Movatterモバイル変換

• 1. Introduction
• 2. Local resolution heterogeneity
• 3. Interpreting EM density maps: fold recognition
• 4. Model building
• 5. Refinement
• 6. Validation
• 7. Discussion
• C1. Pooled-atoms plane
• C2. Angles between planes
• C3. Parallel-plane restraints in Coot
• References

• 1. Introduction
• 2. Local resolution heterogeneity
• 3. Interpreting EM density maps: fold recognition
• 4. Model building
• 5. Refinement
• 6. Validation
• 7. Discussion
• C1. Pooled-atoms plane
• C2. Angles between planes
• C3. Parallel-plane restraints in Coot
• References

Tools for macromolecular model building andrefinement into electron cryo-microscopy reconstructions

The recent rapid development of single-particle electron cryo-microscopy (cryo-EM) now allows structures to be solved by this method at resolutions close to 3 Å. Here, a number of tools to facilitate the interpretation of EM reconstructions with stereochemically reasonable all-atom models are described. TheBALBES database has been repurposed as a tool for identifying protein folds from density maps. Modifications toCoot, including new Jiggle Fit and morphing tools and improved handling of nucleic acids, enhance its functionality for interpreting EM maps.REFMAC has been modified for optimal fitting of atomic models into EM maps. As external structural information can enhance the reliability of the derived atomic models, stabilizerefinement and reduce overfitting,ProSMART has been extended to generate interatomic distance restraints from nucleic acid reference structures, and a new tool,LIBG, has been developed to generate nucleic acid base-pair and parallel-plane restraints. Furthermore, restraint generation has been integrated with visualization and editing inCoot, and these restraints have been applied to both real-space refinement inCoot and reciprocal-spacerefinement inREFMAC.

Keywords:model building;refinement; electron cryo-microscopy reconstructions;LIBG.

4.1. Jiggle Fit

Jiggle Fit is implemented to be used downstream of either rigid-body docking or manual placement of domains and secondary-structure elements (SSEs) to improve the fit to the density. Prior to this work,Coot had an extant simple `Jiggle Fit' system that was designed to optimize the orientation of small ligands (Debreczeni & Emsley, 2012). The atom selection was restricted to a single residue, no map masking was performed and there was no consideration of the neighbouring atoms that might affect the pose. The original system applied a random set of rotations and translations to generate hypotheses, each of which was scored using aZ-weighted sum of the map density at the atom positions. The rotations were selected from a uniform distribution on (0, 2π) for each of the three independent rotation axes, and translations along each of the axes were selected from a uniform distribution on (0,s), wheres is a user-definable distance. The model with the highest scoring fit to the density then underwent real-spacerefinement before the coordinates were updated. This system was extended to make it suitable for optimizing the fit to density for macromolecules as follows.

To test the dependence of Jiggle Fit on map resolution, we created reconstructions of the 54S subunit at multiple resolutions ranging from 3.4 to 6.8 Å (Table 1). Rather than low-pass filtering the maps to lower resolution, we generated maps with subsets of particles usingRELION (Scheres, 2012) to more closely replicate real data sets. The coordinates for a reference molecule (bL9) were agitated as a rigid body by both a random set of unlimited rotations around each axis and a random set of translations that were limited to a defined distance from the final coordinates (0–5 Å). Jiggle Fit was then performed at each resolution for all starting models and the output was assessed by superposition with the reference model. The trials were conducted using complete 54S maps, rather than with segmented maps, to replicate instances in which the boundaries of the protein are not fully known.

From the results, translation had a greater effect on the rate of success than rotation (Figs. 4a and 4b). Jiggle Fit identified the correct solution for each attempt where the coordinates were randomly rotated, or randomly rotated and displaced by up to 1 Å in any direction. As the position of the starting model diverges further from the final location, Jiggle Fit is less able to determine the correct solution. Even at a resolution close to 7 Å and displaced up to 5 Å from the final position, the correct solution is successfully attained in 20% of cases.

