 1. Symbols and abbreviations
 2. Introduction
 3. General features of REMO09
 4. Rotational search when only one monomer lies in the asymmetric unit of the target structure
 5. Translation search when only one monomer lies in the asymmetric unit of the target structure
 6. Rotational search when more than one monomer lies in the asymmetric unit of the target molecule
 7. Translational search when more than one monomer lies in the asymmetric unit of the target molecule
 8. Applications
 9. Conclusions
 Supporting information
 References
 1. Symbols and abbreviations
 2. Introduction
 3. General features of REMO09
 4. Rotational search when only one monomer lies in the asymmetric unit of the target structure
 5. Translation search when only one monomer lies in the asymmetric unit of the target structure
 6. Rotational search when more than one monomer lies in the asymmetric unit of the target molecule
 7. Translational search when more than one monomer lies in the asymmetric unit of the target molecule
 8. Applications
 9. Conclusions
 Supporting information
 References
research papers
How far are we from automatic
solution via molecularreplacement techniques?^{a}Dipartimento di Fisica e Geologia, Università di Perugia, Piazza Università, I06123 Perugia, Italy, and ^{b}Istituto di Cristallografia, CNR, Via Amendola 122/O, I70126 Bari, Italy
^{*}Correspondence email: carmelo.giacovazzo@ic.cnr.it
Although the success of molecularreplacement techniques requires the solution of a sixdimensional problem, this is often subdivided into two threedimensional problems. REMO09 is one of the programs which have adopted this approach. It has been revisited in the light of a new probabilistic approach which is able to directly derive conditional distribution functions without passing through a previous calculation of the joint probability distributions. The conditional distributions take into account various types of prior information: in the rotation step the prior information may concern a nonoriented model molecule alone or together with one or more located model molecules. The formulae thus obtained are used to derive figures of merit for recognizing the correct orientation in the rotation step and the correct location in the translation step. The phases obtained by this new version of REMO09 are used as a starting point for a pipeline which in its first step extends and refines the molecularreplacement phases, and in its second step creates the final electrondensity map which is automatically interpreted by CAB, an automatic modelbuilding program for proteins and DNA/RNA structures.
Keywords: molecular replacement; proteins; nucleic acids; automated model building; structure refinement.
1. Symbols and abbreviations
EDM: electrondensity modification.
C_{s} = (R_{s}, T_{s}), with s = 1, …, m: the symmetry operators of the target structure. R_{s} is the rotational part, T_{s} is the translational part and m is the number of symmetry operators.
t, t_{p}: the numbers of atoms in the asymmetric units of the target and model structure, respectively.
N = mt, N_{p} = mt_{p}: the numbers of atoms in the unit cells of the target structure and model structure, respectively. It is supposed, for the sake of simplicity, that all of the atoms are in general positions. Usually N_{p} ≤ N, but it may also be the case that N_{p} > N.
f_{j}: the of the jth atom (thermal factor included).
F_{p} = = F_{p}exp(iφ_{p}): of the model structure. r_{pj} are the atomic positions of the model structure when it has been well oriented and located.
F = = Fexp(iφ): of the target structure. r_{j} are the true atomic positions. It is supposed that the target and model molecules are isomorphous, so that r_{j} = r_{pj} + Δr_{j}. Δr_{j} is the misfit between the atomic position r_{j} in the target and the corresponding r_{pj} in the model structure.
E = A + iB = Rexp(iφ), E_{p} = A_{p} + iB_{p} = R_{p}exp(iφ_{p}): normalized structure factors F and F_{p}, respectively.
, : the scattering power at a given sinθ/λ for the target and model structure, respectively.
D = 〈cos(2πhΔr_{j})〉. The average is calculated per resolution shell.
σ_{A} = . σ_{A} is a statistical estimate of the correlation between the model and target structures (Srinivasan, 1966). Ideally σ_{A} = 0 for uncorrelated models and σ_{A} = 1 for identical model and target structures.
