research papers
Noncrystallographic symmetryconstrained map obtained by direct density optimization
^{a}Life Science Group, Scientific Research Division, National Synchrotron Radiation Research Center, 101 HsinAnn Road, Hsinchu 30076, Taiwan, ^{b}Institute for Protein Research, Osaka University, 32 Yamadaoka, Suita, Osaka 5650871, Japan, ^{c}Department of Biotechnology and Bioindustry Sciences, National Cheng Kung University, Tainan 701, Taiwan, and ^{d}Department of Physics, National Tsing Hua University, Hsinchu 30043, Taiwan
^{*}Correspondence email: cjchen@nsrrc.org.tw
T = 1 shelldomain subviral particle of Penaeus vannamei nodavirus with data affected by using the REFMAC5 software.
(NCS) averaging following molecularreplacement phasing is generally the major technique used to solve a structure with several molecules in one such as a spherical icosahedral viral particle. As an alternative method to NCS averaging, a new approach to optimize or to refine the electron density directly under NCS constraints is proposed. This method has the same effect as the conventional NCSaveraging method but does not include the process of Fourier synthesis to generate the electron density from amplitudes and the corresponding phases. It has great merit for the solution of structures with limited data that are either twinned or incomplete at low resolution. This method was applied to the case of theKeywords: noncrystallographic symmetry; NCSconstrained map; direct density optimization; twinning; REFMAC5.
1. Introduction
The molecularaveraging method in real space coupled with solvent flattening is powerful in phase determination or phase improvement in protein crystallography. In the et al., 1987). Under the special conditions that the envelope or icosahedral matrices are given with sufficient precision and the of the density are sufficiently small, i.e. a lower crystallographic free fraction, the map can be built ab initio (Yoshimura et al., 2016). Since early in the 1970s, molecular averaging has been performed with iterative calculations of Fourier transformation (FT) and inverse FT between real and inverse space (Buehner et al., 1974; Bricogne, 1976). Many applications and results using iterative molecularaveraging methods have been reviewed by Kleywegt & Read (1997).
of icosahedral viruses, (NCS) averaging with phase extension is a common procedure for phase improvement after initial calculations based on (MR) using a density map from a cryoelectron microscope, a similar structural model or initial experimental phases from or (ArnoldAs an alternative method, we have conceived a method to optimize or refine the density values directly under NCS constraints to reproduce the observables of the amplitude F_{obs}. We know empirically that such a direct densityoptimization (DDO) method has the weakness of a small convergence radius and difficulty in finding the initial conditions to achieve the correct solutions, but once the initial conditions have been obtained, it has a good potential to attain the most reasonable solution. In contrast to the conventional averaging method, the DDO method contains no Fourier synthesis, in which the electron density is calculated from the amplitudes (usually the observables F_{obs}) and the corresponding phases. Consequently, the electrondensity maps do not suffer from incomplete observables, which are typically caused by the experimental setup, such as a cut by the beamstop shadow. Furthermore, in cases of twinned data there is significant merit in avoiding the process of Fourier synthesis, in which detwinning (or deconvoluting) the amplitudes is generally difficult.
In this paper, we report the principles and the application of the DDO method to construct an NCSconstrained density map of the T = 1 shelldomain (Sdomain; Sd) subviral particle (SVP) of Penaeus vannamei nodavirus (PvNV) in order to solve its structure. The biological details and structural results of PvNV have recently been reported (Chen et al., 2019). We obtained two crystal forms, in space groups P2_{1}2_{1}2_{1} and P2_{1}3, for the T = 1 PvNV Sdomain SVP, where the data from the P2_{1}3 crystal were merohedrally twinned. The coordinates of the T = 1 Sdomain SVP of Grouper nervous necrosis virus (GNNV; PDB entry 4rft; Chen et al., 2015) served as an initial model; the NCSconstrained electrondensity maps of the two crystal forms were obtained using the DDO method. Using the NCSconstrained maps, a new structure of the T = 1 PvNV Sdomain SVP was determined. Even though one data set was twinned, the electrondensity map was deduced. In the following, we discuss the DDO method, the conditions for acquiring valid electron density and the convergence radius, and compare the DDO method with other common methods.
