2.1. Test in automated model building

Three protein structures were selected for testing Conditional Optimization for automated model building: (i) the A3-domain from human von Willebrand Factor, vWF-A3 (Huizinga et al., 1997), (ii) outer-membrane protein NspA from Neisseria meningitidis (Vandeputte-Rutten et al., 2003) and (iii) the C-terminal domain of leech anti-platelet protein, LAPP (Huizinga et al., 2001). All three structures were solved in our laboratory at medium to low resolution (2.4–3.0 Å resolution) and with good experimental phases. For all cases, the original models were built manually using the graphics program O (Jones et al., 1991). The main characteristics of these test cases are given in Table 1.

Fig. 1(a) shows the protocol used for automated model building by Conditional Optimization. The process was started by positioning loose and unlabelled atoms in the (m|F_obs|exp{iφ_best}) electron-density map at sites with ρ > 1.0σ, with minimum and maximum interatomic distances of 1.1 and 1.8 Å, respectively, and a maximum of four neighbouring atoms within 1.8 Å. Optimization was performed using the program CNS (Brünger et al., 1998). For the crystallographic target function, we used the phase-restrained maximum-likelihood function (MLHL; Pannu et al., 1998) with phases and figures of merit from the experimental phasing process (Table 1). σ_A values (Read, 1986) were estimated using test-set reflections. For the geometric target functions we used the conditional formulation (Scheres & Gros, 2001, 2003). For each protein, a specific force field was generated using the general parameter list of conditions for α-­helices, β-strands, loops and side chains up to γ-positions (Scheres & Gros, 2003) and an approximate estimate of the secondary-structure content (see Table 1). In the case of LAPP we set the loop content to 0, thereby excluding loop conformations from the force field, to limit computer memory requirements. At the end of each cycle of Conditional Optimization (Fig. 1a), we assigned atom labels based on the implicit assignments used in the Conditional Optimization process. Atoms were labelled with a chemical identity (N, C^α, O, C, C^β, C^γ or S^γ) if the gradient contribution towards that particular atom type was at least twice as large as the second largest contribution. Subsequent cycles of optimization were started from phase-combined maps, combining model and experimental phases. Subsequent starting models consisted of the atoms (but not their labels) selected in our labelling procedure with additional atoms placed in the electron-density map as described above. For vWF-A3 and NspA two cycles and for LAPP four cycles of model building by Conditional Optimization were performed.

Figure 1 Protocols for automated model building and ab initio modelling. (a) Automated model building by Conditional Optimization using the program CNS (Brünger et al., 1998). The standard geometrical target functions were replaced by our conditional formulation (see main text). Electron-density maps were filled with loose atoms (as described in the main text). Refinement cycles used the MLHL crystallographic target function and consisted of 10 000 steps of dynamics (denoted `dyn.') and 200 steps energy minimization (`min.'). Velocities in the dynamics calculations were scaled to a constant temperature of 600 K. Overall anisotropic B-factor scaling, bulk-solvent correction, σA estimation, determination of wa weight and phase combination were performed using standard CNS routines. After each cycle, the positions of atoms that could be assigned to protein fragments (see main text) were maintained in the model. Placing the remaining number of atoms in the phase-combined electron-density map completed the starting model for a next cycle of optimization. (b) Multiple-model optimization protocol starting from N models of randomly distributed atoms. The CNS program was used with conditional formulation for the geometrical target functions. First, a short condensation step was performed for each model with 200 steps of minimization and 1000 steps of dynamics followed by 200 steps minimization using the MLF crystallographic target function. Subsequent optimization cycles consisted of 200 steps of minimization, 10 000 steps of dynamics and 200 steps of minimization using the MLHL crystallographic target function. Phases of the averaged structure factors and figures of merit (ma) were used in the phase restraint (see main text). The phased-translation function was used to put the N models on a common origin prior to averaging. The bulk-solvent model was generated for 20%(v/v); atoms present in solvent regions were given zero occupancy.

