Introduction

Structure determination of larger nucleic acids by conventional solution NMR methods, based on NOEs and J-couplings, can be challenging since these highly helical structures exhibit low proton density and few, if any, NOEs between elements of secondary structure (Allain and Varani 1997). While the NOEs and through-hydrogen-bond J couplings (Dingley et al. 1999) provide valuable information for resonance assignment and identification of base-pair partners, they usually provide little information on the relative positioning of individual helices in a multi-helix system. For larger nucleic acid structures, the use of 13C labeling is limited due to the associated considerable 1H line broadening, arising from the strong 1H–13C dipolar interactions, and the very limited 13C chemical shift dispersion in the helical regions. As a consequence, measurement of an extensive number of 1DCH residual dipolar couplings (RDCs) can be challenging, and often only 1DHN RDCs for base-paired imino protons are readily accessible in such systems.

Since the number of experimental imino 1DNH RDCs restraints is far smaller than the total number of torsion angles in the molecule, de novo structures cannot be determined from these restraints alone. An alternate strategy, which supplements the sparse set of NMR restraints with prior structural information, can yield reasonable nucleic acid models that agree with the experimental NMR restraints (Mollova et al. 2000; Vermeulen et al. 2000, 2005; Lukavsky et al. 2003; D’Souza et al. 2004; Getz et al. 2007).

Here, we report an extension of this approach, which describes a systematic procedure for refining a starting structure, based on homology modeling, by combining a sparse set of experimental 1DHN RDCs, with SAXS data. The idea behind the refinement protocol is to ensure that, at the local level, structural changes relative to the starting model are kept to a minimum by restraining the coordinates of short overlapping sections of the refined model to remain close to those of the starting homology model. At the same time, potentially larger changes in global structure but only smaller local changes along the backbone of the oligonucleotide ensure agreement with experimental RDCs as well as a molecular shape compatible with the SAXS data. Similar approaches, lacking SAXS input, previously have been used successfully for proteins (Chou et al. 2001; Ulmer et al. 2003). We demonstrate the refinement protocol for the 76-nt tRNAVal using only the NMR restraints obtained from 24 imino N–H vectors, measured under two different alignment orientations. This low number of experimental NMR data required restraining the local geometry of the RNA close to that of the X-ray structure (PDB entry 1EHZ, (Shi and Moore 2000)) of the highly homologous tRNAPhe (58% sequence identity). The orientational information contained in the RDCs was complemented by global shape information obtained from SAXS data, recorded under conditions very similar to that of the NMR data acquisition. Several recent studies (Grishaev et al. 2005, 2008; Gabel et al. 2006; Schwieters and Clore 2007; Zuo et al. 2008) have demonstrated the power of combining the molecular shape information, encoded in SAXS data, with global orientational restraints obtained from RDCs. For tRNAVal, the refinement procedure is shown to result in a structure with excellent RDC cross validation statistics (Q free = 14%), compared to 25% for the starting model.

Materials and methods

Sample preparation

Samples of uniformly 15N-enriched native tRNAVal from E. coli were prepared as described previously (Latham et al. 2008). The NMR sample contained 0.5 mM tRNAVal in a buffer containing 10 mM sodium phosphate, pH 6.8, 80 mM NaCl, 5 mM MgCl2, 0.1 mM EDTA, in 10% D2O, 90% H2O. For measurements in the Pf1-aligned phase, ca. 8 mg ml−1 Pf1 was added, yielding a 2H solvent quadrupole splitting of 7.9 Hz. Samples for SAXS data collection were prepared by dialysis against a buffer that contained 10 mM sodium phosphate, 150 mM NaCl, 5 mM MgCl2 and 0.1 mM EDTA at pH 7.0, with the final stock concentration of tRNAVal at 10 mg ml−1.