SI: the sequence identity between model and target molecules.
AMB: automated model building.
2. Introduction
Molecularreplacement (MR) techniques (Rossmann & Blow, 1962; Rossmann, 1972, 1990) aim at phasing an unknown target structure using a known search molecule. The problem to solve is of a sixdimensional nature because it implies the correct orientation and location of the search molecule. Some MR programs face this in sixdimensional space [for example EPMR (Kissinger et al., 1999), SOMoRe (Jamrog et al., 2003) and Queen Of Spades (Glykos & Kokkinidis, 2000); see also Fujinaga & Read (1987)], even if an exhaustive sixdimensional search is generally avoided. Such programs are, in general, very timeconsuming. More frequent is the practice of splitting the MR process into two threedimensional steps: a rotation and a translation step. The most popular related programs are XPLOR/CNS (Brünger, 1992), AMoRe (Navaza, 1994), BEAST (Read, 1999), MOLREP (Vagin & Teplyakov, 2010) and Phaser (McCoy et al., 2007). In BEAST and Phaser, maximumlikelihoodbased conditional distributions are applied (see Read & McCoy, 2016, 2018; McCoy et al., 2018). Comprehensive reviews of the various techniques (updated up to 2007) have been collected in the January 2008 issue of Acta Crystallographica Section D. In recent years, more effort has been dedicated to cases in which the available experimental structures used as search models are only distantly homologous to the target; see, for example, Simpkin et al. (2018), Rigden et al. (2018), Pröpper et al. (2014), Millán et al. (2015) and Cabellero et al. (2018).
In 2009, an MR program (REMO09; Caliandro et al., 2009) was proposed in which a probabilistic approach based on the joint probability distribution method was described. Joint distributions were derived in the absence of or under various prior conditions. For example, in the rotation step the correct rotation of a monomer is found via a figure of merit calculated when other monomers were previously oriented or located, or also when such information is not available. Joint distributions were also derived for the translation step: a monomer is located given its own orientation or the orientations and/or locations of other monomers.
Burla et al. (2017), starting from REMO09 phases, checked the efficiency of a phaserefinement pipeline which synergically combines mainstream techniques (specifically DM; Cowtan, 2001) with outofmainstream techniques [specifically, free lunch (Caliandro et al., 2005a,b), lowdensity Fourier transform (Giacovazzo & Siliqi, 1997), vive la difference (Burla, Caliandro et al., 2010; Burla, Giacovazzo et al., 2010), Phantom derivative (Giacovazzo, 2015b; Carrozzini et al., 2016) and phasedriven model (Giacovazzo, 2015a)]. For simplicity, we will refer to this modulus as SYNERGY. Burla et al. (2017) automatically submitted the protein data obtained by SYNERGY to the AMB procedure CAB (Burla et al., 2017): it applies Buccaneer (Cowtan, 2006) in a cyclic way.
In a recent paper (Giacovazzo, 2019), the standard method of joint probability distribution functions has been revised and updated. In particular, twophase, threephase and fourphase invariants are estimated directly via conditional distributions without passing through a previous calculation of the related joint probability distributions. The probabilistic formulae thus obtained do not coincide, in general, with the corresponding formulae established through the standard study of the joint probability distribution functions. Some of them are immediately applicable to MR, and some others, also suitable for MR, are derived here via this new approach. The formulae thus obtained form the basis for the modified version of REMO09 used in this paper.
In this paper, in accordance with the talk given by one of us at the 2019 CCP4 Study Weekend in Nottingham, England, we show the default results obtained on applying the modified REMO09 → SYNERGY → CAB pipeline to a large set of protein and nucleic acid structures. To obtain these results, we extended CAB to nucleic acid structures (unpublished work) by making the use of Nautilus (Cowtan, 2014) cyclical. The purposes are twofold: to check the efficiency of the new probabilistic formulae used in the modified version of REMO09 and to check how far a modern crystallographic pipeline based on MR phases is from the automatic solution of macromolecules.