All three test cases were also subjected to automated model building by ARP/wARP (version 6.0; Perrakis et al., 1999; Morris et al., 2002) and RESOLVE (version 2.03; Terwilliger, 2003), both using REFMAC version 5.1.24 for refinement (Murshudov et al., 1997). These calculations were performed using default values for all input parameters. In RESOLVE, docking of the primary sequence on the constructed fragments by side-chain modelling was included in the model-building process. Modelling of the side chains was not performed with ARP/wARP, since this option of the program yielded significantly worse results (not shown).

2.2. Testing ab initio modelling

We selected the four-helical bundle Alpha-1 (Privé et al., 1999; PDB code 1byz ) to explore the potentials of Conditional Optimization in ab initio phasing. This structure consists of 396 protein atoms in space group P1 and was originally solved by direct methods using all observed diffraction data to 0.9 Å resolution. Here, we truncated deposited structure-factor amplitudes to 2.0 Å resolution.

The protocol used to refine N multiple models using Conditional Optimization is given in Fig. 1(b). The number of atoms per model was 400. Initial models consisted of randomly distributed atoms. Calculations were performed using the program CNS (Brünger et al., 1998). The target functions from Conditional Optimization replaced the standard geometric target functions. The random models were first subjected to 1000 steps of maximum-likelihood refinement using structure-factor amplitudes (MLF; Pannu & Read, 1996). The σ_A values for this initial cycle were calculated according to σ_A = exp(−150s²). In subsequent cycles, each containing 10 000 steps of dynamics, we used the phase-restrained maximum-likelihood crystallographic restraint (MLHL; Pannu et al., 1998) with target phases and figures of merit derived from averaging the structure-factor sets of the individual models. To this end, all individual structures were first placed on a common origin using the phased-translation function. Figures of merit (m_a) were computed per resolution shell using only test-set reflections: = , where Fⁱ were calculated structure factors from an individual model, and were extrapolated to infinite models by m_a = {[N()² − 1]/(N − 1)}^1/2. For each individual model, we estimated σ_A values per resolution shell. We assumed that the true phase error of a model would be related to phase differences of that model to any other model,

These estimates were calculated using all reflections because the low numbers of reflections in the test set alone yielded unstable results.

Calculations were performed on 667 MHz single-processor Compaq XP1000 workstations with 1–2 Gb of memory. The CPU times for automated model building are given in Table 2. Ab initio modelling calculations took more than 100 d of CPU time.

3.1. Model-building tests

We compared automated model building by Conditional Optimization, which did not include any discrete decision-making steps, with the commonly used automated building programs ARP/wARP and RESOLVE. Models were built for three test cases, vWF-A3, NspA and LAPP, with data to d = 2.4, 2.6 and 3.0 Å resolution, respectively. Criteria for comparison were model completeness, correctness of the trace, accuracy of the positioned fragments assessed by r.m.s. coordinate errors and quality of the phases computed from the models. Statistics of the automatically built models are given in Table 2. C^α traces of the generated models are given in Fig. 2. Coordinate files for the nine generated models and for the three refined models used in the analysis have been submitted as supplementary material.¹

Figure 2 Cα traces of automatically built models (black lines) overlaid with traces of the refined structures (grey lines): models of the vWF-A3 domain (using diffraction data to 2.4 Å resolution) obtained by Conditional Optimization (a), ARP/wARP (b) and RESOLVE (c); models obtained for NspA (with data to 2.6 Å resolution) by Conditional Optimization (d), ARP/wARP (e) and RESOLVE (f); superposition of the three models of three independent molecules of LAPP determined at 3.0 Å resolution by Conditional Optimization (g), ARP/wARP (h) and RESOLVE (i).

