All-atom Monte Carlo simulation of GCAA RNA folding

Petkov P, Milenov TI, Vayssilov GN. Sodium and Magnesium Ion Location at the Backbone and at the Nucleobase of RNA: Ab Initio Molecular Dynamics in Water Solution

The interactions between Na⁺ or Mg²⁺ ions with different parts of single-stranded RNA molecules, namely, the oxygen atoms from the phosphate groups or the guanine base, in water solution have been studied using first-principles molecular dynamics. Sodium ions were found to be much more mobile than Mg²⁺ ions and readily underwent transitions between a state directly bonded to RNA oxygen atoms and a completely solvated state. The inner solvation shell of Na⁺ ions fluctuated stochastically at a femtosecond timescale coordinating on average 5 oxygen atoms for bonded Na⁺ ions and 5.5 oxygen atoms for solvated Na⁺ ions. In contrast, the inner solvation shell of Mg²⁺ ions was stable in both RNA-bonded and completely solvated states. In both cases, Mg²⁺ ions coordinated 6 oxygen atoms from the inner solvation shell. Consistent with their stable solvation shells, Mg²⁺ ions were more effective than Na⁺ ions in stabilizing the RNA backbone conformation. The exclusion zones between the first and second solvation shells, solvation shell widths, and angles for binding to carbonyl oxygen of guanine for solvated Na⁺ or Mg²⁺ ions exhibited a number of quantitative differences when compared with RNA crystallographic data. The presented results support the distinct capacity of Mg²⁺ ions to support the RNA structure not only in the crystal phase but also in the dynamic water environment both on the side of the phosphate moiety and on the side of the nucleobase.

Monovalent ions modulate the flux through multiple folding pathways of an RNA pseudoknot

The assembly mechanism of RNA, vital to describing its functions, depends on both the sequence and the metal ion concentration. How the latter influences the folding trajectories remains an important unsolved problem. Here, we examine the folding pathways of an RNA pseudoknot (PK) with key functional roles in transcription and translation, using a combination of experiments and simulations. We demonstrate that the PK, consisting of two hairpins with differing stabilities, folds by parallel pathways. Surprisingly, the flux between them is modulated by monovalent salt concentration. Our work shows that the order of assembly of PKs is determined by the relative stability of the hairpins, implying that the folding landscape can be controlled by sequence and ion concentration.

The functions of RNA pseudoknots (PKs), which are minimal tertiary structural motifs and an integral part of several ribozymes and ribonucleoprotein complexes, are determined by their structure, stability, and dynamics. Therefore, it is important to elucidate the general principles governing their thermodynamics/folding mechanisms. Here, we combine laser temperature-jump experiments and coarse-grained simulations to determine the folding/unfolding pathways of VPK, a variant of the mouse mammary tumor virus (MMTV) PK involved in ribosomal frameshifting. Fluorescent nucleotide analogs (2-aminopurine and pyrrolocytidine) placed at different stem/loop positions in the PK serve as local probes allowing us to monitor the order of assembly of VPK that has two constituent hairpins with different intrinsic stabilities. We show that at 50 mM KCl, the dominant folding pathway populates only the more stable hairpin intermediate; as the salt concentration is increased, a parallel folding pathway emerges involving the less stable hairpin as an alternate intermediate. Notably, the flux between the pathways is modulated by the ionic strength. Our findings support the principle that the order of PK structure formation is determined by the relative stabilities of the hairpins, which can be altered by sequence variations or salt concentrations. The experimental results of salt effects on the partitioning between the two folding pathways are in remarkable agreement with simulations that were performed with no adjustable parameters. Our study not only unambiguously demonstrates that VPK folds by parallel pathways but also showcases the power of combining experiments and simulations for a more enriched description of RNA self-assembly.