RDC data collection and processing

Resonance assignments are taken from (Vermeulen 2003), but small changes in the resonance positions of the imino resonances of G22, G43, and G53 resulted from slightly different solvent conditions, causing assignment ambiguity for these resonances. Even though for the assignments depicted in Fig. 1 the measured RDCs for these nucleotides are in excellent agreement with predicted data, these RDCs were not used at any stage of the analysis or refinement. 1DHN RDCs in Pf1 medium were collected from a set of interleaved 800 MHz TROSY-HSQC and regular 1H–15N HSQC spectra (Fig. 1), where the frequency difference in the 15N dimension corresponds to (1JNH + 1DNH)/2. Although in principle the (1JNH + 1DNH)/2 splittings can be obtained independently either from the relative peak displacements in the 15N dimension or in the 1H dimension, in practice the measurement is best carried out in the 15N dimension where line widths are narrowest and line shapes are most symmetric. In the 1H dimension, the presence of unresolved 1H–1H dipolar couplings and relaxation interference between 1HN–15N and 1H–1H dipolar interactions can give rise to a slight asymmetry of the line shape that adversely impacts the accuracy of the measured (1JNH + 1DNH)/2 splitting (Fig. 1b). Isotropic 1JNH splittings, which also include a very small dipolar contribution resulting from alignment due to tRNA’s magnetic susceptibility anisotropy (MSA), were measured previously (Ying et al. 2007). For obtaining the Pf1 RDCs, the isotropic (1JNH + 1DNH MSA) splittings, measured previously at the same magnetic field strength, were subtracted from the corresponding splittings, 1JNH + 1DNH, measured in the presence of Pf1. 1DHN Pf1 values thus obtained (Supplementary Material) only include the Pf1-induced RDC contribution and correlate closely (Pearson’s correlation coefficient R P = 0.994) with values measured previously (Ying et al. 2007).

Fig. 1
figure 1

800 MHz 1H–15N shift correlation of tRNAVal. (a) Part of the imino region of the superimposed 2D 15N–1H TROSY-HSQC (blue) and regular HSQC (red) correlation spectra of tRNAVal recorded at 800 MHz, 28°C, showing correlations for the imino groups of G nucleotides. (b) Expanded cross peak for the TROSY imino correlation of U64, which forms a wobble base pair with G50 and experiences extensive homonuclear 1H–1H RDC with adjacent protons. 15N–1H/1H–1H dipolar cross correlated relaxation results in asymmetry of the unresolved 1H–1H multiplet

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SAXS data collection

Solution scattering data were acquired on a SAXSess instrument from Anton-Paar, which includes a Kratky camera equipped with high-flux multilayer optics and a wide angle measurement extension. A sealed fine-focus tube (Princeton Instruments), operating at 40 kV and 50 mA, served as the X-ray source. An elliptically bent multilayer mirror selected radiation at the Cu Kα wavelength (1.542 Å). A 1-mm inner diameter quartz capillary of 10 mm length was used as the sample cell and kept at 25°C with a Peltier element. An X-ray beam of ca. 9 mm width, parallel to the sample capillary, was generated by adjustment of the collimator slit. Data were collected as series of sequential 2 h acquisitions on the tRNAVal sample, followed immediately by the matching dialysate buffer. Due to fast signal relaxation in the first few minutes after exposure, the imaging plates were read out with a 5 min delay after each data collection on a Cyclone Plus scanner from Perkin Elmer. Data at three RNA concentrations (2.29, 4.93 and 10.0 mg ml−1) were acquired in order to evaluate the magnitude of the inter-particle structure factor. The recorded scattering profiles spanned a q-range from ~0.02 to ~2.8 Å−1, where q = 4πsin(θ)/λ, 2θ is the scattering angle, and λ is the wavelength of the incident radiation. The raw 2D images were converted to 1D scattering profiles by radial integration within 5 mm strips aligned at the center of the incident beam. 1D profiles were then mapped onto the q-axis by reference to the position of the primary beam, attenuated by the semi-transparent beam stop of the instrument. The converted profiles were corrected for the scanner readout noise and normalized to the recorded intensities of the primary beam. The scattering curves from the buffer were then subtracted from the scattering curves of the tRNA sample. The resulting scattering intensity curves were averaged over two independent sample/buffer data acquisitions. The line-collimation 1D profiles were desmeared using GNOM software (Svergun 1992; Svergun et al. 2001), taking into account the length and width profiles of the incident beam. The resulting point-collimation-like data were used for the subsequent structural analysis in the q interval from 0.03 to 0.35 Å−1 (crystallographic resolutions between ~200 and ~18 Å). Evaluations of the quality of the fit of the scattering data to the various structural models were made with the program Crysol, version 2.5 (Svergun et al. 1995).