For vWF-A3, automated model building was tested using data to 2.4 Å resolution and experimental phases with an (amplitude-weighted) mean phase error of 35.6°. ARP/wARP and RESOLVE built more complete and less fragmented models than did Conditional Optimization (Figs. 2a–2c). RESOLVE produced the best model that was most complete, missing only one loop and a small α-helix, had the smallest r.m.s. coordinate error and the lowest mean-phase error. The model from ARP/wARP missed one α-helix, one β-strand and two loops. It had a relatively large coordinate error and the phases computed from the model were not better than the experimental phases. Conditional Optimization yielded a fragmented model consisting of almost all α-­helical and β-­strand segments, except the one small α-­helix that was also missed by the other two programs. Only one of the loops was modelled correctly. Another loop was modelled with a reversed chain direction, as was a small β-strand which flanks the central β-­sheet. Notwithstanding these errors, the model from Conditional Optimization was more accurate and yielded better phases than the model from ARP/wARP.

For NspA (Figs. 2d–2f), using data to 2.6 Å resolution and experimental phases with an (amplitude-weighted) mean phase error of 38.3°, Conditional Optimization built most of the strands in the β-barrel, but none of the turns. The largest errors in this model were reverse chain directions for an entire β-­strand and two smaller β-strand fragments. In this case, ARP/wARP produced the most complete model and the lowest phase error, though the model included two β-strands with reversed chain directions. RESOLVE built a smaller portion of the molecule with a strand that crossed over into a neighbouring strand. The mean-phase error using calculated phases from this model was relatively large, possibly reflecting the incompleteness of the model produced by RESOLVE.

The data from LAPP represent a situation in which automated model building generally does not work owing to the limited resolution (3 Å) of the diffraction data. However, solvent flattening and threefold non-crystallographic symmetry averaging yielded high-quality phases with an (amplitude-weighted) mean-phase error of 28.2°. Conditional Optimization (Fig. 2g) yielded a model with partially built β-sheets and most α-­helical segments of the three molecules in the asymmetric unit. Reversed chain directions were observed for some of the β-strands and for one α-helix; one of the loops was also modelled incorrectly by an α-helical turn (note: in this particular case loops were omitted from the force field). This model had a fairly low r.m.s. coordinate error and good phase quality. ARP/wARP built a more complete model with more β-strands and more loops (Fig. 2h). One incorrect main-chain trace from an α-helix to a neighbouring β-strand was observed in this model. This model yielded a phase error comparable to that obtained with conditional optimization. As for NspA, RESOLVE (Fig. 2i) produced the model with the lowest completeness. In addition, this model had a low accuracy of the positioned fragments and a high mean-phase error. It contained more main-chain trace errors than the models built by either Conditional Optimization or ARP/wARP.

These three cases clearly demonstrate that Conditional Optimization can produce models of comparable completeness and comparable phase quality as ARP/wARP and RESOLVE. The models from Conditional Optimization have significantly lower connectivity and contain fewer loops and turns. Similar to ARP/wARP and RESOLVE, an increase in connectivity would be most efficiently achieved by introducing discrete model-completion steps. Moreover, unsuccessful modelling of turns and loops reflects the currently limited conditions defined for turn and loop conformations in the conditional force field. The worst aspect of Conditional Optimization is the excessive amount of CPU time and computer memory required.

3.2. Ab initio modelling

The possibility of phasing protein structures by ab initio modelling was explored using experimental `real' diffraction data of the four-helical bundle Alpha-1 (Privé et al., 1999), cf. the tests of ab initio phasing using artificial data in Scheres & Gros (2001). We chose to perform parallel optimization of multiple random models and used the set of models in two ways: (i) the variation among the multiple models provided an indication of the statistical relevance of the individual models and (ii) averaging the individual structure-factor sets provided phases that were used as phase restraints in the MLHL crystallographic target.

