Topological classification of RNA structures

We present a novel topological classification of RNA secondary structures with pseudoknots. It is based on the topological genus of the circular diagram associated to the RNA base-pair structure. The genus is a positive integer number whose value quantifies the topological complexity of the folded RNA structure. In such a representation, planar diagrams correspond to pure RNA secondary structures and have zero genus, whereas non-planar diagrams correspond to pseudoknotted structures and have higher genus. The topological genus allows for the definition of topological folding motifs, similar in spirit to those introduced and commonly used in protein folding. We analyze real RNA structures from the databases Worldwide Protein Data Bank and Pseudobase and classify them according to their topological genus. For simplicity, we limit our analysis by considering only Watson-Crick complementary base pairs and G-U wobble base pairs. We compare the results of our statistical survey with existing theoretical and numerical models. We also discuss possible applications of this classification and show how it can be used for identifying new RNA structural motifs.     

Relative flexibility of DNA and RNA: a molecular dynamics study

State of the art molecular dynamics simulations are used to study the structure, dynamics, molecular interaction properties and flexibility of DNA and RNA duplexes in aqueous solution. Special attention is paid to the deformability of both types of structures, revisiting concepts on the relative flexibility of DNA and RNA duplexes. Our simulations strongly suggest that the concepts of flexibility, rigidity and deformability are much more complex than usually believed, and that it is not always true that DNA is more flexible than RNA.     

Stability and rigidity/flexibility—Two sides of the same coin?

Protein molecules require both flexibility and rigidity for functioning. The fast and accurate prediction of protein rigidity/flexibility is one of the important problems in protein science. We have determined flexible regions for four homologous pairs from thermophilic and mesophilic organisms by two methods: the fast FoldUnfold which uses amino acid sequence and the time consuming MDFirst which uses three-dimensional structures. We demonstrate that both methods allow determining flexible regions in protein structure. For three of the four thermophile–mesophile pairs of proteins, FoldUnfold predicts practically the same flexible regions which have been found by the MD/First method. As expected, molecular dynamics simulations show that thermophilic proteins are more rigid in comparison to their mesophilic homologues. Analysis of rigid clusters and their decomposition provides new insights into protein stability. It has been found that the local networks of salt bridges and hydrogen bonds in thermophiles render their structure more stable with respect to fluctuations of individual contacts. Such network includes salt bridge triads Agr-Glu-Lys and Arg-Glu-Arg, or salt bridges (such as Arg-Glu) connected with hydrogen bonds. This ionic network connects alpha helices and rigidifies the structure. Mesophiles can be characterized by stand alone salt bridges and hydrogen bonds or small ionic clusters. Such difference in the network of salt bridges results in different flexibility of homologous proteins. Combining both approaches allows characterizing structural features in atomic detail that determine the rigidity/flexibility of a protein structure. This article is a part of a Special Issue entitled: The emerging dynamic view of proteins: Protein plasticity in allostery, evolution and self-assembly.     

Electrostatic effects in a network of polar and ionizable groups in staphylococcal nuclease

His121 and His124 are embedded in a network of polar and ionizable groups on the surface of staphylococcal nuclease. To examine how membership in a network affects the electrostatic properties of ionizable groups, the tautomeric state and the pK_a values of these histidines were measured with NMR spectroscopy in the wild-type nuclease and in 13 variants designed to disrupt the network. In the background protein, His121 and His124 titrate with pK_a values of 5.2 and 5.6, respectively. In the variants, where the network was disrupted, the pK_a values range from 4.03 to 6.46 for His121, and 5.04 to 5.99 for His124. The largest decrease in a pK_a was observed when the favorable Coulomb interaction between His121 and Glu75 was eliminated; the largest increase was observed when Tyr91 or Tyr93 was substituted with Ala or Phe. In all variants, the dominant tautomeric state at neutral pH was the N^ε2 state. At one level the network behaves as a rigid unit that does not readily reorganize when disrupted: crystal structures of the E75A or E75Q variants show that even when the pivotal Glu75 is removed, the overall configuration of the network was unaffected. On the other hand, a few key hydrogen bonds appear to govern the conformation of the network, and when these bonds are disrupted the network reorganizes. Coulomb interactions within the network report an effective dielectric constant of 20, whereas a dielectric constant of 80 is more consistent with the magnitude of medium to long-range Coulomb interactions in this protein. The data demonstrate that when structures are treated as static, rigid bodies, structure-based pK_a calculations with continuum electrostatics method are not useful to treat ionizable groups in cases where pK_a values are governed by short-range polar and Coulomb interactions.     