SAXS refinement using quasi-uniform angular averaging

A new module was developed for fitting RNA SAXS data via XPLOR-NIH or CNS which no longer requires the use of a globbic approximation and its associated correction terms, employed in several of our previous protein studies (Grishaev et al. 2005, 2008; Parsons et al. 2008). The new algorithm, following the standard description (Koch et al. 2003), represents scattering intensity predicted from a structure as

$$ I(q) = \left\langle {\left| {{\mathbf{F}}_{a} ({\mathbf{q}}) - \rho_{o} {\mathbf{F}}_{s} ({\mathbf{q}})} \right|^{2} } \right\rangle_{\Omega } $$
(1)

where F a (q) and F s (q) are the scattering amplitudes for the macromolecule and the excluded volume, respectively, ρ o is the solvent electron density, and 〈 〉Ω denotes the solid angle average over all orientations of the momentum transfer vector q for the fixed norm q.

Using a previously described dummy-solvent approximation, which assumes that displaced solvent resides exactly at the atomic positions in the macromolecule (Fraser et al. 1978), the solvent-subtracted complex scattering amplitude then becomes

$$ {\mathbf{F}}_{d} ({\mathbf{q}}) = \sum\limits_{j = 1}^{N} {g_{j} (q)\exp (i{\mathbf{qr}}_{j} } ) $$
(2)

with g j (q) representing solvent-subtracted atomic scattering amplitudes and the summation extending over all atomic coordinates r j . The advantage of this expression is that it scales linearly with the number of atoms, compared to the quadratic scaling inherent in the Debye formula, used in our earlier work. The calculation is then accelerated by approximating the exact angular average of Eq. 1 by a summation over a finite number of orientations, evenly distributed on the surface of a sphere (Schwieters and Clore 2007). A spiral grid algorithm with a total of 90 angular directions gives a robust representation of the scattering data within the experimental q range (0–0.35 Å−1) used in this study. In order to suppress systematic errors resulting from the finite number of equidistant angular directions, the set of spiral grid vectors is rotated by a random angle around a random axis every 50 time steps of the molecular dynamics trajectory. With this formalism, the force acting on atom m, when the experimental scattering data is I°(q) and the scattering data predicted from the current model is I(q), becomes

$$ \nabla_{m} \chi^{2} = \frac{4}{{N_{\text{dat}} N_{\text{grid}} }}\sum\limits_{j = 1}^{{N_{\text{dat}} }} {c_{j} \frac{{c_{j} I({q}_{j} ) - I^{o} ({q}_{j} )}}{{\sigma_{j}^{2} }}g_{m} ({q}_{j} )} \sum\limits_{k = 1}^{{N_{\text{grid}} }} {{\mathbf{q}}_{jk} \left\{ {\cos ({\mathbf{q}}_{jk} \cdot {\mathbf{r}}_{m} )\text{Im} [F_{d} ({\mathbf{q}}_{jk} )] - \sin ({\mathbf{q}}_{jk} \cdot {\mathbf{r}}_{m} )\text{Re} [F_{d} ({\mathbf{q}}_{jk} )]} \right\}} $$
(3)

where c j are the bound solvent corrections, σ j are the experimental uncertainties, and the sums run over all data points and q vector grid directions. The real and imaginary parts of the scattering amplitude in the above expression are calculated for a particular direction of the q vector on the equi-spaced grid.

tRNAVal structure refinement

Building a homology model for the tRNAVal structure and its further refinement against experimental restraints was carried out in two stages, summarized in Fig. 2. During the first stage, a homology-based model was built starting from the X-ray structure of tRNAPhe (Shi and Moore 2000). In the second stage, this homology model was refined against RDC and solution small angle X-ray scattering data. The two stages are described in detail below.

