research papers
Deformable complex network for refining low-resolution X-ray structures
aApplied Physics Program, Rice University, Houston, TX 77005, USA, bVerna and Marrs McLean Department of Biochemistry and Molecular Biology, Baylor College of Medicine, One Baylor Plaza, Houston, TX 77030, USA, and cDepartment of Bioengineering, Rice University, Houston, TX 77005, USA
*Correspondence e-mail: jpma@bcm.edu
In macromolecular X-ray crystallography, building more accurate atomic models based on lower resolution experimental diffraction data remains a great challenge. Previous studies have used a deformable elastic network (DEN) model to aid in low-resolution structural
In this study, the development of a new algorithm called the deformable complex network (DCN) is reported that combines a novel angular network-based restraint with the DEN model in the target function. Testing of DCN on a wide range of low-resolution structures demonstrated that it constantly leads to significantly improved structural models as judged by multiple criteria, thus representing a new effective tool for low-resolution structural determination.1. Introduction
It is often a challenge to refine the atomic structures of macromolecular assemblies owing to their weak diffraction of X-rays. In order to build better structural models based on limited-resolution experimental data, it is desirable to introduce additional restraints such as the conventional stereochemical potential (Engh & Huber, 1991). In recent studies, following the development of elastic network models (ENMs; Tirion, 1996; Hinsen, 1998; Atilgan et al., 2001; Stember & Wriggers, 2009), Schröder and coworkers proposed a deformable elastic network (DEN) method (Schröder et al., 2007, 2010) for better structural The DEN method utilizes `reference structures' from homology models (Qian et al., 2007; Šali & Blundell, 1993) and a series of virtual `springs' between randomly selected atom pairs with variable equilibrium lengths to guide the process. In principle, any structure with reasonable quality that bears some similarity to the target model (the one to be refined) could be used as a reference structure. Compared with conventional the DEN method delivered substantial improvements for a wide range of low-resolution structures. However, the DEN method only incorporated one-dimensional information on distances between atom pairs, and neglected potentially valuable information from higher dimensions as well as the interdependence of pairs owing to the interaction of more than two atoms, thus limiting the performance of refinement.
To address the weakness in the DEN method in refining macromolecular structures, in this work we introduce a deformable complex network (DCN) method that combines DEN with additional information obtained from a deformable angular network (DAN). While DEN defines virtual `springs' between selected atom pairs in the reference model (Schröder et al., 2007), the DAN defines harmonic angles formed by randomly selected atom triplets. Each atom in a triplet is subject to an angular bending potential. The resultant target function used for includes experimental X-ray diffraction data, the conventional stereochemical potential and the DCN energy that combines DAN and DEN.
DCN is deformable owing to the deformability of both the angular part (DAN) and the distance part (DEN). The direction of deformation at a certain γ, μ and wDCN, where γ and μ control the rate of deformation and wDCN is the weight of the DCN restraint, are determined by a three-dimensional grid search with the lowest Rfree factor of the final structure as an indicator of the best choice.
step is determined based on the current configuration of the target structure together with the reference structure. The three parametersTwo sets of tests have been used to evaluate the performance of the DCN method. The first set is the 1aun ; Koiwa et al., 1999) at three lower resolutions using its homology model from the plant antifungal protein osmotin (PDB entry 1pcv ; Min et al., 2004) as the reference structure. The deposited 1aun structure serves as the `true answer' and enables additional assessments of the refined structural models based on multiple criteria besides the Rfree value (Brünger, 1992), such as the all-atom root-mean-square deviation (r.m.s.d.), the global distance test (GDT) (<1 Å) score (Zemla, 2003) and the template modeling score (TMscore; Zhang & Skolnick, 2004). The second set is a broader test of re-refining 16 randomly selected low-resolution structures to demonstrate generality. The results from this set of tests show that by using DCN, which merges information independently fetched from DEN and DAN, we achieved additional improvements over the existing DEN method (Schröder et al., 2010), with a decrease in Rfree of 0.15–1.95% (0.41–6.75% over conventional refinement). In addition, we obtained constant improvements in terms of mitigated overfitting effects, better Ramachandran statistics and higher-quality electron-density maps.
