Zhi Chen^†, Eric Blanc^†§ and Michael S. Chapman*^†‡

* Department of Chemistry & † Institute of Molecular Biophysics, Florida State University, Tallahassee, FL 32306-4380, USA.

Synopsis: Torsion angle molecular dynamics makes real-space methods the most efficient for all but the final cycles of refinement when excellent (anomalous diffraction) phases are available. Overfitting is reduced and convergence improved by implicit use of phases and use of a local refinement that does not allow remote errors to be mutually compensating.

Abstract

Real-space targets and molecular dynamics search protocols have been combined to improve the convergence of macromolecular atomic refinement. This was accomplished by providing a local real-space target function for the molecular dynamics program X-plor (Brünger, A.T., 1992, X-Plor Version 3.1 A System for X-ray Crystallography and NMR, Yale University Press). With poor isomorphous replacement experimental phases, molecular dynamics does not improve real-space refinement. However, with high quality anomalous diffraction phases, convergence is improved at the start of refinement, and torsion angle real-space molecular dynamics performs better than other available least-squares or maximum likelihood methods in real- or reciprocal-space. It is shown that the improvements result from an optimization method that can escape local minima and from a reduction of overfitting through the implicit use of phases, and through use of a local refinement in which errors in remote parts of the structure can not be mutually compensating.

Introduction

The most common macromolecular crystallographic refinement involves restrained optimization of the agreement between diffraction amplitudes calculated from an atomic model and those that are derived from the experimental data (Jensen, 1985 ). Stereochemical restraints are introduced either by optimizing the agreement with ideal geometries (Engh & Huber, 1991; Hendrickson, 1985; Waser, 1963) or (as in this work) through minimization of an empirical estimate of the configurational potential energy, E_chem. (Brünger et al., 1987; Levitt, 1974). Structural refinement then becomes minimization of the objective function:

, where

Equation 1

where F_o, F_c are the experimental and scaled calculated structure amplitudes for reflection h. Equation 1 shows a least-squares target, but analogous maximum likelihood targets have recently also shown promise (Murshudov et al., 1997; Pannu & Read, 1996). Methods to find an optimal structure are either one of the gradient descent methods (see Tronrud, 1992) or use a protocol such as simulated annealing (SA) optimization that enables the structure to leave a local minimum, combined with molecular dynamics (MD) sampling of conformational space (Brünger et al., 1987). SA and MD methods increase the convergence radius of atomic refinement, because the surface of the objective function has many local minima in which gradient descent methods can stall, because they may be unable to pass through stereochemically unfavorable configurations to explore, for example, alternative rotamers. Constraining bond lengths and angles by changing torsion angles in MD instead of the Cartesian atomic coordinates increases the radius of reciprocal-space refinement to ~1.7 Å (Brünger & Rice, 1997).

Although reciprocal-space E_xray terms are currently preferred, early refinements used a real-space target (Diamond, 1971):

Equation 2

where is it now a difference between observed and calculated electron density over molecular volume, v that is optimized. Although Diamond (Diamond, 1971) noted a potentially wide convergence radius, real-space methods have been largely superseded by reciprocal-space methods due to the implicit dependence of real-space methods upon phases used to calculate the electron density map that are often much less accurate than the diffraction amplitudes. Real-space refinement continued to be applied in several niches, such as the refinement of (virus) structures with excellent phase and map quality due to high non-crystallographic symmetry (Jones & Liljas, 1984), and as a tool to assist the fitting of rigid fragments to electron density during interactive model-building (reviewed in Jones & Kjeldgaard, 1997). Recent methodological advances have expanded its application – as an aide in model-building (Blanc & Chapman, 1997), as a complement to reciprocal-space methods to improve convergence (Chapman & Blanc, 1997) in protein structure refinement, and as an efficient alternative for the complete refinement of virus structures (Chapman & Rossmann, 1996). The improvements include the incorporation of modern stereochemical restraints, and the use of an atomic electron density function that explicitly accounts for the resolution limits of the electron density map (Chapman, 1995):

Equation 3

where O is the occupancy and g is the atomic scattering factor that is dependent on the resolution r* and that incorporates isotropic thermal vibration factors.

Like earlier implementations, the newer real-space refinements optimized the structure by gradient descent methods (Tronrud, 1992). This work examines the potential of combining two methods that have robust convergence properties – namely real-space targets and molecular dynamics optimization. The potential application of such methodology has broadened considerably recently with the availability of accurate macromolecular phases from multiwavelength anomalous diffraction (MAD) methods (Hendrickson & Ogata, 1997).

