Prediction of the three-dimensional structure of Escherichia coli 30S ribosomal subunit: a molecular mechanics approach

Advances in RNA 3D Structure Prediction

RNA molecules carry out various cellular functions, and understanding the mechanisms behind their functions requires the knowledge of their 3D structures. Different types of computational methods have been developed to model RNA 3D structures over the past decade. These methods were widely used by researchers although their performance needs to be further improved. Recently, along with these traditional methods, machine-learning techniques have been increasingly applied to RNA 3D structure prediction and show significant improvement in performance. Here we shall give a brief review of the traditional methods and recent related advances in machine-learning approaches for RNA 3D structure prediction.

3dDNA: A Computational Method of Building DNA 3D Structures

Considerable progress has been made in the prediction methods of 3D structures of RNAs. In contrast, no such methods are available for DNAs. The determination of 3D structures of the latter is also increasingly needed for understanding their functions and designing new DNA molecules. Since the number of experimental structures of DNA is limited at present, here, we propose a computational and template-based method, 3dDNA, which combines DNA and RNA template libraries to predict DNA 3D structures. It was benchmarked on three test sets with different numbers of chains, and the results show that 3dDNA can predict DNA 3D structures with a mean RMSD of about 2.36 Å for those with one or two chains and fewer than 4 Å with three or more chains.

Large-scale simulations of nucleoprotein complexes: ribosomes, nucleosomes, chromatin, chromosomes and CRISPR

Recent advances in biotechnology such as Hi-C, CRISPR/Cas9 and ribosome display have placed nucleoprotein complexes at center stage. Understanding the structural dynamics of these complexes aids in optimizing protocols and interpreting data for these new technologies. The integration of simulation and experiment has helped advance mechanistic understanding of these systems. Coarse-grained simulations, reduced-description models, and explicit solvent molecular dynamics simulations yield useful complementary perspectives on nucleoprotein complex structural dynamics. When combined with Hi-C, cryo-EM, and single molecule measurements, these simulations integrate disparate forms of experimental data into a coherent mechanism.

Computational modeling of RNA 3D structures, with the aid of experimental restraints

In addition to mRNAs whose primary function is transmission of genetic information from DNA to proteins, numerous other classes of RNA molecules exist, which are involved in a variety of functions, such as catalyzing biochemical reactions or performing regulatory roles. In analogy to proteins, the function of RNAs depends on their structure and dynamics, which are largely determined by the ribonucleotide sequence. Experimental determination of high-resolution RNA structures is both laborious and difficult, and therefore, the majority of known RNAs remain structurally uncharacterized. To address this problem, computational structure prediction methods were developed that simulate either the physical process of RNA structure formation ("Greek science" approach) or utilize information derived from known structures of other RNA molecules ("Babylonian science" approach). All computational methods suffer from various limitations that make them generally unreliable for structure prediction of long RNA sequences. However, in many cases, the limitations of computational and experimental methods can be overcome by combining these two complementary approaches with each other. In this work, we review computational approaches for RNA structure prediction, with emphasis on implementations (particular programs) that can utilize restraints derived from experimental analyses. We also list experimental approaches, whose results can be relatively easily used by computational methods. Finally, we describe case studies where computational and experimental analyses were successfully combined to determine RNA structures that would remain out of reach for each of these approaches applied separately.

Computational studies of molecular machines: the ribosome

The past decade has produced an avalanche of experimental data on the structure and dynamics of the ribosome. Groundbreaking studies in structural biology and kinetics have placed important constraints on ribosome structural dynamics. However, a gulf remains between static structures and time dependent data. In particular, X-ray crystallography and cryo-EM studies produce static models of the ribosome in various states, but lack dynamic information. Single molecule studies produce information on the rates of transitions between these states but do not have high-resolution spatial information. Computational studies have aided in bridging this gap by providing atomic resolution simulations of structural fluctuations and transitions between configurations.

