Force unfolding kinetics of RNA using optical tweezers. II. Modeling experiments

By exerting mechanical force, it is possible to unfold/refold RNA molecules one at a time. In a small range of forces, an RNA molecule can hop between the folded and the unfolded state with force-dependent kinetic rates. Here, we introduce a mesoscopic model to analyze the hopping kinetics of RNA hairpins in an optical tweezers setup. The model includes different elements of the experimental setup (beads, handles, and RNA sequence) and limitations of the instrument (time lag of the force-feedback mechanism and finite bandwidth of data acquisition). We investigated the influence of the instrument on the measured hopping rates. Results from the model are in good agreement with the experiments reported in the companion article. The comparison between theory and experiments allowed us to infer the values of the intrinsic molecular rates of the RNA hairpin alone and to search for the optimal experimental conditions to do the measurements. We conclude that the longest handles and softest traps that allow detection of the folding/unfolding signal (handles approximately 5-10 Kbp and traps approximately 0.03 pN/nm) represent the best conditions to obtain the intrinsic molecular rates. The methodology and rationale presented here can be applied to other experimental setups and other molecules.     

Force unfolding kinetics of RNA using optical tweezers. I. Effects of experimental variables on measured results

Experimental variables of optical tweezers instrumentation that affect RNA folding/unfolding kinetics were investigated. A model RNA hairpin, P5ab, was attached to two micron-sized beads through hybrid RNA/DNA handles; one bead was trapped by dual-beam lasers and the other was held by a micropipette. Several experimental variables were changed while measuring the unfolding/refolding kinetics, including handle lengths, trap stiffness, and modes of force applied to the molecule. In constant-force mode where the tension applied to the RNA was maintained through feedback control, the measured rate coefficients varied within 40% when the handle lengths were changed by 10-fold (1.1-10.2 Kbp); they increased by two- to threefold when the trap stiffness was lowered to one-third (from 0.1 to 0.035 pN/nm). In the passive mode, without feedback control and where the force applied to the RNA varied in response to the end-to-end distance change of the tether, the RNA hopped between a high-force folded-state and a low-force unfolded-state. In this mode, the rates increased up to twofold with longer handles or softer traps. Overall, the measured rates remained with the same order-of-magnitude over the wide range of conditions studied. In the companion article on pages 3010-3021, we analyze how the measured kinetics parameters differ from the intrinsic molecular rates of the RNA, and thus how to obtain the molecular rates.     

Unfolding single RNA molecules: bridging the gap between equilibrium and non-equilibrium statistical thermodynamics

During the last 15 years, scientists have developed methods that permit the direct mechanical manipulation of individual molecules. Using this approach, they have begun to investigate the effect of force and torque in chemical and biochemical reactions. These studies span from the study of the mechanical properties of macromolecules, to the characterization of molecular motors, to the mechanical unfolding of individual proteins and RNA. Here I present a review of some of our most recent results using mechanical force to unfold individual molecules of RNA. These studies make it possible to follow in real time the trajectory of each molecule as it unfolds and characterize the various intermediates of the reaction. Moreover, if the process takes place reversibly it is possible to extract both kinetic and thermodynamic information from these experiments at the same time that we characterize the forces that maintain the three-dimensional structure of the molecule in solution. These studies bring us closer to the biological unfolding processes in the cell as they simulate in vitro, the mechanical unfolding of RNAs carried out in the cell by helicases. If the unfolding process occurs irreversibly, I show here that single-molecule experiments can still provide equilibrium, thermodynamic information from non-equilibrium data by using recently discovered fluctuation theorems. Such theorems represent a bridge between equilibrium and non-equilibrium statistical mechanics. In fact, first derived in 1997, the first experimental demonstration of the validity of fluctuation theorems was obtained by unfolding mechanically a single molecule of RNA. It is perhaps a sign of the times that important physical results are these days used to extract information about biological systems and that biological systems are being used to test and confirm fundamental new laws in physics.     