3. General features of REMO09
Various directives allow REMO09 users to choose proper approaches for solving macromolecular structures. In this section, we will summarize the default approach used in all of our applications.
4. Rotational search when only one monomer lies in the of the target structure
The rotational search is performed by locating the model molecule in a P1 cubic According to Rabinovich et al. (1998), the structure factors of the model are calculated only once: fitting to the observed data is obtained by rotating the observed with respect to the model lattice.
The figure of merit designed for picking up the correct orientation of the model molecule is RFOM, the correlation factor between the observed R^{2} and its expected value 〈R^{2}〉 as calculated by the probabilistic approach described by Giacovazzo (2019). RFOM is expected to be maximum for the correct model orientation and 〈R^{2}〉 is the expected value of R^{2} given the prior information on the model stereochemistry:
where
F_{ps} is the contribution to the calculated model arising from the of the model structure, and E_{ps} is its normalized (with respect to the scattering power of the model structure, symmetryequivalent molecules included) form. The E_{ps} are calculated and stored for each reflection via FFT of the electron density of the model structure in the enlarged cubic cell.
(1) has appropriate asymptotic behaviours: i.e. when σ_{A} = 0 then 〈R^{2}〉 = 1, as it should be in the absence of prior information, and when σ_{A} = 1 then 〈R^{2}〉 = . The identity 〈R^{2}〉 = R^{2} may only occur in P1 when the contains only one monomer showing a high similarity index to the target molecule.
Despite its good asymptotic properties, the use of (1) did not lead to a very efficient RFOM. The reason may lie in the mathematical definition of σ_{A}^{2}: according to Carrozzini et al. (2013) it coincides with the correlation factor between F^{2} and the calculated squared In the rotation step the experimental values of σ_{A}^{2} are generally small, mostly because is not the dominant component of the calculated squared Thus, in some resolution shells σ_{A} < 0 (anticorrelation situation), while the σ_{A}^{2} parameter to be used in (1) remains positive. This suggested that we eliminate the calculation of σ_{A} from (1) and simplify it as
The 200 orientations corresponding to the highest values of RFOM are selected for the translation step: this number is enhanced to 300 if more than one monomer is in the target molecule and to 400 if SI < 0.4.
5. Translation search when only one monomer lies in the of the target structure
The orientations selected according to Section 4 are submitted to the translation search one by one. This is performed by using the T2 function of Crowther & Blow (1967) in the form modified by Harada et al. (1981) and by Navaza (1994). T2 is implemented via FFT, as suggested by Vagin & Teplyakov (1997).
Only peaks falling inside the Cheshire F and the structurefactor amplitude F_{p} as calculated for each translation.
are considered. For the same orientation, more peaks can be found: to spare computing time, only the largest five translations per orientation are saved. The selection of the best translations is made via the figure of merit TFOM, coinciding with the correlation factor between the observed amplitude Some further controls modify the simple approach above.
The rototranslation with the highest figure of merit is automatically submitted to the SYNERGY step and to the CAB procedure.
6. Rotational search when more than one monomer lies in the of the target molecule
In the standard REMO09 program, when several monomers with the same stereochemistry are present in the the following threestep approach is used.

This is the case for PDB entries 1lat and 2iff. The first test structure shows two chains of 71 and 74 resideues, respectively, and two identical nucleic acid chains, each with 19 The structure with PDB code 2iff is composed of three protein chains: two with 212 and 214 residues and a third chain with only 129 residues. The model coincides with the third target protein chain.
We then decided to modify the REMO09 approach as follows: when the first molecule has been located, the rotations of the second and the others must be searched for using an ex novo rotation step and, where the case, by using a different model.
In both of the approaches the figures of merit to be used for recognizing the correct rotation must be designed to take into account that one or more monomers have been previously oriented and located. This increases the signal to noise in the search for the new monomer.