RNA and protein 3D structure modeling: similarities and differences

In analogy to proteins, the function of RNA depends on its structure and dynamics, which are encoded in the linear sequence. While there are numerous methods for computational prediction of protein 3D structure from sequence, there have been very few such methods for RNA. This review discusses template-based and template-free approaches for macromolecular structure prediction, with special emphasis on comparison between the already tried-and-tested methods for protein structure modeling and the very recently developed “protein-like” modeling methods for RNA. We highlight analogies between many successful methods for modeling of these two types of biological macromolecules and argue that RNA 3D structure can be modeled using “protein-like” methodology. We also highlight the areas where the differences between RNA and proteins require the development of RNA-specific solutions.

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Accommodation of aminoacyl-tRNA into the ribosome involves reversible excursions along multiple pathways

The ribosome is a massive ribonucleoprotein complex ( approximately 2.4 MDa) that utilizes large-scale structural fluctuations to produce unidirectional protein synthesis. Accommodation is a key conformational change during transfer RNA (tRNA) selection that allows movement of tRNA into the ribosome. Here, we address the structure-function relationship that governs accommodation using all-atom molecular simulations and single-molecule fluorescence resonance energy transfer (smFRET). Simulations that employ an all-atom, structure-based (Gō-like) model illuminate the interplay between configurational entropy and effective enthalpy during the accommodation process. This delicate balance leads to spontaneous reversible accommodation attempts, which are corroborated by smFRET measurements. The dynamics about the endpoints of accommodation (the A/T and A/A conformations) obtained from structure-based simulations are validated by multiple 100-200 ns explicit-solvent simulations (3.2 million atoms for a cumulative 1.4 micros), and previous crystallographic analysis. We find that the configurational entropy of the 3'-CCA end of aminoacyl-tRNA resists accommodation, leading to a multistep accommodation process that encompasses a distribution of parallel pathways. The calculated mechanism is robust across simulation methods and protocols, suggesting that the structure of the accommodation corridor imposes stringent limitations on the accessible pathways. The identified mechanism and observed parallel pathways establish an atomistic framework for interpreting a large body of biochemical data and demonstrate that conformational changes during translation occur through a stochastic trial-and-error process, rather than in concerted lock-step motions.

Determining the architectures of macromolecular assemblies

To understand the workings of a living cell, we need to know the architectures of its macromolecular assemblies. Here we show how proteomic data can be used to determine such structures. The process involves the collection of sufficient and diverse high-quality data, translation of these data into spatial restraints, and an optimization that uses the restraints to generate an ensemble of structures consistent with the data. Analysis of the ensemble produces a detailed architectural map of the assembly. We developed our approach on a challenging model system, the nuclear pore complex (NPC). The NPC acts as a dynamic barrier, controlling access to and from the nucleus, and in yeast is a 50 MDa assembly of 456 proteins. The resulting structure, presented in an accompanying paper, reveals the configuration of the proteins in the NPC, providing insights into its evolution and architectural principles. The present approach should be applicable to many other macromolecular assemblies.

A structural perspective on protein-protein interactions

Structures of macromolecular complexes are necessary for a mechanistic description of biochemical and cellular processes. They can be solved by experimental methods, such as X-ray crystallography, NMR spectroscopy and electron microscopy, as well as by computational protein structure prediction, docking and bioinformatics. Recent advances and applications of these methods emphasize the need for hybrid approaches that combine a variety of data to achieve better efficiency, accuracy, resolution and completeness.

From words to literature in structural proteomics

Technical advances on several frontiers have expanded the applicability of existing methods in structural biology and helped close the resolution gaps between them. As a result, we are now poised to integrate structural information gathered at multiple levels of the biological hierarchy - from atoms to cells - into a common framework. The goal is a comprehensive description of the multitude of interactions between molecular entities, which in turn is a prerequisite for the discovery of general structural principles that underlie all cellular processes.

