force provides both unfolding and folding rates. Force-jumps and force-drops, similar to the temperature jump method, provide direct measurement of reaction rates over a wide range of forces. The advantages of applying force and using single-molecule methods are discussed. These methods, for example, allow reactions to be studied in non-denaturing solvents at physiological temperatures; they also simplify analysis of kinetic mechanisms because only one intermediate at a time is present. Unfolding of RNA in biological cells by helicases, or ribosomes, has similarities to unfolding by force.     

Translocation of structured polynucleotides through nanopores

We investigate theoretically the translocation of structured RNA/DNA molecules through narrow pores which allow single but not double strands to pass. The unzipping of basepaired regions within the molecules presents significant kinetic barriers for the translocation process. We show that this circumstance may be exploited to determine the full basepairing pattern of polynucleotides, including RNA pseudoknots. The crucial requirement is that the translocation dynamics (i.e. the length of the translocated molecular segment) needs to be recorded as a function of time with a spatial resolution of a few nucleotides. This could be achieved, for instance, by applying a mechanical driving force for translocation and recording force-extension curves (FECs) with a device such as an atomic force microscope or optical tweezers. Our analysis suggests that, with this added spatial resolution, nanopores could be transformed into a powerful experimental tool to study the folding of nucleic acids.     

An RNA toolbox for single-molecule force spectroscopy studies

Precise, controllable single-molecule force spectroscopy studies of RNA and RNA-dependent processes have recently shed new light on the dynamics and pathways of RNA folding and RNA-enzyme interactions. A crucial component of this research is the design and assembly of an appropriate RNA construct. Such a construct is typically subject to several criteria. First, single-molecule force spectroscopy techniques often require an RNA construct that is longer than the RNA molecules used for bulk biochemical studies. Next, the incorporation of modified nucleotides into the RNA construct is required for its surface immobilization. In addition, RNA constructs for single-molecule studies are commonly assembled from different single-stranded RNA molecules, demanding good control of hybridization or ligation. Finally, precautions to prevent RNase- and divalent cation-dependent RNA digestion must be taken. The rather limited selection of molecular biology tools adapted to the manipulation of RNA molecules, as well as the sensitivity of RNA to degradation, make RNA construct preparation a challenging task. We briefly illustrate the types of single-molecule force spectroscopy experiments that can be performed on RNA, and then present an overview of the toolkit of molecular biology techniques at one's disposal for the assembly of such RNA constructs. Within this context, we evaluate the molecular biology protocols in terms of their effectiveness in producing long and stable RNA constructs.     

Applications of graph theory to enzyme kinetics and protein folding kinetics. Steady and non-steady-state systems

Graphic methods have proved to be very useful in enzyme kinetics, as reflected in both raising the efficiency of performing calculations and aiding in the analysis of catalytic mechanisms. The kinetic relations among protein folding states are very similar to those between enzyme-catalyzed species. Therefore, it should be equally useful to provide a visually intuitive relation between kinetic calculations and folding mechanisms for protein folding kinetics, as manifested by the graphic rules in enzyme kinetics. It can actually be anticipated that, due to increasing interest in protein folding, the graphic method will become an important tool in folding kinetics as well. Based on the recent progress made in graphic methods of enzyme kinetics, in this review four graphic rules are summarized, which can be used to deal with protein folding systems as well as enzyme-catalyzed systems. Rules 1-3 are established for deriving the kinetic equations for steady-state processes and Rule 4 for those in the case of non-steady-state processes. In comparison with conventional graphic methods, which can only be applied to a steady-state system, the current rules have the following advantages: (1) Complicated and tedious calculations can be greatly simplified. (2) A lot of wasted labor can be turned away. (3) Final results can be double-checked by a formula provided in each of the graphic rules. (4) Transient kinetic systems can also be treated. The mathematical proof of Rules 1-4 is given in appendices A-D, respectively.     

Principles of RNA compaction: insights from the equilibrium folding pathway of the P4-P6 RNA domain in monovalent cations

Counterions are required for RNA folding, and divalent metal ions such as Mg(2+) are often critical. To dissect the role of counterions, we have compared global and local folding of wild-type and mutant variants of P4-P6 RNA derived from the Tetrahymena group I ribozyme in monovalent and in divalent metal ions. A remarkably simple picture of the folding thermodynamics emerges. The equilibrium folding pathway in monovalent ions displays two phases. In the first phase, RNA molecules that are initially in an extended conformation enforced by charge-charge repulsion are relaxed by electrostatic screening to a state with increased flexibility but without formation of long-range tertiary contacts. At higher concentrations of monovalent ions, a state that is nearly identical to the native folded state in the presence of Mg(2+) is formed, with tertiary contacts that involve base and backbone interactions but without the subset of interactions that involve specific divalent metal ion-binding sites. The folding model derived from these and previous results provides a robust framework for understanding the equilibrium and kinetic folding of RNA.     