of a high-resolution structure of tobacco PR-5d protein (PDB entry2. Methods
2.1. Summary
For a macromolecular structure that is to be determined, called the target structure, we first performed a FASTA search (Pearson & Lipman, 1988) for each polypeptide chain with MODELLER (Šali & Blundell, 1993). Templates that shared higher sequence identity with the target structure, were longer in length and had higher resolution would be preferable (Supplementary Table S2). Five homologous structure candidates were built on this chosen template, and that with the lowest discrete optimized protein energy (DOPE) score (Shen & Sali, 2006) was picked as a reference model for this chain. Reference models of different chains can be generated from different sources and adopt any relative positions and orientations, including overlaps. After all or most chains (for multi-chain systems) in the target structure had their reference models constructed, the position and orientation of each of the reference models was determined by using Phaser (McCoy et al., 2007), and the resultant coordinates were merged into one single coordinate file in PDB format and served as the unique reference structure for the whole molecule. DCN excluded interchain interactions in defining deformable angular and elastic network. The DCN model and the corresponding restraints were automatically generated according to a pre-set criteria for angular network triplets and elastic network pairs. These restraints contributed to the term in the total target function described below.
The
target function took the formHere, Estereo is the usual stereochemical energy. The term Eexp is the contribution of the experimental diffraction data and wa is the weight. Typically, the amplitude-based function (MLF) was used for Eexp. In the case where experimental phase information was available, the was carried out with the Hendrickson–Lattman (MLHL) target function, in which experimental phase contributions were incorporated in the form of Hendrickson–Lattman coefficients (Hendrickson & Lattman, 1970). wDCN, the weight of EDCN, was determined by a specific three-dimensional grid-search method (Fig. 1). The last term, EDCN, is the harmonic energy owing to the deviation of selected atom pairs and triplets in the target structure from their corresponding equilibrium values. These values were derived from both the current target structure and the reference structure. Simulated annealing was used as the protocol, with a starting temperature of 3000 K and a cooling rate of 50 K per step. Torsion-angle were used for dynamics simulation. with each parameter group was repeated ten times with different random seeds for initial velocity assignments and DCN restraint selections.
2.2. Brief description of the DEN model
The deformable elastic network (DEN; Schröder et al., 2007, 2010) is a set of randomly chosen atomic pairs subject to a harmonic potential
with the summation taken over all atomic pairs in the restraint list. The term di is the instantaneous distance for the ith pair in the target structure. The di0(γ, n) is related to both the reference structure and the target structure. Here, n denotes the step and γ is a constant determined in the three-dimensional grid-search procedure (Fig. 1).
2.3. Introduction to the DAN and DCN models
method, we first introduced a deformable angular network (DAN) model. The DAN model consists of a series of angles, each spanned by two bonds within an atom triplet. The three atoms, of which one is specified as the vertex and the other two as tail atoms, must be present in both the reference and target structures. They also need to satisfy the following additional criteria: (i) all three atoms should be within the same polypeptide chain, (ii) the first and last atom in the triplet should be within a cutoff distance from the middle vertex atom, which is commonly set to be 15 Å, the same as in DEN, (iii) the vertex atom and the tail atom should be no more than ten residues apart and (iv) the vertex angle spanned should be between 60 and 120°. The final angular restraints for were randomly selected from the shortlist, with the number of restraints set to a multiple (one in our study) of the total number of atoms in the target structure. All of these parameters, including the cutoff, residue separation, angle range and restraint-number multiples, were designed to be customizable and to allow finer tuning. We also provided two modes (directional and arbitrary modes) for constructing the restraint list (Fig. 2The harmonic bending energy in DAN is defined as
here, the summation was taken over all angle triplets, θj is the instantaneous angle for the jth triplet in the target structure and θj0(μ, n) is the corresponding equilibrium angle at a specific (nth) step.
DCN was established by combining DEN and DAN. It should be noted that the reference structures of DEN and DAN can be established independently, for example from different homology models. These restraints were considered as unified DCN restraints for use in the total
target function.The DCN potential is the sum of the harmonic stretching energy of DEN and the harmonic bending energy of DAN,
We set the coefficient k to 0.01 and the unit of angle was the degree.