Figure 4 (a,b) Jiggle Fit improves local fit to density. (a) Randomly rotated and displaced models (by up to 1 Å, left) can be jiggled into their corresponding densities in a manner not dependent on resolution (right). (b) The dependence of Jiggle Fit on resolution and displacement from the correct solution. For clarity, four resolutions are shown: 3.37 Å (unfilled squares), 4.05 Å (triangles), 5.03 Å (circles) and 6.79 Å (filled squares). (c,d) Jiggle Fit coupled to SSE identification. (c) Examples of density for anα-helix at (from left to right) 6.8, 5.0 and 3.2 Å resolution, showing loss of pitch and side-chain densities at lower resolution. (d) Resolution dependence of Jiggle Fit in determining helix orientation.

4.2. Morphing

Often, the initial model used to interpret the density map is similar to the structure to be solved. However, differences, perhaps as the result of conformational changes, the absence of crystal contacts or inaccurately modelled regions, can leave sections of the model outside the density. Additionally, rigid-body docking of multiple components can result in unphysical bonds and steric clashes at the boundaries of domains. To overcome some of these limitations, fitting methods have been described that take into account the dynamic properties of macromolecules. These include normal modes, as implemented iniMODFIT (Lopéz-Blanco & Chacón, 2013), deformable elastic networks, as inDireX (Wang & Schröder, 2012), and molecular-dynamics flexible fitting (MDFF; Trabucoet al., 2008).

Model morphing inCoot is designed to take advantage of the local similarity of the template and target structures. EM maps are sufficiently noisy and low resolution that a rigid-body fit of individual residues would result in a model with severe geometric problems. The model-morphing tool was designed to make local shifts that reduce geometric distortions. The method takes each residue in turn and constructs a (by default) five-residue fragment based around this central residue (using two residues upstream and downstream of the central residue). Each five-residue fragment is fitted to density by a rigid-body fit, which provides a rotation–translation operator for each residue. Each residue has a local environment (i.e. residues which have atoms that are within a user-specified distance, typically 10 Å, of the atoms of the central residue). The rotation–translation operators of the residues of the environment are sorted by how much they move their atoms and robustly averaged, with the top and bottom 25% discarded to provide a rotation–translation operator for the central residue. This process is repeated for each residue in the chain and can be carried out recursively. Indeed, serial application of morphing is often required for convergence. The larger the averaging radius, the smaller the local shifts that are applied, which increases the number of times that this morphing procedure has to be executed to reach convergence.

To illustrate morphing, the structure of bacterial 23SrRNA (PDB entry3v2d , chainA) was fitted by global rigid-body docking to the density of half maps from 54S reconstructions at resolutions ranging from 3.4 to 6.8 Å (Table 1). The core regions of rRNA from bacterial and mitochondrial ribosomes are structurally conserved but divergent in sequence, and display local conformational changes at the periphery. There are several regions where the bacterial rRNA model and mitochondrial rRNA density do not correspond, but it is clear that with a relatively small local rotation–translation the residues of the model could be made to fit the map. The bacterial structure was morphed, using a local environment set at 7 Å, for four iterations (Fig. 5). The progress of morphing was followed by calculating FSC curves for the starting bacterial model, the morphed model and the final fully refined mitochondrial rRNA against the half map used for morphing (FSC_work). To confirm that morphing was not resulting in overfitting (see below), the FSC was also calculated against the half map that had not been used for morphing (FSC_test; Fig. 6).

Figure 5 Example of model morphing. (a) Section of RNA taken from the complete rigid-body docking of bacterial rRNA into the mitochondrial ribosome map (morph 0) and morphed inCoot for three iterations. (b) The final refined structure of mitochondrial rRNA.

Figure 6 FSC curves following the progress of morphing at 3.37, 4.05, 5.03 and 6.79 Å resolution. Black lines represent the fit of mitochondrial rRNA to both mitochondrial half maps at the given resolution. Dark blue lines represent the initial fit of bacterial rRNA to both mitochondrial half maps. The bacterial rRNA was morphed four times: iterations 1 (light blue), 2 (green), 3 (orange) and 4 (red). Excluding the fourth iteration at 6.79 Å resolution, the FSC curves for both half maps overlap, demonstrating that morphing does not result in overfitting.

A similar approach to morphing has been reported (Terwilligeret al., 2012) for improving crystallographic models, particularly for molecular-replacement solutions that are not close enough to the target structure for automated building, using electron-density maps. Morphing inCoot can be used in a similar way.

