research papers
Shiftfield
of macromolecular atomic models^{a}Department of Chemistry, University of York, York, United Kingdom, and ^{b}Derpartment of Mechanical Engineering, McGill University, Montréal, Canada
^{*}Correspondence email: kevin.cowtan@york.ac.uk
The aim of crystallographic structure solution is typically to determine an atomic model which accurately accounts for an observed diffraction pattern. A key step in this process is the
of the parameters of an initial model, which is most often determined by using another structure which is broadly similar to the structure of interest. In macromolecular crystallography, the resolution of the data is typically insufficient to determine the positional and uncertainty parameters for each individual atom, and so stereochemical information is used to supplement the observational data. Here, a new approach to is evaluated in which a `shift field' is determined which describes changes to model parameters affecting whole regions of the model rather than individual atoms only, with the size of the affected region being a key parameter of the calculation which can be changed in accordance with the resolution of the data. It is demonstrated that this approach can improve the radius of convergence of the calculation while also dramatically reducing the calculation time.Keywords: refinement; low resolution; computational methods.
1. Introduction
Crystallographic R factor and free R factor (Brünger, 1992), which provide a measure of reliability for the resulting model.
in which the parameters of an atomic model are optimized to best explain an observed diffraction pattern, is an important stage in the structuresolution process. In addition to improving the correspondence between the atomic model and the scattering matter of the crystal, the step typically incorporates the calculation of statistics such as theCrystallographic et al., 1989). In the case of macromolecular disorder and thermal motion limit the resolution of the diffraction pattern and thus the number of available observations; this means that the data are insufficient to allow us to uniquely determine all of the atomic parameters. Geometrical restraints on the stereochemistry are therefore incorporated in order to ensure that the equations are overdetermined. The diffraction observations and stereochemical restraints introduce very different patterns of correlation among model parameters: each diffraction observation impacts every positional coordinate, whereas any given geometric restraint affects only a small number of atoms, introducing strong correlations between the coordinate parameters of those atoms. The use of stereochemical restraints therefore leads to a substantial increase in the complexity of the topology of the landscape, leading to slower convergence; this is only partially mitigated by use of higher order derivatives of the target function (Murshudov et al., 2011).
has traditionally involved the optimization of positional, thermal and other parameters that describe each individual atom in the (DriessenAn alternative approach to this problem is to reduce the number of model parameters to better reflect the amount of information that is present in the data. In particular, if the finescale features of the model (including the desired bond lengths and angles, but also less conserved features such as sidechain rotamers) are assumed to be correct, then coarsescale shifts can be applied to the model in order to better account for features of the diffraction pattern, for example domain motions or differences in chain placement between a close structural homologue and the experimental target. Terwilliger et al. (2013) proposed a method for doing this by performing a local search for electrondensity features which match a given model chain fragment for consecutive fragments extracted from the atomic model. Other approaches explored in the field of cryoelectron microscopy include the use of normalmode analysis to explore difference in model conformation (Tama et al., 2004; Suhre et al., 2006) and the hierarchical of rigid model domains (Joseph et al., 2016).
An alternative approach was proposed by Cowtan & Agirre (2018) in which parameter shifts are calculated, based on the modification of existing algorithms, which are applicable to arbitrarily sized regions of the map rather than to individual atoms only. A preliminary test demonstrated the ability of this approach to refine isotropic thermal parameters for an atomic model given data to limited resolution, and is now available as a tool for Bfactor in the Coot modelbuilding software (Emsley & Cowtan, 2004).
The advantage of seeking coarsescale model shifts to explain the observations is that the resulting shifts largely preserve the finescale features of the model (such as bond lengths) and can therefore be performed without the introduction of stereochemical restraints. The simplicity of the
landscape, coupled with the ability to refine at low resolution, ultimately leads to faster calculations and faster convergence.1.1. Crystallographic refinement
The crystallographic et al., 1989; Sheldrick & Schneider, 1997; Henderson & Moffat, 1971). In modern implementations, calculations include likelihood weighting to make optimal use of the information available (Read, 1986; Murshudov et al., 1997; Blanc et al., 2004; Afonine et al., 2012).