The GAAA tetraloop-receptor interaction contributes differentially to folding thermodynamics and kinetics for the P4-P6 RNA domain

Tetraloops with the generic sequence GNRA are commonly found in RNA secondary structure, and interactions of such tetraloops with "receptors" elsewhere in RNA play important roles in RNA structure and folding. However, the contributions of tetraloop-receptor interactions specifically to the kinetics of RNA tertiary folding, rather than the thermodynamics of maintaining tertiary structure once folded, have not been reported. Here we investigate the role of the key GAAA tetraloop-receptor motif in folding of the P4-P6 domain of the Tetrahymena group I intron RNA. Insertions of one or more nucleotides into the tetraloop significantly disrupt the thermodynamics of tertiary folding; single-nucleotide insertions shift the folding free energy by 2-4 kcal/mol (DeltaDeltaG(o)'). The folding kinetics of several modified P4-P6 domains were determined by stopped-flow fluorescence spectroscopy, using an internally incorporated pyrene residue as the chromophore. In contrast to the thermodynamic results, the kinetics of Mg(2+)-induced folding were barely affected by the tetraloop modifications, with a DeltaDeltaG(++) of 0.2-0.4 kcal/mol and a Phi value (ratio of the kinetic and thermodynamic contributions) of <0.1. These data indicate an early transition state for folding of P4-P6 with respect to forming the tetraloop-receptor contact, consistent with previous results for modifications elsewhere in P4-P6. We conclude that the GAAA tetraloop-receptor motif contributes little to the stabilization of the transition state for Mg(2+)-induced P4-P6 folding. Rather, the tetraloop-receptor motif acts to clamp the RNA once folding has occurred. This is the first report to correlate the kinetic and thermodynamic contributions of an important RNA tertiary motif, the GNRA tetraloop-receptor. The results are related to possible models for the Mg(2+)-induced folding of the P4-P6 RNA, including a model invoking rapid nonspecific electrostatic collapse.     

Understanding the Role of Three-Dimensional Topology in Determining the?Folding Intermediates of Group I Introns

Many RNA molecules exert their biological function only after folding to unique three-dimensional structures. For long, noncoding RNA molecules, the complexity of finding the native topology can be a major impediment to correct folding to the biologically active structure. An RNA molecule may fold to a near-native structure but not be able to continue to the correct structure due to a topological barrier such as crossed strands or incorrectly stacked helices. Achieving the native conformation thus requires unfolding and refolding, resulting in a long-lived intermediate. We investigate the role of topology in the folding of two phylogenetically related catalytic group I introns, the Twort and Azoarcus group I ribozymes. The kinetic models describing the Mg²⁺-mediated folding of these ribozymes were previously determined by time-resolved hydroxyl (⋅OH) radical footprinting. Two intermediates formed by parallel intermediates were resolved for each RNA. These data and analytical ultracentrifugation compaction analyses are used herein to constrain coarse-grained models of these folding intermediates as we investigate the role of nonnative topology in dictating the lifetime of the intermediates. Starting from an ensemble of unfolded conformations, we folded the RNA molecules by progressively adding native constraints to subdomains of the RNA defined by the ⋅OH time-progress curves to simulate folding through the different kinetic pathways. We find that nonnative topologies (arrangement of helices) occur frequently in the folding simulations despite using only native constraints to drive the reaction, and that the initial conformation, rather than the folding pathway, is the major determinant of whether the RNA adopts nonnative topology during folding. From these analyses we conclude that biases in the initial conformation likely determine the relative flux through parallel RNA folding pathways.     