We updated di0 and θj0 every six torsion-angle molecular-dynamics (MD) steps in simulated annealing (when the temperature also drops by 50 K) according to the following equations:
The equilibrium values in the next step for distance and angle, di0(γ, n + 1) and θj0(μ, n + 1), were functions of their current equilibrium values, di0(γ, n) and θj0(μ, n), their actual instantaneous values, di and θj, and the values of the equivalent triplet and pair in the reference model, diref and θjref. Typically, the initial equilibrium values of the atom pair di0(γ, 0) and the triplet θj0(μ, 0) were set to these values in the starting structure. The coefficients κ and φ are weights controlling the rate of change between consecutive equilibria. For initial relaxation, κ and φ were set to 0 during the first three macrocycles (refinement protocol). After that, κ and φ were set to a fixed value of 0.1. The values of γ and μ were optimized, together with the weight of the DCN potential wDCN (1), via the three-dimensional grid search (Fig. 1). The value of wDCN was reset to 0 during the last two macrocycles to reduce the bias of the minimum of the target function.
2.4. A three-dimensional grid-search scheme for optimizing the parameter set (γ, wDCN, μ)
The parameter set (γ, wDCN, μ) was optimized via a three-dimensional grid search through 180 grid points: (0, 0.2, 0.4, 0.6, 0.8, 1) for γ, (3, 10, 30, 100, 300) for wDCN and (0, 0.2, 0.4, 0.6, 0.8, 1) for μ (Fig. 1) At each point, ten refinements with different random seeds were carried out and the result with the lowest Rfree represented the final refined structure at that grid point. The seed controlled the assignment of the initial velocities in dynamics simulation for atoms as well as the selection of DCN restraints from the pair and triplet pool. It should be noted that the final results can depend on the choice of random-number seeds; thus, to ensure consistency, we used the exact integers from 1 to 10 as the ten random seeds throughout this work.
2.5. protocol
Torsion-angle ) combined with traditional simulated annealing (Kirkpatrick et al., 1983) was used as the main protocol (Schröder et al., 2010). The time step of dynamics simulation was 4 fs. For the annealing process, the initial temperature was set to 3000 K, with a decreasing rate of 50 K per six TAMD steps. Every six TAMD steps can be defined as a `microcycle', which determined the update frequency for both the annealing temperature and the equilibrium values of the DCN restraints. The period in which the temperature decreased from 3000 to 0 K formed a `macrocycle'. Each task in this work, including conventional DEN and DCN used eight macrocycles. During the first three of them, φ and κ were set to zero rather than 0.1 to allow initial relaxation. The van der Waals radii were decreased to 75% of the original value during several initial macrocycles, together with a reduced van der Waals force constant to facilitate sampling, and were thereafter fully restored in the last two macrocycles. Moreover, the DCN restraint weight was set to zero in the last two macrocycles to reduce the bias in the global minimum of the target function.
(TAMD; Rice & Brünger, 1994Anisotropic overall B-factor correction and bulk-solvent correction (Jiang & Brünger, 1994; Brünger et al., 1998) were applied to all refinements and no positional minimization was used. For the 16 re-refinement tasks, 50 steps of group B-factor minimization with a tenfold increase of the target σ values of the B-factor main-chain/side-chain bond/angle restraints were performed, and initial values of B factors were reset to 50 Å2. Ligands that were not recognized by default by CNS (Brünger et al., 1998; Brunger, 2007) were explicitly defined as groups for group B-factor minimization. For the purposes of appropriate comparison, all parameter settings were kept identical across all test systems. It should be noted that certain parameters, such as the initial annealing temperature, the cooling rate or the multiples of the target σ value for group B factors, can also possibly be further optimized for better Upon the completion of a all refined structures were sorted according to their values of Rfree and that with the lowest value was chosen for subsequent analysis.
2.6. Computation
Source codes for this approach (DCN_REF) were developed under the framework of the Crystallography and NMR System (CNS; v.1.3; Brunger, 2007; Brünger et al., 1998). Computation was carried out on the Shared University Grid at Rice (SUG@R) cluster platform of the Shared Computing Resources (ShareCoRe).