4.3. Identification of secondary-structure elements (SSEs)

At subnanometre resolutions, SSEs are discernible in density maps:α-helices appear as long cylinders andβ-sheets as continuous and somewhat flat expanses of density. As SSEs can reliably be identified from protein amino-acid sequences, locating these in the density map is critical for initiatingde novo model building. SSE localization has been implemented in bothGorgon (Bakeret al., 2012) andChimera (Pettersenet al., 2004) through a graphical version ofSSEHunter (Bakeret al., 2007). A similar function, the `Find Secondary Structure' tool inCoot, performs a six-dimensional rotation and translation search to find the likely positions of bothα-helices andβ-strands within the density map (Emsleyet al., 2010).

However, this tool had been tuned to fit to electron-density maps from X-ray crystallography, where there is little variation in theZ-score (the number of standard deviations) of the electron density of secondary-structure main-chain atoms. The density maps obtained in cryo-EM reconstructions can have substantially largerZ-values owing to the typically larger box size, much of which is filled with zero, or near-zero, density values (a result of putting the EM reconstruction density in an empty box and normalizing). Thus, the calculation of map statistics from EM maps has been changed; instead of simply summing the density values and their squares to generate the mean and variance, the values are now added into finely sampled bins. The peak of this histogram is determined and the corresponding density points are discarded from the calculation of the mean and variance. This results in an estimation of the mean and variance of the map that is more consistent with those from X-ray data and allows the fitting of SSEs, without user intervention, in maps from both X-ray crystallography and cryo-EM.

For nucleic acid macromolecules,Coot can generate idealized atomic models with canonical Watson–Crickbase pairing of single-stranded or double-stranded A-form or B-form DNA or RNA given a nucleotide sequence. Alternatively, RNA motifs can be obtained fromRNA Looplib or modelled usingAssemble2 (Jossinetet al., 2010) and imported intoCoot. These can all act as starting points forde novo building.

After the localization of SSEs and/or idealized nucleic acid helices, Jiggle Fit can be used to improve the fit to density and to correctly orientateα-helices. To demonstrate this, we placed polyalanine helices in both orientations in density corresponding to the mitoribosomal protein bL27 (PDB entry3j6b , chainR). Each helix was subjected to Jiggle Fit and scored for correct orientation against the final structure for a range of resolutions. The results (Fig. 4) show that at up to 4 Å resolution helix identification followed by Jiggle Fit invariably finds the correct orientation; even at close to 7 Å resolution, where helices predominantly appear as featureless tubes (Fig. 4), the correct orientation is identified 75% of the time.

4.4.De novo building

Coot offers many tools forde novo model building. C^α baton mode allows the path of a protein to be traced by placing correctly spaced C atoms that can then be converted into a main chain and the sequence assigned (Emsleyet al., 2010). Alternatively, residues can be added to the N-termini and C-termini of chains one residue at a time. For building nucleic acids,RCrane (Keating & Pyle, 2012) allows users to trace the backbone by placing phosphates into density and then automatically constructs all-atom models of the nucleotides. Once an initial model has been built,Coot has a suite of tools for moving atoms to optimize the fit and stereochemistry, alongside methods of validation (Emsley & Cowtan, 2004; Emsleyet al., 2010).

5.1. Similarity of real-space and reciprocal-space refinements

There is some debate in the structural biology community as to whether real-space or reciprocal-space (Fourier space)refinement should be used for optimizing the fit of atomic models into EM maps. Both have their advantages, and in essence refinements in real andreciprocal space are similar (AppendixA). The advantages of using reciprocal-space refinement are as follows.

However, real-spacerefinement also offers many attractive features.

It has been shown that real-spacerefinement as a supplement to reciprocal-space methods improves protein models more than the exclusive use ofreciprocal space (Chapman & Blanc, 1997). Therefore, we advocate a strategy that utilizes both real-space refinement tools inCoot and reciprocal-spacerefinement withREFMAC (Fig. 7).

Figure 7 Flowchart showing the overall scheme for restrained refinement of models against EM data.ProSMART generates three classes of restraint: (i) reference restraints, (ii) helical fragment restraints and (iii) secondary-structure hydrogen-bond restraints (which include helix, sheet and loop restraints). Alongside reciprocal-spacerefinement inREFMAC, real-space refinement tools inCoot can be used to optimize the fit to density.