Fig. 2
figure 2

Flow diagram of the two-stage process to generate the refined tRNAVal structure

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Generation of tRNAPhe-based stage 1 model

A regularized tRNAVal model was built on the basis of the 1.93 Å resolution X-ray structure of yeast tRNAPhe, PDB code 1EHZ (Shi and Moore 2000), with hydrogens added with the program Reduce (Word et al. 1999). Generation of this “first stage” model comprised a Cartesian simulated annealing protocol, performed using XPLOR-NIH (Schwieters et al. 2003), including active energy terms for bonds, angles, impropers, repulsive-only non-bonded interactions, non-crystallographic symmetry (NCS) terms, base pairing planarity restraints (Kuszewski et al. 1997), as well as database potentials of mean force (PMF) for base stacking, pairing, and backbone dihedral angle correlations (Cai et al. 2003). The NCS module of the XPLOR-NIH program, with a force constant of 10 kcal Å−2, was used to keep the structures of the 1EHZ X-ray reference structure and the coordinates of the tRNAVal working model very close to one another. A single NCS term included all non-hydrogen atoms in the ribose-phosphate backbone and base non-hydrogen atoms for the 44 nucleotides which are identical in tRNAPhe and tRNAVal. Specifics of the NCS terms and their violation statistics are listed in the Supplementary Material section. The force constants for the database potentials were adjusted to yield matching PMF energies for the tRNAVal homology model and the 1EHZ tRNAPhe structure. The empirical force field terms for the bonds, angles and impropers were used at their defaults settings. Non-bonded interactions were modeled by a repulsive-only quartic term with the van der Waals radii scaled by a factor of 0.85 and a standard force constant multiplier of 4.0 kcal Å−4.

Structure refinement with RDC and SAXS data

During the “stage 2” refinement, the above derived homology model was refined against the experimental data, comprising 24 Pf1 RDCs, 20 MSA RDCs, and the SAXS profile. To allow moderate reorientation of the helices relative to one another, while preserving the relative geometries of the stacked and base-paired bases as much as possible, a large number of local NCS terms were defined by reference to their respective parts in the rigidly held stage 1 model. The terms included all sequential pairs of nucleotides, except for the connections between the acceptor stem/D arm and anticodon stem-loop/TψC arm which were kept flexible, as well as all base pairs within the helices, and long-range interactions that define the three-dimensional tRNA fold. Specifics of the NCS terms are listed, along with their violation statistics, in the Supplementary Material section. The refinement calculations were carried out using a CNS (Brunger et al. 1998) torsion angle dynamics simulated annealing protocol, with the temperature ramped in 80 steps from 2001 to 1 K, and one thousand 2-fs integration steps at each temperature stage. The experimental SAXS data extended from 0.03 to 0.35 Å−1 and were sparsened to 33 data points, prior to input for the refinement calculations.

Results and discussion

The accuracy of experimental data is a key consideration during any structure refinement, and this aspect becomes particularly critical when the number of experimental observables is well below the number of degrees of freedom in generating the structure, as applies to the current study. We therefore first discuss the uncertainties in the experimental input data, prior to evaluating the final structures.