2. Concept and methods
2.1. Concept
Molecular averaging in real space is generally performed with an iterative calculation between real and inverse space linked by Fourier transformation (FT) and inverse FT (FT^{−1}). A schematic diagram of this common method is shown in Fig. 1(a). Here, we propose a new method according to which the density of the minimal region for a target molecule is first postulated and copies of NCSequivalent molecule densities are then generated. The calculated amplitudes (F_{calc}) produced by FT are eventually compared with the observed amplitudes (F_{obs}), as shown in Fig. 1(b). To maximize the agreement between the calculated F_{calc} and the data F_{obs}, the density of the molecule is hence treated as a parameter and optimized. This method is free from the Fourier synthesis (FT^{−1}) process that uses the data amplitude F_{obs} (Fig. 1). It has the merit of treating twinned data and solving the structure without difficulty (Fig. 2). This concept can be recognized as a `refinement' technique that is directly applied to the electron density. To obtain valid electron density, the correct region of the target molecule, the correct NCS matrices and a sufficiently small crystallographic free fraction (ff; Yoshimura et al., 2016), which denotes the ratio of the unconstrained density region to the unitcell volume, are essential. These conditions are described in Section 4. Table 1 lists notations for and definitions of the terms and abbreviations used in this work.

2.2. Method for DDO calculations
As there was no suitable DDO software for us to directly optimize or to refine the electrondensity values in this work, we utilized the popular REFMAC5 (Murshudov et al., 2011) to perform and test the DDO method in an indirect way. REFMAC5 refines the coordinate parameters and temperature factors in seeking the to reproduce the amplitude observables. The solution from the MR method was used for the initial density. The solution using the GNNV SVP retained the icosahedral relation. An icosahedral unit of the molecule was selected; the other molecules were expressed with each NCS matrix, and the NCS matrices were used as the NCSconstraint parameters for the In the case of our icosahedral virus structures, strict icosahedral matrices were constantly applied as the NCS matrices. Using REFMAC5, the coordinates and B factors of the initial model were refined under unrestrained while imposing NCS constraints using the ncsconstraint option. Using unrestrained that is free of any restriction by local geometry restraints, such as bond lengths and bond angles, the common case is to move atom positions that are meaninglessly dispersed because of the overfitting condition. In our case of unrestrained while imposing many NCS constraints, atomic coordinates were also dispersed such that the atoms were no longer present at the actual atomic positions. Nevertheless, when the R factor was decreased sufficiently even under many NCS constraints, we obtained interpretable electron density that was generated with the dispersed coordinates and B factors of atoms. The obtained density map, which was generated with calculated amplitudes F_{calc} and phases, was determined as an `NCSconstrained map'. If necessary, the parameters of the NCS matrices can be refined by monitoring the decreasing R factors.
program2.3. Method for purification, crystallization and data collection
Orthorhombic (P2_{1}2_{1}2_{1}) and cubic (P2_{1}3) crystals of ΔNARM T = 1 PvNV Sdomain SVP, hereafter denoted T = 1 PvNVSd, were obtained by the hangingdrop vapourdiffusion method. Drops containing the T = 1 Sdomain subparticle (30 mg ml^{−1} in 50 mM HEPES pH 7.4, 300 mM NaCl, 5 mM CaCl_{2}) were allowed to equilibrate at 18°C against reservoirs consisting of (i) 0.1 M Tris pH 8.0, 20%(w/v) poly(acrylic acid sodium salt) 5100 or (ii) 0.1 M Tris pH 7.8, 0.2 M Larginine, 8%(w/v) polyγglutamic acid. All crystals grew after two weeks. Xray diffraction data from these two crystal forms, cryoprotected with 20% glycerol, were collected at 100 or 110 K on beamline BL44XU at SPring8, Harima, Japan and beamline TPS 05A at NSRRC, Hsinchu, Taiwan. The data were indexed and processed using HKL2000 (Otwinowski & Minor, 1997). The data resolutions for the P2_{1}2_{1}2_{1} and P2_{1}3 crystals was 3.38 and 3.30 Å, respectively, in the initial phasing stage. The data with the best resolution of 3.12 Å for the P2_{1}2_{1}2_{1} crystal form were obtained later during the final structurerefinement stage for the structure report (Chen et al., 2019). The data statistics are shown in Table 2.