5.2. Electron scattering

Although the density distributions obtained from X-ray crystallography (electron density) and EM (Coulomb potential) both originate from scattering events on the atoms within macromolecules, they are not equivalent. Electrons are scattered by the charge on the nucleus screened by the electron shell of atoms and, unlike the scattering of X-rays, their scattering is affected by local electric charges and ionization states. To take this into consideration,REFMAC was modified so that in EM mode it switches to a five-Gaussian approximation for electron scattering factors taken from Cowleyet al. (2006).

5.3. Refinement against averaged, composite and segmented maps

Sample heterogeneity can result in multiple maps being calculated from a single data set, with each map displaying discrepancies in both resolution and occupancy (Fernándezet al., 2014; Unverdorbenet al., 2014). The resolution of defined regions within the maps (for example a bound factor or an individual ribosomal subunit) can be improved by focusing particle classification/alignment on this particular region through the application of soft masks during EM data processing (Amuntset al., 2014; Fernándezet al., 2014). This further expands the collection of maps that can be utilized for model building, refinement and biological interpretation. Multiple maps can be used in refinement to improve the quality of the data to which the model is fitted. Therefore,REFMAC has been adapted to handle, and refine against, multiple input maps. Averaging maps will improve the signal-to-noise ratio by increasing the strength of the signal relative to noise. However, in the case of maps generated through focused alignments, averaging may not be desirable as this would negate the advantage introduced by masking. Therefore,REFMAC can generate and refine against composite maps formed by combining maps, with averaging only at the interfaces between the maps.

Forrefinement,REFMAC can calculate structure factors for only the section of the map explained by the input model. These are complex structure factors and not just amplitudes, so phase information is not discarded. It is against these structure factors that the model is refined rather than the complete map. This strategy can be used to refine individual components within a larger reconstruction or repeat units of symmetric macromolecules, and requires the model to be placed in aunit cell with the same dimensions as the box size used for the EM reconstruction.

5.4. External restraints

Including chemical and structural information as restraints inrefinement reduces the effective number of parameters, thus increasing the effective residualdegrees of freedom. Restraints can increase the consistency of the derived atomic models with the available prior knowledge, help to preserve the correct geometry in cases where local structures would otherwise be distorted duringrefinement, stabilizerefinement and reduce overfitting. We have previously demonstrated the value of distance restraints generated from homologous reference structures and structural fragments in improving the quality of protein structures from crystallographic data (Nichollset al., 2012, 2013). It has recently become apparent that their application to EM data is just as valuable (Amuntset al., 2014). To improve the geometry of nucleic acids during refinement, we have modifiedProSMART to generate nucleic acid reference restraints, and provide a new toolLIBG to generate base-pair and parallel-plane restraints.

5.5.ProSMART

Restraints generated using external structural information should help the macromolecule underrefinement to adopt a conformation that is more consistent with previous observations. If the reference and target models share a high degree of structural similarity, then we might expect their local inter­atomic distances to be approximately equal. Such information is exploited byProSMART, which generates local interatomic distance restraints that can then be used to aid the refinement of the lower resolution structure in reciprocal space withREFMAC or in real space withCoot.ProSMART only generates restraints with objective values less than a given threshold (typically 4.2 Å) to maintain a degree of global conformational independence between the target and reference structures. Indeed, external restraints are designed to be longer range than chemical bond and angle restraints, while being sufficiently short to be resistant to differences in global conformation. This allows external restraints to be used even when the target and reference structures are, for example, in different bound states, display large-scale domain movements or when crystal contact-induced conformations have resulted in differences between the X-ray and EM structures.

Structurally similar models that can act as reference structures can be identified from the PDB using services such asPDBeFold (Krissinel & Henrick, 2004) orDALI (Holm & Rosenström, 2010). The modifications toProSMART allow reference structures to be either protein and/or nucleic acid macromolecules. As the usefulness of external restraints is limited by the quality of the prior information, reference-model reliability should be considered. The reference structure should be solved experimentally at a higher resolution than the current model and the potential for reference-model errors should not be overlooked. Alongside manual checking of the fit of the model to the density, it may be sensible to attempt re-refinement, and even manual rebuilding, of any reference structure before restraint generation. This might be performed manually or automatically, for example withPDB_REDO (Joostenet al., 2009). Such approaches may reduce error propagation from reference to target models.

