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 Physics , 2004, DOI: 10.1140/epjb/e2004-00201-3 Abstract: We consider a simple model for the unfolding of RNA tertiary structure under dynamic loading. The opening of such a structure is regarded as a two step process, each corresponding to the overcoming of a single energy barrier. The resulting two-barrier energy landscape accounts for the dependence of the unfolding kinetics on the pulling rate. Furthermore at intermediate force, the two barriers cannot be distinguished by the analysis of the opening kinetic, which turns out to be dominated by a single macro-barrier, whose properties depend non-trivially on the two single barriers. Our results suggest that in pulling experiments on RNA molecule containing tertiary structures, the details of the single kinetic barriers can only be obtained using a low pulling rate value, or in the high force regime.
 Quantitative Biology , 2003, DOI: 10.1073/pnas.0330720100 Abstract: The energy landscapes of proteins have evolved to be different from most random heteropolymers. Many studies have concluded that evolutionary selection for rapid and reliable folding to a given structure that is stable at biological temperatures leads to energy landscapes having a single dominant basin and an overall funnel topography. We show here that, while such a landscape topography is indeed a sufficient condition for folding, another possibility also exists, giving a new class of foldable sequences. These sequences have landscapes that are only weakly funneled in the conventional thermodynamic sense, but have unusually low kinetic barriers for reconfigurational motion. Traps have been specifically removed by selection. Here we examine the possibility of folding on these "buffed" landscapes, by mapping the determination of statistics of pathways for the heterogeneous nucleation processes involved in escaping from traps to the solution of an imaginary time Schroedinger equation. This equation is solved analytically in adiabatic and soft-wall'' approximations, and numerical results are shown for the general case. The fraction of funneled vs. buffed proteins in sequence space is estimated, suggesting the statistical dominance of the funneling mechanism for achieving foldability.
 Wilfred Ndifon Physics , 2005, DOI: 10.1016/j.biosystems.2005.08.004 Abstract: The kinetic folding of RNA sequences into secondary structures is modeled as a complex adaptive system, the components of which are possible RNA structural rearrangements (SRs) and their associated bases and base pairs. RNA bases and base pairs engage in local stacking interactions that determine the probabilities (or fitnesses) of possible SRs. Meanwhile, selection operates at the level of SRs; an autonomous stochastic process periodically (i.e., from one time step to another) selects a subset of possible SRs for realization based on the fitnesses of the SRs. Using examples based on selected natural and synthetic RNAs, the model is shown to qualitatively reproduce characteristic (nonlinear) RNA folding dynamics such as the attainment by RNAs of alternative stable states. Possible applications of the model to the analysis of properties of fitness landscapes, and of the RNA sequence to structure mapping are discussed.
 Quantitative Biology , 2012, DOI: 10.1021/jz301537t Abstract: Non-coding RNA sequences play a great role in controlling a number of cellular functions, thus raising the need to understand their complex conformational dynamics in quantitative detail. In this perspective, we first show that single molecule pulling experiments when combined with with theory and simulations can be used to quantitatively explore the folding landscape of nucleic acid hairpins, and riboswitches with tertiary interactions. Applications to riboswitches, which are non-coding RNA elements that control gene expression by undergoing dynamical conformational changes in response to binding of metabolites, lead to an organization principle that assembly of RNA is determined by the stability of isolated helices. We also point out the limitations of single molecule pulling experiments, with molecular extension as the only accessible parameter, in extracting key parameters of the folding landscapes of RNA molecules.
 PLOS Computational Biology , 2010, DOI: 10.1371/journal.pcbi.1000835 Abstract: The folding pathway and rate coefficients of the folding of a knotted protein are calculated for a potential energy function with minimal energetic frustration. A kinetic transition network is constructed using the discrete path sampling approach, and the resulting potential energy surface is visualized by constructing disconnectivity graphs. Owing to topological constraints, the low-lying portion of the landscape consists of three distinct regions, corresponding to the native knotted state and to configurations where either the N or C terminus is not yet folded into the knot. The fastest folding pathways from denatured states exhibit early formation of the N terminus portion of the knot and a rate-determining step where the C terminus is incorporated. The low-lying minima with the N terminus knotted and the C terminus free therefore constitute an off-pathway intermediate for this model. The insertion of both the N and C termini into the knot occurs late in the folding process, creating large energy barriers that are the rate limiting steps in the folding process. When compared to other protein folding proteins of a similar length, this system folds over six orders of magnitude more slowly.
 Algorithms , 2009, DOI: 10.3390/a2010200 Abstract: Unlike protein folding, the process by which a large RNA molecule adopts a functionally active conformation remains poorly understood. Chemical mapping techniques, such as Hydroxyl Radical (·OH) footprinting report on local structural changes in an RNA as it folds with single nucleotide resolution. The analysis and interpretation of this kinetic data requires the identification and subsequent optimization of a kinetic model and its parameters. We detail our approach to this problem, specifically focusing on a novel strategy to overcome a factorial explosion in the number of possible models that need to be tested to identify the best fitting model. Previously, smaller systems (less than three intermediates) were computationally tractable using a distributed computing approach. However, for larger systems with three or more intermediates, the problem became computationally intractable. With our new enumeration strategy, we are able to significantly reduce the number of models that need to be tested using non-linear least squares optimization, allowing us to study systems with up to five intermediates. Furthermore, two intermediate systems can now be analyzed on a desktop computer, which eliminates the need for a distributed computing solution for most mediumsized data sets. Our new approach also allows us to study potential degeneracy in kinetic model selection, elucidating the limits of the method when working with large systems. This work establishes clear criteria for determining if experimental ·OH data is sufficient to determine the underlying kinetic model, or if other experimental modalities are required to resolve any degeneracy.