Comparative photocross-linking analysis of the tertiary structures of Escherichia coli and Bacillus subtilis RNase P RNAs

Bacterial ribonuclease P contains a catalytic RNA subunit that cleaves precursor sequences from the 5' ends of pre-tRNAs. The RNase P RNAs from Bacillus subtilis and Escherichia coli each contain several unique secondary structural elements not present in the other. To understand better how these phylogenetically variable elements affect the global architecture of the ribozyme, photoaffinity cross-linking studies were carried out. Photolysis of photoagents attached at homologous sites in the two RNAs results in nearly identical cross-linking patterns, consistent with the homology of the RNAs and indicating that these RNAs contain a common, core tertiary structure. Distance constraints were used to derive tertiary structure models using a molecular mechanics-based modeling protocol. The resulting superimposition of large sets of equivalent models provides a low resolution (5-10 A) structure for each RNA. Comparison of these structure models shows that the conserved core helices occupy similar positions in space. Variably present helical elements that may play a role in global structural stability are found at the periphery of the core structure. The P5.1 and P15.1 helical elements, unique to the B.subtilis RNase P RNA, and the P6/16/17 helices, unique to the E.coli RNA, occupy similar positions in the structure models and, therefore, may have analogous structural function.

Modeling the structure of the ribosome

Considering the size and complexity of the ribosome and the growing body of data from a wide range of experiments on ribosomal structure, it is becoming increasingly important to develop tools that facilitate the development of reliable models for the ribosome. We use a combination of manual and computer-based approaches for building and refining models of the ribosome and other RNA-protein complexes. Our methods are aimed at determining the range of models compatible with the data, making quantitative statements about the positional uncertainties (resolution) of different regions, identifying conflicts in the data, establishing which regions of the ribosome need further experimental exploration, and, where possible, predicting the outcome of future experiments. Our previous low-resolution model for the small subunit of the Escherichia coli ribosome is briefly reviewed, along with progress on atomic resolution modeling of the mRNA-tRNA complex and its interaction with the decoding site of the 16S RNA.

A small modified hammerhead ribozyme and its conformational characteristics determined by mutagenesis and lattice calculation

A prototypic hammerhead ribozyme has three helices that surround an asymmetrical central core loop. We have mutagenized a hammerhead type ribozyme. In agreement with previous studies, progressive removal of stem-loop II from a three stemmed ribozyme showed that this region is not absolutely critical for catalysis. However, complete elimination of stem II and its loop did reduce, but did not eliminate, function. In a stem-loop II-deleted ribozyme, activity was best preserved when a purine, preferably a G, was present at position 10.1. This G contributed to catalysis irregardless of its role as either one part of a canonical pair with a C residue at 11.1 or a lone nucleotide with C (11.1) deleted. Computational methods using lattices generated 87 million three-dimensional chain forms for a stem-loop II-deleted RNA complex that preserved one potential G.C base pair at positions 10.1 and 11.1. This exhaustive set of chain forms included one major class of structures with G(10.1) being spatially proximal to the GUCX cleavage site of the substrate strand. Strong correlations were observed between colinear arrangement of stems I and III, constraints of base-pairing in the central core loop, and one particular placement of G(10.1) relative to the cleavage site. Our calculations of a stem-loop II-deleted ribozyme indicate that without needing to invoke any other constraints, the inherent asymmetry in the lengths of the two loop strands (3 nt in one and 7 nt in the other) that compose the core and flank G10.1-C11.1 stipulated strongly this particular G placement. This suggests that the hammerhead ribozyme maintains an asymmetry in its internal loop for a necessary structure/function reason.