calculation is characterized by optimization of the parameters of the atomic model which, in most implementations, involves trying to minimize the features of the difference map that arises from the disagreement between the observed and calculated structure factors (DriessenThe disagreement between the model and the data may be expressed in et al., 1981; Bricogne, 2001). In the Lifchitz formulation, the calculated electrondensity map is differentiated with respect to the (positional or thermal) parameters of interest (Bricogne, 2001); a positive correlation between this gradient map and the difference map, when integrated over the volume of an atom at a given position in the map, implies that the agreement between the model and observations can be improved by modifying the corresponding parameters of the atom at that position.
as differences between observed and calculated structure factors or in real space though a difference map calculated from the structurefactor differences, with the two formulations being mathematically indistinguishable (AgarwalThe
of atomic coordinates converges quickly when the current atomic model coordinates are close to the true values (assuming that the model is in all other respects a good description of the crystal cell) but often fails to converge to a global minimum when parts of the initial model are far from the true coordinates: this is often the case when starting from a molecularreplacement model which shares only modest sequence identity with the structure of interest.1.2. Shiftfield refinement
Cowtan & Agirre (2018) made two modifications to this approach. Firstly, instead of integrating over the volume of a single atom, the agreement between the gradient map and the difference map is integrated over a much larger spherical region whose volume is determined by the resolution of the data. Secondly, agreement is determined by linear regression, where the gradient maps with respect to each type of model parameter are used as predictors of the difference map. This regression calculation, which is performed via Fourier transforms (Bricogne, 2001), produces a spatial field of shifts for each model parameter and is referred to as a shift field.
Shiftfield B factors), are described by the following system of equations,
therefore attempts to account for features of the difference map through adjustments to the calculated electrondensity map, which may or may not be determined from a model. These adjustments, which may include moving the density (for example by moving atoms) or changing the density peak heights (for example by adjustingwhere x_{j} is some parameter of the electron density (such as the position of a grid point in the map), Δx_{j} is the shift to be applied to that parameter, ρ_{i} is a calculated electrondensity value determined from the current model through a position in the map close to x_{j}, and D_{i} is the differencemap value at that position in the map. In order to determine the coordinate shifts to be applied to the model density at any point in the map without the calculation becoming underdetermined, we accumulate information from a spherical region of many electrondensity points (denoted by the index j) about the point denoted by index i. By choosing a large enough sphere (and therefore a large enough number of electrondensity points), the problem is always well determined and can be solved by leastsquares or, better, by weighted linear leastsquares regression in which density values closer to the point denoted by the index i are given greater weight (Cowtan & Agirre, 2018). In addition to the conventional parameters x, y, z and B, an additional constant term can be included in the regression calculation to mop up a roughly constant offset between the model density and the observations, which may arise from errors in lowresolution terms.
The method as described performs a single step of shiftfield ρ is calculated, a likelihoodweighted difference map D is determined using the observed structurefactor amplitudes and a shift field is determined. The initial stages of the calculation are performed at low resolution to allow large shifts to whole domains and the resolution limit is increased at each step.
As in existing strategies, the calculation does not typically converge in a single cycle and so the calculation must be applied iteratively. At each step, a new model electrondensity map1.3. Comparison to previous work
The method presented here is essentially the same as the proofofconcept calculation of Cowtan & Agirre (2018), except that we now explore the application of the method to coordinate rather than Bfactor and test whether the inclusion of an additional constant parameter provides any benefits. We have introduced the term `shiftfield in this paper to distinguish the method described here from conventional calculations and to highlight the applicability of the method to the of a map against observations, although we do not utilize this feature in the current work.
While shiftfield et al. (2013), there are significant underlying differences. Terwilliger and coworkers calculate gradients for individual atoms based on their local environment and then apply a moving average of the shifts along the chain, whereas shiftfield averages shifts over a spherical region around each map grid point without reference to a model. On theoretical grounds, we would therefore expect the two methods to be complementary: model morphing should be capable of sliding one chain longitudinally with respect to a neighbouring chain because shifts are only propagated along chains, while shiftfield should be capable of capturing large domain shifts because the shifts can capture information from neighbouring chains, and do so more quickly by working at lower resolution and not requiring geometrical restraints.