Methods

Real space MD was implemented by programming an alternative target function for X-plor (Brünger, 1992b) that provides the option of substituting Equation 2 for E_xray in Equation 1. A prior implementation of a real-space target (Chapman, 1995) was adapted so that input and output was compatible with X-plor programs, data files and control scripts. The target value and its derivatives with respect to the atomic parameters are calculated with the new module and passed back to X-plor. Thus, all methods of optimization that have been applied in reciprocal-space (Brünger et al., 1997) can now be applied in real-space, including torsion angle or Cartesian MD (and conjugate gradient optimization).

Initial tests used simulated structure amplitudes, phases and maps calculated from -amylase inhibitor (Pflugrath et al., 1989) between 17 and 2 Å resolution. Starting models were perturbed by varying amounts using molecular dynamics at 600K followed by energy minimization, all in the absence of a E_xray term. The test refinement protocol involved 4 ps of torsion angle MD at 8000 K (now including an E_xray term), 0.2 ps of quenching at 300 K in Cartesian space, then conjugate gradient energy minimization. Slow-cooling protocols were tested but, as in reciprocal space torsion angle refinements (Rice & Brünger, 1994), they proved to be inferior to rapid quenching and are not considered further. With phases calculated directly from the correct structure, a starting model with 1.43 Å rms backbone error is refined to an error of 0.1 Å. The convergence radius is 3.6 Å, as defined by the maximal backbone perturbation that can still be refined to approximate the correct structure. With an omit map (Bhat, 1988 ) calculated from the perturbed model, the radius of convergence was 0.6 Å, indicating dependence of the method upon the availability of experimental phases.

Real-space torsion angle molecular dynamics (RSTAMD) was tested in two systems with actual crystallographic data. HMG CoA reductase represented poor experimental phases and mannose binding protein A (MBPA) represented high quality experimental phases. HMG CoA reductase exemplified a large protein structure determination in which poor multiple isomorphous replacement (MIR) phases were improved by the application of non-crystallographic symmetry (NCS) in the actual structure determination (Lawrence et al., 1995). For the current tests, the NCS was ignored and the unrefined model of Lawrence et al. was refined against a barely interpretable MIR map.

MBPA exemplified high quality MAD phasing (Burling et al., 1996). The starting model was based on a 2.3 Šresolution homologous complex with a different lanthanide ion (Weis et al., 1991), with remodeling of seven disordered terminal residues into the 1.8 Šresolution MAD map (Burling et al., 1996), deletion of solvent molecules, resetting all B-factors to 15 Ų, stereochemical regularization, and rigid body reciprocal-space refinement against the 1.8 Šcryo-diffraction data (Burling et al., 1996). Additional tests started with the 1.8 Šresolution final refined structure (Burling et al., 1996).

The refinement methods that were compared using MBPA included the following. All methods were as implemented in X-plor (Brünger, 1992b). When molecular dynamics was used, the torsion angle implementation (Rice & Brünger, 1994) was used and was followed by conjugate gradient minimization of the objective function:

1. Real-space least squares MD refinement of E_xray(r ) (Equation 2). The real-space targets used here, and in methods 2 & 5 are described in detail in the Introduction.

1b. Further locally refined the amino acids with the worst agreement with the electron density, as indicated by correlation coefficient (Jones et al., 1991; calculated according to Zhou et al., 1998).

2. Real-space least squares conjugate gradient refinement of E_xray(r ) (Equation 2) without MD.
3. Reciprocal-space least squares MD refinement of E_xray(F) (Equation 1). This corresponds to the mode of refinement for which Xplor is most commonly used (Brünger et al., 1987).
4. Reciprocal-space least squares conjugate gradient refinement of E_xray(F) (Equation 1) without MD. This would correspond to a "conventional" non-dynamics refinement.
5. Real-space least squares MD refinement of E_xray(r ) (Equation 2) followed by reciprocal-space least squares MD refinement of E_xray(F) (Equation 1).
6. Vector residual (Arnold & Rossmann, 1988) least squares MD refinement:

Equation 4

where (A, B) are the real and imaginary components of the structure factor, and m is the figure of merit.
7. Phase-(f ) restrained (Rees & Lewis, 1983) least squares MD refinement:

Equation 5

(Methods 6 and 7 can be considered pseudo-real-space methods in that they approximate the effect of a real-space target with computation in reciprocal-space.)
8. Maximum likelihood (Pannu & Read, 1996) MD refinement (Adams et al., 1997):

Equation 6

where P is the probability of observing Fo given a model Fc,  is the variance of P and denotes the expected value for F.
1. maximum likelihood MD refinement with additional phase restraints analogous to method 7 (Pannu & Read, 1997).