3. Results
3.1. Singlecrystal data in P2_{1}2_{1}2_{1}
Initial phasing of the data in P2_{1}2_{1}2_{1} was performed by the MR method using the structure of T = 1 GNNV SVP (PDB entry 4rft) as the initial model (Chen et al., 2015) in Phaser (McCoy et al., 2007). One T = 1 SVP, which consists of 60 copies of the Sdomain, was estimated to be included in the NCS unit. Phaser found an MR solution that showed a loglikelihood gain (LLG) score of 570 (the LLG before was 35) on inputting an identity parameter (IDENT) of 0.5. The initial weighted (2F_{o} − F_{c}) map with the MR phases was completely uninterpretable (Supplementary Fig. S1). From the initial MR solution of the SVP of GNNV, we obtained the initial 60fold NCS matrices that maintained the strict icosahedral form. Using the REFMAC5 software (Murshudov et al., 2011), the atomic coordinates and temperature factors of a single Sdomain (a minimal icosahedral unit) were optimized by `unrestrained while imposing 60fold `constrained' NCS matrices. The number of nonH atoms in the GNNV Sdomain was 1248. After the unrestrained the atomic coordinates of the GNNV Sdomain completely dispersed: almost all of the atomic coordinates did not reflect the new correct atom positions. Despite the dispersed atomic coordinates, REFMAC5 showed a reasonable R factor; in this case it was less than 25%. The R factor typically shows the agreement of the calculated amplitudes from the atomic model with the diffraction data. In this work, we name the R factor the `R_{d} factor' to represent the agreement of the calculated amplitudes from the electron density, instead of the atomic model, with the observed data.
The number of unrestrained R_{d} factor decreased and converged to a minimum value. The number of cycles was usually less than 200. On monitoring the R_{d} factors, initial parameters, such as NCS matrices, were refined with perturbation trials and as few as ten cycles. In Fig. 3, these R_{d} factors with initial NCS matrices and with refined NCS matrices are shown against the cycle number of the REFMAC5 When of the NCS matrices was performed, the 60fold NCS matrices were maintained to retain a strict icosahedral relation to each other; i.e. only six parameters for rotations and positions of the centre of the icosahedron to generate NCS matrices were refined. After the NCS matrices had been refined, we obtained an NCSconstrained map with an R_{d} factor of 23%, which was generated from `calculated amplitudes F_{calc}' and `calculated phases'. Using the data with the highest resolution of 3.12 Å, the NCS matrix of the icosahedron was refined again and the final R_{d} factor reached 21%. As the refined atomic coordinates did not reflect the actual atomic positions of the structure, we call these refined atoms `dummy atoms'. The dispersed dummy atoms that served the original MR model are shown in Fig. 4(a) with the obtained NCSconstrained map. With the NCSconstrained map, the new structure of the T = 1 PvNVSd particle in the icosahedral unit could be manually built with Coot (Emsley et al., 2010). The built model in the icosahedral unit underwent under the final NCS constraints that maintained strict icosahedral relations (PDB entry 5yl1). The R_{work} and R_{free} of the final model were 21.9% and 22.1%, respectively (Chen et al., 2019; Table 2). The main chains of the initial MR model (T = 1 GNNV SVP; PDB entry 4rft), the final refined structure model (T = 1 PvNVSd) and the NCSconstrained map are shown in Fig. 4(b).
