Shape changes and cooperativity in the folding of the central domain of the 16S ribosomal RNA.

Both the small and large subunits of the ribosome, the molecular machine that synthesizes proteins, are complexes of ribosomal RNAs (rRNAs) and a number of proteins. In bacteria, the small subunit has a single 16S rRNA whose folding is the first step in its assembly. The central domain of the 16S rRNA folds independently, driven either by Mg2+ ions or by interaction with ribosomal proteins. To provide a quantitative description of ion-induced folding of the ∼350-nucleotide rRNA, we carried out extensive coarse-grained molecular simulations spanning Mg2+ concentration between 0 and 30 mM. The Mg2+ dependence of the radius of gyration shows that globally the rRNA folds cooperatively. Surprisingly, various structural elements order at different Mg2+ concentrations, indicative of the heterogeneous assembly even within a single domain of the rRNA. Binding of Mg2+ ions is highly specific, with successive ion condensation resulting in nucleation of tertiary structures. We also predict the Mg2+-dependent protection factors, measurable in hydroxyl radical footprinting experiments, which corroborate the specificity of Mg2+-induced folding. The simulations, which agree quantitatively with several experiments on the folding of a three-way junction, show that its folding is preceded by formation of other tertiary contacts in the central junction. Our work provides a starting point in simulating the early events in the assembly of the small subunit of the ribosome.

Ion Condensation onto Ribozyme Is Site Specific and Fold Dependent.

The highly charged RNA molecules, with each phosphate carrying a single negative charge, cannot fold into well-defined architectures with tertiary interactions in the absence of ions. For ribozymes, divalent cations are known to be more efficient than monovalent ions in driving them to a compact state, although Mg2+ ions are needed for catalytic activities. Therefore, how ions interact with RNA is relevant in understanding RNA folding. It is often thought that most of the ions are territorially and nonspecifically bound to the RNA, as predicted by the counterion condensation theory. Here, we show using simulations of Azoarcus ribozyme, based on an accurate coarse-grained three-site interaction model with explicit divalent and monovalent cations, that ion condensation is highly specific and depends on the nucleotide position. The regions with high coordination between the phosphate groups and the divalent cations are discernible even at very low Mg2+ concentrations when the ribozyme does not form tertiary interactions. Surprisingly, these regions also contain the secondary structural elements that nucleate subsequently in the self-assembly of RNA, implying that ion condensation is determined by the architecture of the folded state. These results are in sharp contrast to interactions of ions (monovalent and divalent) with rigid charged rods, in which ion condensation is uniform and position independent. The differences are explained in terms of the dramatic nonmonotonic shape fluctuations in the ribozyme as it folds with increasing Mg2+ or Ca2+ concentration.

Molecular Simulations of Ion Effects on the Thermodynamics of RNA Folding.

How ions affect RNA folding thermodynamics and kinetics is an important but a vexing problem that remains unsolved. Experiments have shown that the free-energy change, Δ G( c), of RNA upon folding varies with the salt concentration ( c) as, Δ G( c) = k c ln c + const, where the coefficient k c is proportional to the difference in the ion preferential coefficient, ΔΓ. We performed simulations of a coarse-grained model, by modeling electrostatic interactions implicitly and with explicit representation of ions, to elucidate the molecular underpinnings of the relationship between Δ G and ΔΓ. The simulations quantitatively reproduce the heat capacity for a pseudoknot, thus validating the model. We show that Δ G( c), calculated directly from ΔΓ, varies linearly with ln c ( c < 0.2 M), for a hairpin and the pseudoknot, demonstrating a molecular link between the two quantities. Explicit ion simulations also show the linear dependence of Δ G( c) on ln c at all c with k c = 2 kB T, except that Δ G( c) values are shifted by ∼2 kcal/mol higher than experiments. The discrepancy is due to an underestimation of Γ for both the folded and unfolded states while giving accurate values for ΔΓ. The predictions for the salt dependence of ΔΓ are amenable to test using single-molecule pulling experiments. The framework provided here can be used to obtain accurate thermodynamics for other RNA molecules as well.

Frictional Effects on RNA Folding: Speed Limit and Kramers Turnover.

