Section 18.2.5. Examples

Many examples have shown that simulated-annealing refinement starting from initial models (obtained by standard crystallographic techniques) produces significantly better final models compared to those produced by least-squares or conjugate-gradient minimization (Brünger et al., 1987; Brünger, 1988; Fujinaga et al., 1989; Kuriyan et al., 1989; Rice & Brünger, 1994; Adams et al., 1997). In another realistic test case (Adams et al., 1999), a series of models for the aspartic proteinase penicillopepsin were generated from homologous structures present in the Protein Data Bank. The sequence identity among these structures ranged from 100% to 25%, thus providing a set of models with increasing coordinate error compared to the refined structure of penicillopepsin. These models, after truncation of all residues to alanine, were all used as search models in molecular replacement against the native penicillopepsin diffraction data. In all cases, the correct placement of the model in the penicillopepsin unit cell was found.

Both conjugate-gradient minimization and simulated annealing were carried out in order to compare the performance of the least-squares residual [equation (18.2.3.2)], MLF (the maximum-likelihood target using amplitudes) and MLHL (the maximum-likelihood target using amplitudes and experimental phase information). In the latter case, phases from single isomorphous replacement (SIR) were used. A very large number of conjugate-gradient cycles were carried out in order to make the computational requirements equivalent for both minimization and simulated annealing. The conjugate-gradient minimizations were converged, i.e. there was no change when further cycles were carried out.

For a given target function, simulated annealing always outperformed minimization (Fig. 18.2.5.1). For a given starting model, the maximum-likelihood targets outperformed the least-squares-residual target for both minimization and simulated annealing, producing models with lower phase errors and higher map correlation coefficients when compared with the published penicillopepsin crystal structure (Fig. 18.2.5.1). This improvement is illustrated in -weighted electron-density maps obtained from the resulting models (Fig. 18.2.5.2). The incorporation of experimental phase information further improved the refinement significantly despite the ambiguity in the SIR phase probability distributions. Thus, the most efficient refinement will make use of simulated annealing and phase information in the MLHL maximum-likelihood target function.

Figure 18.2.5.1 | top | pdf | Simulated annealing produces better models than extensive conjugate-gradient minimization. Map correlation coefficients were computed before and after refinement against the native penicillopepsin diffraction data (Hsu et al., 1977) for the polyalanine model derived from Rhizopuspepsin (Suguna et al., 1987, PDB code 2APR). Correlation coefficients are between -weighted maps calculated from each model and from the published penicillopepsin structure. The observed penicillopepsin diffraction data were in space group with cell dimensions , ,  Å and . All refinements were carried out using diffraction data from the lowest-resolution limit of 22.0 Å up to 2.0 Å. The MLHL refinements used single isomorphous phases from a K3UO2F5 derivative of the penicillopepsin crystal structure, which covered a resolution range of 22.0 Å to 2.8 Å. Simulated-annealing refinements were repeated five times with different initial velocities. The numerical averages of the map correlation coefficients for the five refinements are shown as hashed bars. The best map correlation coefficients from simulated annealing are shown as white bars.

Figure 18.2.5.2 | top | pdf | Maximum-likelihood targets significantly decrease model bias in simulated-annealing refinement. -weighted electron-density maps contoured at 1.25σ for models from simulated-annealing refinement with different targets are shown. Residues 233 to 237 are shown for the published penicillopepsin crystal structure (Hsu et al., 1977) as solid lines, and for the model with the lowest free R value from five independent refinements as dashed lines.

Cross validation is essential in the calculation of the maximum-likelihood target (Kleywegt & Brünger, 1996; Pannu & Read, 1996; Adams et al., 1997). Maximum-likelihood refinement without cross validation gives much poorer results, as indicated by higher free R values, differences and phase errors (Adams et al., 1997). It should be noted that the final normal R value is in general increased, compared to refinements with the least-squares target, when using the cross-validated maximum-likelihood formulation. This is a consequence of the reduction of overfitting by this method.