RDC data quality

The close correlation relative to previously measured RDCs in Pf1 medium for tRNAVal (pairwise rmsd 1.1 Hz after scaling by a factor of 0.77 to account for differences in Pf1 concentration) suggests a random error ≤1 Hz in either set of values. The measurement error in the MSA RDCs, reported here as the difference between isotropic 1JNH splittings at 800 and 500 MHz, previously was estimated to be 0.3 Hz (Ying et al. 2007). The final values of the magnitude and rhombicity of the two corresponding alignment tensors, optimized during structure refinement, are 16.2 Hz and 0.570 for the Pf1 data and 0.908 Hz and 0.195 for the MSA alignment data. A ratio of 10:1 between the force constants for the MSA and Pf1 data was used in refinement, which reflects the much higher relative error of the MSA data. Note that a ratio in force constants equal to ca. 400:1 would be needed to give the two types of RDCs equal importance if their relative uncertainties would have been the same. Despite the higher relative uncertainty of the MSA data, they have a beneficial albeit small impact on the structure refinement of tRNAVal.

SAXS data quality

The scattering data recorded at concentrations of 2.29, 4.93 and 10.0 mg ml−1 show the presence of a non-negligible structure factor at all concentrations, presumably due to the high charge carried by the RNA, even though relatively high salt concentration (150 mM NaCl) was used in the buffer. In order to remove the effects of interparticle interference from the data, a linear extrapolation to zero concentration was performed based on the three measured concentration points. Briefly, the scattering curves at all three concentrations were aligned using the data from 0.10 to 0.35 Å−1, where the effects of structure factor are negligible. The aligned data (see inset to Fig. 3a) shows the presence of structure factor at q ≤ 0.07 Å−1. Therefore, linear extrapolation was performed point-by-point below 0.07 Å−1 and the extrapolated data were then merged with the 10 mg ml−1 data above that value. None of the collected data show any indication of aggregation, as evidenced by P(r) distributions that decay smoothly at the highest inter-atomic vector values. The extrapolated data were used for all subsequent structure analyses, with d max set to 95 Å for GNOM (Svergun 1992; Svergun et al. 2001) desmearing (Fig. 3b). The uncertainty of the scattering data, evaluated from the photon counting statistics, ranges from ~0.4% at q = 0.03 Å−1 to ~18% at q ~ 0.35 Å−1.

Fig. 3
figure 3

SAXS data for tRNAVal. (a) Desmeared experimental scattering data extrapolated to zero concentration, and their fit to the starting (stage 1; χ ~ 2.8) and refined (stage 2; χ ~ 0.9) tRNAPhe-based models. The inset shows the effect of the structure factor on the raw line-smeared data as a function of concentration. (b) Pairwise distance distributions, P(r), obtained from the Fourier transforms of the zero-concentration-extrapolated data, and the scattering data simulated from the final stage 2 model (red)

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Refinement of the homology model

The regularized homology model exhibits a 0.3 Å backbone rmsd to the tRNAPhe 1EHZ X-ray structure, with virtually unchanged relative positions of the four individual helices. This stage 1 model represents a near-optimal starting point for deriving a refined tRNAVal structure because the 58% sequence identity with tRNAPhe is the highest among tRNAs for which complete coordinates are available; moreover, all nucleotides involved in the long-range interactions responsible for the tertiary fold of tRNAPhe are strictly conserved between tRNAPhe and tRNAVal. For nucleotides lacking identity to the corresponding one in tRNAPhe, application of the empirical database potentials (Clore and Kuszewski 2003) which impact both sequential base stacking and relative positions of the base-paired elements is used to optimize the quality of the starting model.