cycles was extended as long as the3.2. Twinned crystal data in P2_{1}3
According to the statistics of the amplitudes and the icosahedralrelated peaks from the selfrotation function, the data in P2_{1}3 were expected to be merohedrally twinned. All estimators of obtained using TRUNCATE in the CCP4 software package (Winn et al., 2011) showed that the data were completely twinned; for example, an L statistic of 0.339 (Padilla & Yeates, 2003). The selfrotation function showed two directions of icosahedral symmetry, with each orientation related by a 90° rotation. From the crystal symmetry and packing analysis, we located the centres of four presumed particles at (1/4, 1/4, 1/4), (3/4, 3/4, 1/4), (1/4, 3/4, 3/4) and (3/4, 1/4, 3/4). Onethird of the T = 1 SVP, which contains 20 copies of PvNVSd, was present in the NCS unit.
Using the refined dummy atoms, which contained the information from the electron density of PvNVSd but in which the atomic positions seemed to be dispersed, the initial phases from MR for the twinned data from the P2_{1}3 crystal were obtained with a high LLG score (IDENT = 0.8) that was as large as 10 000 on refining the rotations and translations of 20 initial molecules with and phasing by Phaser. As a trial, using the T = 1 GNNV SVP as the initial structural model, Phaser picked very low LLG scores of ∼20 (IDENT = 0.8). One dummyatom molecule of PvNVSd from the MR solution was selected; its NCS matrix was determined as a unit matrix. Using the other 19 NCSrelated molecules of the solution, the 19 NCS matrices were calculated. The DDO method only has a convolution process of amplitude calculation for twinned observables. REFMAC5 has a function for the fraction (see Section 2.3 of Murshudov et al., 2011); this showed that the operator was (k, h, −l) against (h, k, l). Using REFMAC5, the dummy atoms were subjected to unrestrained with 20 NCS constraints and with twinningfraction The R_{d} factor of the unrestrained reached 22.4% (Fig. 3). The fractions were 0.69 and 0.31 in the final cycle. The NCSconstrained maps for twinned data with the final refined model of PvNVSd are shown in Fig. 5.
4. Discussion
4.1. Direct density optimization and REFMAC5
Since the early years, NCS has been known to be powerful in phase improvement (Main & Rossmann, 1966; Bricogne, 1976; Arnold & Rossmann, 1986; Rossmann, 1990), and has assisted in many cases of phase improvement using several programs, such as DM (Cowtan & Main, 1996) in the CCP4 suite and RESOLVE (Terwilliger, 2002) in Phenix (Liebschner et al., 2019). Recent attention has been paid to realspace because of the requirements of cryoEM structure (Afonine et al., 2018). In almost all cases, the imposition of NCS has been performed more or less using the iterative or averaged method shown in Fig. 1(a). Fourier synthesis calculations (or inverse Fourier transformations; FT^{−1}) are required to process the map, whereas the DDO method has no Fourier synthesis calculation (Fig. 1b). A map generated using Fourier synthesis calculations could be affected by defects in the data. The strength of the DDO method is its robustness against defects in the data F_{obs}, such as statistical errors, systematic noise and incompleteness. Especially in cases of the decomposition of amplitudes is a difficult task in the Fourier synthesis calculations in the common iterativeaveraging method. The DDO method has a great strength in avoiding this difficulty in the decomposition of observables.