Forced-unfolding and force-quench refolding of RNA hairpins

Nanomanipulation of individual RNA molecules, using laser optical tweezers, has made it possible to infer the major features of their energy landscape. Time-dependent mechanical unfolding trajectories, measured at a constant stretching force (f_S) of simple RNA structures (hairpins and three-helix junctions) sandwiched between RNA/DNA hybrid handles show that they unfold in a reversible all-or-none manner. To provide a molecular interpretation of the experiments we use a general coarse-grained off-lattice Gō-like model, in which each nucleotide is represented using three interaction sites. Using the coarse-grained model we have explored forced-unfolding of RNA hairpin as a function of f_S and the loading rate (r_f). The simulations and theoretical analysis have been done both with and without the handles that are explicitly modeled by semiflexible polymer chains. The mechanisms and timescales for denaturation by temperature jump and mechanical unfolding are vastly different. The directed perturbation of the native state by f_S results in a sequential unfolding of the hairpin starting from their ends, whereas thermal denaturation occurs stochastically. From the dependence of the unfolding rates on r_f and f_S we show that the position of the unfolding transition state is not a constant but moves dramatically as either r_f or f_S is changed. The transition-state movements are interpreted by adopting the Hammond postulate for forced-unfolding. Forced-unfolding simulations of RNA, with handles attached to the two ends, show that the value of the unfolding force increases (especially at high pulling speeds) as the length of the handles increases. The pathways for refolding of RNA from stretched initial conformation, upon quenching f_S to the quench force f_Q, are highly heterogeneous. The refolding times, upon force-quench, are at least an order-of-magnitude greater than those obtained by temperature-quench. The long f_Q-dependent refolding times starting from fully stretched states are analyzed using a model that accounts for the microscopic steps in the rate-limiting step, which involves the trans to gauche transitions of the dihedral angles in the GAAA tetraloop. The simulations with explicit molecular model for the handles show that the dynamics of force-quench refolding is strongly dependent on the interplay of their contour length and persistence length and the RNA persistence length. Using the generality of our results, we also make a number of precise experimentally testable predictions.     

Monte Carlo simulation for single RNA unfolding by force

Using polymer elastic theory and known RNA free energies, we construct a Monte Carlo algorithm to simulate the single RNA folding and unfolding by mechanical force on the secondary structure level. For the constant force ensemble, we simulate the force-extension curves of the P5ab, P5abc deltaA, and P5abc molecules in equilibrium. For the constant extension ensemble, we focus on the mechanical behaviors of the RNA P5ab molecule, which include the unfolding force dependence on the pulling speed, the force-hysteresis phenomenon, and the coincidence of stretching-relaxing force-curves in thermal equilibrium. We particularly simulate the time traces of the end-to-end distance of the P5ab under the constant force in equilibrium, which also have been recorded in the recent experiment. The reaction rate constants for the folding and unfolding are calculated. Our results show that the agreement between the simulation and the experimental measurements is satisfactory.     

Thermodynamics and kinetics of the hairpin ribozyme from atomistic folding/unfolding simulations

We report a set of atomistic folding/unfolding simulations for the hairpin ribozyme using a Monte Carlo algorithm. The hairpin ribozyme folds in solution and catalyzes self-cleavage or ligation via a specific two-domain structure. The minimal active ribozyme has been studied extensively, showing stabilization of the active structure by cations and dynamic motion of the active structure. Here, we introduce a simple model of tertiary-structure formation that leads to a phase diagram for the RNA as a function of temperature and tertiary-structure strength. We then employ this model to capture many folding/unfolding events and to examine the transition-state ensemble (TSE) of the RNA during folding to its active “docked” conformation. The TSE is compact but with few tertiary interactions formed, in agreement with single-molecule dynamics experiments. To compare with experimental kinetic parameters, we introduce a novel method to benchmark Monte Carlo kinetic parameters to docking/undocking rates collected over many single molecular trajectories. We find that topology alone, as encoded in a biased potential that discriminates between secondary and tertiary interactions, is sufficient to predict the thermodynamic behavior and kinetic folding pathway of the hairpin ribozyme. This method should be useful in predicting folding transition states for many natural or man-made RNA tertiary structures.     

The effects of macromolecular crowding on the mechanical stability of protein molecules

Macromolecular crowding, a common phenomenon in the cellular environments, can significantly affect the thermodynamic and kinetic properties of proteins. A single-molecule method based on atomic force microscopy (AFM) was used to investigate the effects of macromolecular crowding on the forces required to unfold individual protein molecules. It was found that the mechanical stability of ubiquitin molecules was enhanced by macromolecular crowding from added dextran molecules. The average unfolding force increased from 210 pN in the absence of dextran to 234 pN in the presence of 300 g/L dextran at a pulling speed of 0.25 microm/sec. A theoretical model, accounting for the effects of macromolecular crowding on the native and transition states of the protein molecule by applying the scaled-particle theory, was used to quantitatively explain the crowding-induced increase in the unfolding force. The experimental results and interpretation presented could have wide implications for the many proteins that experience mechanical stresses and perform mechanical functions in the crowded environment of the cell.     