36 models with randomly distributed atoms were first subjected to 1000 steps of Conditional Optimization with MLF refinement (see Fig. 1b). In this cycle, the random atoms of most models condensed into four rod-like structures corresponding to the lowest resolution features of the diffraction data (see Fig. 3a). Of these 36 initial optimization runs, three runs did not finish owing to the formation of extensively branched structures requiring more computer memory than was available. 17 models were selected which appeared to have a common hand in an analysis based on the phased translation function. However, a posteriori analysis showed that these models did not yet have a significant handedness to allow a useful selection to be made. Subsequently, the selected 17 models were subjected to 25 optimization cycles of 10 000 steps each using a MLHL crystallographic target function (Fig. 1b). Initially, the estimated values of the figures of merit (m_a) for the averaged structure factors and σ_A values for the individual models behaved well when analyzed using the phases of the refined model (see Fig. 4). However, after seven cycles the figures of merit m_a were increasingly overestimated. Similarly, the σ_A values were overestimated from cycle ten onwards. Since the overestimation of m_a appeared to coincide with a drop in convergence (as judged by the map-correlation coefficients depicted in Fig. 5a), we decided to continue from cycle 7 with fixed figures of merit m_a. Fixed and low values for m_a also avoided subsequent overestimation of the σ_A values. Under these conditions, we observed a slow but steady convergence, as indicated by a ∼0.005 increase in map-correlation coefficient per cycle and an overall decrease in (amplitude-weighted) mean-phase error of the average structure factors by ∼10° over 25 cycles (Figs. 5a and 5b). The mean phase error after 25 cycles was 76.3° and the map-correlation coefficient was 0.37 for all data to 2 Å resolution. Inspection of the models indicated the significance of this gain in phasing quality. An overlay of all 17 models showed that right-handed helices have developed to a reasonable extent (Fig. 3b). The best model, as identified by the highest σ_A value, had three and a half helices formed (see Fig. 3c).

Figure 3 Structures of Alpha-1 obtained by ab initio modelling against 2 Å resolution data. (a) Stereoview of a ball-and-stick representation of an optimized model after the initial condensation using 1000 steps of conditional dynamics using the MLF target function; (b) Cα trace of 17 models obtained after 25 cycles of optimization (black lines) with the Cα trace of the refined structure overlaid (grey lines); (c) stereoview of the Cα trace of the model with the highest σA value. From left to right the helices are oriented down, up, down and up in the refined structure (the position of the N-terminus is indicated by a ball). The chain directionality in the depicted model is therefore from left to right: incorrect, incorrect, correct and correct. Assignment of atom labels (for b and c) was based on the gradient contributions in Conditional Optimization (as described in the main text for automated model building).

Figure 4 Figures of merit and σA values derived from multiple models. Figures of merit ma (a) and σA values for one of the individual models (b) computed for data up to 2 Å resolution over 15 cycles of optimization. Solid lines represent the estimates used in our calculation; dashed lines indicate the corresponding values, 〈cos(φave − φcalc)〉 and σA = 〈|Eobs|2〉〈|Ei|2〉, where φave is the phase of the average structure factor, φcalc is the phase of the structure factors computed from the published structure and φi is the phase of the individual model.

Figure 5 Convergence in ab initio modelling of multiple models starting from randomly distributed atoms. Map-correlation coefficients (a) and overall Fobs-weighted phase errors (b) to 2.0 Å resolution of the average structure factors with respect to structure factors calculated from the published structure. Solid lines show the results for the optimization cycles with updated figures of merit (cycles 1–15). Dashed lines show the results for the optimization cycles with fixed figures of merit (cycles 7–25).

This test of ab initio modelling was computationally demanding, which seriously limited the testing of relevant parameters. Nonetheless, the preliminary results presented here clearly indicated the most prominent shortcomings of our current approach. Throughout these calculations, the average structure factors scored among the top three of the individual sets of structure factors (i.e. top ∼20%) with respect to phase quality. This justifies the idea of using these phases as phase restraints. Obviously, applying this information through the MLHL target function introduces serious bias into the calculation. Though our σ_A estimates were without thorough theoretical foundation, we observed a significant correlation (0.77 after cycle 25) between the estimated σ_A values and the map-correlation coefficients of the individual models. This indicates that the use of multiple models for estimating the statistical relevance holds promise.