Although this stage 1 model fits both Pf1 RDCs (rmsd 3.02 Hz; Q = 0.191) and MSA RDCs (rmsd 0.34 Hz; Q = 0.401) very well, it should be borne in mind that such fitting, carried out by singular value decomposition (SVD), includes five adjustable parameters (Losonczi et al. 1999; Sass et al. 1999) and therefore underestimates the true error, especially when only 20–24 RDCs are being fitted. A fairer way to evaluate the errors for a set of N RDCs, which also makes comparison to the analogous results on the refined model more straightforward, uses N − 1 RDCs to fit the alignment tensor by SVD, and calculates the difference between the remaining observed and predicted RDC. This procedure is repeated N times, each time leaving out a different RDC, and the rmsd between the observed and predicted non-fitted couplings is then evaluated. This jack-knifing procedure results in an rmsd of 4.03 (Pf1) and 0.45 Hz (MSA) for these two sets of data (Fig. 4), corresponding to jack-knifed Q factors of 25% and 53%, respectively.

Fig. 4
figure 4

Agreement between experimental RDCs and tRNAVal models. RDCPf1 (a), or RDCMSA (b) versus stage 1 model, generated without SAXS or RDC input data. (c) RDCPf1 versus refined (stage 2) model; (d) RDCMSA versus refined model. In all cases, filled symbols represent RDCs included when carrying out the SVD fit between the experimental data and the structure; open symbols are (a, b) jack-knifed RDC values, predicted from the alignment tensor when that particular RDC was not included in the SVD fit, and (c) cross-validated RDC values, predicted for the refined model calculated without that particular RDC

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Refinement of the stage 1 model against both RDC and SAXS data resulted in a narrow bundle of structures (coordinate rmsd to average of 0.3 Å) that exhibit a ~2.8 Å rmsd (nt. 1–72) relative to the stage 1 model (Table 1). These refined structures are characterized by an increase in the angle between the two arms of the L-shaped tRNA from ~81 to ~98° (Fig. 5). An analogous, slightly larger increase in the angle between these two arms previously was obtained by rigid-body optimization against the Pf1 RDCs alone (Vermeulen et al. 2005). Although application of only the SAXS restraints results in a slightly smaller increase in the angle between the two arms than application of just the RDC restraints (Table 1), this smaller change in global structure simply reflects the minimum change needed to get acceptable agreement with the SAXS data. The fit to the SAXS data remains equally good when the SAXS terms and RDC restraints are applied simultaneously, even though the inter-arm angles for the SAXS-only and SAXS + RDC structures differ by about 6°. Addition of the SAXS data also does not significantly impact the Q free factors of the resulting structures over the use of RDC restraints alone. On the other hand, it is important to note that when omitting the RDC restraints from the stage 2 structure refinement, inclusion of the SAXS data results in improved agreement with the RDCs, as manifested in a decrease of the jack-knifed Q value from 25% to 21% (Table 1).

Table 1 Impact of different types of restraints on tRNAVal structure during structure refinement
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Fig. 5
figure 5

Backbone structures of tRNAVal obtained by refinement of a tRNAPhe-based homology model. The regularized starting model (stage 1) is shown in blue; the model refined against Pf1 and MSA RDCs in green, and the model that includes both the RDC and SAXS data in red. The models are superimposed by best-fitting nucleotides 1–7 and 66–72. The absence of restraints for nucleotides 73–76 results in their disorder during the refinement simulated annealing protocol when using only RDC data, whereas they adopt a more compact shape when SAXS data are active

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The final rmsd between the experimental and best-fitted RDCs is ~1.2 Hz for Pf1 data and ~0.34 Hz for the MSA data, comparable to their estimated experimental uncertainties. Bound surface water corrections, which include the scattering by counter-ions, used during the SAXS data fits required six cycles for convergence to a final value of 0.066 eÅ−3. This is about two-fold higher than the typical 10% solvent density increase often seen in the water layer surrounding proteins, and may reflect the presence of counter-ions associated with the high charge density of RNA.