3. Results
3.1. of the tobacco PR-5d protein (PDB entry 1aun ) at three lower resolutions
In this test, we used the 1aun , 1.8 Å resolution; Koiwa et al., 1999) to allow a systematic assessment of the DCN approach. Its full diffraction data were obtained from the PDB and were then truncated using CCP4 (Winn et al., 2011) to give three lower resolution sets at 3.5, 4.0 and 4.5 Å. These three sets were treated as independent original low-resolution experimental data for subsequent A homology model (PDB entry 1pcv ; 2.3 Å resolution; Min et al., 2004) was used as the starting structure, the position and orientation of which were determined by using Phaser (McCoy et al., 2007) against each of the three low-resolution data sets. The solutions from the served as the reference structure in the DCN refinement.
of the tobacco PR-5d protein (PDB entryTo evaluate the performance of the DCN method, we also conducted two other refinements against these three low-resolution data sets. One used the conventional target function combining a stereochemistry potential (Engh & Huber, 1991) term with the experimental data term (in the form of energy; Bricogne & Gilmore, 1990). The other used the conventional target function in addition to the DEN potential.
In terms of Rfree values, which measure the agreement between the structural model and X-ray diffraction data (Fig. 3a), DCN achieved substantial improvements over the DEN method: DCN-refined structural models have a 0.94 and 1.12% lower Rfree than those refined by DEN at 3.5 and 4.0 Å resolution, respectively. At 4.5 Å resolution, the structural models refined by DCN and DEN have similar Rfree values (with the DCN-refined structure having a value 0.24% higher than that of the DEN structure). Compared with the structures refined by the conventional method, the DCN-refined structures are lower in Rfree by 2.21, 6.85 and 13.16% at 3.5, 4.0 and 4.5 Å resolution, respectively (Table 1, Fig. 3a).
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In addition to the Rfree values, with the 1.8 Å resolution 1aun as the `true answer', additional criteria can be used to assess the quality of the refined structure including the all-atom r.m.s.d., the GDT (<1 Å) score (Zemla, 2003) and the TMscore (Zhang & Skolnick, 2004). In terms of r.m.s.d. (Fig. 3b, Table 1), DCN always outperformed DEN at all three resolutions. For the GDT (<1 Å) score (Fig. 3c, Table 1) and the TMscore (Fig. 3d, Table 1), DCN consistently delivered the most favorable value among all three approaches. It is important to note that in general the largest improvements provided by DCN are observed at the lowest resolution (4.5 Å); thus, DCN is expected to perform the best for against X-ray data at a resolution limit of 4.0 Å or lower (Table 1).
3.2. Re-refinement of 16 randomly selected low-resolution structures
We also randomly selected 16 low-resolution all-atom structures (4.0–4.51 Å resolution, 1–14 polypeptide chains, 304–10 941 observed residues; Supplementary Tables S1 and S2) and performed re-refinements. For some structures, the topologies and parameter files of nonstandard ligands, ions and modified residues were obtained from the Hetero-compound Information Center, Uppsala, Sweden (HIC-Up; Kleywegt & Jones, 1998). To test the performance of DCN, we carried out automatic re-refinements without any manual adjustments. In order to minimize the bias, we reset the DCN potential to zero in the last two of the total of eight macrocycles (see §2). As a control, identical protocol and settings were used in DEN and conventional refinements of each of the 16 re-refinements. They generally resulted in better structural models in this work compared with previous work (Supplementary Table S3). These re-refinements serve as the basis for evaluating the performance of our new DCN method.
3.2.1. The Rfree values
Rfree (Brünger, 1992) was introduced as a cross-validation parameter for the fit between the experimental data and the refined structure, and is ubiquitously used as the primary measure of structure quality in macromolecular crystallography. In our tests of 16 randomly selected low-resolution structures, all of the final Rfree values obtained by DCN were substantially lower than those obtained using the standalone DEN method (ranging between 0.15–1.95%) and more so than those obtained using the conventional method (by 0.41–6.75%) (Table 2, Fig. 4a).
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3.2.2. Overfitting
The degree of overfitting can be assessed by the difference between the absolute values of Rfree and Rwork. The latter is calculated using the reflections that are involved in the process and is therefore typically smaller than Rfree. In most of our test cases (14 of 16), DCN consistently delivered the smallest Rfree − Rwork among all three methods (Table 2, Fig. 4). As shown in Table 2, the case with the most favorable value of Rfree − Rwork for DCN was 0.3% (PDB entry 2i37 ), whereas for DEN and the conventional method the best cases were 1.2% (PDB entry 2i37 ) and 1.9% (PDB entry 3alz ), respectively. In addition, the averaged Rfree − Rwork from DCN for all 16 test cases is 5.8%, which is 0.6 and 1.5% lower than that from DEN and conventional respectively (Table 2).