Structures of small subunit ribosomal RNAs in situ from Escherichia coli and Thermomyces lanuginosus

Small ribosomal subunits from the prokaryote Escherichia coli and the eukaryote Thermomyces lanuginosus were imaged electron spectroscopically, and single particle analysis used to yield three-dimensional reconstructions of the net phosphorus distribution representing the nucleic acid (RNA) backbone. This direct approach showed both ribosomal RNAs to have a three domain structure and other characteristic morphological features. The eukaryotic small ribosomal subunit had a prominent bill present in the head domain, while the prokaryotic subunit had a small vestigial bill. Both ribosomal subunits contained a thick 'collar' central domain which correlates to the site of the evolutionarily conserved ribosomal RNA core, and the location of the majority of ribosomal RNA bases that have been implicated in translation. The reconstruction of the prokaryotic subunit had a prominent protrusion extending from the collar, forming a channel approximately 1.5 nm wide and potentially representing a 'bridge' to the large subunit in the intact monosome. The basal domain of the prokaryotic ribosomal subunit was protein free. In this region of the eukaryotic subunit, there were two basal lobes composed of ribosomal RNA, consistent with previous hypotheses that this is a site for the 'non-conserved core' ribosomal RNA.

A genetic algorithm based molecular modeling technique for RNA stem-loop structures

A new modeling technique for arriving at the three dimensional (3-D) structure of an RNA stem-loop has been developed based on a conformational search by a genetic algorithm and the following refinement by energy minimization. The genetic algorithm simultaneously optimizes a population of conformations in the predefined conformational space and generates 3-D models of RNA. The fitness function to be optimized by the algorithm has been defined to reflect the satisfaction of known conformational constraints. In addition to a term for distance constraints, the fitness function contains a term to constrain each local conformation near to a prepared template conformation. The technique has been applied to the two loops of tRNA, the anticodon loop and the T-loop, and has found good models with small root mean square deviations from the crystal structure. Slightly different models have also been found for the anticodon loop. The analysis of a collection of alternative models obtained has revealed statistical features of local variations at each base position.

A model for the stabilities of RNA hairpins based on a study of the sequence dependence of stability for hairpins of six nucleotides

Thermodynamic parameters are reported for hairpin formation in 1 M NaCl by RNA sequences of the type GGCXUAAUYGCC, where XY is the set of 10 possible mismatch base pairs. A nearest neighbor analysis of the data indicates the free energy for loop formation at 37 C varies from 2.9 to 4.5 kcal/mol. Thermodynamic parameters are also reported for hairpin formation by RNA sequences of the type GGXGUAAUAYCC (where XY are CG, GC, AU, UA, GU, and UG), with the common naturally occurring GA first mismatch (45% of small and large subunit rRNA loops of six). These results allow the development of a model to predict the stability of RNA hairpin loops. The model includes the size of the loop, the identity of the closing base pair, the free energy increment (delta G zero 37MM) for interaction of the closing base pair with the first mismatch, and an additional stabilization term for GA and UU first mismatches. delta G zero 37L(n) = delta G zero 37i(n) + delta G zero 37MM + 0.4 (if closed by AU or UA) -0.7 (if first mismatch is GA or UU). Here delta G zero 37i(n) is the free energy for initiating a loop of n nucleotides. delta G zero 37i(n) for n = 4-9 is 4.9, 4.4, 5.0, 5.0, 5.1, and 5.2 kcal/mol, respectively. The delta G zero 37MM is derived from measurements of model duplexes with terminal mismatches. The model gives good agreement when tested against four naturally occurring hairpin sequences.

