Towards an action principle governing biopolymer foldingin vitro

The exploration of conformation space performed by biopolymers is biased towards a confined region. This property is paramount in providing theoretical underpinnings of the time constrained nature of folding. By introducing anaction principle in the space of folding pathways, we show how the folding process is guided expeditiously within realistic time frames.     

Prediction of conformationally dependent atomic multipole moments in carbohydrates

The conformational flexibility of carbohydrates is challenging within the field of computational chemistry. This flexibility causes the electron density to change, which leads to fluctuating atomic multipole moments. Quantum Chemical Topology (QCT) allows for the partitioning of an “atom in a molecule,” thus localizing electron density to finite atomic domains, which permits the unambiguous evaluation of atomic multipole moments. By selecting an ensemble of physically realistic conformers of a chemical system, one evaluates the various multipole moments at defined points in configuration space. The subsequent implementation of the machine learning method kriging delivers the evaluation of an analytical function, which smoothly interpolates between these points. This allows for the prediction of atomic multipole moments at new points in conformational space, not trained for but within prediction range. In this work, we demonstrate that the carbohydrates erythrose and threose are amenable to the above methodology. We investigate how kriging models respond when the training ensemble incorporating multiple energy minima and their environment in conformational space. Additionally, we evaluate the gains in predictive capacity of our models as the size of the training ensemble increases. We believe this approach to be entirely novel within the field of carbohydrates. For a modest training set size of 600, more than 90% of the external test configurations have an error in the total (predicted) electrostatic energy (relative to ab initio) of maximum 1 kJ mol^?1 for open chains and just over 90% an error of maximum 4 kJ mol^?1 for rings. © 2015 Wiley Periodicals, Inc.     

Direct measurement of tertiary contact cooperativity in RNA folding

All structured biological macromolecules must overcome the thermodynamic folding problem to populate a unique functional state among a vast ensemble of unfolded and alternate conformations. The exploration of cooperativity in protein folding has helped reveal and distinguish the underlying mechanistic solutions to this folding problem. Analogous dissections of RNA tertiary stability remain elusive, however, despite the central biological importance of folded RNA molecules and the potential to reveal fundamental properties of structured macromolecules via comparisons of protein and RNA folding. We report a direct quantitative measure of tertiary contact cooperativity in a folded RNA. We precisely measured the stability of an independently folding P4-P6 domain from the Tetrahymena thermophila group I intron by single molecule fluorescence resonance energy transfer (smFRET). Using wild-type and mutant RNAs, we found that cooperativity between the two tertiary contacts enhances P4-P6 stability by 3.2 +/- 0.2 kcal/mol.     

Simulation of chaperonin effect on protein folding: a shift from nucleation-condensation to framework mechanism

The iterative annealing mechanism (IAM) of chaperonin-assisted protein folding is explored in a framework of a well-established coarse-grained protein modeling tool, which enables the study of protein dynamics in a time-scale well beyond classical all-atom molecular mechanics. The chaperonin mechanism of action is simulated for two paradigm systems of protein folding, B domain of protein A (BdpA) and B1 domain of protein G (GB1), and compared to chaperonin-free simulations presented here for BdpA and recently published for GB1. The prediction of the BdpA transition state ensemble (TSE) is in perfect agreement with experimental findings. It is shown that periodic distortion of the polypeptide chains by hydrophobic chaperonin interactions can promote rapid folding and leads to a decrease in folding temperature. It is also demonstrated how chaperonin action prevents kinetically trapped conformations and modulates the observed folding mechanisms from nucleation-condensation to a more framework-like.     

Optimum folding pathways of proteins: their determination and properties

We develop a dynamic optimization technique for determining optimum folding pathways of proteins starting from different initial configurations. A coarse-grained Go model is used. Forces acting on each bead are (i) the friction force, (ii) forces from bond length constraints, (iii) excluded volume constraints, and (iv) attractive forces between residue pairs that are in contact in the native state. An objective function is defined as the total attractive energy between nonbonded residues, which are neighbors in the native state. The objective function is minimized over all feasible paths, satisfying bond length and excluded volume constraints. The optimization problem is nonconvex and contains a large number of constraints. An augmented Lagrangian method with a penalty barrier function was used to solve the problem. The method is applied to a 36-residue protein, chicken villin headpiece. Sequences of events during folding of the protein are determined for various pathways and analyzed. The relative time scales are compared and scaled according to experimentally measured events. Formation times of the helices, turn, and the loop agree with experimental data. We obtain the overall folding time of the protein in the range of 600 ns-1.2 micros that is smaller than the experimental result of 4-5 micros, showing that the optimal folding times that we obtain may be possible lower bounds. Time dependent variables during folding and energies associated with short- and long-range interactions between secondary structures are analyzed in modal space using Karhunen-Loeve expansion.     