addresses a similar problem to the modelmorphing approach of Terwilliger2. Methods
To evaluate the effectiveness of shiftfield et al. (2020). Bond and coworkers identified 1351 well refined structures from the Protein Data Bank (Berman et al., 2007) for which experimental data were available, which uniformly sample the resolution range 1.0–3.5 Å. The structures were selected at random, but had to be good quality as judged by five validation metrics: DCC R_{free}, clashscore, Ramachandran outliers, sidechain outliers and RSRZ outliers. They also had to be diverse, containing no protein chains with a sequence identity of 50% or more. Bond et al. (2020) also produced molecularreplacement models which lead to initial maps that sample a range of map qualities. They performed sequence alignment of the search model on the deposited model using GESAMT (Krissinel, 2012) before (MR) using Sculptor and Phaser (Bunkóczi & Read, 2011) and a short conventional using REFMAC5. For many of these structures, either MR did not successfully find all copies of the molecule, some copies of the molecule were incorrectly positioned or multiple chain sequences were present in the structure, so this set of structures was reduced to a subset for which there was only one chain in both the MR structure and the deposited structure, leaving a total of 452 models. The free R factors for the models, after preliminary against the deposited reflection data, range from 22 to 56%. The target structures, search models and relevant quality metrics are listed in the supporting information to this paper.
for atomic coordinates, was attempted using 452 molecularreplacement problems from a set compiled by BondEach model was then refined using one of the following four procedures. The choices of parameters for these procedures will be discussed below.
The results and computational requirements of a REFMAC5 uses data to the resolution limit by default and runs for a userdetermined number of cycles rather than using a convergence criterion. The conventional calculation is performed here using the commandline defaults, with the exception of the number of cycles, which was increased from the CCP4 graphical user interface default of 10 (Potterton et al., 2003) to 20 on the basis of tests with automated model building. For jellybody the default parameters were taken from the CCP4 graphical user interface, and the number of cycles was set to 200 to address the slower convergence of this method (Kovalevskiy et al., 2018); inspection of the with cycle number suggests that the has essentially converged by the end of the computation. For shiftfield the resolution and number of cycles were determined by a coarse search to optimize the results using a smaller data set; we found that running for significantly more than 12 cycles can distort the model geometry to the point where the final conventional cannot restore it.
calculation are influenced by the choice of program parameters and the number of cycles run.2.1. Metrics for evaluating results
In order to evaluate the behaviour of the different
protocols, quality metrics are required. Given the size of the test set, these must be quantitative and not require manual evaluation on a perstructure basis. We consider two types of metrics: those which depend on some estimate of the `true' structure (which we refer to as extrinsic metrics) and those which do not (which we refer to as intrinsic metrics).We calculate extrinsic metrics based on the structure deposited in the PDB (Berman et al., 2007). Common metrics include coordinate differences and phase errors or map correlations. Given that the aim of this work is to improve the preliminary of MR models, we assume that the deposited structures are a better description of the data than the refined search models and will use C^{α}atom r.m.s.d. to evaluate mainchain fit. This does depend on the deposited model being largely correct, and on establishing which atoms of the MR model are structurally homologous to corresponding atoms in the true structure.
Intrinsic metrics include the crystallographic R factors and model geometry metrics such as bondlength variability, Ramachandran and rotamer outliers and clashscore. The agreement between observed and calculated structurefactor amplitudes is particularly useful because the residual error in the calculated complex is typically independent in phase from the itself, and thus the error in the amplitude provides an intrinsic estimator of phase error (Srinivasan & Parthasarathy, 1976; Murshudov et al., 1997). This assumption is however biased by of the model against the structurefactor amplitudes, which can be mitigated by use of the crystallographic free R factor (Brünger, 1992; Lunin & Skovoroda, 1995), calculated in this case by REFMAC5 (Murshudov et al., 2011). The utility of the free rather than work R factor in this context is also supported by comparison with the phase errors (Supplementary Table S2). We also evaluate geometry statistics determined by REFMAC5 and MolProbity (Chen et al., 2010). These metrics are calculated for each of the 452 test structures after of the MR solutions using each of the four procedures outlined above.
2.2. Shiftfield implementation
The mathematical details of the shiftfield method have been described by Cowtan & Agirre (2018); however, in order to facilitate the reimplementation of the method by other authors we outline it in more detail here. The steps of the calculation are as follows.
A simplified process diagram is shown in Supplementary Fig. S3. The whole procedure has been implemented in C++ using the Clipper crystallographic libraries (Cowtan, 2003).
