Mechanically probing the folding pathway of single RNA molecules

We study theoretically the denaturation of single RNA molecules by mechanical stretching, focusing on signatures of the (un)folding pathway in molecular fluctuations. Our model describes the interactions between nucleotides by incorporating the experimentally determined free energy rules for RNA secondary structure, whereas exterior single-stranded regions are modeled as freely jointed chains. For exemplary RNA sequences (hairpins and the Tetrahymena thermophila group I intron), we compute the quasiequilibrium fluctuations in the end-to-end distance as the molecule is unfolded by pulling on opposite ends. Unlike the average quasiequilibrium force-extension curves, these fluctuations reveal clear signatures from the unfolding of individual structural elements. We find that the resolution of these signatures depends on the spring constant of the force-measuring device, with an optimal value intermediate between very rigid and very soft. We compare and relate our results to recent experiments by Liphardt et al. (2001).     

On the occurrence of the T-loop RNA folding motif in large RNA molecules

The T-loop RNA folding motif may be considered as a five-nucleotide motif composed of a U-turn flanked by a noncanonical base pair. It was recently proposed that the flanking noncanonical base pair is always a UA trans Watson-Crick/Hoogsteen base pair stacked on a Watson-Crick base pair on one side. Here we show that structural analysis of several large RNA molecules, including the recently solved crystal structure of the specificity domain of Bacillus subtilis RNase P, combined with sequence analysis, indicates a broader sequence consensus for the motif. Additionally, we show that the flanking base pair does not necessarily stack on a Watson-Crick base pair and the 3' terminus of the five-nucleotide motif is often followed by a sharp turn in the phosphate backbone rather than just a bulged base or bases.     

Force spectroscopy of single DNA and RNA molecules

Experiments in which single molecules of RNA and DNA are stretched, and the resulting force as a function of extension is measured have yielded new information about the physical, chemical and biological properties of these important molecules. The behavior of both single-stranded and double-stranded nucleic acids under changing solution conditions, such as ionic strength, pH and temperature, has been studied in detail. There has also been progress in using these techniques to study both the kinetics and equilibrium thermodynamics of DNA-protein interactions. These studies generate unique insights into the functions of these proteins in the cell.     

Practicality and time complexity of a sparsified RNA folding algorithm

Commonly used RNA folding programs compute the minimum free energy structure of a sequence under the pseudoknot exclusion constraint. They are based on Zuker's algorithm which runs in time O(n(3)). Recently, it has been claimed that RNA folding can be achieved in average time O(n(2)) using a sparsification technique. A proof of quadratic time complexity was based on the assumption that computational RNA folding obeys the "polymer-zeta property". Several variants of sparse RNA folding algorithms were later developed. Here, we present our own version, which is readily applicable to existing RNA folding programs, as it is extremely simple and does not require any new data structure. We applied it to the widely used Vienna RNAfold program, to create sibRNAfold, the first public sparsified version of a standard RNA folding program. To gain a better understanding of the time complexity of sparsified RNA folding in general, we carried out a thorough run time analysis with synthetic random sequences, both in the context of energy minimization and base pairing maximization. Contrary to previous claims, the asymptotic time complexity of a sparsified RNA folding algorithm using standard energy parameters remains O(n(3)) under a wide variety of conditions. Consistent with our run-time analysis, we found that RNA folding does not obey the "polymer-zeta property" as claimed previously. Yet, a basic version of a sparsified RNA folding algorithm provides 15- to 50-fold speed gain. Surprisingly, the same sparsification technique has a different effect when applied to base pairing optimization. There, its asymptotic running time complexity appears to be either quadratic or cubic depending on the base composition. The code used in this work is available at: .     