Application of Monte Carlo simulation in addressing key issues of complex coacervation formed by polyelectrolytes and oppositely charged colloids

This paper reviews the recent advance of Monte Carlo (MC) simulation in addressing key issues of complex coacervation between polyelectrolytes and oppositely charged colloids. Readers were first supplied with a brief overview of current knowledge and experimental strategies in the study of complex coacervation. In the next section, the general MC simulation procedures as well as representative strategies applied in complex coacervation were summarized. The unique contributions of MC simulation in either capturing delicate features, easing the experimental trials or proving the concept were then elucidated through the following aspects: i) identify phase boundary and decouple interaction contributions; ii) clarify composition distribution and internal structure; iii) predict the influences of physicochemical conditions on complex coacervation; iv) delineate the mechanisms for "binding on the wrong side of the isoelectric point". Finally, current challenges as well as prospects of MC simulation in complex coacervation are also discussed. The ultimate goal of this review is to provide readers with basic guideline for synergistic design of experiments in combination with MC simulation, and deliver convincing interpretation and reliable prediction for the structure and behavior in polyelectrolyte-macroion complex coacervation.

MINT: software to identify motifs and short-range interactions in trajectories of nucleic acids

Structural biology experiments and structure prediction tools have provided many high-resolution three-dimensional structures of nucleic acids. Also, molecular dynamics force field parameters have been adapted to simulating charged and flexible nucleic acid structures on microsecond time scales. Therefore, we can generate the dynamics of DNA or RNA molecules, but we still lack adequate tools for the analysis of the resulting huge amounts of data. We present MINT (Motif Identifier for Nucleic acids Trajectory) — an automatic tool for analyzing three-dimensional structures of RNA and DNA, and their full-atom molecular dynamics trajectories or other conformation sets (e.g. X-ray or nuclear magnetic resonance-derived structures). For each RNA or DNA conformation MINT determines the hydrogen bonding network resolving the base pairing patterns, identifies secondary structure motifs (helices, junctions, loops, etc.) and pseudoknots. MINT also estimates the energy of stacking and phosphate anion-base interactions. For many conformations, as in a molecular dynamics trajectory, MINT provides averages of the above structural and energetic features and their evolution. We show MINT functionality based on all-atom explicit solvent molecular dynamics trajectory of the 30S ribosomal subunit.

A kinetic zipper model with intrachain interactions applied to nucleic acid hairpin folding kinetics

Single-stranded DNA and RNA hairpin structures with 4-10 nucleotides (nt) in the loop and 5-8 basepairs (bp) in the stem fold on 10-100 μs timescale. In contrast, theoretical estimate of first contact time of two ends of an ideal semiflexible polymer of similar lengths (with persistence length ~2-nt) is 10-100 ns. We propose that this three-orders-of-magnitude difference between these two timescales is a result of roughness in the folding free energy surface arising from intrachain interactions. We present a statistical mechanical model that explicitly includes all misfolded microstates with nonnative Watson-Crick (WC) and non-WC contacts. Rates of interconversion between different microstates are described in terms of two adjustable parameters: the strength of the non-WC interactions (ΔG(nWC)) and the rate at which a basepair is formed adjacent to an existing basepair (k(bp)(+)). The model accurately reproduces the temperature and loop-length dependence of the measured relaxation rates in temperature-jump studies of a 7-bp stem, single-stranded DNA hairpin with 4-20-nt-long poly(dT) loops, with ΔG(nWC) ≈ -2.4 kcal/mol and k(bp)(+) ≥ (1 ns)(-1), in 100 mM NaCl. Thus, our model provides a microscopic interpretation of the slow hairpin folding times as well as an estimate of the strength of intrachain interactions.

Role of loop entropy in the force induced melting of DNA hairpin

Dynamics of a single stranded DNA, which can form a hairpin have been studied in the constant force ensemble. Using Langevin dynamics simulations, we obtained the force-temperature diagram, which differs from the theoretical prediction based on the lattice model. Probability analysis of the extreme bases of the stem revealed that at high temperature, the hairpin to coil transition is entropy dominated and the loop contributes significantly in its opening. However, at low temperature, the transition is force driven and the hairpin opens from the stem side. It is shown that the elastic energy plays a crucial role at high force. As a result, the force-temperature diagram differs significantly with the theoretical prediction.

