Structural parameters affecting the kinetics of RNA hairpin formation

There is little experimental knowledge on the sequence dependent rate of hairpin formation in RNA. We have therefore designed RNA sequences that can fold into either of two mutually exclusive hairpins and have determined the ratio of folding of the two conformations, using structure probing. This folding ratio reflects their respective folding rates. Changing one of the two loop sequences from a purine- to a pyrimidine-rich loop did increase its folding rate, which corresponds well with similar observations in DNA hairpins. However, neither changing one of the loops from a regular non-GNRA tetra-loop into a stable GNRA tetra-loop, nor increasing the loop size from 4 to 6 nt did affect the folding rate. The folding kinetics of these RNAs have also been simulated with the program ‘Kinfold’. These simulations were in agreement with the experimental results if the additional stabilization energies for stable tetra-loops were not taken into account. Despite the high stability of the stable tetra-loops, they apparently do not affect folding kinetics of these RNA hairpins. These results show that it is possible to experimentally determine relative folding rates of hairpins and to use these data to improve the computer-assisted simulation of the folding kinetics of stem–loop structures.     

Cooperativity in two-state protein folding kinetics

We present a solvable model that predicts the folding kinetics of two-state proteins from their native structures. The model is based on conditional chain entropies. It assumes that folding processes are dominated by small-loop closure events that can be inferred from native structures. For CI2, the src SH3 domain, TNfn3, and protein L, the model reproduces two-state kinetics, and it predicts well the average Phi-values for secondary structures. The barrier to folding is the formation of predominantly local structures such as helices and hairpins, which are needed to bring nonlocal pairs of amino acids into contact.     

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.     

Exploring the complex folding kinetics of RNA hairpins: II. Effect of sequence, length, and misfolded states

The complexity of RNA hairpin folding arises from the interplay between the loop formation, the disruption of the slow-breaking misfolded states, and the formation of the slow-forming native base stacks. We investigate the general physical mechanism for the dependence of the RNA hairpin folding kinetics on the sequence and the length of the hairpin loop and the helix stem. For example, 1), the folding would slow down when a stable GC basepair moves to the middle of the stem; 2), hairpin with GC basepair near the loop would fold/unfold faster than the one with GC near the tail of the stem; 3), within a certain range of the stem length, a longer stem can cause faster folding; and 4), certain misfolded states can assist folding through the formation of scaffold structures to lower the entropic barrier for the folding. All our findings are directly applicable and quantitatively testable in experiments. In addition, our results can be useful for molecular design to achieve desirable fast/slow-folding hairpins, hairpins with/without specific misfolded intermediates, and hairpins that fold along designed pathways.     

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.     

Transition states in protein folding kinetics: modeling phi-values of small beta-sheet proteins

Small single-domain proteins often exhibit only a single free-energy barrier, or transition state, between the denatured and the native state. The folding kinetics of these proteins is usually explored via mutational analysis. A central question is which structural information on the transition state can be derived from the mutational data. In this article, we model and structurally interpret mutational Φ-values for two small β-sheet proteins, the PIN and the FBP WW domains. The native structure of these WW domains comprises two β-hairpins that form a three-stranded β-sheet. In our model, we assume that the transition state consists of two conformations in which either one of the hairpins is formed. Such a transition state has been recently observed in molecular dynamics folding-unfolding simulations of a small designed three-stranded β-sheet protein. We obtain good agreement with the experimental data 1), by splitting up the mutation-induced free-energy changes into terms for the two hairpins and for the small hydrophobic core of the proteins; and 2), by fitting a single parameter, the relative degree to which hairpins 1 and 2 are formed in the transition state. The model helps us to understand how mutations affect the folding kinetics of WW domains, and captures also negative Φ-values that have been difficult to interpret.     

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.     

The early folding kinetics of apomyoglobin.