Structural statistics of the final refined models are summarized in Table 2. Objective evaluation of the quality of the models requires cross-validation statistics where the refinement is repeated, with a given Pf1 RDC left out of the refinement, and an SVD fit to the remaining ones is used to predict the value of the RDC not used during refinement. This refinement protocol is then repeated 24 times for each of the Pf1 RDCs. When SAXS data are not being fitted, such jack-knifed cross-validation yields a Q free of 13.9% when MSA RDCs are included and 14.3% without them. The small magnitude of the difference between the two Q free values results from the relatively high uncertainty of the MSA RDC data, and their correspondingly weak weighting factor. The final structure shows very low interatomic clashing scores (~5 clashes >0.4 Å per 1,000 atoms, versus ~18 for the stage 1 model, and ~23 for 1EHZ), as evaluated by Molprobity (Davis et al. 2004).

Table 2 Structural statistics for the RDC- and SAXS-refined tRNAVal homology model
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Concluding remarks

SAXS is increasingly being used to provide structural information in RNAs (Lipfert and Doniach 2007; Putnam et al. 2007). Recent applications include yeast tRNAPhe, the P4–P6 domain of the Tetrahymena ribozyme, a glycine riboswitch and a SAM riboswitch (Lipfert et al. 2007a, b; Putnam et al. 2007). However, in the absence of other structural restraints, the scattering data only provide low-resolution structural information and cannot uniquely define 3D structure. The goal of the present study is to extend these studies by combining the SAXS data with RDC and structural restraints to the homologous tRNAPhe to generate a refined model for tRNAVal. The RDC and SAXS data are largely complementary, where SAXS reflects the overall molecular shape and the RDCs provide orientational constraints for the helical domains. In our study, the RDCs tightly constrain the possible orientations of the helical arms, but provide no translation information on the distance between these arms. On the other hand, the SAXS data tightly constrain the distance between the two arms, and are less sensitive to small changes in interhelical angle or twisting about a helical axis.

Refinement of any structural model on the basis of limited experimental data, each with their own inherent uncertainty, can be challenging. For example, if NOE data were to be used for such refinement, calibration of the reference distance used for extracting distances from NOE intensities can cause systematic errors. Similarly, when using SAXS data, unrecognized interparticle interference effects or transient aggregation could result in a systematic bias during refinement. For RDCs, a potential systematic problem can arise when the magnitude or rhombicity of the alignment tensor used during refinement deviates from its true value. In our refinement procedure, these values as well as the orientation of the alignment tensor were allowed to float to give the best agreement with all the experimental data (Sass et al. 2001). Furthermore, the first five RDCs have no restraining value as there are five independent parameters required to define the alignment tensor, or three parameters if the system under study were to exhibit three-fold or higher axial symmetry.

The improved cross-validation statistics obtained upon inclusion of the RDC restraints indicates higher quality of the refined model compared to the starting structure. Clearly, however, when using a very small set of experimental restraints the cross-validation statistics attainable for the refined model strongly depend on the quality of the starting structure. For example, starting from a homology model that is based on the 1.9-Å X-ray structure of yeast tRNAPhe (58% identity) yields better statistics than starting from a more general model, generated on the assumption of idealized A-form helices (Supplementary Material). Starting from this latter model, which yields relatively poor agreement with the RDCs (Q free = 53%), refinement against RDC and SAXS data again yields considerable improvement. The final refined structure falls close (2.1 Å coordinate rsmd) to that of our refined homology model (PDB entry 2K4C), albeit with less favorable cross validation statistics (Q free = 28% instead of 14%; see Supplementary Material). In this respect, it is important to note that Q free simply reports on the orientations of imino N–H vectors relative to the alignment frame, which are impacted by both the global structure (e.g. interhelical angles) and by local structural noise (Zweckstetter and Bax 2002). Because the number of experimental restraints is far smaller than the number of parameters that define N–H vector orientations, the refinement protocol is fundamentally limited in its ability to remove local structural noise. Improvements in cross validation are therefore dominated by the more global changes in structure.

It is interesting that even the use of only SAXS data during refinement already results in a significant improvement in the fit of the high precision Pf1 RDC data to the model. A similar improvement in the fit of the Pf1 RDC data occurs when using only the MSA RDC values during refinement (Table 1). Due to the relatively large fractional measurement error in the very small MSA RDCs, they are only enforced with a weak force constant to prevent introduction of local distortions, and their impact on changing the global structure during refinement is therefore limited.