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.

Laser-induced temperature jump infrared measurements of RNA folding

Probing a sample using infrared spectroscopy following a laser-induced temperature jump is a powerful method to monitor fast relaxation kinetics. Here, we describe how this approach is used to study the kinetics of RNA folding. We begin with a concise summary of the infrared spectral properties of RNA in the 1500-1800cm(-1) region. The infrared transitions in this region are directly related to the double bond stretching vibrations and ring modes of the nucleotide bases. When RNA undergoes a conformational change, the local environments of the nucleotides are altered. Consequently, the changes in the corresponding infrared spectrum are associated with the structural changes. Experimentally, temperature is used to systematically vary the RNA structure. When a short laser pulse is used to produce a rapid temperature increase in the sample, the structural changes that ensue can be followed in real time. In this contribution, we discuss experimental methods including sample preparation, instrumentation, and data analysis. We conclude with several experimental examples that highlight usefulness of the technique.

Methods for Monte Carlo simulations of biomacromolecules

The state-of-the-art for Monte Carlo (MC) simulations of biomacromolecules is reviewed. Available methodologies for sampling conformational equilibria and associations of biomacromolecules in the canonical ensemble, given a continuum description of the solvent environment, are reviewed. Detailed sections are provided dealing with the choice of degrees of freedom, the efficiencies of MC algorithms and algorithmic peculiarities, as well as the optimization of simple movesets. The issue of introducing correlations into elementary MC moves, and the applicability of such methods to simulations of biomacromolecules is discussed. A brief discussion of multicanonical methods and an overview of recent simulation work highlighting the potential of MC methods are also provided. It is argued that MC simulations, while underutilized biomacromolecular simulation community, hold promise for simulations of complex systems and phenomena that span multiple length scales, especially when used in conjunction with implicit solvation models or other coarse graining strategies.

Probing the structural hierarchy and energy landscape of an RNA T-loop hairpin

The T-loop motif is an important recurrent RNA structural building block consisting of a U-turn sub-motif and a UA trans Watson–Crick/Hoogsteen base pair. In the presence of a hairpin stem, the UA non-canonical base pair becomes part of the UA-handle motif. To probe the hierarchical organization and energy landscape of the T-loop, we performed replica exchange molecular dynamics (REMD) simulations of the T-loop in isolation and as part of a hairpin. Our simulations reveal that the isolated T-loop adopts coil conformers stabilized by base stacking. The T-loop hairpin shows a highly rugged energy landscape featuring multiple local minima with a transition state for folding consisting of partially zipped states. The U-turn displays a high conformational flexibility both when the T-loop is in isolation and as part of a hairpin. On the other hand, the stability of the UA non-canonical base pair is enhanced in the presence of the UA-handle. This motif is apparently a key component for stabilizing the T-loop, while the U-turn is mostly involved in long-range interaction. Our results suggest that the stability and folding of small RNA motifs are highly dependent on local context.

Explicit factorization of external coordinates in constrained statistical mechanics models

If a macromolecule is described by curvilinear coordinates or rigid constraints are imposed, the equilibrium probability density that must be sampled in Monte Carlo simulations includes the determinants of different mass-metric tensors. In this work, the authors explicitly write the determinant of the mass-metric tensor G and of the reduced mass-metric tensor g, for any molecule, general internal coordinates and arbitrary constraints, as a product of two functions; one depending only on the external coordinates that describe the overall translation and rotation of the system, and the other only on the internal coordinates. This work extends previous results in the literature, proving with full generality that one may integrate out the external coordinates and perform Monte Carlo simulations in the internal conformational space of macromolecules.