3. Results
Each of the four shows a comparison between the final free R factors for models refined using conventional alone, jellybody plus conventional and shiftfield plus conventional (with the omission of the constant term). Fig. 1(a) compares conventional with jellybody In the majority of cases, the inclusion of jellybody causes a very small reduction in the free R factor compared with conventional alone. However, there are a subset of cases where the reduction in free R factor is far more substantial, with the extreme case being PDB entry 4c2q, where the inclusion of jellybody leads to a reduction in the free R factor of 15%. Fig. 1(b) compares conventional and shiftfield with the results showing a similar pattern of improvement to the jellybody method. The largest improvement is again for PDB entry 4c2q, with an improvement in free R factor of 13%. Fig. 1(c) compares jellybody and shiftfield confirming that while there are outliers for which either method may lead to a better reduction in free R factor, shiftfield provides the lowest free R factor for a larger number of test cases and offers the greatest benefits for the challenging cases where the free R factor is highest. Comparison of the R factors for the work rather than free sets yields similar results (see supporting information).
procedures described above were applied to each of the 452 molecularreplacement models in turn. Fig. 1The MR search models often represent only part of the structure owing to sequence nonhomology, pruning, omission of waters and ligands, and in some cases significant conformational differences, and as a result the R factors for the refined MR models are substantially higher than the deposited structures (an average of 42% versus 17%). This shows that the benefits of shiftfield are not contingent on having a substantially complete model.
Fig. 2 shows a comparison of free R factor after shiftfield plus conventional where the constant term is either omitted or included. Inclusion of the constant term appears to offer no significant benefit yet significantly increases the number of fast Fourier transforms required, so it is not recommended. We hypothesize that the principal benefit of including the constant term would be to address the case of missing lowresolution reflections which lead to longrange ripples across the map. However, these are already removed in practice by the omission of these terms from the differencemap calculation when using difference map coefficients (Murshudov et al., 2011).
Geometry validation indicators for models from the various . The differences are minor and there is no clear pattern; for example, jellybody produces the lowest proportion of Ramachandran outliers but the highest proportion of rotamer outliers.
procedures are given in Table 1

We also examined the stability of the shiftfield method when applied to the deposited structures in order to determine the extent to which the method is able to preserve the features of the most correct model available. Shiftfield REFMAC5 and a zerocycle run was then used to evaluate the 12 cycles of shiftfield were applied to the resulting model, and another zerocycle run was used to determine the without performing any further conventional Application of shiftfield caused an increase in the average of the crystallographic R factor over the 452 test cases from 17.3% to 18.5%, while the free R factor increased from 20.7% to 21.8%. The results show no evidence of overfitting (since the free R factor increases by no more than the work R factor), and the models suffer only minor distortions, which are small in comparison to the convergence radius of conventional refinement.
is performed with no stereochemical restraints, and so it is possible that the model may be significantly distorted or overfitted. For each case, the deposited structure was subjected to 20 cycles of inFig. 3 shows comparisons between the same methods as in Fig. 1 but for the rootmeansquared C^{α} coordinate difference (r.m.s.d.) in comparable regions of the refined and deposited models. To ensure a valid comparison, the coordinate differences must be calculated over the same subset of atoms for each case, but the MR models may have substantial insertions and deletions compared with the target structure. Therefore, a list of comparable residues was first identified for each case by performing a sequence alignment between the MR and deposited models. We then select from the successfully aligned residues those for which the C^{α} atom of any one of the refined models lies within 3 Å of the deposited coordinate in order to exclude outlier residues which otherwise dominate the r.m.s.d. Fig. 3 shows a very similar pattern to Fig. 1, with both jellybody and shiftfield leading to an improvement in the r.m.s.d. of the deposited model, but with shiftfield generally offering larger benefits for the poorest models.
Fig. 4(a) shows the distribution of r.m.s.d.s between each of the 452 initial MR models and models refined by conventional, jellybody or shiftfield Jellybody moves the initial model substantially further than conventional with shiftfield moving the model further still. Fig. 4(b) shows a similar comparison of all of the models with the corresponding deposited structure. The conventionally refined models are on average the furthest from the deposited structure and the shiftfieldrefined models are closest, which implies that the larger movements shown in Fig. 4(a) primarily move the models towards the deposited structures.