3.2.3. Ramachandran statistics
To further evaluate the quality of the refined structures, we carried out structure validation using MolProbity (Chen et al., 2010). Compared with the structures refined by the conventional method, 15 of the 16 DCN-refined structures have a higher percentage of residues that fall in the favored regions of the Ramachandran plot, with a largest increase of 16.9% and an average increase of 9.6% for all 16 cases. Relative to the structures refined by the DEN method, 13 of the 16 DCN-refined structures exhibit a larger percentage of residues in the favored regions, with a largest increase of 9.0% and an average increase of 1.9% (Table 2, Fig. 4c). These data collectively suggest greatly enhanced Ramachandran statistics compared with structures refined by the conventional method or the DEN method.
3.2.4. Electron-density maps
Re-refinement by the DCN method also resulted in improved electron-density maps (Fig. 5). The phase-combined σ-weighted 2Fo − Fc electron-density maps calculated from the experimental amplitudes and model phases are shown in Fig. 5 for two examples: PDB entries 1jl4 (Figs. 5a, 5b and 5c) and 2bf1 (Figs. 5d, 5e and 5f). In the example of PDB entry 1jl4 , the σ-weighted 2Fo − Fc electron-density maps from the structural models refined by the conventional method (Fig. 5a) or the DEN method (Fig. 5b) both exhibit broken densities around the main-chain atoms of Thr23. In sharp contrast, the map from the structural model refined by the DCN method has clear density around Thr23 (Fig. 5c). In the second example, PDB entry 2bf1 , DEN resulted in an Rfree value that is 4.35% lower than that from the conventional method and the DEN-refined structure displayed large positional shifts in several places on main-chain atoms relative to the structure refined by the conventional method. However, there are regions where the large structural shifts are not supported by electron-density maps (compare Figs. 5d and Fig. 5e). In marked contrast, the DCN-refined structure, with an additional decrease in Rfree (by 1.65%) over the DEN method, showed a much better map-coordinate consistency (Fig. 5f).
4. Discussion
In macromolecular X-ray crystallography, structural et al., 2007, 2010). In this study, we developed a new algorithm, DCN, that combines the DEN model with a novel angular network-based DAN restraint that exploits higher-dimension interaction networks among atoms. Test of DCN on a wide range of low-resolution structures demonstrated the power of this new method in delivering significant improvements by multiple measures, thus representing a new effective tool for low-resolution structural determination.
based on lower-resolution experimental diffraction data remains a major challenge where new and efficient algorithms are urgently needed. Previous studies have used a DEN model to aid in low-resolution structural (SchröderFor generality, it was our intention to fix many parameters at their default values without any adjustment in this work. We expect that finer tuning of DCN settings will further enhance the performance and robustness of this method. For instance, restraints can be established only for certain regions of the molecule that have sufficiently reliable reference structures available; several angle criteria for the DCN model can be more elaborately tailored to account for the characteristics of individual macromolecular systems. As an example, choosing DAN sequence separation limits of 5 and 8, respectively, delivered an Rfree of 20.75% for PDB entry 1isr and of 28.97% for PDB entry 1ye1 , which are about 0.4% lower than using the default value of 10 as shown in Table 2. The method can also be extended: in cases where the best homology model found in the database does not possess satisfactory sequence identity or resolution, DAN and DEN information for a single chain can be derived from different homology sources. Also, the deformations of angular network and distance network do not need to be synchronized. A more favorable for a given system may emerge when the two networks deform alternatively or with uneven frequencies. Moreover, DCN can be easily implemented in grid computing servers with an online GUI (O'Donovan et al., 2012), allowing interested users to use it via a web portal with ease.
Acknowledgements
JM is grateful for support from the National Institutes of Health (R01-GM067801), the National Science Foundation (MCB-0818353) and a Simmons Collaborative Research Fund Award from the Gulf Coast Consortium and the Welch Foundation (Q-1512). QW is grateful for support from the National Institutes of Health (R01-AI067839), the Gillson–Longenbaugh Foundation, a Simmons Collaborative Research Fund Award from the Gulf Coast Consortium and The Welch Foundation (Q-1826).
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