A large collapsed-state RNA can exhibit simple exponential single-molecule dynamics

Intermolecular structural correlations in model globular and unconcatenated ring polymer liquids

We employ the field theoretic polymer integral equation theory to construct a segment-level theory for the thermodynamics and pair structure of dense liquids of interpenetrating ring polymers and a simple globule model. The latter is defined by a fractal mass distribution on all internal length scales with an exponent equal to the spatial dimension (dF = ds = 3). In an isochoric ensemble the dimensionless compressibility and pressure is predicted to vary exponentially with macromolecular volume fraction. An intermolecular correlation hole exists down to small length scales. This model appears to be useful for a recently studied experimental soft nanoparticle suspension, and also serves as a reference system for our analysis of ring liquids. Motivated by simulations, a two-fractal exponent ring model is adopted for the intramolecular structure factor. At smaller lengths it describes chain-like macromolecules, while on larger scales it corresponds to a space-filling object in the sense that dF = ds = 3. The crossover between these two regimes is of order the entanglement length of the linear chain analog. Based on a constant compressibility ensemble, the effective volume fraction grows at intermediate values of degree of polymerization (N), and crosses over to a very slow logarithmic growth at large N. A weaker intermolecular correlation hole is predicted. The number of nearest neighbor rings increases dramatically at small N, akin to linear chain melts, but then tends to saturate at large N, in accord with simulations. The tools developed may be relevant for other partially interpenetrating soft objects such as core-shell nanogels or microgels.

The role of counterion valence and size in GAAA tetraloop-receptor docking/undocking kinetics

For RNA to fold into compact, ordered structures, it must overcome electrostatic repulsion between negatively charged phosphate groups by counterion recruitment. A physical understanding of the counterion-assisted folding process requires addressing how cations kinetically and thermodynamically control the folding equilibrium for each tertiary interaction in a full-length RNA. In this work, single-molecule FRET (fluorescence resonance energy transfer) techniques are exploited to isolate and explore the cation-concentration-dependent kinetics for formation of a ubiquitous RNA tertiary interaction, that is, the docking/undocking of a GAAA tetraloop with its 11-nt receptor. Rate constants for docking (k(dock)) and undocking (k(undock)) are obtained as a function of cation concentration, size, and valence, specifically for the series Na(+), K(+), Mg(2+), Ca(2+), Co(NH(3))(6)(3+), and spermidine(3+). Increasing cation concentration acceleratesk(dock)dramatically but achieves only a slight decrease in k(undock). These results can be kinetically modeled using parallel cation-dependent and cation-independent docking pathways, which allows for isolation of the folding kinetics from the interaction energetics of the cations with the undocked and docked states, respectively. This analysis reveals a preferential interaction of the cations with the transition state and docked state as compared to the undocked RNA, with the ion-RNA interaction strength growing with cation valence. However, the corresponding number of cations that are taken up by the RNA upon folding decreases with charge density of the cation. The only exception to these behaviors is spermidine(3+), whose weaker influence on the docking equilibria with respect to Co(NH(3))(6)(3+) can be ascribed to steric effects preventing complete neutralization of the RNA phosphate groups.

Optimal use of data in parallel tempering simulations for the construction of discrete-state Markov models of biomolecular dynamics

Parallel tempering (PT) molecular dynamics simulations have been extensively investigated as a means of efficient sampling of the configurations of biomolecular systems. Recent work has demonstrated how the short physical trajectories generated in PT simulations of biomolecules can be used to construct the Markov models describing biomolecular dynamics at each simulated temperature. While this approach describes the temperature-dependent kinetics, it does not make optimal use of all available PT data, instead estimating the rates at a given temperature using only data from that temperature. This can be problematic, as some relevant transitions or states may not be sufficiently sampled at the temperature of interest, but might be readily sampled at nearby temperatures. Further, the comparison of temperature-dependent properties can suffer from the false assumption that data collected from different temperatures are uncorrelated. We propose here a strategy in which, by a simple modification of the PT protocol, the harvested trajectories can be reweighted, permitting data from all temperatures to contribute to the estimated kinetic model. The method reduces the statistical uncertainty in the kinetic model relative to the single temperature approach and provides estimates of transition probabilities even for transitions not observed at the temperature of interest. Further, the method allows the kinetics to be estimated at temperatures other than those at which simulations were run. We illustrate this method by applying it to the generation of a Markov model of the conformational dynamics of the solvated terminally blocked alanine peptide.

Splitting probabilities as a test of reaction coordinate choice in single-molecule experiments

To explain the observed dynamics in equilibrium single-molecule measurements of biomolecules, the experimental observable is often chosen as a putative reaction coordinate along which kinetic behavior is presumed to be governed by diffusive dynamics. Here, we invoke the splitting probability as a test of the suitability of such a proposed reaction coordinate. Comparison of the observed splitting probability with that computed from the kinetic model provides a simple test to reject poor reaction coordinates. We demonstrate this test for a force spectroscopy measurement of a DNA hairpin.

Flow-Dependent Unfolding and Refolding of an RNA by Nonequilibrium Umbrella Sampling

Nonequilibrium experiments of single biomolecules such as force-induced unfolding reveal details about a few degrees of freedom of a complex system. Molecular dynamics simulations can provide complementary information, but exploration of the space of possible configurations is often hindered by large barriers in phase space that separate metastable regions. To solve this problem, enhanced sampling methods have been developed that divide a phase space into regions and integrate trajectory segments in each region. These methods boost the probability of passage over barriers and facilitate parallelization since integration of the trajectory segments does not require communication, aside from their initialization and termination. Here, we present a parallel version of an enhanced sampling method suitable for systems driven far from equilibrium: nonequilibrium umbrella sampling (NEUS). We apply this method to a coarse-grained model of a 262-nucleotide RNA molecule that unfolds and refolds in an explicit flow field modeled with stochastic rotation dynamics. Using NEUS, we are able to observe extremely rare unfolding events that have mean first passage times as long as 45 s (1.1 × 10(15) dynamics steps). We examine the unfolding process for a range of flow rates of the medium, and we describe two competing pathways in which different intramolecular contacts are broken.

