- Email: [email protected]
Crystallographic phasing and refinement of macromolecules
Crystallographic phasing and refinement of macromolecules Axel T. Briinger Yale University, New Haven, Connecticut, USA Recent developments in crystallographic phasing and refinement are discussed with particular emphasis on ab initio phasing, molecular replacement, the efficiency of crystallographic refinement, thermal motion, disorder of macromolecules, and bulk solvent. The past year has seen a number of developments in these areas which were made possible by the availability of increased computing power and the application of novel mathematical techniques. Current Opinion in Structural Biology 1991, 1:1016-1022
Introduction X-ray crystallography (for reviews, see [1,2]) is becoming an increasingly important tool for understanding the structure, function and control of biological macromolecules. Developments in genetics, data collection and computer hardware have produced an unprecedented growth in macromolecular crystallographic studies. X-ray crystallography produces large amounts of diffraction data whose interpretation is entirely dependent upon the availability of powerful computers and sophisticated gorithms. After crystallization and data collection, a number of computational procedures are required in order to solve and refine the structure. These procedures include methods of phasing, density modification, chain tracing, refinement and correction of errors. Many of these computational procedures can be formulated as non-linear optimization problems, i.e. one is trying to optimize a target function, usually the discrepancy between observed and computed diffraction data, as a function of certain parameters, such as phases, scale factors between structure factors, or parameters of an atomic model (for a review, see [3 o]). Current research has two primary goals: to increase the efficiency and extend the applicability of phasing and refinement procedures; and, to improve the modelling of crystal structures in order to obtain a better understanding of thermal motion, disorder of macromolecules and bulk solvent. The past year has seen a number of important developments in both areas.
reflections: yet phases are required to compute electrondensity maps by Fourier transformation of the structure factor, which is described by a set of complex numbers for each reflection. Although the phase problem has been solved in the case of small molecules (up to a few hundred atoms in the unit cell) by the use of so-called direct methods [5], the application of these methods to macromolecules has not yet succeeded and one has to resort to time-consuming and sometimes unsuccessful experimental methods. Current efforts focus on new mathematical approaches to ab initio phasing and to knowledge-based phasing. (Recent developments in experimental phasing techniques are not discussed in this review.)
Ab initio
phasing
Phasing
The past year has seen a number of interesting results in the area of maximum entropy methods [6,7]. A new multi-solution method for direct phase determination by combined maximization of entropy and likelihood has been proposed [8.] and successfully applied in the solution of hard-to-solve small-molecule crystal structures. The relationship between entropy maximization and the spin-glass problem has been discussed and it has been shown that the entropy 'landscape' in phase space contains many local minima which may correspond to fragments of a crystal structure [9°]. Very encouraging is the successful test of a maximum entropy approach to the phasing of crystals of recombinant bovine chymosin [10o], the structure of which was previously solved by molecular replacement. It remains to be seen whether or not this result represents a breakthrough that is generally applicable.
The phase problem [4] arises from the fact that a monochromatic diffraction experiment on single crystals provides only the amplitudes, but not the phases, of the
The direct-methods approach to the phase problem has recently, been formulated as an optimization problem of a minimal function which essentially consists of the
Abbreviation SA~-simulated annealing.