Counterion charge density determines the position and plasticity of RNA folding transition states

The self-assembly of RNA structure depends on the interactions of counterions with the RNA and with each other. Comparison of various polyamines showed that the tertiary structure of the Tetrahymena ribozyme is more stable when the counterions are small and highly charged. By monitoring the folding kinetics of the ribozyme as a function of polyamine concentration, we now find that the charge density of the counterions determines the positions of the folding transition states. The transition state ensemble (TSE) between U and N moves away from the native state as the counterion valence and charge density increase, as predicted by the Hammond postulate. The TSE is broader and less structured when the RNA is refolded in polyamines rather than Mg2+. That the charge density of the counterions determines the plasticity of the TSE demonstrates the importance of interactions among condensed counterions for the self-assembly of RNA structures. We propose that the major barrier to RNA folding is dominated by entropy changes when counterion charge density is low and enthalpy differences when it is high.     

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.     

An early transition state for folding of the P4-P6 RNA domain

Tertiary folding of the 160-nt P4-P6 domain of the Tetrahymena group I intron RNA involves burying of substantial surface area, providing a model for the folding of other large RNA domains involved in catalysis. Stopped-flow fluorescence was used to monitor the Mg2+-induced tertiary folding of pyrene-labeled P4-P6. At 35 degrees C with [Mg2+] approximately 10 mM, P4-P6 folds on the tens of milliseconds timescale with k(obs) = 15-31 s(-1). From these values, an activation free energy deltaG(double dagger) of approximately 8-16 kcal/mol is calculated, where the large range for deltaG(double dagger) arises from uncertainty in the pre-exponential factor relating k(obs) and delta G(double dagger). The folding rates of six mutant P4-P6 RNAs were measured and found to be similar to that of the wild-type RNA, in spite of significant thermodynamic destabilization or stabilization. The ratios of the kinetic and thermodynamic free energy changes phi = delta deltaG(double dagger)/delta deltaG(o') are approximately 0, implying a folding transition state in which most of the native-state tertiary contacts are not yet formed (an early folding transition state). The k(obs) depends on the Mg2+ concentration, and the initial slope of k(obs) versus [Mg2+] suggests that only approximately 1 Mg2+ ion is bound in the rate-limiting folding step. This is consistent with an early folding transition state, because folded P4-P6 binds many Mg2+ ions. The observation of a substantial deltaG(double dagger) despite an early folding transition state suggests that a simple two-state folding diagram for Mg2+-induced P4-P6 folding is incomplete. Our kinetic data are some of the first to provide quantitative values for an activation barrier and location of a transition state for tertiary folding of an RNA domain.     

Salt dependence of the elasticity and overstretching transition of single DNA molecules

As double-stranded DNA is stretched to its B-form contour length, models of polymer elasticity can describe the dramatic increase in measured force. When the molecule is stretched beyond this contour length, it shows a highly cooperative overstretching transition. We have measured the elasticity and overstretching transition as a function of monovalent salt concentration by stretching single DNA molecules in an optical tweezers apparatus. As the sodium ion concentration was decreased from 1000 to 2.57 mM, the persistence length of DNA increased from 46 to 59 nm, while the elastic stretch modulus remained approximately constant. These results are consistent with the model of Podgornik, et al. (2000, J. Chem. Phys. 113:9343-9350) using an effective DNA length per charge of 0.67 nm. As the monovalent salt concentration was decreased over the same range, the overstretching transition force decreased from 68 to 52 pN. This reduction in force is attributed to a decrease in the stability of the DNA double helix with decreasing salt concentration. Although, as was shown previously, the hydrogen bonds holding DNA strands in a helical conformation break as DNA is overstretched, these data indicate that both DNA strands remain close together during the transition.     

Stochastic dynamic study of optical transition properties of single GFP-like molecules

Due to high fluctuations and quantum uncertainty, the processes of single-molecules should be treated by stochastic methods. To study fluorescence time series and their statistical properties, we have applied two stochastic methods, one of which is an analytic method to study the off-time distributions of certain fluorescence transitions and the other is Gillespie’s method of stochastic simulations. These methods have been applied to study the optical transition properties of two single-molecule systems, GFPmut2 and a Dronpa-like molecule, to yield results in approximate agreement with experimental observations on these systems. Rigorous oscillatory time series of GFPmut2 before it unfolds in the presence of denaturants have not been obtained based on the stochastic method used, but, on the other hand, the stochastic treatment puts constraints on the conditions under which such oscillatory behavior is possible. Furthermore, a sensitivity analysis is carried out on GFPmut2 to assess the effects of transition rates on the observables, such as fluorescence intensities.     