Let us consider the simplest case: the first monomer has been located and we want to orient the second monomer (no other monomers are supposed to lie in the asymmetric unit). Appendix A suggests that RFOM may still be the correlation factor between the observed R^{2} and its expected value 〈R^{2}〉, but now
where R^{2}_{p1} is the squared amplitude of the normalized model corresponding to the already located first model monomer (normalized with respect to the scattering power of the structure containing the first monomer and its symmetry equivalents) and σ_{A1} is the σ_{A} value corresponding to the pairs (R, R_{p1}). The last term on the righthand side of (4) corresponds to the contribution of the second model monomer (the correct orientation of which we are searching for). σ_{A2} is the σ_{A} value corresponding to the pairs (R, 〈R_{2}^{2}〉^{1/2}), where
Let us briefly discuss the expected behaviour of (4).
The probabilistic approach used to derive (4) excludes the existence of a mixed nonzero term relating the monomer already positioned to the monomer for which the orientation is searched. Thus, the two contributions are simply additive.
When the first monomer is badly oriented and/or located σ^{2}_{A1} is expected to be close to zero. Since σ^{2}_{A2} is always expected to be a small value (at least for nonP1 space groups; see Section 4), RFOM is expected to be small. When the first monomer is well located and the second is well oriented then RFOM is expected to be larger. However, values of σ^{2}_{A1} and σ^{2}_{A2} that are both close to unity are not expected because Σ_{p1}/Σ_{N} and Σ_{p2}/Σ_{N} values that are both close to unity are not allowed. Sections 4 and 5 suggest avoiding the use of σ_{A} values so that 〈R^{2}〉 reduces to
The final RFOM is the R^{2} and its expected value 〈R^{2}〉. Let us now generalize (6) to the case in which three monomers are contained in the under the condition that the first and second monomers have already been oriented and located. The expression (6) is still valid; we only have to change the meaning of the symbols. R_{p1} will represent the normalized amplitude of the model structure corresponding to the first and second monomers (symmetry equivalents included), will represent the contribution arising from the monomer for which the correct orientation is searched.
between the observedThe procedure is now cyclic: the same equation may be applied to any number of monomers.
7. Translational search when more than one monomer lies in the of the target molecule
Let us first suppose that one monomer has already been oriented and located (F_{1} is its generic structure factor) and that a second monomer has been oriented. If we use the Crowther T2 function to locate the second monomer in the translation step then the expected squared of the structure constituted by the two monomers and their symmetry equivalents in correct positions is
This is a weak relation owing to the fact that 〈F^{2}〉 does not include the mixed term F_{1}F_{2}.
A better approach is that using the translation function involving F instead than its square. Let r_{pj} be the current positional vector of the jth atom of the second model monomer: the of the structure constituted by the second monomer and its symmetry equivalents in correct positions is then
where Δr is a suitable unknown positional shift,
and
is the component of the current model structure factor.
The algorithm is very simple. F_{2} is calculated for each active reflection only once, in the initial position of the second monomer. The second monomer is then moved by the shift Δr on all of the grid points of the where F_{2} is calculated via (7) and summed with F_{1} to obtain
The correct grid position is expected to be that for which TFOM, the correlation factor between the observed amplitude F and the structurefactor amplitude 〈F〉, is a maximum.
The method is simply generalized to locate an nth well oriented monomer when the first n − 1 monomers have been well oriented and located.
8. Applications
We applied the automatic modified pipeline REMO09 → SYNERGY → CAB to an extended set of test structures, proteins and We used 80 protein and 38 nucleic acid test structures, the PDB codes of which are reported in Tables 1 and 2. The first 34 protein test structures had previously been used by Burla et al. (2017) to check the SYNERGY process on standard REMO09 phases. Proteins 25–34 belong to the set of 13 structures studied by DiMaio et al. (2011) and characterized by an SI between the model and target structures of lower than 0.30. The experimental data and models for the remaining 46 protein test structures had been deposited in the PDB by the Joint Centre for Structural Genomics, Wilson Laboratory, Scripps Institute: they were used to verify the efficiency of our pipeline on a larger number of test structures (most of them were not originally solved by MR).