The reweighted path ensemble

We introduce a reweighting scheme for the path ensembles in the transition interface sampling framework. The reweighting allows for the analysis of free energy landscapes and committor projections in any collective variable space. We illustrate the reweighting scheme on a two dimensional potential with a nonlinear reaction coordinate and on a more realistic simulation of the Trp-cage folding process. We suggest that the reweighted path ensemble can be used to optimize possible nonlinear reaction coordinates.     

A measure of folding complexity for d-dimensional polymers

A measure of folding characterizes aspects of the instantaneous organization of a polymer chain in space. For three-dimensional polymers (D = 3), one such measure is the mean overcrossing number. An intuitively similar property, the radial intersection number, has been proposed as a tool to characterize "folding features" in two-dimensional polymers (D = 2). In this work, we show rigorously that these measures are indeed related and that they can be derived as particular cases within a single, unified formulation. The present approach provides an analytical expression for a measure of folding complexity that can be applied to generic D-dimensional polymers. In the case D = 2, we show results for models derived from experimental structures by using optimized multidimensional scaling transformations for data compression.     

Discrete structure of an RNA folding intermediate revealed by cryo-electron microscopy

RNA folding occurs via a series of transitions between metastable intermediate states. It is unknown whether folding intermediates are discrete structures folding along defined pathways or heterogeneous ensembles folding along broad landscapes. We use cryo-electron microscopy and single-particle image reconstruction to determine the structure of the major folding intermediate of the specificity domain of a ribonuclease P ribozyme. Our results support the existence of a discrete conformation for this folding intermediate.     

Folding of human telomerase RNA pseudoknot using ion-jump and temperature-quench simulations

Globally RNA folding occurs in multiple stages involving chain compaction and subsequent rearrangement by a number of parallel routes to the folded state. However, the sequence-dependent details of the folding pathways and the link between collapse and folding are poorly understood. To obtain a comprehensive picture of the thermodynamics and folding kinetics we used molecular simulations of coarse-grained model of a pseudoknot found in the conserved core domain of the human telomerase (hTR) by varying both temperature (T) and ion concentration (C). The phase diagram in the [T,C] plane shows that the boundary separating the folded and unfolded state for the finite 47-nucleotide system is relatively sharp, implying that from a thermodynamic perspective hTR behaves as an apparent two-state system. However, the folding kinetics following single C-jump or T-quench is complicated, involving multiple channels to the native state. Although globally folding kinetics triggered by T-quench and C-jump are similar, the kinetics of chain compaction are vastly different, which reflects the role of initial conditions in directing folding and collapse. Remarkably, even after substantial reduction in the overall size of hTR, the ensemble of compact conformations are far from being nativelike, suggesting that the search for the folded state occurs among the ensemble of low-energy fluidlike globules. The rate of unfolding, which occurs in a single step, is faster upon C-decrease compared to a jump in temperature. To identify "hidden" states that are visited during the folding process we performed simulations by periodically interrupting the approach to the folded state by lowering C. These simulations show that hTR reaches the folded state through a small number of connected clusters that are repeatedly visited during the pulse sequence in which the folding or unfolding is interrupted. The results from interrupted folding simulations, which are in accord with non-equilibrium single-molecule folding of a large ribozyme, show that multiple probes are needed to reveal the invisible states that are sampled by RNA as it folds. Although we have illustrated the complexity of RNA folding using hTR as a case study, general arguments and qualitative comparisons to time-resolved scattering experiments on Azoarcus group I ribozyme and single-molecule non-equilibrium periodic ion-jump experiments establish the generality of our findings.     

Hidden protein folding pathways in free-energy landscapes uncovered by network analysis

A network analysis is used to uncover hidden folding pathways in free-energy landscapes usually defined in terms of such arbitrary order parameters as root-mean-square deviation from the native structure, radius of gyration, etc. The analysis has been applied to molecular dynamics (MD) trajectories of the B-domain of staphylococcal protein A, generated with the coarse-grained united-residue (UNRES) force field in a broad range of temperatures (270K ≤ T ≤ 325K). Thousands of folding pathways have been identified at each temperature. Out of these many folding pathways, several most probable ones were selected for investigation of the conformational transitions during protein folding. Unlike other conformational space network (CSN) methods, a node in the CSN variant implemented in this work is defined according to the nativelikeness class of the structure, which defines the similarity of segments of the compared structures in terms of secondary-structure, contact-pattern, and local geometry, as well as the overall geometric similarity of the conformation under consideration to that of the reference (experimental) structure. Our previous findings, regarding the folding model and conformations found at the folding-transition temperature for protein A (Maisuradze et al., J. Am. Chem. Soc. 132, 9444, 2010), were confirmed by the conformational space network analysis. In the methodology and in the analysis of the results, the shortest path identified by using the shortest-path algorithm corresponds to the most probable folding pathway in the conformational space network.     