To execute the DDO method, we used the REFMAC5, which refines the atomic coordinates and their temperature factors. Our initial intent was to seek to undertake of the temperature factors or occupancies of fixed coordinate atoms, which served to refine densities on the map grid positions. We found, unexpectedly, that flexible atomic coordination was not a problem in obtaining correct densities under the conditions described in the next section. We previously reported that the envelope of the region is important for NCS averaging (Yoshimura et al., 2016). The condition that we can refine the coordinates of density grid points means that we can refine the envelope itself with increased flexibility.
function of4.2. Conditions for obtaining a correct map
In our previous work (Yoshimura et al., 2016), we introduced the crystallographic free fraction (ff), which gives the ratio of the unconstrained density region to the When ff is sufficiently small, which is the condition of overdetermination, the density of the unconstrained region is determined uniquely. We showed an empirical ff value of <0.1 to be the criterion for a feasible ab initio condition using the iterationaveraging method. In this work, we show the same situation for the feasible direct determination of the density. The values of ff are <1/60 for the P2_{1}2_{1}2_{1} data and <1/20 for the twinned P2_{1}3 data; the inequality arises from considering the constraint on the solvent region. Both ff values are sufficiently small. Even the number of dummy atoms (1248), which was the original number of nonH atoms in the GNNV Sdomains, has an additional degree of freedom of the x, y and z coordinates in the REFMAC5 The condition of overdetermination remained satisfied when the number of variables (4996; 1248 × 4) was 1/21 times the number of unique observables (206 891) in the P2_{1}2_{1}2_{1} data. Two further conditions, the correct NCS matrices and NCS minimal regions and the acquisition of a global minimum unique solution, are also required to derive a correct map. We summarize these three points as

When the value of ff is not sufficiently small but satisfies ff < 0.5, even on obtaining a global minimum solution or a unique solution (Millane & Arnal, 2015) care should be taken in obtaining the density map in order to avoid it being illposed through errors in the data.
To examine whether a global minimum solution can be obtained, we refer to the metric of agreement between the calculated amplitudes and observables as the R_{d} factor; we can then evaluate the solution by inspecting whether or not the electrondensity map is interpretable. A starting point or an initial condition from which the global minimum can be attained is referred to as `in the convergence radius'. When the parameter optimization falls into a local minimum or a shortage of the degree of freedom of the parameters occurs, the third condition of the acquisition of a global minimum solution is not satisfied.
4.3. Convergence radius of the DDO method
In contrast to the iterationaveraging methods, the convergence radius of direct density P2_{1}2_{1}2_{1} singlecrystal data beginning from the initial condition of the GNNV Sdomain particles. So far, for the twinned P2_{1}3 data we only obtained a correct solution when using PvNV dummy atoms as the starting point, for which the initial R_{d} factor was 35% (Fig. 3). Our few trials using GNNV Sdomain particles as a starting point did not succeed for the twinned data, which might reflect a small convergence radius for the DDO method. The other reason is that the twinningfraction parameter led to a moderate increase in ff (<1/20) because of the additional of the variables. Phasing of the twinned crystal is a major problem to be overcome (Ginn & Stuart, 2016; Sabin & Plevka, 2016); we need to further investigate how to increase the convergence radius to obtain a global minimum solution.
is empirically expected to be small. The small values of ff and the precise icosahedral NCS matrices guided the derivation of the correct density. Fortunately, we attained a global minimum solution for the4.4. Comparison with the common iterativeaveraging method
In Fig. 6, we compare the conventional NCSaveraging map from the DM software with the map from the DDO method. There is no critical difference in quality between the two maps. However, in Fig. 6(a) there are missing densities for the DM map on the outside surface of the T = 1 PvNVSd particle which might not be covered by the NCS mask. For DM, it is difficult to generate the NCS (solvent or averaging) mask at the boundary regions between neighbouring particles. The DDO method has the same function of solvent flattening as DM in placing no dummy atoms in the solvent region. Such a vacant space for dummy atoms is fixed to a density value of zero. The DDO method has flexibility in the boundaryregion determination because of the atom mobilities. DM includes an additional function of histogram matching, but the DDO method does not apply such a process. Furthermore, DM can execute averaging in a phaseextension manner to guide towards a global minimum solution, whereas no such equivalent function can be applied in the DDO method. The incorporation of phase extension into the DDO method will be our next work, in which the convergence radius will be expected to increase.