The procedure used in our refinement aims to keep local structure close to that of the starting model by requiring similar geometries for short, overlapping segments in the polymer, and conserved hydrogen bonding where indicated by homology. At the same time, these local geometries are not completely frozen and permit gradual changes along the polymer backbone. In principle, more abrupt changes are also easily accommodated, and this may be appropriate when indicated by a lack of homology in a certain region or by another perturbation such as a ligand binding event, marked by a chemical shift change. Although computationally quite demanding, the refinement approach used in our study strikes a balance between full-fledged structure calculations, which would require far more experimental input parameters, and the widely used procedure of rigid body refinement (Wang et al. 2000; Clore and Bewley 2002; Cai et al. 2003; Jain et al. 2004; Vermeulen et al. 2005; Tang et al. 2006; Bhatnagar et al. 2007).

Our refinement procedure relies on the use of a large number of NCS terms that serve to minimize local structural changes. Therefore, the refinement procedure reaches the solution closest to the starting structure (in terms of local rmsd) that is in satisfactory agreement with the experimental data. These NCS terms also extend to tertiary interactions that define pairing of the helices and tertiary structure of the tRNA. In cases where the secondary information is available but the tertiary fold is not known, a similar approach may be applicable, but whether or not a unique (and correct) solution can be obtained ultimately depends on the amount and quality of the available data and the specifics of the particular structure. In such cases, the NCS terms can be applied in the same way for the helical segments, but not for segments involving any unknown long-range tertiary interaction. Serious complications can arise when the inter-helical linkages are flexible, resulting in the absence of fixed orientations and/or translations between the individual helical units. In favorable cases, where a sufficient number of RDCs is available for each helical segment to determine an alignment tensor, such flexibility can be recognized if the alignment strengths of the helices differ, or it may manifest itself by different relaxation characteristics of the helical segments (Zhang et al. 2006). Although detailed information regarding such flexible structures can be obtained from NMR data by resorting to cleverly chosen modifications of the molecular system (Zhang et al. 2006; Bailor et al. 2007), data collected for a single molecule in a single liquid crystalline medium generally will be insufficient to uniquely define average orientations.

For a rigid system consisting of N helical segments, the degeneracy of the RDCs with respect to 180° rotations around each of the three principal axes of the alignment tensor results in 4N−1 distinct conformations (Al-Hashimi et al. 2000; Latham et al. 2008). Whether or not a unique solution can be selected from such a set depends on whether all but one of the 4N−1 conformations can be ruled out due to steric clashes, linkage strain, etc. SAXS data will also aid in filtering out incorrect conformations, but there is no guarantee that a unique solution will emerge. In either case, such a solution would have to be validated by the analysis of additional data which might include observed NOEs or comparison between the alignment tensor parameters predicted from a procedure such as PALES (Zweckstetter et al. 2004) and the experimentally observed ones. Although NOE analysis at the early stages of structure determination is often hampered by extensive resonance overlap in A-form RNA, once the set of solutions is restricted to a small number of structures, identification of long-range NOEs can become much easier. These considerations suggest that our “hybrid” approach to refining a multi-helical A-form RNA structure against a small number of RDCS and SAXS data may be applicable even in cases where the inter-helical connections are not known a priori.

Supplementary information available

Description of the NCS terms used in the refinement and their violation statistics; description of the refinement procedure starting from the idealized A-form tRNA model and its results; table with RDCs observed in tRNAVal.

Coordinates deposited to the RCSB Protein Data Bank under reference number 2K4C.

Software available

Scripts used for model refinement and source code for Xplor-NIH and CNS modules for SAXS data refinement via procedures described in this paper can be downloaded from http://spin.niddk.nih.gov/bax/software/.