Ribosomal components neighboring the conserved 518-533 loop of 16S rRNA in 30S subunits

We report the synthesis of a radioactive, photolabile oligodeoxyribonucleotide probe complementary to 16S rRNA nucleotides 518-526 and its exploitation in identifying 30S ribosomal subunit components neighboring its target site in 16S rRNA. Nucleotides 518-526 lie within an almost universally conserved single-stranded loop that has been linked to the decoding region of Escherichia coli ribosomes. On photolysis in the presence of activated 30S ribosomes, the probe site-specifically incorporates into proteins S3, S4, S7, and S12 (identified by SDS-PAGE, RP-HPLC, and immunological analysis); nucleotides C525, C526, and G527 adjacent to its target binding site; and the 3'-terminus of 16S rRNA. When the probe is photoincorporated into 30S subunits subjected to brief cold inactivation (SI subunits), S7 labeling is increased compared to activated subunit incorporation, while S3, S4, and S12 labeling is decreased, as is labeling to nucleotides C525, C526, and G527; labeling at the 16S rRNA 3'-terminus appears unchanged. Longer cold inactivation of the 30S subunits (LI subunits) leads to decreases in the labeling of all components. These results provide clear evidence that C526 lies within 24 A (the distance between C526 and the photogenerated nitrene) of proteins S3, S4, S7, and S12 and the 3'-terminus of 16S rRNA. The identity of the tryptic digestion patterns of S7 labeled with the probe complementary to 16S rRNA nucleotides 518-526 and with a probe complementary to nucleotides 1397-1405 [Muralikrishna, P., & Cooperman, B. S. (1994) Biochemistry 33, 1392-1398] also provides evidence for proximity between C526 and G1405. Our results support the conclusion of Dontsova et al. [Dontsova, O., et al. (1992) EMBO J. 11, 3105-3116] in placing the 530 loop in close proximity to the decoding center of the 30S subunit but are apparently inconsistent with some protein-protein distances determined by neutron diffraction [Capel, M. S., et al. (1988) J. Mol. Biol. 200, 65-87]. This inconsistency suggests that a multistate model of subunit conformation may be required to account for the totality of results pertaining to the internal structure of the 30S subunit.

Conformations of t-RNA: base pairing and stacking

The Phe t-RNA structure can be fit with one point per nucleotide to lattice models, and a fit for the 76 points to a face-centered cubic lattice is achieved with an RMS of 1.76 A. There are 32 chain folds possible upon these points. Because it is impossible to calculate directly all combinations of potential base pairs for these cases, an alternative is to determine low energy secondary structures and subsequently the tertiary pairs. For each lattice fold, the low energy secondary structures are generated from a list of proximal bases. From the lists of remaining possible tertiary pairs, all combinations are generated, and these include 2,365,440 allowed conformers. Among the possible types of non-native conformational variations observed is slip pairing, accompanied by a bulge, at the end of a stem. Small changes in secondary structure can result in different tertiary pairs. Other calculations, not constrained to the t-RNA shape, are presented that involve the packing of rigid stems on a flexible internal loop. For a simple cubic lattice there are 36,484,128 lattice folds for the sixteen bases enclosing the internal loop. By attaching rigid stems and accounting for their excluded volume these are reduced to only 258,979 possible configurations. The most common stacking arrangements involve the usual two pairs of stacked stems indicated in the crystal structure. The present enumerations suggest that a completely thorough exploration of three dimensional RNA structures is feasible only with prior specification of restrictions on conformational freedom, such as those given by secondary structures.

Modeling large RNAs and ribonucleoprotein particles using molecular mechanics techniques

There is a growing body of low-resolution structural data that can be utilized to devise structural models for large RNAs and ribonucleoproteins. These models are routinely built manually. We introduce an automated refinement protocol to utilize such data for building low-resolution three-dimensional models using the tools of molecular mechanics. In addition to specifying the positions of each nucleotide, the protocol provides quantitative estimates of the uncertainties in those positions, i.e., the resolution of the model. In typical applications, the resolution of the models is about 10-20 A. Our method uses reduced representations and allows us to refine three-dimensional structures of systems as big as the 16S and 23S ribosomal RNAs, which are about one to two orders of magnitude larger than nucleic acids that can be examined by traditional all-atom modeling methods. Nonatomic resolution structural data--secondary structure, chemical cross-links, chemical and enzymatic footprinting patterns, protein positions, solvent accessibility, and so on--are combined with known motifs in RNA structure to predict low-resolution models of large RNAs. These structural constraints are imposed on the RNA chain using molecular mechanics-type potential functions with parameters based on the quality of experimental data. Surface potential functions are used to incorporate shape and positional data from electron microscopy image reconstruction experiments into our models. The structures are optimized using techniques of energy refinement to get RNA folding patterns. In addition to providing a consensus model, the method finds the range of models consistent with the data, which allows quantitative evaluation of the resolution of the model. The method also identifies conflicts in the experimental data. Although our protocol is aimed at much larger RNAs, we illustrate these techniques using the tRNA structure as an example and test-bed.