Protein folding at single-molecule resolution

The protein folding reaction carries great significance for cellular function and hence continues to be the research focus of a large interdisciplinary protein science community. Single-molecule methods are providing new and powerful tools for dissecting the mechanisms of this complex process by virtue of their ability to provide views of protein structure and dynamics without associated ensemble averaging. This review briefly introduces common FRET and force methods, and then explores several areas of protein folding where single-molecule experiments have yielded insights. These include exciting new information about folding landscapes, dynamics, intermediates, unfolded ensembles, intrinsically disordered proteins, assisted folding and biomechanical unfolding. Emerging and future work is expected to include advances in single-molecule techniques aimed at such investigations, and increasing work on more complex systems from both the physics and biology standpoints, including folding and dynamics of systems of interacting proteins and of proteins in cells and organisms. This article is part of a Special Issue entitled: Protein Dynamics: Experimental and Computational Approaches.     

Time-resolved hydroxyl-radical footprinting of RNA using Fe(II)-EDTA

Chemical footprinting methods have been used extensively to probe the structures of biologically important RNAs at nucleotide resolution. One of these methods, hydroxyl-radical footprinting, has recently been employed to study the kinetics of RNA folding. Hydroxyl radicals can be generated by a number of different methods, including Fe(II)-EDTA complexes, synchrotron radiation, and peroxynitrous acid disproportionation. The latter two methods have been used for kinetic studies of RNA folding. We have taken advantage of rapid hydroxyl-radical generation by Fe(II)-EDTA-hydrogen peroxide solutions to develop a benchtop method to study folding kinetics of RNA complexes. This technique can be performed using commercially available chemicals, and can be used to accurately define RNA folding rate constants slower than 6 min(-1). Here we report the method and an example of time-resolved footprinting on the hairpin ribozyme, a small endoribonuclease and RNA ligase.     

Protein solubility and folding enhancement by interaction with RNA

While basic mechanisms of several major molecular chaperones are well understood, this machinery has been known to be involved in folding of only limited number of proteins inside the cells. Here, we report a chaperone type of protein folding facilitated by interaction with RNA. When an RNA-binding module is placed at the N-terminus of aggregation-prone target proteins, this module, upon binding with RNA, further promotes the solubility of passenger proteins, potentially leading to enhancement of proper protein folding. Studies on in vitro refolding in the presence of RNA, coexpression of RNA molecules in vivo and the mutants with impaired RNA binding ability suggests that RNA can exert chaperoning effect on their bound proteins. The results suggest that RNA binding could affect the overall kinetic network of protein folding pathway in favor of productive folding over off-pathway aggregation. In addition, the RNA binding-mediated solubility enhancement is extremely robust for increasing soluble yield of passenger proteins and could be usefully implemented for high-throughput protein expression for functional and structural genomic research initiatives. The RNA-mediated chaperone type presented here would give new insights into de novo folding in vivo.     

Predicting Secondary Structural Folding Kinetics for Nucleic Acids

We report a new computational approach to the prediction of RNA secondary structure folding kinetics. In this approach, each elementary kinetic step is represented as the transformation between two secondary structures that differ by a helix. Based on the free energy landscape analysis, we identify three types of dominant pathways and the rate constants for the kinetic steps: 1), formation; 2), disruption of a helix stem; and 3), helix formation with concomitant partial melting of a competing (incompatible) helix. The third pathway, termed the tunneling pathway, is the low-barrier dominant pathway for the conversion between two incompatible helices. Comparisons with experimental data indicate that this new method is quite reliable in predicting the kinetics for RNA secondary structural folding and structural rearrangements. The approach presented here may provide a robust first step for further systematic development of a predictive theory for the folding kinetics for large RNAs.     

Slow folding kinetics of RNase P RNA.

Understanding the folding mechanisms of large, highly structured RNAs is important for understanding how these molecules carry out their function. Although models for the three-dimensional architecture of several large RNAs have been constructed, the process by which these structures are formed is only now beginning to be explored. The kinetic folding pathway of the Tetrahymena ribozyme involves multiple intermediates and both Mg2+-dependent and Mg2+-independent steps. To determine whether this general mechanism is representative of folding of other large RNAs, a study of RNase P RNA folding was undertaken. We show, using a kinetic oligonucleotide hybridization assay, that there is at least one slow step on the folding pathway of RNase P RNA, resulting in conformational changes in the P7 helix region on the minute timescale. Although this folding event requires the presence of Mg2+, the slow step itself does not involve Mg2+ binding. The P7 and P2 helix regions exhibit distinctly different folding behavior and ion dependence, implying that RNase P folding is likely to be a complex process. Furthermore, there are distinct similarities in the folding of RNase P RNA from both Bacillus subtilis and Escherichia coli, indicating that the folding pathway may also be conserved along with the final structure. The slow folding kinetics, Mg2+-independence of the rate, and existence of intermediates are basic features of the folding mechanism of the Tetrahymena group I intron that are also found in RNase P RNA, suggesting these may be general features of the folding of large RNAs.     