 Journal of Nucleic Acids Investigation , 2012, DOI: 10.4081/2652 Abstract: The concept of a free energy (FE) landscape, in which the surface spirals down like a funnel, has long been viewed as the reason why a complex protein structure forms so rapidly compared to the number of conformations available to it. On the other hand, this landscape picture is less clear with RNA due to the multiplicity of conformations and the uncertainties in the current thermodynamics. It is therefore sometimes proposed that the ensemble average is the main factor deciding the structure. However, calculations of the free energy of observed structures often suggest that this ensemble is far from the minimum FE, particularly in the case of long sequences. If so, then such a FE surface is unlikely to be funnel shaped. We have been developing a version of vsfold that can evaluate the suboptimal structures of the FE surface (vs_subopt). Here we show that the ensemble for a number of known RNA structures can actually be both close to the minimum FE and also be the dominant observed structure when a proper Kuhn length is selected. Two state aptamers known as riboswitches can show neighboring FE states in the suboptimal structures that match the observed structures and their relative difference in FE is well within the range of the binding free energy of the metabolite. For the riboswitches and other short RNA sequences (less than 250 nt), the flow of the suboptimal structures (including pseudoknots) tended to resemble a rock rolling down a hill along the reaction coordinate axis.
 PLOS ONE , 2012, DOI: 10.1371/journal.pone.0012953 Abstract: Due to the energetic frustration of RNA folding, tertiary structured RNA is typically characterized by a rugged folding free energy landscape where deep kinetic barriers separate numerous misfolded states from one or more native states. While most in vitro studies of RNA rely on (re)folding chemically and/or enzymatically synthesized RNA in its entirety, which frequently leads into kinetic traps, nature reduces the complexity of the RNA folding problem by segmental, co-transcriptional folding starting from the 5′ end. We here have developed a simplified, general, nondenaturing purification protocol for RNA to ask whether avoiding denaturation of a co-transcriptionally folded RNA can reduce commonly observed in vitro folding heterogeneity. Our protocol bypasses the need for large-scale auxiliary protein purification and expensive chromatographic equipment and involves rapid affinity capture with magnetic beads and removal of chemical heterogeneity by cleavage of the target RNA from the beads using the ligand-induced glmS ribozyme. For two disparate model systems, the Varkud satellite (VS) and hepatitis delta virus (HDV) ribozymes, we achieve >95% conformational purity within one hour of enzymatic transcription, without the need for any folding chaperones. We further demonstrate that in vitro refolding introduces severe conformational heterogeneity into the natively-purified VS ribozyme but not into the compact, double-nested pseudoknot fold of the HDV ribozyme. We conclude that conformational heterogeneity in complex RNAs can be avoided by co-transcriptional folding followed by nondenaturing purification, providing rapid access to chemically and conformationally pure RNA for biologically relevant biochemical and biophysical studies.
 Quantitative Biology , 2010, DOI: 10.1021/jp107495f Abstract: The influence of the salts KCl, NaCl, and NaI at molar concentrations on the {\alpha}-helical folding kinetics of the alanine-based oligopeptide Ace-AEAAAKEAAAKA-Nme is investigated by means of (explicit-water) molecular dynamics simulations and a diffusional analysis. The mean first passage times for folding and unfolding are found to be highly salt-specific. In particular, the folding times increase about one order of magnitude for the sodium salts. The drastic slowing down can be traced back to long-lived, compact configurations of the partially folded peptide, in which sodium ions are tightly bound by several carbonyl and carboxylate groups. This multiple trapping is found to lead to a non-exponential residence time distribution of the cations in the first solvation shell of the peptide. The analysis of {\alpha}-helical folding in the framework of diffusion in a reduced (one-dimensional) free energy landscape further shows that the salt not only specifically modifies equilibrium properties, but also induces kinetic barriers due to individual ion binding. In the sodium salts, for instance, the peptide's configurational mobility (or "diffusivity") can decrease about one order of magnitude. This study demonstrates the highly specific action of ions and highlights the intimate coupling of intramolecular friction and solvent effects in protein folding.
 T. C. B. McLeish Physics , 2003, Abstract: We explore the consequences of very high dimensionality in the dynamical landscape of protein folding. Consideration of both typical range of stabilising interactions, and folding rates themselves, leads to a model of the energy hypersurface that is characterised by the structure of diffusive "hypergutters" as well as the familiar "funnels". Several general predictions result: (1) intermediate subspaces of configurations will always be visited; (2) specific but non-native interactions are important in stabilising these low-dimensional diffusive searches on the folding pathway; (3) sequential barriers will commonly be found, even in "two-state"proteins; (4) very early times will show charactreristic departures from single-exponential kinetics; (5) contributions of non-native interactions to phi-values are calculable, and may be significant. The example of a three-helix bundle is treated in more detail as an illustration. The model also shows that high-dimensional structures provide conceptual relations between the "folding funnel", "diffusion-collision", "nucleation-condensation" and "topomer search" models of protein folding. It suggests that kinetic strategies for fast folding may be encoded rather generally in non-native, rather than native interactions. The predictions are related to very recent findings in experiment and simulation.
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