We investigated frictional effects on the folding rates of a human telomerase hairpin (hTR HP) and H-type pseudoknot from the Beet Western Yellow Virus (BWYV PK) using simulations of the Three Interaction Site (TIS) model for RNA. The heat capacity from TIS model simulations, calculated using temperature replica exchange simulations, reproduces nearly quantitatively the available experimental data for the hTR HP. The corresponding results for BWYV PK serve as predictions. We calculated the folding rates ( kF) from more than 100 folding trajectories for each value of the solvent viscosity (η) at a fixed salt concentration of 200 mM. By using the theoretical estimate (∝ √N where N is the number of nucleotides) for folding free energy barrier, kF data for both the RNAs are quantitatively fit using one-dimensional Kramers's theory with two parameters specifying the curvatures in the unfolded basin and the barrier top. In the high-friction regime (η ≳ 10-5 Pa·s), for both HP and PK, kF values decrease as 1/η, whereas in the low friction regime, kF values increase as η increases, leading to a maximum folding rate at a moderate viscosity (∼10-6 Pa·s), which is the Kramers turnover. From the fits, we find that the speed limit to RNA folding at water viscosity is between 1 and 4 µs, which is in accord with our previous theoretical prediction as well as results from several single molecule experiments. Both the RNA constructs fold by parallel pathways. Surprisingly, we find that the flux through the pathways could be altered by changing solvent viscosity, a prediction that is more easily testable in RNA than in proteins.

Salt Effects on the Thermodynamics of a Frameshifting RNA Pseudoknot under Tension.

Because of the potential link between -1 programmed ribosomal frameshifting and response of a pseudoknot (PK) RNA to force, a number of single-molecule pulling experiments have been performed on PKs to decipher the mechanism of programmed ribosomal frameshifting. Motivated in part by these experiments, we performed simulations using a coarse-grained model of RNA to describe the response of a PK over a range of mechanical forces (fs) and monovalent salt concentrations (Cs). The coarse-grained simulations quantitatively reproduce the multistep thermal melting observed in experiments, thus validating our model. The free energy changes obtained in simulations are in excellent agreement with experiments. By varying f and C, we calculated the phase diagram that shows a sequence of structural transitions, populating distinct intermediate states. As f and C are changed, the stem-loop tertiary interactions rupture first, followed by unfolding of the 3'-end hairpin (I⇌F). Finally, the 5'-end hairpin unravels, producing an extended state (E⇌I). A theoretical analysis of the phase boundaries shows that the critical force for rupture scales as (logCm)(α) with α=1(0.5) for E⇌I (I⇌F) transition. This relation is used to obtain the preferential ion-RNA interaction coefficient, which can be quantitatively measured in single-molecule experiments, as done previously for DNA hairpins. A by-product of our work is the suggestion that the frameshift efficiency is likely determined by the stability of the 5'-end hairpin that the ribosome first encounters during translation.

How do metal ions direct ribozyme folding?

Ribozymes, which carry out phosphoryl-transfer reactions, often require Mg(2+) ions for catalytic activity. The correct folding of the active site and ribozyme tertiary structure is also regulated by metal ions in a manner that is not fully understood. Here we employ coarse-grained molecular simulations to show that individual structural elements of the group I ribozyme from the bacterium Azoarcus form spontaneously in the unfolded ribozyme even at very low Mg(2+) concentrations, and are transiently stabilized by the coordination of Mg(2+) ions to specific nucleotides. However, competition for scarce Mg(2+) and topological constraints that arise from chain connectivity prevent the complete folding of the ribozyme. A much higher Mg(2+) concentration is required for complete folding of the ribozyme and stabilization of the active site. When Mg(2+) is replaced by Ca(2+) the ribozyme folds, but the active site remains unstable. Our results suggest that group I ribozymes utilize the same interactions with specific metal ligands for both structural stability and chemical activity.

Coarse-grained model for predicting RNA folding thermodynamics.

We present a thermodynamically robust coarse-grained model to simulate folding of RNA in monovalent salt solutions. The model includes stacking, hydrogen bond, and electrostatic interactions as fundamental components in describing the stability of RNA structures. The stacking interactions are parametrized using a set of nucleotide-specific parameters, which were calibrated against the thermodynamic measurements for single-base stacks and base-pair stacks. All hydrogen bonds are assumed to have the same strength, regardless of their context in the RNA structure. The ionic buffer is modeled implicitly, using the concept of counterion condensation and the Debye-Hückel theory. The three adjustable parameters in the model were determined by fitting the experimental data for two RNA hairpins and a pseudoknot. A single set of parameters provides good agreement with thermodynamic data for the three RNA molecules over a wide range of temperatures and salt concentrations. In the process of calibrating the model, we establish the extent of counterion condensation onto the single-stranded RNA backbone. The reduced backbone charge is independent of the ionic strength and is 60% of the RNA bare charge at 37 °C. Our model can be used to predict the folding thermodynamics for any RNA molecule in the presence of monovalent ions.

Entropic stabilization of the folded states of RNA due to macromolecular crowding.