Ant colony optimization for predicting RNA folding pathways

RNA folding dynamics plays important roles in various functions of RNAs. To date, coarse-grained modeling has been successfully employed to simulate RNA folding dynamics on the energy landscape composed of secondary structures. In such a modeling, the energy barrier height between metastable structures is a key parameter that crucially affects the simulation results. Although a number of approaches ranging from the exact method to heuristic ones are available to predict the barrier heights, developing an efficient heuristic for this purpose is still an algorithmic challenge.We developed a novel RNA folding pathway prediction method, ACOfoldpath, based on Ant Colony Optimization (ACO). ACO is a widely used powerful combinatorial optimization algorithm inspired from the food-seeking behavior of ants. In ACOfoldpath, to accelerate the folding pathway prediction, we reduce the search space by utilizing originally devised structure generation rules. To evaluate the performance of the proposed method, we benchmarked ACOfoldpath on the known nineteen conformational RNA switches. As a result, ACOfoldpath successfully predicted folding pathways better than or comparable to the previous heuristics. The results of RNA folding dynamics simulations and pseudoknotted pathway predictions are also presented.     

Kinetic partitioning mechanism as a unifying theme in the folding of biomolecules

We describe a unified approach to describe the kinetics of protein and RNA folding. The underlying conceptual basis for this framework relies on the notion that biomolecules are topologically frustrated due to their polymeric nature and due to the presence of conflicting energies. As a result, the free energy surface (FES) has, in addition to the native basin of attraction (NBA), several competing basins of attraction. A rough FES results in direct and indirect pathways to the NBA, i.e., a kinetic partitioning mechanism (KPM). The KPM leads to a foldability principle according to which fast folding sequences are characterized by the folding transition temperature T _F being close to the collapse transition temperature T _θ, at which a transition from the random coil to the compact structure takes place. Biomolecules with T _θ ≈ T _F , such as small proteins and tRNAs, are expected to fold rapidly with two-state kinetics. Estimates for the multiple time scales in KPM are also given. We show that experiments on proteins and RNA can be understood semi-quantitatively in terms of the KPM. Received: 14 January 1997 / Accepted: 23 January 1997     

Tertiary interactions determine the accuracy of RNA folding

RNAs must fold into unique three-dimensional structures to function in the cell, but how each polynucleotide finds its native structure is not understood. To investigate whether the stability of the tertiary structure determines the speed and accuracy of RNA folding, docking of a tetraloop with its receptor in a bacterial group I ribozyme was perturbed by site-directed mutagenesis. Disruption of the tetraloop or its receptor destabilizes tertiary interactions throughout the ribozyme by 2-3 kcal/mol, demonstrating that tertiary interactions form cooperatively in the transition from a native-like intermediate to the native state. Nondenaturing PAGE and RNase T1 digestion showed that base pairs form less homogeneously in the mutant RNAs during the transition from the unfolded state to the intermediate. Thus, tertiary interactions between helices bias the ensemble of secondary structures toward native-like conformations. Time-resolved hydroxyl radical footprinting showed that the wild-type ribozyme folds completely within 5-20 ms. By contrast, only 40-60% of a tetraloop mutant ribozyme folds in 30-40 ms, with the remainder folding in 30-200 s via nonnative intermediates. Therefore, destabilization of tetraloop-receptor docking introduces an alternate folding pathway in the otherwise smooth energy landscape of the wild-type ribozyme. Our results show that stable tertiary structure increases the flux through folding pathways that lead directly and rapidly to the native structure.     

Darobactin B Stabilises a Lateral-Closed Conformation of the BAM Complex in E. coli Cells

The β-barrel assembly machinery (BAM complex) is essential for outer membrane protein (OMP) folding in Gram-negative bacteria, and represents a promising antimicrobial target. Several conformational states of BAM have been reported, but all have been obtained under conditions which lack the unique features and complexity of the outer membrane (OM). Here, we use Pulsed Electron-Electron Double Resonance (PELDOR, or DEER) spectroscopy distance measurements to interrogate the conformational ensemble of the BAM complex in E. coli cells. We show that BAM adopts a broad ensemble of conformations in the OM, while in the presence of the antibiotic darobactin B (DAR-B), BAM′s conformational equilibrium shifts to a restricted ensemble consistent with the lateral closed state. Our in-cell PELDOR findings are supported by new cryoEM structures of BAM in the presence and absence of DAR-B. This work demonstrates the utility of PELDOR to map conformational changes in BAM within its native cellular environment.     