Force unfolding single RNAs

We develop a continue time Monte Carlo algorithm to simulate single RNAs unfolded by a time-dependent external force on the secondary structure level. Two recent unfolding RNA experiments carried out by Bustamante group are mainly investigated. We find that, in contrast to popular two-state assumption about the RNAs free energy landscape along the molecular extension, the molecules used in the experiments do not present apparent energy barriers. The strong cooperative folding and unfolding transitions of the RNAs observed in the experiments and in our simulations arise from the interaction of the molecules and the light trap. In addition, we also investigate the properties of Jarzynski's remarkable equality, whose experimental test has received considerable attention.     

83 Application of differential scanning calorimetry to measure the differential binding of ions,water, and protons in the unfolding of DNA molecules

The overall stability of DNA molecules globally depends on base-pair stacking, base pairing, polyelectrolyte effect, and hydration contributions. In order to improve our understanding of the role of ions, water, and protons in the stability and melting behavior of DNA structures, we report an experimental approach to determine the differential binding of ions (Δn _ion), water (Δn _W), and protons (Δn _H+) in the helix-coil transition of DNA molecules. A combination of differential scanning calorimetry (DSC) and temperature-dependent UV and CD spectroscopic techniques to investigate the unfolding of a variety of DNA molecules: S.T. DNA, two dodecamers, one undecamer, nine short hairpins as a function of the GC content of their stem, and two triplexes. We determine complete thermodynamic profiles, including all the three linking numbers, for the unfolding of each molecule. The CD spectra indicated that all molecules adopted the B-conformation at low temperatures. Thermodynamic profiles obtained from the DSC curves indicate that the favorable folding of each molecule results from the typical compensation of favorable enthalpy and unfavorable entropy contributions, and negligible heat capacity effects. UV and DSC melting curves as a function of salt, osmolyte, and proton concentrations yielded releases of ions, water, and protons (for the triplex with C⁺GC base triplets). Therefore, the favorable folding of each DNA molecule results from the formation of base-pair stacks and uptake of water and counterions. The thermodynamic data will be discussed in terms of the effects of DNA length, loop contributions and type of water molecules.     

Comparing proteins by their unfolding pattern

Single molecule force spectroscopy has evolved into an important and extremely powerful technique for investigating the folding potentials of biomolecules. Mechanical tension is applied to individual molecules, and the subsequent, often stepwise unfolding is recorded in force extension traces. However, because the energy barriers of the folding potentials are often close to the thermal energy, both the extensions and the forces at which these barriers are overcome are subject to marked fluctuations. Therefore, force extension traces are an inadequate representation despite widespread use particularly when large populations of proteins need to be compared and analyzed. We show in this article that contour length, which is independent of fluctuations and alterable experimental parameters, is a more appropriate variable than extension. By transforming force extension traces into contour length space, histograms are obtained that directly represent the energy barriers. In contrast to force extension traces, such barrier position histograms can be averaged to investigate details of the unfolding potential. The cross-superposition of barrier position histograms allows us to detect and visualize the order of unfolding events. We show with this approach that in contrast to the sequential unfolding of bacteriorhodopsin, two main steps in the unfolding of the enzyme titin kinase are independent of each other. The potential of this new method for accurate and automated analysis of force spectroscopy data and for novel automated screening techniques is shown with bacteriorhodopsin and with protein constructs containing GFP and titin kinase.     

The secondary structure of guide RNA molecules from Trypanosoma brucei.

RNA editing in kinetoplastid organisms is a mitochondrial RNA processing phenomenon that is characterized by the insertion and deletion of uridine nucleotides into incomplete mRNAs. Key molecules in the process are guide RNAs which direct the editing reaction by virtue of their primary sequences in an RNA-RNA interaction with the pre-edited mRNAs. To understand the molecular details of this reaction, especially potential RNA folding and unfolding processes as well as assembly phenomena with mitochondrial proteins, we analyzed the secondary structure of four different guide RNAs from Trypanosoma brucei at physiological conditions. By using structure-sensitive chemical and enzymatic probes in combination with spectroscopic techniques we found that the four molecules despite their different primary sequences, fold into similar structures consisting of two imperfect hairpin loops of low thermodynamic stability. The molecules melt in two-state monomolecular transitions with Tms between 33 and 39 degrees C and transition enthalpies of -32 to -38 kcal/mol. Both terminal ends of the RNAs are single-stranded with the 3' ends possibly adopting a single-stranded, helical conformation. Thus, it appears that the gRNA structures are fine tuned to minimize stability for an optimal annealing reaction to the pre-mRNAs while at the same time maximizing higher order structural features to permit the assembly with other mitochondrial components into the editing machinery.     