The variations in calculation time as a function of unitcell volume for shiftfield and jellybody computations are shown in Fig. 5 for highresolution (better than 1.5 Å) and lowresolution (worse than 2.5 Å) subsets of the data. In all cases the calculation becomes slower with increasing structure size. For conventional and jellybody higher data resolutions lead to longer calculation times. Since in these tests the shiftfield calculation is performed at the same resolution for each case, the calculation time is independent of resolution. For the least favourable case of a large (6 × 10^{6} Å^{3}) and low (3 Å) resolution, the shiftfield calculation is still four times faster than conventional and 40 times faster than jellybody with these numbers increasing by a further factor of five for higher resolution data sets. The additional time penalty for jellybody arises primarily from slower convergence and thus a greater number of required cycles, while the speed of the shiftfield calculation arises in part from being able to perform the calculation at lower resolution, varying from 6 to 3 Å over the course of the calculation. The disparities with conventional and jellybody can be somewhat reduced by reducing the number of REFMAC5 cycles; for example, Kovalevskiy et al. (2018) use only 100 cycles of jellybody in favourable cases. Further largescale testing is required to optimize computational cost against model quality for all of the procedures.
We further investigated the disparities in computational cost by performing REFMAC5 and shiftfield at the same resolution (3.5 Å) for each structure. In this case the percycle calculation time is about equal for the two methods (about 1 s per cycle using our current hardware), confirming that the ability to run at lower resolution and for fewer cycles accounts for almost all of the difference in computation. Startup time also plays a role, being negligible for shiftfield but an average of 30 s for REFMAC5. The similar percycle costs at comparable resolutions are unsurprising since both approaches to require an electrondensity calculation and a few tens of fast Fourier transforms, although the shiftfield algorithm may be amenable to some optimization. Despite this similarity, the numbers of density grid points which contribute to gradient determination at a given atom or grid point are very different: in the case of conventional the contributing volume covers the volume of an atom: usually of the order of ten grid points for an individual atom in the case of 3 Å resolution data and thus a 1 Å grid. In the case of shiftfield the gradient at a point in a 3 Å resolution map will be determined using all of the density points in a sphere of 12 Å radius: more than 5000 grid points.
calculations using bothWhen developing software tools for unknown problems, our principal focus is on the results of the automatic application of the method to large systematically chosen ensembles of test structures because this provides the most useful predictor of the distribution of results that a nonspecialist user might obtain when applying the method using default options to an unknown structure. Nonetheless, it is occasionally possible to obtain insights into the functioning of different methods by examining individual models. We examined cases for which either shiftfield or jellybody
substantially outperformed the other method and highlight two such examples below.For PDB entry 4l9m, the R factor after shiftfield plus conventional was 40%, compared with 43% for jellybody plus conventional A section through the C^{α} traces of the two models is compared with the deposited structure in Fig. 6. While the lower domain is largely the same in all of the models, the upper domain shows a significant rotation, leading to a displacement of the main chain by up to 4.5 Å. This displacement is largely corrected by shiftfield but jellybody fails to correct the structure in this case.
By contrast, for PDB entry 2d66 the R factor after shiftfield plus conventional was 37%, compared with 32% for jellybody plus conventional The C^{α} traces of the two models are compared with the deposited structure in Fig. 7. The centre of the helix at the top of the molecule is displaced by about 3 Å : this is largely corrected by jellybody but not by shiftfield A likely factor in this behaviour is the presence of part of a neighbouring symmetry molecule close to the surface helix: this portion of the neighbouring molecule does not need to move. Shiftfield looks for coordinated motions over large regions and so cannot reconcile the change in behaviour across a molecular boundary, whereas jellybody can make this distinction through the explicit use of stereochemical restraints. It is likely that the modelmorphing technique of Terwilliger et al. (2013) would also work in this case.