RNA folding during transcription

The evolution of RNA sequence needs to satisfy three requirements: folding, structure, and function. Studies on folding during transcription are related directly to folding in the cell. Understanding RNA folding during transcription requires the elucidation of structure formation and structural changes of the RNA, and the consideration of intrinsic properties of the RNA polymerase and other proteins that interact with the RNA. This review summarizes the research progress in this area and outlines the enormous challenges facing this field. Significant advancement requires the development of new experimental methods and theoretical considerations in all aspects of transcription and RNA folding.

Single-molecule RNA folding

Single-molecule experiments significantly expand our capability to characterize complex dynamics of biological processes. This relatively new approach has contributed significantly to our understanding of the RNA folding problem. Recent single-molecule experiments, together with structural and biochemical characterizations of RNA at the ensemble level, show that RNA molecules typically fold across a highly rugged energy landscape. As a result, long-lived folding intermediates, multiple folding pathways, and heterogeneous conformational dynamics are commonly found for RNA enzymes. While initial results have suggested that stable secondary structures are partly responsible for the rugged energy landscape of RNA, a complete mechanistic understanding of the complex folding behavior has not yet been obtained. A combination of single-molecule experiments, which are well suited to analyze transient and heterogeneous dynamic behaviors, with ensemble characterizations that can provide structural information at a superior resolution will likely provide more answers.

Mapping of Steroids Binding to 17β-Hydroxysteroid Dehydrogenase Type 1 Using Monte Carlo Energy Minimization Reveals Alternative Binding Modes

Crystallographic studies of ligand-protein complexes reveal most preferable ligand binding modes, but do not show less populated modes that may contribute to measurable biochemical and biophysical characteristics of the complexes. In some cases, a ligand may bind a protein in essentially different modes. An example is 17beta-hydroxysteroid dehydrogenase type 1 (17beta-HSD1), a steroidogenic enzyme that catalyzes reduction of estrone to estradiol in gonadal and peripheral tissues. The enzyme exhibits a high specificity for estrogens which bind with their C17 atom in the proximity of the NADP(H) cofactor. 17Beta-HSD1 can also bind androgens, but in a reverse binding mode, in which the steroid C3 atom is the closest carbon atom to the cofactor. Here we map the interaction energy of estradiol and dihydrotestosterone binding to 17beta-HSD1. Positions and orientations of the steroids in the ligand-binding tunnel were sampled systematically, and at each combination of these generalized coordinates, the energy was Monte Carlo minimized. The computed maps show energy minima corresponding to the X-ray structures and predict alternative binding modes, in particular, an upside-down orientation in which steroidal face alpha is exposed to protein residues that normally interact with face beta. The methodology can be used for mapping ligand-receptor interactions in various systems, for example, in ion channels and G-protein-coupled receptors that bind elongated ligands in confined space between transmembrane helices.

Does water play a structural role in the folding of small nucleic acids?

Nucleic acid structure and dynamics are known to be closely coupled to local environmental conditions and, in particular, to the ionic character of the solvent. Here we consider what role the discrete properties of water and ions play in the collapse and folding of small nucleic acids. We study the folding of an experimentally well-characterized RNA hairpin-loop motif (sequence 5'-GGGC[GCAA]GCCU-3') via ensemble molecular dynamics simulation and, with nearly 500 micros of aggregate simulation time using an explicit representation of the ionic solvent, report successful ensemble folding simulations with a predicted folding time of 8.8(+/-2.0) micros, in agreement with experimental measurements of approximately 10 micros. Comparing our results to previous folding simulations using the GB/SA continuum solvent model shows that accounting for water-mediated interactions is necessary to accurately characterize the free energy surface and stochastic nature of folding. The formation of the secondary structure appears to be more rapid than the fastest ionic degrees of freedom, and counterions do not participate discretely in observed folding events. We find that hydrophobic collapse follows a predominantly expulsive mechanism in which a diffusion-search of early structural compaction is followed by the final formation of native structure that occurs in tandem with solvent evacuation.