The folding pathway of apomyoglobin has been experimentally shown to have early kinetic intermediates involving the A, B, G, and H helices. The earliest detected kinetic events occur on a ns to micros time scale. We show that the early folding kinetics of apomyoglobin may be understood as the association of nascent helices through a network of diffusion-collision-coalescence steps G + H <--> GH + A <--> AGH + B <--> ABGH obtained by solving the diffusion-collision model in a chemical kinetics approximation. Our reproduction of the experimental results indicates that the model is a useful way to analyze folding data. One prediction from our fit is that the nascent A and H helices should be relatively more helix-like before coalescence than the other apomyoglobin helices.     

Probing the kinetics of single molecule protein folding

We propose an approach to integrate the theory, simulations, and experiments in protein-folding kinetics. This is realized by measuring the mean and high-order moments of the first-passage time and its associated distribution. The full kinetics is revealed in the current theoretical framework through these measurements. In the experiments, information about the statistical properties of first-passage times can be obtained from the kinetic folding trajectories of single molecule experiments (for example, fluorescence). Theoretical/simulation and experimental approaches can be directly related. We study in particular the temperature-varying kinetics to probe the underlying structure of the folding energy landscape. At high temperatures, exponential kinetics is observed; there are multiple parallel kinetic paths leading to the native state. At intermediate temperatures, nonexponential kinetics appears, revealing the nature of the distribution of local traps on the landscape and, as a result, discrete kinetic paths emerge. At very low temperatures, exponential kinetics is again observed; the dynamics on the underlying landscape is dominated by a single barrier. The ratio between first-passage-time moments is proposed to be a good variable to quantitatively probe these kinetic changes. The temperature-dependent kinetics is consistent with the strange kinetics found in folding dynamics experiments. The potential applications of the current results to single-molecule protein folding are discussed.     

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.     

Simulating RNA folding kinetics on approximated energy landscapes

We present a general computational approach to simulate RNA folding kinetics that can be used to extract population kinetics, folding rates and the formation of particular substructures that might be intermediates in the folding process. Simulating RNA folding kinetics can provide unique insight into RNA whose functions are dictated by folding kinetics and not always by nucleotide sequence or the structure of the lowest free-energy state. The method first builds an approximate map (or model) of the folding energy landscape from which the population kinetics are analyzed by solving the master equation on the map. We present results obtained using an analysis technique, map-based Monte Carlo simulation, which stochastically extracts folding pathways from the map. Our method compares favorably with other computational methods that begin with a comprehensive free-energy landscape, illustrating that the smaller, approximate map captures the major features of the complete energy landscape. As a result, our method scales to larger RNAs. For example, here we validate kinetics of RNA of more than 200 nucleotides. Our method accurately computes the kinetics-based functional rates of wild-type and mutant ColE1 RNAII and MS2 phage RNAs showing excellent agreement with experiment.     

Role of unfolded state heterogeneity and en-route ruggedness in protein folding kinetics

In order to improve our understanding of the physical bases of protein folding, there is a compelling need for better connections between experimental and computational approaches. This work addresses the role of unfolded state conformational heterogeneity and en-route intermediates, as an aid for planning and interpreting protein folding experiments. The expected kinetics were modeled for different types of energy landscapes, including multiple parallel folding routes, preferential paths dominated by one primary folding route, and distributed paths with a wide spectrum of microscopic folding rate constants. In the presence of one or more preferential routes, conformational exchange among unfolded state populations slows down the observed rates for native protein formation. We find this to be a general phenomenon, taking place even when unfolded conformations interconvert much faster than the "escape" rate constants to folding. Dramatic kinetic deceleration is expected in the presence of an increasing number of folding-incompetent unfolded conformations. This argues for the existence of parallel folding paths involving several folding-competent unfolded conformations, during the early stages of protein folding. Deviations from single-exponential behavior are observed for unfolded conformations exchanging at comparable rates or more slowly than folding events. Analysis of the effect of en-route (on-path) intermediate formation and landscape ruggedness on folding kinetics leads to the following unexpected conclusions: (1) intermediates, which often retard native state formation, may in some cases accelerate folding, and (2) rugged landscapes, usually associated with stretched exponentials, display single-exponential behavior in the presence of late high-friction paths.     