4. Discussion
We have shown that shiftfield
can complement conventional modelrefinement methods in two major ways: firstly, the calculation can be conducted at low resolution, with benefits in terms of both speed and radius of convergence; secondly, the calculation is very fast, usually converging in fewer cycles than conventional or jellybody The improvement in speed arises largely from the application of coarsegrained shifts to correct largescale errors in the model, which largely preserves local geometry, thus avoiding the need for stereochemical restraints which can complicate the topology of the target function, and in part from the being performed at low resolution with fewer data.The lack of stereochemical restraints in the shiftfield method does allow gradual distortion of the model geometry to occur as the calculation progresses, so the shiftfield REFMAC5, offsetting some of the speed benefit. Nevertheless, the combined procedure still has a larger radius of convergence than conventional alone and is significantly faster than the jellybody method, which is often used to increase the radius of convergence (Murshudov et al., 2011). The method also appears to provide an increased radius of convergence compared with jellybody but performs less well than jellybody when different shifts are required across the boundaries between closely neighbouring molecules.
calculation must be complemented by conventional in a program such asFuture work will investigate whether alternating cycles of shiftfield ab initio modelling solutions, this may offer benefits for largescale screening calculations in which a large portion of the Protein Data Bank is searched to try and explain the observed diffraction pattern (Rodríguez et al., 2012; Simpkin et al., 2018). A second aim will be to apply the shiftfield approach to the of one map against another or against a set of diffraction observations, which has applications both in the use of and in the use of cryoelectron microscopy reconstructions to explain Xray or other diffraction data.
and regularization could allow the full speed of shiftfield to be exploited without the overhead of a conventional step. Since is a useful tool in the evaluation of or4.1. Data and methods
The computer code and data sets used in this paper are available at https://doi.org/10.15124/5d8e73077bde4e47875d5f15f30177bd. The methods described here are also distributed with version 7.1 of the CCP4 software suite in the sheetbend software.
Supporting information
Link https://doi.org/10.15124/5d8e73077bde4e47875d5f15f30177bd
Computer code and data sets used.
Description of data sets used along with additional metrics and figures. DOI: https://doi.org/10.1107/S2059798320013170/di5041sup1.pdf
Acknowledgements
The authors thank G. Bricogne for suggesting that we evaluate the contribution of the constant term to the regression, and a referee for suggesting that we investigate the stability of the method when applied to deposited structures.
Funding information
This work was supported by the BBSRC through grants BB/L006383/1 and BB/S005099/1, by the White Rose BBSRC DTP in Mechanistic Biology (BB/M011151/1) and by Collaborative Computational Project, Number 4: Software for Macromolecular Xray Crystallography (CCP4).
References
Afonine, P. V., GrosseKunstleve, R. W., Echols, N., Headd, J. J., Moriarty, N. W., Mustyakimov, M., Terwilliger, T. C., Urzhumtsev, A., Zwart, P. H. & Adams, P. D. (2012). Acta Cryst. D68, 352–367. Web of Science CrossRef CAS IUCr Journals Google Scholar
Agarwal, R., Lifchitz, A. & Dodson, E. (1981). In Proceedings of the CCP4 Study Weekend. Refinement of Protein Structures, edited by P. A. Machin, J. W. Campbell & M. Elder. Warrington: Daresbury Laboratory. Google Scholar
Berman, H., Henrick, K., Nakamura, H. & Markley, J. (2007). Nucleic Acids Res. 35, D301–D303. Web of Science CrossRef PubMed CAS Google Scholar