ProSMART is also able to generate restraints based on generic hydrogen-bond patterns and idealized structural fragments (Fig. 7). These can help to stabilize protein secondary structure and might be applied when a suitable reference structure is not available, or when the reference is itself not sufficiently well resolved. For example, an idealα-helix may be used to generate restraints that will keep helical structures intact. Such helical restraints are different to generic hydrogen-bond helical restraints, since they include restraints between all sufficiently close backbone atoms. Also, the fragment-based helical restraints do not require strict compliance with ideal secondary-structure conformation in order to be detected. This is particularly relevant at lower resolutions, where secondary structure may not be sufficiently well formed to be detected from predicted hydrogen-bonding patterns.

The exact usage of external restraints tends to vary between cases and at different stages of the structure-determination process. For example, restraints can be used to temporarily force the maintenance of sensible conformations during the earlier stages of structure determination, and then subsequently to stabilize refinement in later stages. However, it should be acknowledged that such an approach can introduce bias, resulting in the model adopting a conformation that is less consistent with the observed data. However, the use of external restraints can result in a model adopting a conformation very similar to a high-resolution homologue, ideally resulting in an improved model. We suggest that external restraints should only be used if the benefits of any improvements in reliability are deemed to outweigh the negative effects.

5.6.LIBG

LIBG produces restraints to maintain nucleic acid geometry using information extracted directly from a model, similar to that described forCNS andphenix.refine (Laurberget al., 2008). These restraints are applicable to all DNA/RNA molecules and can be applied in conjunction with reference restraints. Putative base pairs are identified by inspecting the local neighbourhood around the N and O atoms of a base for hydrogen-bond candidates in an adjacent base. A base pair is selected if the combination of hydrogen-bonding patterns between two bases satisfies the preset patterns of hydrogen bonding between DNA/RNA base pairs and the values of the hydrogen-bonding lengths, torsion angles and features of chirality are within the allowed deviation ranges from the corresponding reference values, which are estimated statistically from the database of high-resolution X-ray and neutron crystal structures (Clowneyet al., 1996; Xin & Olson, 2009). Users can adjust these criteria by changing the allowed deviations.

Currently,LIBG generates restraints for canonical Watson–Crick and noncanonical G:U base pairs. Since noncanonicalbase pairing allows multiple pairing of bases (for example, wobble and reverse wobble G:U pairs),REFMAC was adjusted to refine against multiple distance and torsion-angle targets (Fig. 8). During refinement, in every cycle, the best agreeing target is selected as the `ideal' parameter.

Figure 8 Restraint visualization inCoot. (a) Restraints were generated usingProSMART for an initial model of the mitoribosome (yellow) using the bacterial ribosome (purple) as a reference and were visualized inCoot. There are conformational differences between the two rRNA chains despite the sequence identity in the displayed region. Consequently, the local interatomic distances are conserved along the chain (grey) but are shorter across the chain (blue). Interatomic vectors coloured red indicate that the distances in the target structure are longer than in the reference structure. (b) Visualization ofProSMART hydrogen-bond restraints inCoot. (c) G:U base pair shown in (top) wobble and (bottom) reverse wobble configuration. (d) A G:U base pair with both pairs ofLIBG restraints displayed inCoot. Only the distance restraints that best describe the orientation of the bases (grey, G:U wobble) are used as targets during refinement. Restraints for the reverse wobble configuration are shown in red. Parallel-plane restraints are shown in yellow.

LIBG also generates restraints to preserve stacking interactions between nucleic acid bases and planar side chains of protein amino acids (parallel-plane restraints). The definition of a plane by a set of atoms is given in AppendixC. The atom sets appearing in each of all possible planes in DNA/RNA bases and protein residues are also pre-defined (Vaginet al., 2004). The possible pair of stacking planes is determined by calculating the angle between the normals of two atom planes in different DNA/RNA bases or protein amino acids, angles between the normal of one plane and the vector linking the two `gravity' centres of planar atoms, and the distance between those two `gravity' centres. If the calculated values are within pre-defined ranges, which can be varied by the user, then the two planes are selected as candidates for stacking.

5.7. Coupling restraint weight with local resolution

Unlike the globalrefinement weight applied duringrefinement withREFMAC, external restraints operate locally. This is of particular use in refinement against EM data, where the most appropriate refinement strategy should be selected based on local resolution. For regions at lower resolution it may be necessary to increase the contribution (weight) of the external restraints in order to restrict overfitting or distorting geometry, whereas for regions of higher resolution the contribution of the external restraints can be reduced to limit model bias. Resolution can be quantified on a local basis usingResMap (Kucukelbiret al., 2013) or by calculating the `gold-standard' FSC while applying a soft mask over the required region. For this purpose, we provide a script that usesRELION (Scheres, 2012) to calculate the local resolution for every chain in a given PDB entry. This information can then be used to select appropriate external restraint weights.