Statics of the Ribosomal Exit Tunnel: Implications for Cotranslational Peptide Folding, Elongation Regulation, and Antibiotics Binding

A sophisticated interplay between the static properties of the ribosomal exit tunnel and its functional role in cotranslational processes is revealed by constraint counting on topological network representations of large ribosomal subunits from four different organisms. As for the global flexibility characteristics of the subunit, the results demonstrate a conserved stable structural environment of the tunnel. The findings render unlikely that deformations of the tunnel move peptides down the tunnel in an active manner. Furthermore, the stable environment rules out that the tunnel can adapt widely so as to allow tertiary folding of nascent chains. Nevertheless, there are local zones of flexible nucleotides within the tunnel, between the peptidyl transferase center and the tunnel constriction, and at the tunnel exit. These flexible zones strikingly agree with previously identified folding zones. As for cotranslational elongation regulation, flexible residues in the β-hairpin of the ribosomal L22 protein were verified, as suggested previously based on structural results. These results support the hypothesis that L22 can undergo conformational changes that regulate the tunnel voyage of nascent polypeptides. Furthermore, rRNA elements, for which conformational changes have been observed upon interaction of the tunnel wall with a nascent SecM peptide, are less strongly coupled to the subunit core. Sequences of coupled rigid clusters are identified between the tunnel and some of these elements, suggesting signal transmission by a domino-like mechanical coupling. Finally, differences in the flexibility of the glycosidic bonds of bases that form antibiotics-binding crevices within the peptidyl transferase center and the tunnel region are revealed for ribosomal structures from different kingdoms. In order to explain antibiotics selectivity, action, and resistance, according to these results, differences in the degrees of freedom of the binding regions may need to be considered.     

Determining and visualizing flexibility in protein structures

How to compare the structures of an ensemble of protein conformations is a fundamental problem in structural biology. As has been previously observed, the widely used RMSD measure due to Kabsch, in which a rigid‐body superposition minimizing the least‐squares positional deviations is performed, has its drawbacks when comparing and visualizing a set of flexible protein structures. Here, we develop a method, fleximatch, of protein structure comparison that takes flexibility into account. Based on a distance matrix measure of flexibility, a weighted superposition of distance matrices rather than of atomic coordinates is performed. Subsequently, this allows a consistent determination of (a) a superposition of structures for visualization, (b) a partitioning of the protein structure into rigid molecular components (core atoms), and (c) an atomic mobility measure. The method is suitable for highlighting both particularly flexible and rigid parts of a protein from structures derived from NMR, X‐ray diffraction or molecular simulation. Proteins 2015; 83:820–826. © 2015 Wiley Periodicals, Inc.     

Multiscale modeling of macromolecular conformational changes combining concepts from rigidity and elastic network theory

The development of a two-step approach for multiscale modeling of macromolecular conformational changes is based on recent developments in rigidity and elastic network theory. In the first step, static properties of the macromolecule are determined by decomposing the molecule into rigid clusters by using the graph-theoretical approach FIRST and an all-atom representation of the protein. In this way, rigid clusters are not limited to consist of residues adjacent in sequence or secondary structure elements as in previous studies. Furthermore, flexible links between rigid clusters are identified and can be modeled as such subsequently. In the second step, dynamical properties of the molecule are revealed by the rotations-translations of blocks approach (RTB) using an elastic network model representation of the coarse-grained protein. In this step, only rigid body motions are allowed for rigid clusters, whereas links between them are treated as fully flexible. The approach was tested on a data set of 10 proteins that showed conformational changes on ligand binding. For efficiency, coarse-graining the protein results in a remarkable reduction of memory requirements and computational times by factors of 9 and 27 on average and up to 25 and 125, respectively. For accuracy, directions and magnitudes of motions predicted by our approach agree well with experimentally determined ones, despite embracing in extreme cases >50% of the protein into one rigid cluster. In fact, the results of our method are in general comparable with when no or a uniform coarse-graining is applied; and the results are superior if the movement is dominated by loop or fragment motions. This finding indicates that explicitly distinguishing between flexible and rigid regions is advantageous when using a simplified protein representation in the second step. Finally, motions of atoms in rigid clusters are also well predicted by our approach, which points to the need to consider mobile protein regions in addition to flexible ones when modeling correlated motions.     