The 38 nucleic acid structures were selected from the PDB: we downloaded the observed diffraction data, information on the Supplementary Tables S1 and S2.
spacegroup symmetry, published sequences and MR models. 20 of them are DNA and the remaining 18 are RNA fragments. Additional information on all of the test structures is given inFor all of the test structures the same small set of directives was used (coinciding with our default set) such as those shown in Table 3 for PDB entry 1xyg.

The experimental results are reported in Tables 1 and 2. For each test structure PDB is the PDB code, MRP° is the average phase error in degrees at the end of the REMO09 step and SYN° is the average phase error in degrees at the end of the SYNERGY step. For proteins, MA is the ratio `number of C^{α} atoms within 0.6 Å distance from the published positions/number of C^{α} atoms in the as obtained by CAB. For MA is the ratio `number of residues with P atoms within 1.3 Å distance from the published positions/number of residues in the in accordance with CAB interpretation. We will assume that good models are obtained by CAB when MA is sufficiently large: as a rough rule of thumb, we will assume that a good solution has been automatically found when MA > 0.5.
For proteins we observe the following.

The panorama is different for ; Murray et al., 2003). Our experimental results may be summarized as follows: of the 38 nucleic acid structures only 24 are routinely solved. Ten of the 14 failures may be ascribed to REMO09 (i.e. for these MRP° ≥ 77°). Four of the remaining five failures are owing to CAB failures (CAB is unable to interpret the electrondensity maps of PDB entries 3tok, 4gsg, 4xqz and 5ihd, for which SYN° ≤ 51°).
Such behaviour is in part expected because of the special stereochemistry of DNA/RNA structures. They have a large number of rotatable bonds in the main chain (six, while there are two for proteins); consequently, the conformation at low resolution is often ambiguous (Keating & Pyle, 2012SYNERGY is again efficient (MPR° values of >70° are broken down to values smaller than 40°).
The above experimental tests indicate that the application of REMO09 and CAB to DNA/RNA are the weakest points of the pipeline. On the contrary, SYNERGY, applied to both and to proteins, and the application of CAB to proteins are particularly efficient. The existence of weak points in the pipeline do not allow us to positively answer the question in the title of this paper. There are three simple ways to improve the present situation.

Modifications (i) and (iii) would require supplementary and probably lengthy work which is beyond the purpose of the present paper. For suggestion (ii) the easiest choice would be to replace REMO09 by a popular and documented MR tool to check whether the conclusions suggested by the results obtained via our pipeline are confirmed by the inclusion of a better updated MR program. MOLREP (Vagin & Teplyakov, 2010) was our choice: it is also preferred amongst others because of its simple use and its possible automation. Our default MOLREP procedure corresponds to the following directives (i.e. such as those shown below for PDB entry 1xyg):
A better default can probably be provided by expert users; therefore, the potential of MOLREP is certainly much greater than that corresponding to the naïve default we choose. However, the experimental results obtained by the pipeline MOLREP → SYNERGY → CAB, shown in Tables 4 and 5, help to better answer the general question regarding automatic solution via MR.


The results in Table 4 for proteins may be summarized as follows.

The experimental results in Table 5 for nucleic acid structures may be summarized as follows.