Click Chemistry Assisted Single-Molecule Fingerprinting Reveals a 3D Biomolecular Folding Funnel

A 3D folding funnel was proposed in the 1990s to explain the fast kinetics exhibited by a biomacromolecule in presence of seemingly unlimited folding pathways. Over the years, numerous simulations have been performed with this concept; however, experimental verification is yet to be attained even for the simplest proteins. Here, we have used a click chemistry based strategy to introduce six pairs of handles in a human telomeric DNA sequence. A laser-tweezers-based, single-molecule structural fingerprinting on the six inter-handle distances reveals the formation of a hybrid-1 G-quadruplex in the sequence. Kinetic and thermodynamic fingerprinting on the six trajectories defined by each handle-pair depict a 3D folding funnel and a kinetic topology in which the kinetics pertaining to each handle residue is annotated for this G-quadruplex. We anticipate the methods and the concepts developed here are well applicable to other biomacromolecules, including RNA and proteins.     

Perturbation theory of Phi-value analysis of two-state protein folding: relation between p fold and Phi values

In protein folding, the transition state ensemble is defined as the set of conformations with p(fold)=12, where the p(fold) of a conformation is the probability that starting from this conformation the protein folds before it unfolds. Experimentally, this ensemble is probed by the Phi-value analysis, where Phi is the ratio of the changes in the logarithms of the folding rate and the equilibrium constant when the system is perturbed by a mutation. We show that for a two-state protein the Phi value can be expressed in terms of the perturbation and only the first two eigenfunctions of the evolution operator (e.g., a rate matrix) of the wild-type protein. The first eigenfunction is the equilibrium probability distribution while the second is proportional to p(fold), thus establishing a formal relation between p(fold) and Phi values. In addition to providing insight into the theoretical foundation of the Phi-value analysis, our results may prove practically useful in performing such analyses within the framework of models containing a large number of states.     

Fast folding of RNA pseudoknots initiated by laser temperature-jump

RNA pseudoknots are examples of minimal structural motifs in RNA with tertiary interactions that stabilize the structures of many ribozymes. They also play an essential role in a variety of biological functions that are modulated by their structure, stability, and dynamics. Therefore, understanding the global principles that determine the thermodynamics and folding pathways of RNA pseudoknots is an important problem in biology, both for elucidating the folding mechanisms of larger ribozymes as well as addressing issues of possible kinetic control of the biological functions of pseudoknots. We report on the folding/unfolding kinetics of a hairpin-type pseudoknot obtained with microsecond time-resolution in response to a laser temperature-jump perturbation. The kinetics are monitored using UV absorbance as well as fluorescence of extrinsically attached labels as spectroscopic probes of the transiently populated RNA conformations. We measure folding times of 1-6 ms at 37 °C, which are at least 100-fold faster than previous observations of very slow folding pseudoknots that were trapped in misfolded conformations. The measured relaxation times are remarkably similar to predictions of a computational study by Thirumalai and co-workers (Cho, S. S.; Pincus, D.L.; Thirumalai, D. Proc. Natl. Acad. Sci. U. S. A. 2009, 106, 17349-17354). Thus, these studies provide the first observation of a fast-folding pseudoknot and present a benchmark against which computational models can be refined.     

Exploring the free-energy landscape of a short peptide using an average force

The reversible folding of deca-alanine is chosen as a test case for characterizing a method that uses an adaptive biasing force (ABF) to escape from the minima and overcome the barriers of the free-energy landscape. This approach relies on the continuous estimation of a biasing force that yields a Hamiltonian in which no average force is exerted along the ordering parameter xi. Optimizing the parameters that control how the ABF is applied, the method is shown to be extremely effective when a nonequivocal ordering parameter can be defined to explore the folding pathway of the peptide. Starting from a beta-turn motif and restraining xi to a region of the conformational space that extends from the alpha-helical state to an ensemble of extended structures, the ABF scheme is successful in folding the peptide chain into a compact alpha helix. Sampling of this conformation is, however, marginal when the range of xi values embraces arrangements of greater compactness, hence demonstrating the inherent limitations of free-energy methods when ambiguous ordering parameters are utilized.