Results & Discussion

Application to the least favorable case of poor MIR phases in HMGCoA reductase showed that MD and conjugate gradient real-space performed near-identically. It was shown previously (Chapman & Blanc, 1997) that for poor phases, real-space refinement enhances but does not supersede reciprocal-space methods when the two are alternated in the initial cycles. Here, it is confirmed that real-space refinement is limited by the poor MIR phases and not the optimization algorithm. Thus, least-squares and MD are equally effective.

The potential with good phases is graphically illustrated in Figure 1. Real-space MD, unlike its gradient descent counterpart, is able to pass through an unfavorable configuration to find the best fit to the electron density. Quantitatively, it is clear from Table 1 that real-space molecular dynamics is the most powerful method for initial refinement when the map quality is good. R^free drops about twice as much as with any of the currently available reciprocal-space methods, including maximum likelihood. (Smaller differences between maximum likelihood and least squares targets are seen here compared with those observed earlier (Pannu & Read, 1996) due to the high quality 2.3 Å MBPA starting model.) Coordinate error drops about 60% farther for RSTAMD than for the reciprocal-space methods. Real-space gradient descent is intermediate in performance between RSTAMD and reciprocal-space methods. The RSTAMD-refined model has an R^free about twice as close to the target model as those produced by reciprocal-space refinements, indicating that some, but not all of the changes normally made by manual intervention have been accomplished automatically. Unlike HMGCoA reductase, the benefit of following initial real-space refinement with reciprocal-space refinement is at most marginal, presumably because the phase quality is not limiting at this stage of the MBPA refinement.

At the end of refinement, the indications are different (Table 2). Starting with the published, fully refined 1.8 Å MBPA structure (Burling et al., 1996), additional refinement in real-space using the MAD phases yields a structure with R^free slightly higher than in reciprocal space (0.217 vs. 0.207). It is the MAD phases that are limiting, as is shown through substitution of phases calculated from the final model which allows real-space refinement to equal reciprocal-space (R^free = 0.206, Table 2). Thus, there comes a point in refinement when the model errors become low enough that phase error limits real-space refinement, and reciprocal-space methods are indicated. With high quality MAD phases, this point is reached only for the final cycles.

While improvements of R^free and backbone coordinate error during initial refinement are appreciable, substantial errors remain in side chain coordinates with all refinement methods (Table 1). Some improvement is made with a new algorithm that is possible with a local method of real-space refinement. The worst amino acids are identified according to correlation between model and experimental electron density (Zhou et al., 1998) and, as individual amino acids, are given additional cycles of RSTAMD. Applied to the 13 worst amino acids, the overall rms coordinate error is reduced from 0.64 to 0.52 Å (Table 1), but there is little change to the R-factors, because it is a small fraction of the weakest-scattering part of the molecule that is improved. The remaining error is mostly because of the selection of incorrect rotamers, due to the lack of ordered water molecules in the model, and due to the remaining need to make some corrections interactively rather than through the automatic refinement methods used. The local procedure helps at the initial stages of refinement when all B-factors are uniform, and the local method enables more appropriate scaling (and refinement) between the more disordered parts of the model and their weak electron density. Such a procedure could also be used to try to fix automatically some of the most egregious errors of a model as highlighted by other (stereochemical) indicators, although there will still be a need for visual inspection to correct, for example, register errors.

Table 1 gives some indications of how real-space refinement helps in all but the final cycles. Although the greatest drop in R^free is seen with real-space refinement, the drop in crystallographic R^work is less than or equal to the drop by the various reciprocal-space refinements, and substantially less than the commonly used amplitude-based MD refinement. Thus, in real-space, overfitting is substantially reduced. Overfitting, early in refinement, can also be reduced by appropriately accounting for model and data errors in the maximum likelihood formulation (Pannu & Read, 1996). The source of the improvement with real-space methods is fundamentally different. It is the improvement of data:parameter ratio through the use of implicit phase information, and also the use of a local refinement method (see later). In the case of MBPA, with low model error and high phase accuracy, the reduction of overfitting is greater with real-space methods than with maximum likelihood methods. The pseudo-real-space methods (numbers 6, 6b & 7) also incorporate phase information, are the best of the reciprocal-space methods, and reduce overfitting somewhat, but not as much as the true real-space algorithm. This may seem counter-intuitive, due to at least a superficial correspondence of the real- and reciprocal-space operations (Diamond, 1971; Silva & Rossmann, 1985). However, there are differences in the weighting and in the local / global nature of refinement. The closest correspondence is between E_xray(r ) and E_xray(A, B) (methods 1 & 6b), when both incorporate figure of merit weighting. The remaining difference between these targets is presumably due to the local nature of real-space refinement versus the global nature of all reciprocal space and pseudo-real-space methods. In global methods, all parts of the model move to decrease the discrepancy between F_o and F_c. Atoms may be moved away from their correct locations to reduce discrepancies due to remote errors in the model (Hodel et al., 1992), an incomplete description of solvent or missing macromolecular atoms. With a local real-space method, these components to the overfitting are eliminated, leading to improved refinement at the early stages.

