While increasingly complex macromolecules or assemblies have been successfully crystallized, such crystals often diffract weakly due to limited crystal growth, high crystal mosaicity or high sensitivity to radiation damage. Underlying causes can be inherent flexibility, inhomogeneity, or disordered solvent components that prove difficult to overcome. Nevertheless, the interpretation of low-resolution diffraction is often desirable as it provides information about the interaction of individual components in the system or insights about large-scale conformational changes between different states of the system. In addition, macromolecular data collection continues to evolve, notably with microdiffraction synchrotron facilities (Sanishvili et al., 2008
) and hard X-ray free electron lasers (FEL) (Chapman et al., 2011
It is a well-known principle in crystallography that the accuracy of determined atomic positions exceeds the resolution limit of the diffraction data. At atomic resolution (around 1.2 Å), this arises from the excluded volumes of atoms: electron cloud repulsion keeps the scattering objects further apart than half the wavelength of the X-ray radiation used (1 to 2 Å resolution) allowing the centroids of the atomic electron density to be typically determined to better than 0.1 Å accuracy. At moderate resolution (up to about 4 Å), knowledge of the stereochemistry of the system (bond lengths, bond angles, fixed torsion angles, chirality) allows this principle to be applied to the majority of macromolecular crystal structures. At even lower resolution (4 to 5 Å), DEN refinement (Schröder et al., 2007
; Schröder et al., 2010
) further extends this principle. New refinement methods based on physical energy functions such as Rosetta (DiMaio et al., 2011
), are complementary to DEN refinement, and are expected to further improve the accuracy of low-resolution crystal structures. Other recent methods may also be useful at low resolution, including LSSR in Buster (Smart et al., 2008
), external structure restraints or jelly body refinement in REFMAC (Murshudov et al., 2011
), restraints in torsion angle space based on a reference model (Headd et al., 2012
), and normal mode refinement (Kidera and Go, 1992
; Delarue, 2008
). It should be noted that the principle of achieving higher accuracy of positional information than the diffraction limit is referred to as “super-resolution” in optical microscopy (Moerner, 2007
; Pertsinidis et al., 2010
). We have therefore suggested adoption of the same term in X-ray crystallography (Schröder et al., 2010
Here, we explore whether one can obtain more accurate structures than naively suggested by the minimum Bragg spacing of a crystal that diffracts to around 7 Å resolution. This resolution is close to the determinacy point for backbone torsion angles of protein crystal structures, i.e. it is the resolution at which the number of independent Bragg reflections is equal to the number of backbone torsion angles. This relationship (Table S1
, W. A. Hendrickson, personal communication) shows that it is reasonable to expect that the secondary structure and tertiary fold of a macromolecule can be determined at around 7 Å resolution. Furthermore, the average X-ray diffraction intensities of a typical macromolecular crystal structure have a characteristic resolution dependence with a local maximum between 6 and 15 Å that is determined by the fold of the molecule; at lower resolution, the intensity distribution is dominated by the envelope of the crystallized molecular entity, and at higher resolution it is determined by the packing of atoms with a maximum at around 5 Å. Thus, the determinacy point for backbone torsion angles is close to the local maximum in X-ray diffraction intensity around 7 Å. The coincidence of high diffraction intensity and determinacy of backbone torsion angles suggests that a reasonable degree of success might be achievable even at such low resolution.
DEN refinement consists of torsion angle refinement interspersed with B
-factor refinement in the presence of a sparse set of distance restraints that are initially obtained from a reference model (Schröder et al., 2010
). Typically, one randomly selected distance restraint is used per atom. The reference model can be simply the starting model for refinement, or it can be a homology or predicted model that provides external information. In this work, the reference model was the search model used for molecular replacement, and only an overall anisotropic B
-factor refinement was performed as appropriate at very low resolution. During the process of torsion angle refinement with a slow-cooling simulated annealing scheme, the DEN distance restraints were slowly adjusted in order to fit the diffraction data. The magnitude of this adjustment of the initial distance restraints is controlled by an adjustable parameter, γ. The weight of the DEN distance restraints is controlled by another adjustable parameter, wDEN
. For the success of DEN refinement it is essential to perform a global search for an optimum parameter pair (γ, wDEN
). Furthermore, for each adjustable parameter pair tested, multiple refinements should be performed with different initial random number seeds for the velocity assignments of the torsion angle molecular dynamics method and different randomly selected DEN distance restraints. The globally optimal model (in terms of minimum Rfree
), possibly augmented by geometric validation criteria, is then used for further analysis. By default, the last two macrocycles of the DEN refinement protocol are performed without any DEN restraints. However, for the low-resolution refinements presented in this paper, the restraints were kept throughout the entire refinement process in keeping with a low ratio of number of observables to number of torsion angle degrees of freedom.
This study was motivated by the recent availability of low-resolution diffraction data of the Photosystem I (PSI) complex collected on a synchrotron light source (the Advanced Light Source, ALS at Lawrence Berkeley National Laboratory, LBL) (Chapman et al., 2011
). The synchrotron data were collected on a single crystal and had a limiting resolution of 6 Å, making them comparable to diffraction data obtained at the first hard X-ray FEL light source (the Linac Coherent Light Source, LCLS, at the SLAC National Accelerator Laboratory) with a minimum Bragg spacing of 7.4 Å (limited in resolution by the wavelength of the FEL photons of 6.9 Å used in this study). The availability of a high-resolution (dmin
= 2.5 Å) crystal structure of PSI (PDB ID 1jb0) (Jordan et al., 2001
) enabled an objective assessment of the accuracy of structures refined by various methods.
Here, we compared DEN refinement of PSI using the ALS diffraction data at 7.4 Å resolution to overall rigid-body refinement, segmented rigid-body refinement, standard refinement consisting of positional minimization, and torsion angle simulated annealing.
We also tested combinations of segmented rigid-body refinement with DEN refinement, and with secondary structure and reference model restrained positional minimization. We assessed the performance of the refinements by (a) Rfree, (b) the root mean square difference (RMSD) to the 2.5 Å resolution crystal structure of PSI, and (c) the significance of features observed in difference maps that were not part of the model used for molecular replacement and refinement. We generated a series of initial models with increasing distance to the 2.5 Å resolution crystal structure, all of which produced a molecular replacement solution. DEN refinement performed better than other methods for all initial models. The most powerful protocol was DEN refinement with initial segmented rigid-body refinement. We also found a good correlation between Rfree and model accuracy among DEN refinements with different adjustable parameters, suggesting that cross-validation is useful even at such low resolution.