The flexibility and dynamics of protein disulfide isomerase

We have studied the mobility of the multidomain folding catalyst, protein disulfide isomerase (PDI), by a coarse‐graining approach based on flexibility. We analyze our simulations of yeast PDI (yPDI) using measures of backbone movement, relative positions and orientations of domains, and distances between functional sites. We find that there is interdomain flexibility at every interdomain junction but these show very different characteristics. The extent of interdomain flexibility is such that yPDI's two active sites can approach much more closely than is found in crystal structures—and indeed hinge motion to bring these sites into proximity is the lowest energy normal mode of motion of the protein. The flexibility predicted for yPDI (based on one structure) includes the other known conformation of yPDI and is consistent with (i) the mobility observed experimentally for mammalian PDI and (ii) molecular dynamics. We also observe intradomain flexibility and clear differences between the domains in their propensity for internal motion. Our results suggest that PDI flexibility enables it to interact with many different partner molecules of widely different sizes and shapes, and highlights considerable similarities of yPDI and mammalian PDI. Proteins 2016; 84:1776–1785. © 2016 Wiley Periodicals, Inc.     

Protein flexibility using constraints from molecular dynamics simulations

Proteins are held together in the native state by hydrophobic interactions, hydrogen bonds and interactions with the surrounding water, whose strength as well as spatial and temporal distribution affects protein flexibility and hence function. We study these effects using 10 ns molecular dynamics simulations of pure water and of two proteins, the glutamate receptor ligand binding domain and barnase. We find that most of the noncovalent interactions flicker on and off over typically nanoseconds, and so we can obtain good statistics from the molecular dynamics simulations. Based on this information, a topological network of rigid bonds corresponding to a protein structure with covalent and noncovalent bonds is constructed, with account being taken of the influence of the flickering hydrogen bonds. We define the duty cycle for the noncovalent interactions as the percentage of time a given interaction is present, which we use as an input to investigate flexibility/rigidity patterns, in the algorithm FIRST which constructs and analyses topological networks.     

What Makes Ras an Efficient Molecular Switch: A Computational, Biophysical, and Structural Study of Ras-GDP Interactions with Mutants of Raf

Ras is a small GTP-binding protein that is an essential molecular switch for a wide variety of signaling pathways including the control of cell proliferation, cell cycle progression and apoptosis. In the GTP-bound state, Ras can interact with its effectors, triggering various signaling cascades in the cell. In the GDP-bound state, Ras looses its ability to bind to known effectors. The interaction of the GTP-bound Ras (Ras^GTP) with its effectors has been studied intensively. However, very little is known about the much weaker interaction between the GDP-bound Ras (Ras^GDP) and Ras effectors. We investigated the factors underlying the nucleotide-dependent differences in Ras interactions with one of its effectors, Raf kinase. Using computational protein design, we generated mutants of the Ras-binding domain of Raf kinase (Raf) that stabilize the complex with Ras^GDP. Most of our designed mutations narrow the gap between the affinity of Raf for Ras^GTP and Ras^GDP, producing the desired shift in binding specificity towards Ras^GDP. A combination of our best designed mutation, N71R, with another mutation, A85K, yielded a Raf mutant with a 100-fold improvement in affinity towards Ras^GDP. The Raf A85K and Raf N71R/A85K mutants were used to obtain the first high-resolution structures of Ras^GDP bound to its effector. Surprisingly, these structures reveal that the loop on Ras previously termed the switch I region in the Ras^GDP·Raf mutant complex is found in a conformation similar to that of Ras^GTP and not Ras^GDP. Moreover, the structures indicate an increased mobility of the switch I region. This greater flexibility compared to the same loop in Ras^GTP is likely to explain the natural low affinity of Raf and other Ras effectors to Ras^GDP. Our findings demonstrate that an accurate balance between a rigid, high-affinity conformation and conformational flexibility is required to create an efficient and stringent molecular switch.     