We review the effects of macromolecular crowding on the folding of RNA by considering the simplest scenario when excluded volume interactions between crowding particles and RNA dominate. Using human telomerase enzyme as an example, we discuss how crowding can alter the equilibrium between pseudoknot and hairpin states of the same RNA molecule-a key aspect of crowder-RNA interactions. We summarize data showing that the crowding effect is significant only if the size of the spherical crowding particle is smaller than the radius of gyration of the RNA in the absence of crowding particles. The implication for function of the wild type and mutants of human telomerase is outlined by using a relationship between enzyme activity and its conformational equilibrium. In addition, we discuss the interplay between macromolecular crowding and ionic strength of the RNA buffer. Finally, we briefly review recent experiments which illustrate the connection between excluded volume due to macromolecular crowding and the thermodynamics of RNA folding.

Noise associated with nonconservative forces in optical traps.

It is known that for a particle held in an optical trap the interaction of thermal fluctuations with a nonconservative scattering force can cause a persistent nonequilibrium probability flux in the particle position. We investigate position fluctuations associated with this nonequilibrium flux analytically and through simulation. We introduce a model which reproduces the nonequilibrium effects, and in which the magnitude of additional position fluctuations can be calculated in closed form. The ratio of additional nonconservative fluctuations to direct thermal fluctuations scales inversely with the square root of trap power, and is small for typical experimental parameters. In a simulated biophysical experiment the nonconservative scattering force does not significantly increase the observed fluctuations in the length of a double-stranded DNA tether.

Crowding promotes the switch from hairpin to pseudoknot conformation in human telomerase RNA.

Formation of a pseudoknot (PK) in the conserved RNA core domain in the ribonucleoprotein human telomerase is required for function. In vitro experiments show that the PK is in equilibrium with an extended hairpin (HP) structure. We use molecular simulations of a coarse-grained model, which reproduces most of the salient features of the experimental melting profiles of PK and HP, to show that crowding enhances the stability of PK relative to HP in the wild type and in a mutant associated with dyskeratosis congenita. In monodisperse suspensions, small crowding particles increase the stability of compact structures to a greater extent than larger crowders. If the sizes of crowders in a binary mixture are smaller than that of the unfolded RNA, the increase in melting temperature due to the two components is additive. In a ternary mixture of crowders that are larger than the unfolded RNA, which mimics the composition of ribosome, large enzyme complexes and proteins in Escherichia coli , the marginal increase in stability is entirely determined by the smallest component. We predict that crowding can partially restore telomerase activity in mutants with decreased PK stability.

Equilibrium and nonequilibrium effects in the collapse of a model polypeptide.

We present results of molecular simulations of a model polypeptide whose hydrophobic collapse proceeds as a cascade of downhill transitions between distinct intermediate states. Different intermediates are stabilized by means of appropriate harmonic constraints, allowing explicit calculation of the equilibrium free energy landscape. Nonequilibrium collapse trajectories are simulated independently and compared to diffusion on the calculated free energy surface. We find that collapse generally adheres to this surface, but quantitative agreement is complicated by nonequilibrium effects and by dependence of the diffusion coefficient on position on the surface.

A new approach for efficient simulation of Coulomb interactions in ionic fluids.

We propose a simplified version of local molecular field (LMF) theory to treat Coulomb interactions in simulations of ionic fluids. LMF theory relies on splitting the Coulomb potential into a short-ranged part that combines with other short-ranged core interactions and is simulated explicitly. The averaged effects of the remaining long-ranged part are taken into account through a self-consistently determined effective external field. The theory contains an adjustable length parameter sigma that specifies the cutoff distance for the short-ranged interaction. This can be chosen to minimize the errors resulting from the mean-field treatment of the complementary long-ranged part. Here we suggest that in many cases an accurate approximation to the effective field can be obtained directly from the equilibrium charge density given by the Debye theory of screening, thus eliminating the need for a self-consistent treatment. In the limit sigma-->0, this assumption reduces to the classical Debye approximation. We examine the numerical performance of this approximation for a simple model of a symmetric ionic mixture. Our results for thermodynamic and structural properties of uniform ionic mixtures agree well with similar results of Ewald simulations of the full ionic system. In addition, we have used the simplified theory in a grand-canonical simulation of a nonuniform ionic mixture where an ion has been fixed at the origin. Simulations using short-ranged truncations of the Coulomb interactions alone do not satisfy the exact condition of complete screening of the fixed ion, but this condition is recovered when the effective field is taken into account. We argue that this simplified approach can also be used in the simulations of more complex nonuniform systems.