Conclusion

Real-space refinement can be significantly enhanced with the addition of molecular dynamics optimization methods. With high quality phases and maps, the improvement over other refinement methods is substantial, until phase errors dominate over model errors in the final cycles. With low quality maps and phases, the method is benign. This suggests a pragmatic approach when the experimental phases are of uncertain quality – do all possible refinement in real-space, then complete refinement in reciprocal space. With the high quality phases and maps that are increasingly available with anomalous diffraction, the results reported here suggest that real-space refinement will become an increasingly important part of the efficient structure determination of a significant proportion of protein structures.

Acknowledgements

We thank Temple Burling, Axel Brünger, Martin Lawrence and Cynthia Stauffacher for access to the MBPA and HMGCoA reductase coordinates and diffraction data. This work was supported by the National Science Foundation (BIR94-18741). Modifications to Xplor that enable real-space molecular dynamics refinement will be available directly from the authors and/or in an upcoming version of the Molecular Simlations Inc. implementation of Xplor.

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Captions

Figure 1: Rotamer correction by real-space molecular dynamics refinement. Ile 147 of Mannose Binding Protein A (Burling et al., 1996) was perturbed to an incorrect rotamer using the modeling program O (Kleywegt & Jones, 1997). Refinement against the MAD experimental electron density (shown) corrects the rotamer. While this particular type of error might be correctable with the tools in O (Kleywegt & Jones, 1997), quick application of the refinement to the whole protein can substantially reduce the number of corrections that need to be made with an interactive modeling program.

Figures

Figure 1

Tables
 

Method	Target		Equation	Optimization		Rms deviation (Å )		Rwork	Rfree
				Gradient descent	Molecular dynamics	backbone	overall
Target: published 1.8 Å structure, less solvent & with B = 15.						0.000	0.000	0.299	0.289
Starting model (modified 2.3 Å structure).						0.282	0.792	0.349	0.345
1	Exray(r )	Equation 2			3	0.123	0.640	0.313†	0.312†
1b	Exray(r )	With local improvement*			3	0.123	0.524	0.313	0.311
2	Exray(r )	Equation 2		3		0.153	0.706	0.317	0.311
3	Exray(F)	Equation 1			3	0.183	0.704	0.289	0.329
4	Exray(F)	Equation 1		3		0.185	0.733	0.304	0.325
5	Exray(r ) then Exray(F)	Equation 2, Equation 1			3	0.112	0.627	0.302	0.313
6	Exray(A,B)	Equation 4			3	0.165	0.669	0.316	0.323
6b	Exray(A,B)	With fom-weighting			3	0.156	0.657	0.310	0.318
7	Exray(F,f)	Equation 5			3	0.175	0.689	0.286	0.327
8		Equation 6			3	0.167	0.716	0.313	0.325
9		Equation 6		3		0.173	0.730	0.315	0.327
10		+ phase restraints			3	0.127	0.688	0.313	0.319

Table 1: Refinement of a 2.3 &Ar(derived from Weis et al., 1991)m \Weis, 1991 #1636] against 1.8 &Arin(Burling et al., 1996)[Burling, 1996 #991] by several methods further described in the text. When used, phase restraints were given weight equal to amplitude restraints. * The 13 regions with worst real-space correlation factor were refined further separately with a local real-space protocol. † Standard deviations for R^work and R^free were ± 0.001 and ± 0.004 respectively, as calculated for 10 repeated refinements with different non-overlapping test sets.
 

Method	Target	Equation	Optimization		Phases	Correlation coefficient	Rwork	Rfree
			Gradient descent	Molecular dynamics
Starting model (published 1.(Burling et al., 1996)[Burling, 1996 #991]).						0.917	0.196	0.216
1	Exray(r )	Equation 2		3	MAD	0.929	0.216	0.217
3	Exray(F)	Equation 1		3	N / A		0.202	0.207
1	Exray(r )	Equation 2		3	calculated		0.203	0.206

Table 2: Further refinement of the 1.8 Å refined MBPA structure in real- and reciprocal-space.