Protein flexibility predictions using graph theory

Techniques from graph theory are applied to analyze the bond networks in proteins and identify the flexible and rigid regions. The bond network consists of distance constraints defined by the covalent and hydrogen bonds and salt bridges in the protein, identified by geometric and energetic criteria. We use an algorithm that counts the degrees of freedom within this constraint network and that identifies all the rigid and flexible substructures in the protein, including overconstrained regions (with more crosslinking bonds than are needed to rigidify the region) and underconstrained or flexible regions, in which dihedral bond rotations can occur. The number of extra constraints or remaining degrees of bond-rotational freedom within a substructure quantifies its relative rigidity/flexibility and provides a flexibility index for each bond in the structure. This novel computational procedure, first used in the analysis of glassy materials, is approximately a million times faster than molecular dynamics simulations and captures the essential conformational flexibility of the protein main and side-chains from analysis of a single, static three-dimensional structure. This approach is demonstrated by comparison with experimental measures of flexibility for three proteins in which hinge and loop motion are essential for biological function: HIV protease, adenylate kinase, and dihydrofolate reductase.     

Hierarchical domain‐motion analysis of conformational changes in sarcoplasmic reticulum Ca2+‐ATPase

Sarco(endo)plasmic reticulum Ca²⁺‐ATPase transports two Ca²⁺ per ATP‐hydrolyzed across biological membranes against a large concentration gradient by undergoing large conformational changes. Structural studies with X‐ray crystallography revealed functional roles of coupled motions between the cytoplasmic domains and the transmembrane helices in individual reaction steps. Here, we employed “Motion Tree (MT),” a tree diagram that describes a conformational change between two structures, and applied it to representative Ca²⁺‐ATPase structures. MT provides information of coupled rigid‐body motions of the ATPase in individual reaction steps. Fourteen rigid structural units, “common rigid domains (CRDs)” are identified from seven MTs throughout the whole enzymatic reaction cycle. CRDs likely act as not only the structural units, but also the functional units. Some of the functional importance has been newly revealed by the analysis. Stability of each CRD is examined on the morphing trajectories that cover seven conformational transitions. We confirmed that the large conformational changes are realized by the motions only in the flexible regions that connect CRDs. The Ca²⁺‐ATPase efficiently utilizes its intrinsic flexibility and rigidity to response different switches like ligand binding/dissociation or ATP hydrolysis. The analysis detects functional motions without extensive biological knowledge of experts, suggesting its general applicability to domain movements in other membrane proteins to deepen the understanding of protein structure and function. Proteins 2015; 83:746–756. © 2015 Wiley Periodicals, Inc.     

The paradoxical behavior of a highly structured misfolded intermediate in RNA folding

Like many structured RNAs, the Tetrahymena group I ribozyme is prone to misfolding. Here we probe a long-lived misfolded species, referred to as M, and uncover paradoxical aspects of its structure and folding. Previous work indicated that a non-native local secondary structure, termed alt P3, led to formation of M during folding in vitro. Surprisingly, hydroxyl radical footprinting, fluorescence measurements with site-specifically incorporated 2-aminopurine, and functional assays indicate that the native P3, not alt P3, is present in the M state. The paradoxical behavior of alt P3 presumably arises because alt P3 biases folding toward M, but, after commitment to this folding pathway and before formation of M, alt P3 is replaced by P3. Further, structural and functional probes demonstrate that the misfolded ribozyme contains extensive native structure, with only local differences between the two states, and the misfolded structure even possesses partial catalytic activity. Despite the similarity of these structures, re-folding of M to the native state is very slow and is strongly accelerated by urea, Na+, and increased temperature and strongly impeded by Mg2+ and the presence of native peripheral contacts. The paradoxical observations of extensive native structure within the misfolded species but slow conversion of this species to the native state are readily reconciled by a model in which the misfolded state is a topological isomer of the native state, and computational results support the feasibility of this model. We speculate that the complex topology of RNA secondary structures and the inherent rigidity of RNA helices render kinetic traps due to topological isomers considerably more common for RNA than for proteins.     