Lagrangian molecular dynamics using selected conformational degrees of freedom, with application to the pseudorotation dynamics of furanose rings

Using internal conformational degrees of freedom for biopolymers as natural variables, and introducing a Lagrangian dynamics approach, one can simulate time-dependent processes over a much longer time scale than in classical Newtonian molecular dynamics (MD) techniques. Two factors contribute to this: a substantial reduction in the number of degrees of freedom and a very large increase in the size of the time step. We present the Lagrangian equations of motion for repuckering transitions in model furanose (F), ribose (R), and 2'-deoxyribose (dR) ring systems using the pseudorotation phase angle as the single dynamic variable. As in most Lagrangian analyses, the effective masses for the R and dR models are dependent on conformation, and we test the behavior of this variable mass (VM) model. Since the variation in effective mass is small, the VM model is compared with a simplified constant mass (CM) model, which is shown to be an excellent approximation. The equations of motion for the CM and VM models are integrated with the leapfrog and the iterative leapfrog algorithms, respectively. The Lagrangian dynamics approach reduces the number of degrees of freedom from about 40 to 1, and allows the use of time steps on the order of 20 fs, about an order of magnitude greater than is used in conventional MD simulations.

Reproducing the three-dimensional structure of a tRNA molecule from structural constraints

The three-dimensional structure of yeast tRNA(Phe) was reproduced at atomic resolution with the automated RNA modeling program MC-SYM, which is based on a constraint-satisfaction algorithm. Structural constraints used in the modeling were derived from the secondary structure, four tertiary base pairs, and other information available prior to the determination of the x-ray crystal structure of the tRNA. The program generated 26 solutions (models), all of which had the familiar "L" form of tRNA and root-mean-square deviations from the crystal structure in the range of 3.1-3.8 A. The interaction between uridine-8 and adenosine-14 was crucial in the modeling procedure, since only this among the tertiary pairs is necessary and sufficient to reproduce the L form of tRNA. Other tertiary interactions were critical in reducing the number of solutions proposed by the program.

Thermodynamic parameters of cooperative helix-to-coil transitions from synthetic A-U-rich oligoribonucleotides up to fourteen basepairs

Heat induced helix-to-coil transitions are studied in the form of ultraviolet-hypochromicity profiles by absorbance spectroscopy, and delta Cp-curves by differential scanning calorimetry of self-complementary ribonucleotides. The results are analyzed and compared. Van 't Hoff transition enthalpies derived by UV-experiments incorporating concentration variations are found to differ from six-parameter and two-parameter Marquardt-fits on the melting profiles. A measure for the maximum number of nucleotides in intermediate states is obtained from a statistical deconvolution. It yields a range from 12.5% for the shortest nucleotide up to 31.5% for r(UA)7. Model independent calorimetric data are reported. A limit for intra-molecular loop-formation preference is reached by rG(UA)6C within this sequence.

Teaching macromolecular modeling

Training newcomers to the field of macromolecular modeling is as difficult as is training beginners in x-ray crystallography, nuclear magnetic resonance, or other methods in structural biology. In one or two lectures, the most that can be conveyed is a general sense of the relationship between modeling and other structural methods. If a full semester is available, then students can be taught how molecular structures are built, manipulated, refined, and analyzed on a computer. Here we describe a one-semester modeling course that combines lectures, discussions, and a laboratory using a commercial modeling package. In the laboratory, students carry out prescribed exercises that are coordinated to the lectures, and they complete a term project on a modeling problem of their choice. The goal is to give students an understanding of what kinds of problems can be attacked by molecular modeling methods and which problems are beyond the current capabilities of those methods.