1016
(~) Current Biology Lid ISSN 0959-440X
Crystallographic phasing and refinement of macromolecules Bringer difference between the actual and expected values of certain structure invariants (DY Guo, HA Hauptman: abstract T02, American Crystallographic Association, 40th Anniversary Meeting, April 8-13, 1990, New Orleans, Louisiana, USA, 18:59). Clearly, powerful optimization algorithms will be needed to solve this problem for macromolecular structures. In fact, one such algorithm, simulated annealing (SA) [11], has already been applied to a number of phasing techniques during the past year. SA has been successfully used for ab initio phasing of small molecules by direct methods [12o]. The optimized target for direct methods consists of Cochran and Woolfson's [13] formula for centrosymmetric structures or the tangent formula [14] for non-centrosymmetric cases. SA was implemented as a Monte Carlo simulation in phase space. Examples showed that SA extends the applicability of direct methods to several hundred atoms as long as atomic resolution data to at least 1.1A are available [12.]. An SA-like procedure was applied to phase improvement through density modification [15"]. The objective of phase improvement is to modify a given set of phases until the electron density computed with this set of phases and observed amplitudes satisfies the nonnegativity rule [5,16]. A cycle of conventional phase improvement [17] consists of the computation of an electron-density map using the current phases and obsewed structure factor amplitudes, truncation of the negative portions of the map, and Fourier transformation of the truncated map. This yields somewhat modified phases which are inputed into the next cycle. This fixedpoint procedure converges to a certain set of phases but is subject to the multiple minimum problem. The new SA procedure includes random phase shifts as an intervening step in the iterative phase-improvement procedure in order to overcome barriers between local minima [15"]. Approximate molecular envelopes for macromolecules using low-resolution data (typically > 8 A.) have been obtained by Monte Carlo optimization [18.]. The strategy consisted of carrying out a Monte Carlo simulation of a hard-sphere gas of point scatterers where the target is given by the Patterson correlation coefficient. The number and size of the hard-sphere atoms is derived empirically from the number of atoms of the macromolecule. In the few cases studied, phase separation occurred, i.e. regions of higher and lower density emerged. It was ob served that regions of higher density corresponded to bulk-solvent portions or, altematively, the macromolecular portion of the unit cell. The method appears promising in being able to define an approximate low-resolution molecular envelope. This is a powerful constraint which can improve the phases through solvent flattening [19] or can significantly reduce the number of orientations and positions that need to be checked in Patterson searches. It should be noted that the separation of point scatterers provides no phase information p e r se, i.e. phases computed from the distribution of scatterers are random (AT Brfinger, unpublished data).
Knowledge-based phasing (molecular replacement) By Fourier transformation of the observed diffraction intensities, one obtains the autocorrelation function of the electron density, called a Patterson map. Although direct visual interpretation of the Patterson map is very difficult, the initial determination of phases by Patterson searches [20-24] is often attempted if the structure of a similar or homologous molecule ('search model') is known. Patterson searches, which are also referred to as molecular replacement, involve the placement (i.e. rotation and translation) of the search model in the unit cell of the target crystal in order to obtain the best agreement between calculated and observed diffraction data. The optimally placed search model is used to obtain initial phases for crystallographic structure fitting and refinement. The method has been applied to both small molecules and macromolecules. It was recently proposed that an intervening step might be introduced between the rotation function and the translation function [23]. The purpose of this step would be to: improve the discrimination between correct and incorrect orientations for the rotation function; and, improve the accuracy of the search model before the translation function is applied. Firstly, a conventional rotation function is evaluated. A large number of orientations corresponding to the highest peaks of the rotation function are selected. Here, the a d hoc assumption is made that the correct orientation is included within this selected subset. Secondly, a small number of parameters that describe the most dominant differences that are expected to occur between the crystal structure and the search model are introduced. Thirdly, and most importantly, refinements are made to the l~arameters against the neg ative correlation, PC, between normalized structure-factor intensities. A necessary condition for the correct solution of the crystal structure is that PC assumes a maximum. Thus, PC-refinement is used as a filter for the highest rotation-function peaks. The final step of the strategy consists of translation functions based on the PC-refined search models for this subset of orientations. The net result of the strategy is to reduce the number of possible orientations to be checked by subsequent translation functions [25"] and to improve the accuracy of the search model before the translation function is applied [25",26"]. Phases for several new crystal structures have been obtained by the generalized Patterson search strategy [27"-29"]. The new method was particularly successful for Fab fragments of antibodies. In retrospect, what "all of these cases have in common is that the inaccuracy of the model, in combination with the presence of non-crystallographic symmetry or the high crystal symmetry of the space group, made it difficult to identify the correct orientation of the molecule(s) or single domains by conventional rotation functions, whereas the new method clearly identified the correct orientation. It was recently proposed that the new method enhances the power of a rotation function by searching for all non-crystallographic syrmnetry operations simultaneously [30°]. In the case of
1017
1018
Biophysicalmethods the crystal structure of the 26-10 Fab, however, which contained a non-crystallographic twofold operation, this would not have produced the correct orientation [27°].