The ensemble folding kinetics of the FBP28 WW domain revealed by an all-atom Monte Carlo simulation in a knowledge-based potential

In this work, we apply a detailed all‐atom model with a transferable knowledge‐based potential to study the folding kinetics of Formin‐Binding protein, FBP28, which is a canonical three‐stranded β‐sheet WW domain. Replica exchange Monte Carlo simulations starting from random coils find native‐like (Cα RMSD of 2.68 Å) lowest energy structure. We also study the folding kinetics of FBP28 WW domain by performing a large number of ab initio Monte Carlo folding simulations. Using these trajectories, we examine the order of formation of two β‐hairpins, the folding mechanism of each individual β‐hairpin, and transition state ensemble (TSE) of FBP28 WW domain and compare our results with experimental data and previous computational studies. To obtain detailed structural information on the folding dynamics viewed as an ensemble process, we perform a clustering analysis procedure based on graph theory. Further, a rigorous P_fold analysis is used to obtain representative samples of the TSEs showing good quantitative agreement between experimental and simulated Φ values. Our analysis shows that the turn structure between first and second β strands is a partially stable structural motif that gets formed before entering the TSE in FBP28 WW domain and there exist two major pathways for the folding of FBP28 WW domain, which differ in the order and mechanism of hairpin formation. Proteins 2011. © 2011 Wiley‐Liss, Inc.     

Osmolyte effects on kinetics of FKBP12 C22A folding coupled with prolyl isomerization

Unfolding and refolding kinetics of human FKBP12 C22A were monitored by fluorescence emission over a wide range of urea concentration in the presence and absence of protecting osmolytes glycerol, proline, sarcosine and trimethylamine-N-oxide (TMAO). Unfolding is well described by a mono-exponential process, while refolding required a minimum of two exponentials for an adequate fit throughout the urea concentration range considered. The bi-exponential behavior resulted from complex coupling between protein folding, and prolyl isomerization in the denatured state in which the urea-dependent rate constant for folding was greater than, equal to, and less than the rate constants for prolyl isomerization within the urea concentration range of zero to five molar. Amplitudes and the observed folding and unfolding rate constants were fitted to a reversible three-state model composed of two sequential steps involving the native state and a folding-competent denatured species thermodynamically linked to a folding-incompetent denatured species. Excellent agreement between thermodynamic parameters for FKBP12 C22A folding calculated from the kinetic parameters and those obtained directly from equilibrium denaturation assays provides strong support for the applicability of the mechanism, and provides evidence that FKBP12 C22A folding/unfolding is two-state, with prolyl isomer heterogeneity in the denatured ensemble. Despite the chemical diversity of the protecting osmolytes, they all exhibit the same kinetic behavior of increasing the rate constant of folding and decreasing the rate constant for unfolding. Osmolyte effects on folding/unfolding kinetics are readily explained in terms of principles established in understanding osmolyte effects on protein stability. These principles involve the osmophobic effect, which raises the Gibbs energy of the denatured state due to exposure of peptide backbone, thereby increasing the folding rate. This effect also plays a key role in decreasing the unfolding rate when, as is often the case, the activated complex exposes more backbone than is exposed in the native state.     

Diffusive motions control the folding and unfolding kinetics of the apomyoglobin pH 4 molten globule intermediate