Three-dimensional reconstruction of the 70S Escherichia coli ribosome in ice: the distribution of ribosomal RNA

A reconstruction, at 40 A, of the Escherichia coli ribosome imaged by cryo-electron microscopy, obtained from 303 projections by a single-particle method of reconstruction, shows the two subunits with unprecedented clarity. In the interior of the subunits, a complex distribution of higher mass density is recognized, which is attributed to ribosomal RNA. The masses corresponding to the 16S and 23S components are linked in the region of the platform of the small subunit. Thus the topography of the rRNA regions responsible for protein synthesis can be described.

Assembly of the Escherichia coli 30S ribosomal subunit reveals protein-dependent folding of the 16S rRNA domains

Protein-nucleic acid interactions involved in the assembly process of the Escherichia coli 30S ribosomal subunit were quantitatively analyzed by high-resolution scanning transmission electron microscopy. The in vitro reconstituted ribonucleoprotein (core) particles were characterized by their morphology, mass, and radii of gyration. During the assembly of the 30S subunit, the 16S rRNA underwent significant conformational changes that were governed by the cooperative interactions of the ribosomal proteins. The sequential association of the first 12 proteins with the 16S rRNA resulted in the formation of core particles containing up to three mass centers at distinct stages of the assembly process. These globular mass centers may correspond to the three major domains (5', central, and 3') of the 16S rRNA. Through the subsequent interactions of the late assembly proteins with the 16S rRNA, two of the three domains merge, yielding the basic structural traits of the native 30S subunit. The fine morphological features of the native 30S subunit became distinctly resolved only after the addition of the full complement of proteins. The fully reconstituted 30S subunits are active in polyphenylalanine synthesis assays. Visualization of the assembly mechanism of the E. coli 30S ribosomal subunit revealed domain-specific folding of the 16S rRNA through the formation of distinct intermediate core particles hitherto not observed.

The combination of symbolic and numerical computation for three-dimensional modeling of RNA

Three-dimensional (3-D) structural models of RNA are essential for understanding of the cellular roles played by RNA. Such models have been obtained by a technique based on a constraint satisfaction algorithm that allows for the facile incorporation of secondary and other structural information. The program generates 3-D structures of RNA with atomic-level resolution that can be refined by numerical techniques such as energy minimization. The precision of this technique was evaluated by comparing predicted transfer RNA loop and RNA pseudoknot structures with known or consensus structures. The root-mean-square deviation (2.0 to 3.0 angstroms before minimization) between predicted and control structures reveal this system to be an effective method in modeling RNA.

Explicit distance geometry: identification of all the degrees of freedom in a large RNA molecule

An alternative approach to distance geometry ("explicit" distance geometry) is being developed for problems, such as the modeling of RNA folding in the ribosome, where relatively few distances are known. The approach explicitly identifies minimal sets of additional distances that can be added to a distance matrix in order to calculate structures that are consistent with all the known information without distorting the original input data. These additional distances are bounded to the extent possible by the known distances. These explicitly added distances can be treated as degrees of freedom and used to explore the full range of alternative foldings consistent with the original input in an organized way. The present paper establishes that it is practical to explicitly determine such degrees of freedom for even very large RNAs. To demonstrate the feasibility of the approach tRNA was represented as a simple undirected graph containing all relevant information represented in the usual cloverleaf secondary structure and nine base-base tertiary interactions. Using a three atom representation for each residue a total of 206 degrees of freedom are explicitly identified. To accomplish this a graph theoretic approach was used in which a minimal covering cycle basis was determined.