5.8. Visualizing external restraints withCoot

Before and afterrefinement, it is important to manually inspect the model alongside the density map to ensure the local suitability of the use of external restraints.ProSMART comparative structural analysis (Nichollset al., 2014) can be used to quickly and easily visualize the extent of local conformational changes that occur during refinement. This can provide information regarding stability during refinement, the effect of different refinement protocols and the degree of influence of any external restraints used. If there are any serious artefacts that arise owing to bias towards reference structures, it may be appropriate to re-attempt refinement excluding particular restraints.Coot can help to facilitate such manual intervention in the external restraint-generation and restraint-application procedure. BothProSMART andLIBG have been integrated withCoot.ProSMART can be executed directly from withinCoot, requiring both the target and reference structures to be specified. Any set of externally generated restraints can be visualized and applied inCoot, with options for manual editing (Fig. 8). Restraints corresponding to interatomic distances that are reasonably similar in both models will aid refinement by acting as regularisers, while those exhibiting large differences will have little effect on refinement owing to being weighted down by the use of the Geman–McClure robust estimation function (Geman & McClure, 1987) inREFMAC.

5.9. Refinement of symmetric particles

For symmetric macromolecules, the signal-to-noise ratio can be greatly improved by averaging symmetry-related projections. This typically results in higher resolution reconstructions than can be achieved for asymmetric molecules. By applying symmetry during particle averaging, eachasymmetric unit is considered to be identical. It is therefore necessary to refine only a singleasymmetric unit against a masked (segmented) map and then apply symmetry operators to generate the complete structure. However,refinement must take symmetry into consideration in order to optimize the contacts at the interface between asymmetric units. Symmetry operators can be given either as a set of operators that generates the whole symmetry group of a molecule or by specifying polar angles, Euler angles or matrix vectors. Once alllocal symmetry operators are known they are used to generate the symmetry-related atoms that can make nonbonded interactions with the refined molecule, and their contributions to therefinement procedure are included. If the whole map is used forrefinement then symmetry-related atoms are used both for map calculation and for the contribution of the fit to the experimental map.

5.10. Monitoring fit to density during refinement

In X-ray crystallography, theR factor is a measure of the agreement between the structure amplitudes calculated from a model and those from the data. It is an important global measure characterizing the quality of an X-ray structure for a given set of experimental data. WeightedR factors (1) are often used to control behaviour duringrefinement. However, when weights inrefinement change these indicators may not comparable, as demonstrated in AppendixB. For example, using map sharpening during refinement is equivalent to multiplication of the structure factorF_h by exp(−Bs²/4). Therefore, care should be taken when using overallR factors, or overall weighted FSCs, as a global measure of fit to density. In order to maintain consistency with crystallographicrefinement,R factors are calculated using amplitudes of structure factors only, whereas FSC is calculated using complex Fourier coefficients; thus, FSC carries more information about the fit of atomic model parameters into the EM map.

To avoid this dependence on weight, we prefer to use FSC_average,

whereN_shell is the number of resolution shells used to calculate FSC, FSC_i is for theith shell andN_i is the number of structure factors in theith shell. FSC_average is therefore independent of weight if the resolution shells are sufficiently thin that the weights on all structure factors within each shell are approximately equal. AverageR factors would also be less dependent on weight than overallR factors; however, they would also be, in general, larger than overallR factors. Therefore, to avoid improper usage and comparison between the two values, it would be desirable for FSC_average to be adopted as the preferred metric for monitoring the progress ofrefinement and comparison between structures solved by EM. It should be noted that FSC_average is not meant as a replacement for a plot of the FSC between map and modelversus resolution.