Single-molecule Förster resonance energy transfer studies of RNA structure, dynamics and function

Single-molecule fluorescence microscopy experiments on RNA molecules brought to light the highly complex dynamics of key biological processes, including RNA folding, catalysis of ribozymes, ligand sensing of riboswitches and aptamers, and protein synthesis in the ribosome. By using highly advanced biophysical spectroscopy techniques in combination with sophisticated biochemical synthesis approaches, molecular dynamics of individual RNA molecules can be observed in real time and under physiological conditions in unprecedented detail that cannot be achieved with bulk experiments. Here, we review recent advances in RNA folding and functional studies of RNA and RNA-protein complexes addressed by using single-molecule Förster (fluorescence) resonance energy transfer (smFRET) technique.

Models of single-molecule experiments with periodic perturbations reveal hidden dynamics in RNA folding

Traditionally, microscopic fluctuations of molecules have been probed by measuring responses of an ensemble to perturbations. Now, single-molecule experiments are capable of following fluctuations without introducing perturbations. However, dynamics not readily sampled at equilibrium should be accessible to nonequilibrium single-molecule measurements. In a recent study [Qu, X. et al. Proc. Natl. Acad. Sci. U.S.A. 2008, 105, 6602-6607], the efficiency of fluorescence resonance energy transfer (FRET) between probes on the L18 loop and 3' terminus of the 260 nucleotide RNase P RNA from Bacillus stearothermophilus was found to exhibit complex kinetics that depended on the (periodically alternating) concentration of magnesium ions ([Mg2+]) in solution. Specifically, this time series was found to exhibit a quasi-periodic response to a square-wave pattern of [Mg2+] changes. Because these experiments directly probe only one of the many degrees of freedom in the macromolecule, models are needed to interpret these data. We find that Hidden Markov Models are inadequate for describing the nonequilibrium dynamics, but they serve as starting points for the construction of models in which a discrete observable degree of freedom is coupled to a continuously evolving (hidden) variable. Consideration of several models of this general form indicates that the quasi-periodic response in the nonequilibrium experiments results from the switching (back and forth) in positions of the minima of the effective potential for the hidden variable. This switching drives oscillation of that variable and synchronizes the population to the changing [Mg2+]. We set the models in the context of earlier theoretical and experimental studies and conclude that single-molecule experiments with periodic peturbations can indeed yield qualitatively new information beyond that obtained at equilibrium.

Single-Molecule Fluorescence Studies of RNA: A Decade's Progress

Over the past decade, single-molecule fluorescence studies have elucidated the structure-function relationship of RNA molecules. The real-time observation of individual RNAs by single-molecule fluorescence has unveiled the dynamic behavior of complex RNA systems in unprecedented detail, revealing the presence of transient intermediate states and their kinetic pathways. This review provides an overview of how single-molecule fluorescence has been used to explore the dynamics of RNA folding and catalysis.

Energy barriers, pathways, and dynamics during folding of large, multidomain RNAs

Large, multidomain RNA molecules are generally thought to fold following multiple pathways down rugged landscapes populated with intermediates and traps. A challenge to understanding RNA folding reactions is the complex relationships that exist between the structure of the RNA and its folding landscape. The identification of intermediate species that populate folding landscapes and characterization of elements of their structures are the key components to solving the RNA folding problem. This review explores recent studies that characterize the dominant pathways by which RNA folds, structural and dynamic features of intermediates that populate the folding landscape, and the energy barriers that separate the distinct steps of the folding process.

Single-molecule nonequilibrium periodic Mg2+-concentration jump experiments reveal details of the early folding pathways of a large RNA

The evolution of RNA conformation with Mg(2+) concentration ([Mg(2+)]) is typically determined from equilibrium titration measurements or nonequilibrium single [Mg(2+)]-jump measurements. We study the folding of single RNA molecules in response to a series of periodic [Mg(2+)] jumps. The 260-residue catalytic domain of RNase P RNA from Bacillus stearothermophilus is immobilized in a microfluidic flow chamber, and the RNA conformational changes are probed by fluorescence resonance energy transfer (FRET). The kinetics of population redistribution after a [Mg(2+)] jump and the observed connectivity of FRET states reveal details of the folding pathway that complement and transcend information from equilibrium or single-jump measurements. FRET trajectories for jumps from [Mg(2+)] = 0.01 to 0.1 mM exhibit two-state behavior whereas jumps from 0.01 mM to 0.4 mM exhibit two-state unfolding but multistate folding behavior. RNA molecules in the low and high FRET states before the [Mg(2+)] increase are observed to undergo dynamics in two distinct regions of the free energy landscape separated by a high barrier. We describe the RNA structural changes involved in crossing this barrier as a "hidden" degree of freedom because the changes do not alter the detected FRET value but do alter the observed dynamics. The associated memory prevents the populations from achieving their equilibrium values at the end of the 5- to 10-sec [Mg(2+)] interval, thereby creating a nonequilibrium steady-state condition. The capability of interrogating nonequilibrium steady-state RNA conformations and the adjustable period of [Mg(2+)]-jump cycles makes it possible to probe regions of the free energy landscape that are infrequently sampled in equilibrium or single-jump measurements.