The sperm whale apomyoglobin pH 4 folding intermediate exists in two forms, Ia and Ib, that mimic transient kinetic intermediates in the folding of the native protein at pH 6. To characterize the nature of the kinetic barrier that controls the formation of the earliest intermediate Ia, we have investigated the effects of small viscogenic cosolvents on its folding and unfolding kinetics. The kinetics are measurable by stopped-flow fluorescence and follow a cooperative two-state model in the absence and presence of cosolvents. Small cosolvents stabilize Ia, but, by applying the isostability test to separate the viscogenic effect of the cosolvent from its stabilizing effect, we found that, in both folding and unfolding conditions, the apparent rate constant decreases when solvent viscosity increases. The unitary inverse dependence of the apparent rate constant on solvent viscosity indicates a diffusion-controlled reaction. This result is consistent with the hypothesis that folding of the apomyoglobin pH 4 intermediate obeys a diffusion-collision model. Additionally, the temperature dependence of the reaction rate at constant viscosity indicates that the formation of Ia is also controlled by an energy barrier. Linear free energy relationships show that the transition state of the U <==> Ia reaction is compact and buries 45% of the surface area that is buried in native apomyoglobin. We conclude that the transition state of the U <==> Ia reaction resembles that for the formation of native proteins; namely, it is dry and its compactness is closer to that of the folded (Ia) form than of the unfolded form.     

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.     

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.     

Predicting repeat protein folding kinetics from an experimentally determined folding energy landscape

The Notch ankyrin domain is a repeat protein whose folding has been characterized through equilibrium and kinetic measurements. In previous work, equilibrium folding free energies of truncated constructs were used to generate an experimentally determined folding energy landscape (Mello and Barrick, Proc Natl Acad Sci USA 2004;101:14102–14107). Here, this folding energy landscape is used to parameterize a kinetic model in which local transition probabilities between partly folded states are based on energy values from the landscape. The landscape‐based model correctly predicts highly diverse experimentally determined folding kinetics of the Notch ankyrin domain and sequence variants. These predictions include monophasic folding and biphasic unfolding, curvature in the unfolding limb of the chevron plot, population of a transient unfolding intermediate, relative folding rates of 19 variants spanning three orders of magnitude, and a change in the folding pathway that results from C‐terminal stabilization. These findings indicate that the folding pathway(s) of the Notch ankyrin domain are thermodynamically selected: the primary determinants of kinetic behavior can be simply deduced from the local stability of individual repeats.     

Folding and aggregation kinetics of a beta-hairpin

We have investigated the solution structure, equilibrium properties, and folding kinetics of a 17-residue beta-hairpin-forming peptide derived from the protein ubiquitin. NMR experiments show that at 4 degrees C the peptide has a highly populated beta-hairpin conformation. At protein concentrations higher than 0.35 mM, the peptide aggregates. Sedimentation equilibrium measurements show that the aggregate is a trimer, while NMR indicates that the beta-hairpin conformation is maintained in the trimer. The relaxation kinetics in nanosecond laser temperature-jump experiments reveal a concentration-independent microsecond phase, corresponding to beta-hairpin unfolding-refolding, and a concentration-dependent millisecond phase due to oligomerization. Kinetic modeling of the relaxation rates and amplitudes yields the folding and unfolding rates for the monomeric beta-hairpin, as well as assembly and disassembly rates for trimer formation consistent with the equilibrium constant determined by sedimentation equilibrium. When the net charge on the peptides and ionic strength were taken into account, the rate of trimer assembly approaches the Debye-Smoluchowski diffusion limit. At 300 K, the rate of formation of the monomeric hairpin is (17 micros)(-1), compared to rates of (0.8 micros)(-1) to (52 micros)(-1) found for other peptides. After using Kramers theory to correct for the temperature dependence of the pre-exponential factor, the activation energy for hairpin formation is near zero, indicating that the barrier to folding is purely entropic. Comparisons with previously measured rates for a series of hairpins are made to distinguish between zipper and hydrophobic collapse mechanisms. Overall, the experimental data are most consistent with the zipper mechanism in which structure formation is initiated at the turn, the mechanism predicted by the Ising-like statistical mechanical model that was developed to explain the equilibrium and kinetic data for the beta-hairpin from protein GB1. In contrast, the majority of simulation studies favor a hydrophobic collapse mechanism. However, with few exceptions, there is little or no quantitative comparison of the simulation results with experimental data.