Navigating the RNA folding landscape

Analysis of individual RNA folding reactions reveals that, as in proteins, cooperative interactions selectively drive RNA toward its biologically active, native conformation. This new work establishes a platform for future investigations of the physical principles underlying the assembly of large RNA enzymes.     

Two energetically disparate folding pathways of alpha-lytic protease share a single transition state

The Lysobacter enzymogenes alpha-lytic protease (alphaLP) is synthesized with a 166 amino acid pro region (Pro) that catalyzes the folding of the 198 amino acid protease into its native conformation. An extraordinary feature of this system is the very high energy barrier (DeltaG = 30 kcal mol-1) that effectively prevents alphaLP from folding in the absence of Pro (t1/2 = 1800 years). A pair of mutations has been isolated in the protease that completely suppresses the catalytic defect incurred in Pro by truncation of its last three amino acids. These mutations also accelerate the folding of alphaLP in the absence of Pro by 400-fold. An energetic analysis of the two folding reactions indicates that the mutations stabilize the transition states of both the catalyzed and uncatalyzed folding reactions by 3 kcal mol-1. This finding points to a single transition state for these two distinct and energetically disparate folding pathways, and raises the possibility that all alphaLP folding pathways share the same transition state.     

On the optimal folding of protein molecules

The process of formation of a globular structure by a long molecular chain has been examined. In this process, various regions of the chain interact with one another. We classify the contacts thus formed as “correct” and “erroneous” ones. The correct contacts are those characteristic of the final native globular structure. All other contacts can be treated as erroneous. It is demonstrated that globule formation may proceed actually without formation and subsequent decay of erroneous contacts. Our model permits avoiding examination of numerous erroneous variants inasmuch as the regions of the chain that form correct contacts enter “long-range” interactions that at the same time can be highly selective. The existence of interactions of this kind facilitates the mutual approach and interaction of just those regions of the chain that yield correct contacts. Based on database analysis, it is shown that the model is valid not only for abstract structures but also for real polypeptide chains capable of forming protein globules and helical fibrils.     

RNA folding and catalysis

Catalysis in RNA is intimately connected to the folding. The small nucleolytic ribozymes function by a nucleophilic attack of the 2-oxygen on the 3-phosphate, in an S_N2 mechanism. This requires an alignment of the 2-O, 3-P and 5-O, that does not occur in normal A-form RNA. It is therefore likely that structural distortion plays a major role in the enhancement of the reaction rate, facilitating the trajectory into the in-line transition state. Given the polyelectrolyte nature of nucleic acids, metal ions are critical to folding processes in RNA. We have shown that two small nucleolytic ribozymes, the hammerhead and hairpin ribozymes, undergo metal ion-induced folding processes. The hammerhead ribozyme folds in two stages, each of which is induced by the binding of a single structural ion. The first corresponds to the formation of the ribozyme scaffold, while the second is the formation of the catalytic core of the ribozyme. By contrast, the hairpin ribozyme undergoes a single folding event induced by the binding of at least two metal ions, and involves the close interaction between two internal loops to form the active ribozyme.This revised version was published online in October 2005 with corrections to the Cover Date.     

RNA folding and unfolding

Single-molecule studies of RNA folding and unfolding are providing impressive details of the intermediates that occur and their rates of interconversion. The folding and unfolding of RNA are controlled by varying the concentration of magnesium ions and measuring fluorescence energy transfer, or by applying force to the RNA and measuring the end-to-end distance. The hierarchical nature of RNA folding - first secondary structure, then tertiary structure - makes the process susceptible to analysis and prediction.