9. Conclusions
The REMO09 → SYNERGY → CAB and MOLREP → SYNERGY → CAB to 80 protein structures and 38 nucleic acid structures. Only nine of the 80 protein structures remained unsolved by both of the pipelines; most of the failures occurred when the SI was extremely low (below 0.30). The increasing availability of better models, the selection of improved default procedures for REMO09 and MOLREP, and the possible use of more efficient MR programs (e.g. SYNERGY and CAB may use Phaser) suggest that automatic solution is close for proteins. The situation for nucleic acid structures is different: 14 of the 38 nucleic acid structures remained unsolved by both of the pipelines. Further efforts are therefore necessary to obtain their automatic solution: the necessary improvements involve the MR programs (in particular the treatment of ligands, which may be a nonnegligible part of the structure) and the AMB section.
for small molecules is considered to be universally solved in practice. The main purpose of this paper is to check whether a similar situation is, or will soon be, available for macromolecules if MR techniques are used. We applied the two pipelinesAPPENDIX A
On the orientation of a second monomer
The problem that we will treat in this appendix is the following: if the first monomer has been correctly oriented and located, how do we fix the orientation of a second monomer? To answer this question, in the following probabilistic approach we will explicitly consider the case in which the orientation of the second monomer has been fixed while its location is unknown. We will see that the conclusive formulae thus obtained may be applied to fix the orientation of the second monomer.
Let t_{1} and t_{p1} be the number of nonH atoms of the first target monomer and of its model molecule, respectively: for simplicity, we are supposing that t_{1} ≥ t_{p1}. t_{2} and t_{p2} are the equivalent numbers for the second target monomer and for its model molecule. We order the atoms in the target so that its may be represented as
where t = t_{1} + t_{2} is the number of nonH atoms in the target In our probabilistic approach h is fixed while the positional vectors are the primitive random variables. U is an overall free translation vector that is necessary to locate the second monomer in the correct position and Δr_{j} are local variables relating the atomic positions of the target monomers to the corresponding positions of the model. In order, (8) may be rewritten as
The atoms contributing to F_{1} are related to the atoms of the model molecule of the first monomer via the local shift vectors Δr_{j} only (the first monomer has been already located). The atoms contributing to F_{2} are related to the atoms of the model molecule of the second monomer through the local shift vectors Δr_{j} and through the unknown overall translation vector U (indeed, the second monomer has not been located). The coordinates of the atoms contributing to F_{q1} and F_{q2} are not related to the atoms of the model molecules; they may be thought of as unconstrained unknown variables.
We now calculate the average value of F^{2} given the prior information described above,
The above equation may be more explicitly written if the cases in which i = j and/or s1 = s2 are emphasized. We have
where D_{1} and D_{2} are the D values (see Section 1) calculated for monomers 1 and 2, respectively. Let us now take into account the relations (11), (12) and (14) below.
so that
F_{p1} is the corresponding to the structure constituted of the model molecule that has already been located (and its symmetry equivalents).
where
is the contribution to the , we have
of the model molecule of the second monomer (oriented but not located) arising from the In accordance with (14)Substituting (11), (13) and (15) into (10) gives
Dividing the left and righthand sides of (16) by Σ_{N} leads to
from which
where
R^{2} is normalized with respect to the scattering power of the full target R^{2}_{p1} is normalized with respect to the scattering power of the structure constituted of the oriented and located molecule (symmetry equivalents included) and E_{ps2}^{2} is normalized with respect to the scattering power of the model molecules (symmetry equivalents included) that are oriented but not located.
We can now return to the question: why did we formulate a probabilistic theory for the case in which one monomer is well located and the second well oriented, when we are primarily interested in the case in which one monomer is well located and we are looking for the orientation of the second monomer? The answer is simple. Indeed, when we continuously rotate R^{2} and 〈R^{2}〉 we hope to find a rotation in which the second monomer is well oriented. In this case 〈R^{2}〉 will really be the expected value of R^{2} in accordance with (17), while for all of the other orientations this condition will not be obeyed. Accordingly, the correlation will be a maximum.
and look for the best fit betweenSupporting information
Supplementary Tables. DOI: https://doi.org/10.1107/S2059798319015468/ip5004sup1.pdf
Acknowledgements
We thank K. Cowtan and P. Bond for illuminating discussions and for kindly offering a set of 46 protein test structures for our MR tests.
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