A graph theoretical approach for assessing bio-macromolecular complex structural stability

Fast and proper assessment of bio macro-molecular complex structural rigidity as a measure of structural stability can be useful in systematic studies to predict molecular function, and can also enable the design of rapid scoring functions to rank automatically generated bio-molecular complexes. Based on the graph theoretical approach of Jacobs et al. [Jacobs DJ, Rader AJ, Kuhn LA, Thorpe MF (2001) Protein flexibility predictions using graph theory. Proteins: Struct Funct Genet 44:150–165] for expressing molecular flexibility, we propose a new scheme to analyze the structural stability of bio-molecular complexes. This analysis is performed in terms of the identification in interacting subunits of clusters of flappy amino acids (those constituting regions of potential internal motion) that undergo an increase in rigidity at complex formation. Gains in structural rigidity of the interacting subunits upon bio-molecular complex formation can be evaluated by expansion of the network of intra-molecular inter-atomic interactions to include inter-molecular inter-atomic interaction terms. We propose two indices for quantifying this change: one local, which can express localized (at the amino acid level) structural rigidity, the other global to express overall structural stability for the complex. The new system is validated with a series of protein complex structures reported in the protein data bank. Finally, the indices are used as scoring coefficients to rank automatically generated protein complex decoys.     

Change in protein flexibility upon complex formation: analysis of Ras-Raf using molecular dynamics and a molecular framework approach

Changes in flexibility upon protein-protein complex formation of H-Ras and the Ras-binding domain of C-Raf1 have been investigated using the molecular framework approach FIRST (Floppy Inclusion and Rigid Substructure Topology) and molecular dynamics simulations (MD) of in total approximately 35 ns length. In a computational time of about one second, FIRST identifies flexible and rigid regions in a single, static three-dimensional molecular framework, whose vertices represent protein atoms and whose edges represent covalent and non-covalent (hydrogen bond and hydrophobic) constraints and fixed bond angles within the protein. The two methods show a very good agreement with respect to the identification of changes in flexibility in both binding partners on a local scale. This implies that flexibility can be successfully predicted by identifying which bonds limit motion within a molecule and how they are coupled. In particular, as identified by MD, the beta-sheet in Raf shows considerably more pronounced orientational correlations in the bound state compared to the unbound state. Similarly, FIRST assigns the beta-sheet to the largest rigid cluster of the complex. Interestingly, FIRST allows us to identify that interactions across the interface (but not conformational changes upon complex formation) result in the observed rigidification. Since regions of the beta-sheet of Raf that do not interact directly with Ras become rigidified, this also demonstrates the long-range aspect to rigidity percolation. Possible implications of the change of flexibility of the Ras-binding domain of Raf on the activation of Raf upon complex formation are discussed. Finally, the sensitivity of FIRST results with respect to the representation of non-covalent interactions used as constraints is probed.     

Extended Structures in RNA Folding Intermediates Are Due to Nonnative Interactions Rather than Electrostatic Repulsion

RNA folding occurs via a series of transitions between metastable intermediate states for Mg²⁺ concentrations below those needed to fold the native structure. In general, these folding intermediates are considerably less compact than their respective native states. Our previous work demonstrates that the major equilibrium intermediate of the 154-residue specificity domain (S-domain) of the Bacillus subtilis RNase P RNA is more extended than its native structure. We now investigate two models with falsifiable predictions regarding the origins of the extended intermediate structures in the S-domains of the B. subtilis and the Escherichia coli RNase P RNA that belong to different classes of P RNA and have distinct native structures. The first model explores the contribution of electrostatic repulsion, while the second model probes specific interactions in the core of the folding intermediate. Using small-angle X-ray scattering and Langevin dynamics simulations, we show that electrostatics plays only a minor role, whereas specific interactions largely account for the extended nature of the intermediate. Structural contacts in the core, including a nonnative base pair, help to stabilize the intermediate conformation. We conclude that RNA folding intermediates adopt extended conformations due to short-range, nonnative interactions rather than generic electrostatic repulsion of helical domains. These principles apply to other ribozymes and riboswitches that undergo functionally relevant conformational changes.     