5.11. General application ofREFMAC to EM structures

We have previously appliedrestrained refinement withREFMAC to ribosome structures solved by cryo-EM (Amuntset al., 2014; Fernándezet al., 2014; Wonget al., 2014). To demonstrate that this approach can be used on a diverse range of structures, EM maps with a reported resolution of 4 Å or better were obtained from the Electron Microscopy Data Bank (EMDB; release 2014-03-26; Lawsonet al., 2011) and the associated models from the PDB (Bermanet al., 2002). Maps not associated with a full-atom model were discarded, and an additional four maps were removed for technical reasons. Higher resolution structures that could act as reference models for restrained refinement were obtained using a search of the PDB for structural similarity (Krissinel & Henrick, 2004). Prior to refinement, each model was inspected for reasonable geometry, conformation and sterics usingMolProbity (Chenet al., 2010) and for fit to density using FSC_average (Fig. 9). Deposited models show a great variation in theMolProbity clashscore, which is the number of clashes per 1000 atoms, with clashes declared at an overlap of ≥0.4 Å. The clashscores are typically worse relative to structures solved by X-ray crystallography within a similar resolution cohort and lie at the 30th percentile. Only 20% of structures are annotated as having undergone any form ofrefinement. Each model was then subjected to two rounds ofrefinement inREFMAC with reference (when applicable) and secondary-structure restraints applied. Each round ofrefinement consisted of 20 cycles with external restraints regenerated between rounds. In cases where the models were of symmetric species, only the repeat unit was refined.Refinement improved the clashscore for all of the structures and improved the fit to density in all but three cases (Fig. 9). These cases were potentially overfitted prior to refinement, or the default refinement procedure was not adequate to improve the fit to density. The clashscore was lowered by a statistically significant average of 69.5 points (p = 6.5 × 10⁻⁶; pairedt-test), with all models occupying a percentile better than 90 (with an average of 98.5). The fit to density, as measured by FSC_average, improved from a mean of 0.58 to 0.67 (p = 6.0 × 10⁻³; pairedt-test). Overfitting could not be examined as it is not yet common practice to deposit half maps.

Figure 9 Box-and-whisker plots for refinement of EM structures at 4 Å resolution or better. (a)MolProbity clashscores before (pre) and after (post)restrained refinement withREFMAC. (b) Improvement in clashscores showing the mean of the differences and 95% confidence intervals. (c) FSCaverage before and afterrestrained refinement. (d) Improvement in FSCaverage showing the mean of the differences and 95% confidence intervals.

As an example, we refined the structure of the heterotrimeric repeat unit of F420-reducing [NiFe] hydrogenase (Frh) from a hydrogenotrophic methanogenic archaeon (PDB entry4ci0 ) against the deposited map at 3.34 Å resolution (EMD-2513; Allegrettiet al., 2014). Reference restraints were generated from other [NiFe] hydrogenases resolved at higher resolution and secondary-structure hydrogen-bond and helical fragment restraints were generated for the complete heterotrimer. The quality of the model was examined before and after refinement usingMolProbity (Chenet al., 2010). All statistics improved (Table 2), with the exception of the Ramachandran outliers, presumably as the dihedral angle restraints applied during model building can position backbones into incorrect local minima.

APPENDIX A

On the similarity of real-space and reciprocal-space refinement

In this appendix, we demonstrate that, in essence,refinement in real space andreciprocal space are similar. We use the following notation: bold letters are vectors,h is a reciprocal-space vector,x is a real-space vector,F is a complex Fourier coefficient,s is the length of the reciprocal-space vector in an orthogonal coordinate frame corresponding to the indexh, denotes the Fourier transformation and denotes the inverse Fourier transformation. To simplify the equations, we assume thatρ₁(x) and its reciprocal-space counterpartF_1h correspond to the observed map and structure factors,ρ₂(x) and its reciprocal-space counterpartF_2h are the map and structure factors corresponding to an atomic model, and all definable parameters including an overall scale are included inρ₂(x) orF_2h.

Let us assume that we have two maps,ρ₁(x) andρ₂(x), in a box with volumeV. Let us denote their Fourier transformationsF_1h = andF_2h =. According to Parseval's theorem (Rudin, 1991),

and for the discrete version of this relationship (in practice we work with discretized versions of maps, so the following relationship is more relevant),

whereN₁,N₂ andN₃ are the number of grid points in three orthogonal directions.