4.5. Comparison with REFMAC5 restrained refinement
In Fig. 7, we compare the DDO map with the resulting map from REFMAC5 with the common use of coordinates restrained by the 60fold NCS constraint. The was performed with sufficient cycles (200). Although there was a structural difference of 2.7 Å on average between the main chains of the initial MR model and the final built model, most fractions of the main chains of the MR model were successfully shifted and adjusted to the new correct positions after The R factor had a smallest value of 0.465 when the ridge distance σ was 0.02 (from the range 0.01–0.30) for the `jellybody' restraint function (see Section 2.6.5 of Murshudov et al., 2011). Most of the main chain seemed to be traceable with the weighted (2F_{o} − F_{c}) map by REFMAC5. Most parts of the main chains were successfully shifted to new mainchain positions, as shown in Fig. 7(a), but some parts failed to be shifted, as shown in Fig. 7(b). The NCSconstrained map from DDO can be seen to easily provide a completely new phase, removing the bias from the initial model phases by completely loosening the restraints on coordinates. In Table 3, the mean phase error and R_{d} factor for each method are summarized, showing that the removal of bias in the DDO method and DM is of comparable quality.

5. Conclusions
The use of NCS is powerful for phasing or phase improvement. NCS has been used as an averaging method in real space. We propose an NCSconstrained map with a new method of direct density optimization (DDO). Whereas the NCSconstrained map is equivalent to an NCSaveraging map, the DDO method requires no Fourier synthesis that calculates a map from F_{obs} and its phases. To perform DDO, we used unrestrained in REFMAC5 while imposing NCS constraints, and applied this method to a new structure of T = 1 PvNVSd. With the condition that the crystallographic ff is sufficiently small, an interpretable map was obtained and the structure of T = 1 PvNVSd was subsequently solved. We further demonstrate the application of the DDO method to other T = 1 PvNVSd data sets that were twinned, in which the DDO method has no difficulty in generating the NCSconstrained map. A comparison of the NCSconstrained map with DDO and the map from DM, which uses the averaging method, shows no critical difference except that the DDO map produces more complete density at the molecular boundary. Compared with the result of REFMAC5 the NCSconstrained map with DDO can easily remove the bias from the initial model with the same effort. By making the convergence radius large in future developments, the method will be more powerful for the solution of structures with a large number of NCS including twinned data.
Supporting information
Supplementary Figure S1. DOI: https://doi.org//10.1107/S2059798319017297/ji5009sup1.pdf
Acknowledgements
We are indebted to the staff of beamlines TPS 05A, TLS 13B1 and 15A1 at the National Synchrotron Radiation Research Center (NSRRC) in Taiwan, Eiki Yamashita at beamline BL44XU and the Taiwan beamline BL12B2 at SPring8 in Japan for technical assistance under proposal Nos. 2015A6600, 2015A4000, 2015B6600, 2015B4004, 2015B4010, 2016A6600, 2016A6659, 201A4012, 2016B4000, 2017A4000, 2017A6600, 2017B6639, 2018A6600, 2018A6864, 2018B6600 and 2018B6864, partly supported by the International Collaborative Research Program of the Institute for Protein Research, Osaka University (ICR1605, ICR1705, ICR1805, ICR1905). We thank Tomitake Tsukihara, Masaki Yamamoto of RIKEN and Takashi Kumasaka of JASRI and their staff for useful discussions.
Funding information
This work was supported in part by National Science Council (NSC) and Ministry of Science and Technology (MOST) grants 1012628B213001MY4, 1022627M213001MY3, 1052311B213001MY3, 1072923B213001MY3 and 1082311B213001MY3 and NSRRC grants to CJC.
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