An efficient method for the prediction of deleterious multiple-point mutations in the secondary structure of RNAs using suboptimal folding solutions

Background  

RNAmute is an interactive Java application which, given an RNA sequence, calculates the secondary structure of all single point mutations and organizes them into categories according to their similarity to the predicted structure of the wild type. The secondary structure predictions are performed using the Vienna RNA package. A more efficient implementation of RNAmute is needed, however, to extend from the case of single point mutations to the general case of multiple point mutations, which may often be desired for computational predictions alongside mutagenesis experiments. But analyzing multiple point mutations, a process that requires traversing all possible mutations, becomes highly expensive since the running time is O(n ^m ) for a sequence of length n with m-point mutations. Using Vienna's RNAsubopt, we present a method that selects only those mutations, based on stability considerations, which are likely to be conformational rearranging. The approach is best examined using the dot plot representation for RNA secondary structure.     

Functionally relevant motions of haloalkane dehalogenases occur in the specificity-modulating cap domains

One-nanosecond molecular dynamics trajectories of three haloalkane dehalogenases (DhlA, LinB, and DhaA) are compared. The main domain was rigid in all three dehalogenases, whereas the substrate specificity-modulating cap domains showed considerably higher mobility. The functionally relevant motions were spread over the entire cap domain in DhlA, whereas they were more localized in LinB and DhaA. The highest amplitude of essential motions of DhlA was noted in the alpha4'-helix-loop-alpha4-helix region, formerly proposed to participate in the large conformation change needed for product release. The highest amplitude of essential motions of LinB and DhaA was observed in the random coil before helix 4, linking two domains of these proteins. This flexibility is the consequence of the modular composition of haloalkane dehalogenases. Two members of the catalytic triad, that is, the nucleophile and the base, showed a very high level of rigidity in all three dehalogenases. This rigidity is essential for their function. One of the halide-stabilizing residues, important for the catalysis, shows significantly higher flexibility in DhlA compared with LinB and DhaA. Enhanced flexibility may be required for destabilization of the electrostatic interactions during the release of the halide ion from the deeply buried active site of DhlA. The exchange of water molecules between the enzyme active site and bulk solvent was very different among the three dehalogenases. The differences could be related to the flexibility of the cap domains and to the number of entrance tunnels.     

Identification of putative, stable binding regions through flexibility analysis of HIV-1 gp120

The acquired-immunodeficiency syndrome has evolved into a major worldwide epidemic. Significant effort has been made in the development of antiviral therapies. A new strategy for vaccine and drug design that complements the existing cocktail therapy is to target entry of the human immunodeficiency virus (HIV). Such an approach provides the advantage of interfering with multiple intermediates in this multi-step process. The extraordinary conformational flexibility, glycosylation, and strain variations of viral glycoprotein gp120 cause general viral evasion of humoral immune response and thus complicate the development of an effective vaccine. Especially difficult to define are the conformation of gp120 before CD4 engagement as well as the relative orientations of the V1/V2 and V3 loops with respect to the inner and outer domains. In this study, we used Floppy Inclusion and Rigid Substructure Topography (FIRST), a program based on graph theory, to analyze the flexibility and rigidity of all known HIV-1 gp120 structures. A flexibility index is used to describe and compare the spatial distribution of protein flexibility and rigidity of these structures in isolation and in complex with CD4, CD4-mimics, and neutralizing antibodies. Using this flexibility analysis, we identified a universal rigid region (the alpha2 helix) as well as the consensus largest rigid cluster involving a beta-sheet located on the coreceptor binding face. Both of these regions may serve as stable targets for vaccine design and drug discovery. Detailed comparisons of the changes in flexibility based on strain variations, stabilizing mutations, binding features of CD4 mimics, and impact of b12 binding are reported.