The left-side summation is over grid points in real space and the right-side summation is over all reciprocal-space vectorsh within the resolution range. Note that the limits ofh are defined by the resolution of the map, whereas in real space the grid sampling can be as fine as desired. Consequently, we can assert that unweighted least-squares minimization in real space is equivalent to least-squares minimization inreciprocal space. One of the advantages of using minimization inreciprocal space is that it is relatively straightforward to design weights,

wherew_h = 1/Σ(h) is a weight,Σ(h) = 〈|F_1h −F_2h|²〉 is the variance of differences between structure factors and 〈.〉 is the expectation operator.

This formulation is essentially equivalent to using the log-likelihood function based on the conditional distribution of observed complex structure factors given calculated structure factors (Luzzati, 1953). Note that weighted least squares in reciprocal space is not directly related to weighted least squares in real space. If we use Parseval's theorem followed by the convolution theorem, we can see that

whereW(x) = is the inverse Fourier transformation of the weights used inreciprocal space and * denotes complex conjugation.

In the summation, bothx andy run over all grid points in the box. This relationship shows that using weighted least squares inreciprocal space is equivalent to using multivariate least squares in real space,i.e. accounting for the correlation between all points in the map. Since Parseval's theorem and the convolution theorem work for forward as well as backward Fourier transformations, it can be seen that using weighted least squares in real space is equivalent to using multivariate least squares inreciprocal space. It seems that although reciprocal-space and real-space refinements are similar, it might sometimes be more convenient to design weights in one space or the other. Iterating between real-space and reciprocal-space weighted least-squares fitting might allow one to derive an optimal model that explains the experimental data.

APPENDIX B

On the dependence of the overallR factor and the Fourier shell correlation onrefinement weights

Notation used in this appendix:h is the reciprocal-space vector with lengths,S is a reciprocal-space sphere with radiuss and dS is an element of this sphere.

It is common practice to controlrefinement behaviour using either correlation orR factors. In this appendix, we demonstrate that one should be careful in using such overall indicators. When weights inrefinement change, these indicators are no longer comparable. Such weights can either be by design or implicit. For example, using different sharpening duringrefinement is equivalent to multiplication of the structure factorsF_h by exp(−Bs²/4). Therefore, calculating overall correlation andR factors is equivalent to using weighted correlation orR factors. The overall weightedR factor is given by

and the overall weighted FSC is given by

When no weights are used thenw_h = 1.

Note that when calculated using different weights these statistics are not equivalent. An extreme case is when the weight corresponding to one reflection (k) is 1 and all others are 0,

In this case the FSC will be cos(φ_1k −φ_2k) and theR factor will depend on only one reflection.

It is easy to see that when using different overallB factors inrefinement (map sharpening or blurring) we are essentially using weights for the calculation of the overallR factor and FSC: for the overallR factor we usew_h = exp(−Bs²/4) and for the overall FSC we usew_h = exp(−Bs²/2). To avoid this dependence on weight, we prefer to use the average FSC,

whereN_shell is the number of resolution shells used to calculate the FSC, FSC_i corresponds to theith shell andN_i is the number of structure factors in theith shell. If the resolution shells are sufficiently thin then the weights of all structure factors within each shell will be roughly equal to each other. Since the same weights are on the denominator and numerator of the expressions for theR factor and correlation, they will cancel out, and FSC_average will be independent of weight. In the limiting case when a shell width goes to 0, and if we assume that the reciprocal-space points are sufficiently dense, then FSC_average would converge to the following integral:

where FSC(s) is calculated on the surface of a reciprocal-space sphere of radiuss,

wheres_min ands_max are the resolution limits used in FSC calculations, integration is over the reciprocal-space surface of the sphereS of radiuss and dS represents a surface element.

If the weights are dependent only on the length of the reciprocal-space vector (as is the case for effective weights owing to the overallB factor) then it is seen that each FSC(s) is independent of the weight, and therefore the average FSC is also independent of such weights.

APPENDIX C

Parallel-plane restraints

Notation used in this appendix: bold letters are three-dimensional vectors,uv^T =u₁v₁ +u₂v₂ +u₃v₃ is the scalar product of two three-dimensional vectors, |u| = (uu^T)^1/2 is the length of the three-dimensional vector, (a,d) defines a plane and the equation of a plane isax^T −d = 0 for allx ∈R³.

Let us assume that we have two sets of atoms {x₁₁,x₁₂ …,x_1n} and {x₂₁,x₂₂ …,x_2m}. We want each set to be on a plane and these planes to be parallel. Mathematically, this can be expressed in various ways, two of which are the following.