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).     

Hopping around an entropic barrier created by force

We use Langevin dynamics to investigate the role played by the recently discovered force-induced entropic energy barrier on the two-state hopping phenomena that has been observed in single RNA, DNA and protein molecules placed under a stretching force. Simple considerations about the free energy of a molecule readily show that the application of force introduces an entropic barrier separating the collapsed state of the molecule, from a force-driven extended conformation. A notable characteristic of the force induced barrier is its long distances to transition state, up to tens of nanometers, which renders the kinetics of crossing this barrier highly sensitive to an applied force. Langevin dynamics across such force induced barriers readily demonstrates the hopping behavior observed for a variety of single molecules placed under force. Such hopping is frequently interpreted as a manifestation of two-state folding/unfolding reactions observed in bulk experiments. However, given that such barriers do not exist at zero force these reactions do not take place at all in bulk.     

Spectrin folding versus unfolding reactions and RBC membrane stiffness

Spectrin (Sp), a key component of the erythrocyte membrane, is routinely stretched to near its fully folded contour length during cell deformations. Such dynamic loading may induce domain unfolding as suggested by recent experiments. Herein we develop a model to describe the folding/unfolding of spectrin during equilibrium or nonequilibrium extensions. In both cases, our model indicates that there exists a critical extension beyond which unfolding occurs. We further deploy this model, together with a three-dimensional model of the junctional complex in the erythrocyte membrane, to explore the effect of Sp unfolding on the membrane's mechanical properties, and on the thermal fluctuation of membrane-attached beads. At large deformations our results show a distinctive strain-induced unstiffening behavior, manifested in the slow decrease of the shear modulus, and accompanied by an increase in bead fluctuation. Bead fluctuation is also found to be influenced by mode switching, a phenomenon predicted by our three-dimensional model. The amount of stiffness reduction, however, is modest compared with that reported in experiments. A possible explanation for the discrepancy is the occurrence of spectrin head-to-head disassociation which is also included within our modeling framework and used to analyze bead motion as observed via experiment.     

Monte Carlo Simulations of the Chemical Potential and Free Energy for Trimer and Hexamer Rings

Abstract

The chemical potential of a trimer and hexamer model ring system was determined by computer simulation over a range of temperatures and densities. Such ring molecules are important as model aromatic and naphthenic hydrocarbons. Thermodynamic integration of the pressure along a reversible path, Widom's ghost particle insertion method and Kirkwood's charging parameter method were used over a molecular density range of 0.05 to 0.30. Data were obtained by Monte Carlo simulation of a 96 molecule system that was modelled with a Lennard-Jones 6-12 truncated potential. The original insertion method, which does not take into account the orientation of the molecule when it is inserted, gives results for the chemical potential which deviate from that obtained using the thermodynamic pressure integration. At high density or temperature the deviation is significant. We have modified the Widom insertion technique to account for this short range orientation and find good agreement between this technique and the thermodynamic integration method for the chemical potential. We also calculated the free energy difference between our model ring molecules and ring molecules made up of hard spheres.     

Temperature‐induced unfolding behavior of proteins studied by tensorial elastic network model

Motivated by single molecule experiments and recent molecular dynamics (MD) studies, we propose a simple and computationally efficient method based on a tensorial elastic network model to investigate the unfolding pathways of proteins under temperature variation. The tensorial elastic network model, which relies on the native state topology of a protein, combines the anisotropic network model, the bond bending elasticity, and the backbone twist elasticity to successfully predicts both the isotropic and anisotropic fluctuations in a manner similar to the Gaussian network model and anisotropic network model. The unfolding process is modeled by breaking the native contacts between residues one by one, and by assuming a threshold value for strain fluctuation. Using this method, we simulated the unfolding processes of four well‐characterized proteins: chymotrypsin inhibitor, barnase, ubiquitein, and adenalyate kinase. We found that this step‐wise process leads to two or more cooperative, first‐order‐like transitions between partial denaturation states. The sequence of unfolding events obtained using this method is consistent with experimental and MD studies. The results also imply that the native topology of proteins “encrypts” information regarding their unfolding process. Proteins 2016; 84:1767–1775. © 2016 Wiley Periodicals, Inc.     