40 Probing single molecule RNA folding using temperature and force

Nucleic acids can be unfolded either by temperature, such as in UV melting, or by mechanical force using optical tweezers. In UV melting experiments, the folding free energy of nucleic acids at mesophilic temperatures are extrapolated from unfolding occurring at elevated temperatures. Additionally, single molecule unfolding experiments are typically performed only at room temperature, preventing calculation of changes in enthalpy and entropy. Here, we present temperature-controlled optical tweezers suitable for studying folding of single RNA molecules at physiological temperatures. Constant temperatures between 22 and 37?°C are maintained with an accuracy of 0.1?°C, whereas the optical tweezers display a spatial resolution of ～1?nm over the temperature range. Using this instrument, we measured the folding thermodynamics and kinetics of a 20-base-pair RNA hairpin by force-ramp and constant force experiments. Between 22 and 37?°C, the hairpin unfolds and refolds in a single step. Increasing temperature decreases the stability of the hairpin and thus decreases the force required to unfold it. The equilibrium force, at which unfolding and refolding rates are equal, drops ～1?pN as temperature increases every 5?°C. At each temperature, the folding energy can be quantified by reversible work done to unfold the RNA and from the equilibrium constant at constant forces. Over the experimental temperature range, the folding free energy of the hairpin depends linearly on temperature, indicating that ΔH is constant. The measured folding thermodynamics are further compared with the nearest neighbor calculations using Turner’s parameters of nucleic acid folding energetics.     

Biphasic folding kinetics of RNA pseudoknots and telomerase RNA activity

Using a combined master equation and kinetic cluster approach, we investigate RNA pseudoknot folding and unfolding kinetics. The energetic parameters are computed from a recently developed Vfold model for RNA secondary structure and pseudoknot folding thermodynamics. The folding kinetics theory is based on the complete conformational ensemble, including all the native-like and non-native states. The predicted folding and unfolding pathways, activation barriers, Arrhenius plots, and rate-limiting steps lead to several findings. First, for the PK5 pseudoknot, a misfolded 5' hairpin emerges as a stable kinetic trap in the folding process, and the detrapping from this misfolded state is the rate-limiting step for the overall folding process. The calculated rate constant and activation barrier agree well with the experimental data. Second, as an application of the model, we investigate the kinetic folding pathways for human telomerase RNA (hTR) pseudoknot. The predicted folding and unfolding pathways not only support the proposed role of conformational switch between hairpin and pseudoknot in hTR activity, but also reveal molecular mechanism for the conformational switch. Furthermore, for an experimentally studied hTR mutation, whose hairpin intermediate is destabilized, the model predicts a long-lived transient hairpin structure, and the switch between the transient hairpin intermediate and the native pseudoknot may be responsible for the observed hTR activity. Such finding would help resolve the apparent contradiction between the observed hTR activity and the absence of a stable hairpin.     

RNA and protein folding: common themes and variations

Visualizing the navigation of an ensemble of unfolded molecules through the bumpy energy landscape in search of the native state gives a pictorial view of biomolecular folding. This picture, when combined with concepts in polymer theory, provides a unified theory of RNA and protein folding. Just as for proteins, the major folding free energy barrier for RNA scales sublinearly with the number of nucleotides, which allows us to extract the elusive prefactor for RNA folding. Several folding scenarios can be anticipated by considering variations in the energy landscape that depend on sequence, native topology, and external conditions. RNA and protein folding mechanism can be described by the kinetic partitioning mechanism (KPM) according to which a fraction (Phi) of molecules reaches the native state directly, whereas the remaining fraction gets kinetically trapped in metastable conformations. For two-state folders Phi approximately 1. Molecular chaperones are recruited to assist protein folding whenever Phi is small. We show that the iterative annealing mechanism, introduced to describe chaperonin-mediated folding, can be generalized to understand protein-assisted RNA folding. The major differences between the folding of proteins and RNA arise in the early stages of folding. For RNA, folding can only begin after the polyelectrolyte problem is solved, whereas protein collapse requires burial of hydrophobic residues. Cross-fertilization of ideas between the two fields should lead to an understanding of how RNA and proteins solve their folding problems.