Blanc, E., Roversi, P., Vonrhein, C., Flensburg, C., Lea, S. M. & Bricogne, G. (2004). Acta Cryst. D60, 2210–2221. Web of Science CrossRef CAS IUCr Journals Google Scholar
Bond, P. S., Wilson, K. S. & Cowtan, K. D. (2020). Acta Cryst. D76, 713–723. Web of Science CrossRef IUCr Journals Google Scholar
Bricogne, G. (2001). International Tables for Crystallography, Vol. B, 2nd ed., edited by U. Shmueli, pp. 25–98. Dordrecht: Kluwer Academic Publishers. Google Scholar
Brünger, A. T. (1992). Nature, 355, 472–475. PubMed Web of Science Google Scholar
Bunkóczi, G. & Read, R. J. (2011). Acta Cryst. D67, 303–312. Web of Science CrossRef IUCr Journals Google Scholar
Chen, V. B., Arendall, W. B., Headd, J. J., Keedy, D. A., Immormino, R. M., Kapral, G. J., Murray, L. W., Richardson, J. S. & Richardson, D. C. (2010). Acta Cryst. D66, 12–21. Web of Science CrossRef CAS IUCr Journals Google Scholar
Cowtan, K. (2003). IUCr Comput. Comm. Newsl. 2, 9. Google Scholar
Cowtan, K. (2008). Acta Cryst. D64, 83–89. Web of Science CrossRef CAS IUCr Journals Google Scholar
Cowtan, K. & Agirre, J. (2018). Acta Cryst. D74, 125–131. Web of Science CrossRef IUCr Journals Google Scholar
Driessen, H., Haneef, M. I. J., Harris, G. W., Howlin, B., Khan, G. & Moss, D. S. (1989). J. Appl. Cryst. 22, 510–516. CrossRef CAS Web of Science IUCr Journals Google Scholar
Emsley, P. & Cowtan, K. (2004). Acta Cryst. D60, 2126–2132. Web of Science CrossRef CAS IUCr Journals Google Scholar
Henderson, R. & Moffat, J. K. (1971). Acta Cryst. B27, 1414–1420. CrossRef CAS IUCr Journals Web of Science Google Scholar
Joseph, A. P., Malhotra, S., Burnley, T., Wood, C., Clare, D. K., Winn, M. & Topf, M. (2016). Methods, 100, 42–49. Web of Science CrossRef CAS PubMed Google Scholar
Kovalevskiy, O., Nicholls, R. A., Long, F., Carlon, A. & Murshudov, G. N. (2018). Acta Cryst. D74, 215–227. Web of Science CrossRef IUCr Journals Google Scholar
Krissinel, E. (2012). J. Mol. Biochem. 1, 76–85. CAS PubMed Google Scholar
Lunin, V. Y. & Skovoroda, T. P. (1995). Acta Cryst. A51, 880–887. CrossRef CAS Web of Science IUCr Journals Google Scholar
Murshudov, G. N., Skubák, P., Lebedev, A. A., Pannu, N. S., Steiner, R. A., Nicholls, R. A., Winn, M. D., Long, F. & Vagin, A. A. (2011). Acta Cryst. D67, 355–367. Web of Science CrossRef CAS IUCr Journals Google Scholar
Murshudov, G. N., Vagin, A. A. & Dodson, E. J. (1997). Acta Cryst. D53, 240–255. CrossRef CAS Web of Science IUCr Journals Google Scholar
Potterton, E., Briggs, P., Turkenburg, M. & Dodson, E. (2003). Acta Cryst. D59, 1131–1137. Web of Science CrossRef CAS IUCr Journals Google Scholar
Read, R. J. (1986). Acta Cryst. A42, 140–149. CrossRef CAS Web of Science IUCr Journals Google Scholar
Rodríguez, D., Sammito, M., Meindl, K., de Ilarduya, I. M., Potratz, M., Sheldrick, G. M. & Usón, I. (2012). Acta Cryst. D68, 336–343. Web of Science CrossRef IUCr Journals Google Scholar
Sheldrick, G. & Schneider, T. (1997). Methods Enzymol. 277, 319–343. CrossRef PubMed CAS Web of Science Google Scholar
Simpkin, A. J., Simkovic, F., Thomas, J. M. H., Savko, M., Lebedev, A., Uski, V., Ballard, C., Wojdyr, M., Wu, R., Sanishvili, R., Xu, Y., Lisa, M.N., Buschiazzo, A., Shepard, W., Rigden, D. J. & Keegan, R. M. (2018). Acta Cryst. D74, 595–605. CrossRef IUCr Journals Google Scholar
Srinivasan, R. & Parthasarathy, S. (1976). Some Statistical Applications in Xray Crystallography. Oxford: Pergamon Press. Google Scholar
Suhre, K., Navaza, J. & Sanejouand, Y.H. (2006). Acta Cryst. D62, 1098–1100. Web of Science CrossRef CAS IUCr Journals Google Scholar
Tama, F., Miyashita, O. & Brooks, C. L. III (2004). J. Struct. Biol. 147, 315–326. Web of Science CrossRef PubMed CAS Google Scholar
Terwilliger, T. C., Read, R. J., Adams, P. D., Brunger, A. T., Afonine, P. V. & Hung, L.W. (2013). Acta Cryst. D69, 2244–2250. Web of Science CrossRef IUCr Journals Google Scholar
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