Thermal unfolding of soybean peroxidase. Appropriate high denaturant concentrations induce cooperativity allowing the correct measurement of thermodynamic parameters

We have earlier reported that both guanidine hydrochloride (GdnHCl)-induced and heat-induced unfolding of seed coat soybean peroxidase (SBP), monitored by far UV CD, show single step transition. However, although GdnHCl-induced unfolding follows a two-state pathway, the heat-induced denaturation proceeds through intermediates as indicated by the very low cooperativity of transition. In the former case, analysis of the data based on the two-state model gives true thermodynamic parameters, whereas underestimated values are obtained in the latter case. Available complex equations also cannot be applied for the analysis of the thermal unfolding of SBP due to the absence of separate transitions for the intermediates. In the present study, we report a method to obtain true thermodynamic parameters from thermal transition curves of SBP using the two-state model. When SBP is subjected to thermal unfolding at high GdnHCl concentrations (5.8-6.9 M), cooperative behavior is observed, which allowed the analysis by the two-state model to determine their thermodynamic parameters. We then obtained the thermodynamic parameters in the absence of GdnHCl by extrapolating the graph of linear dependence of DeltaH(m) on T(m) to the T(m) corresponding to 0 m GdnHCl. Another key point for checking the validity of our method was the fact that the unfolded state of SBP generated by either heat or GdnHCl is the same by which we could cross-check our results with that obtained from GdnHCl unfolding. Having obtained the true thermodynamic parameters, we report a detailed thermodynamic study of SBP. Further we address the effect of heme in the thermal unfolding mechanism of SBP.     

Characterization of the mechanical unfolding of RNA pseudoknots

The pseudoknot is an important RNA structural element that provides an excellent model system for studying the contributions of tertiary interactions to RNA stability and to folding kinetics. RNA pseudoknots are also of interest because of their key role in the control of ribosomal frameshifting by viral RNAs. Their mechanical properties are directly relevant to their unfolding by ribosomes during translation. We have used optical tweezers to study the kinetics and thermodynamics of mechanical unfolding and refolding of single RNA molecules. Here we describe the unfolding of the frameshifting pseudoknot from infectious bronchitis virus (IBV), three constituent hairpins, and three mutants of the IBV pseudoknot. All four pseudoknots cause −1 programmed ribosomal frameshifting. We have measured the free energies and rates of mechanical unfolding and refolding of the four frameshifting pseudoknots. Our results show that the IBV pseudoknot requires a higher force than its corresponding hairpins to unfold. Furthermore, its rate of unfolding changes little with increasing force, in contrast with the rate of hairpin unfolding. The presence of Mg²⁺ significantly increases the kinetic barriers to unfolding the IBV pseudoknot, but has only a minor effect on the hairpin unfolding. The greater mechanical stability of pseudoknots compared to hairpins, and their kinetic insensitivity to force supports the hypothesis that −1 frameshifting depends on the difficulty of unfolding the mRNA.     

The role of water in the reversibility of thermal denaturation of lysozyme in solid and liquid states

Although unfolding of protein in the liquid state is relatively well studied, its mechanisms in the solid state, are much less understood. We evaluated the reversibility of thermal unfolding of lysozyme with respect to the water content using a combination of thermodynamic and structural techniques such as differential scanning calorimetry, synchrotron small and wide-angle X-ray scattering (SWAXS) and Raman spectroscopy. Analysis of the endothermic thermal transition obtained by DSC scans showed three distinct unfolding behaviors at different water contents. Using SWAXS and Raman spectroscopy, we investigated reversibility of the unfolding for each hydration regime for various structural levels including overall molecular shape, secondary structure, hydrophobic and hydrogen bonding interactions. In the substantially dehydrated state below 37 wt% of water the unfolding is an irreversible process and can be described by a kinetic approach; above 60 wt% the process is reversible, and the thermodynamic equilibrium approach is applied. In the intermediate range of water contents between 37 wt% and 60 wt%, the system is phase separated and the thermal denaturation involves two processes: melting of protein crystals and unfolding of protein molecules. A phase diagram of thermal unfolding/denaturation in lysozyme - water system was constructed based on the experimental data.