Crystallographic refinements The high dimensionality of the parameter space of the atomic model (typically three times the number of atoms) introduces many local minima into the target function; thus, gradient-descent methods, such as conjugate gradient minimization or least-squares methods [31], do not normally achieve shifts of atomic positions large enough to fully refine the structure. In a sense, the difficulty of refinement arises from the crystallographic phase problem. Phases for new crystal structures are usually obtained from experimental methods such as multiple isomorphous replacement [32], and such phases are, in most cases, rather imprecise. Electron-density maps computed by a combination of native crystal amplitudes and multiple isomorphous replacement phases are then sometimes insufficient to allow a complete and unambiguous tracing of the macromolecule. Furthermore, electron-density maps for macromolecules are usually obtained at lower than atomic resolution and are thus prone to human errors when interpreted [33°]. Thus, initial atomic models are likely to contain (partially) incorrect regions and require refinement with a large radius of convergence. The average phase difference between the initial and a refined atomic model can be as large as 54* [34]. Up to now, even the best protein crystal structures produce relatively large final R factors of 10-20% [35] compared with the noise in the data which is typically ,,, 5%. This is probably the result of an inadequate description of thermal motion, disorder and bulk solvent.
able to Langevin dynamics in the context of SA as the constant friction term of the latter slows the atomic motions. Slow-cooling protocols (typically a decrease in temperature of 25 K every 25 fs) produce lower R factors than faster-cooling protocols. So far, slow cooling appears to be the best compromise for crystallographic refinement in terms of generality and efficiency [44.]. The result of crystallographic refinement is not very sensitive to the accuracy of the geometric or empirical energy function employed. For instance, the root mean square deviation for backbone atoms between a structure of crambin refined by PROLSQ [37] using a geometric energy function, and a structure refined by conjugate gradient minimization at 2A resolution using X-PLOR [41] was 0.05A [41]. However, the geometry statistics (deviations of bond lengths and bond angles from ideality) are clearly dependent of the parameters of the energy function. Recently, it was suggested that small-molecule structural databases might be used to derive equilibrium geometry and force constants based on average values and their observed variability [46°].
Improved convergence
SA has greatly improved the efficiency of crystallographic refinement. However, automated algorithms alone are still insufficient to fully refine a crystal structure at present [34,43°,47]. Thus, crystallographic refinement of macromolecules frequently proceeds in a series of steps, each of which consists of a round of SA or minimization followed by re-fitting of the model structure to difference electron-density maps using interactive computer graphics. Improved graphics programs for automated building of protein models in electron-density maps and the location of errors through a real space R factor are important recent developments [48°]. Methods to improve phases and minimize model bias are necessary when interpreting electron-density maps. As was recently shown, weighting schemes based on probability relationships between structure factors can play an important role in this regard [49°]. The automatic detection and correction of errors in crystal structures is an area that will need further investigation.
Several algorithms for the refinement of macromolecular crystal structures have been developed over the past 20 years [14]. These algorithms can be generally classified into constrained or restrained least-squares optimization [36-38], conjugate gradient minimization [39,40] and SA refinement [41]. The two implementations of SA refinement reported make use of molecular dynamics [41,42]. The first implementation [ 41 ] involved heating and cooling the system whereas the second implementation [42] maintained a constant temperature while modifying the relative weighting between diffraction data and chemical restraints. Both methods can be viewed as spedal cases of more general annealing schedules [43°]. The influence of the temperature-variation method, weighting of the energy terms, cooling rate, and duration of the heating stage on the success of SA refinement has been studied [44°]. It was concluded that a temperature-coupling method [45] is preferable to velocity scaling as the latter sometimes causes large temperature fluctuations during high-temperature molecular dynamics. It is also prefer-
Water constitutes a large portion of the volume in water-soluble macromolecule crystals [50]. The macromolecule is surrounded by a tighdy bound layer of water molecules; the remaining bulk water is disordered, and its contribution in refinement procedures is usually neglected or approximated by a 'flat' solvent model. Structure factors of the flat solvent model have been computed using Babinet's principle [51,52] or by construction of a solvent mask [53]. This produces discrepancies between observed and calculated structure factors, especially at low and medium resolution. Recently, a more detailed description of the solvent has been achieved by dividing the solvent volume into shells extending outward from the surface of the protein and by the refinement of two parameters for each shell, the solvent-scattering density and an isotropic temperature factor [54°]. The method is implemented on a finely spaced lattice of scat-
Bulk solvent
Crystallographic phasing and refinement of macromolecules BrOnger 1019 tering centers and produces a radial.distribution function of water shells around protein crystals. Two hydration layers in myoglobin crystals were observed [54"]. In order to obtain more detailed information about the threedimensional structure of bulk solvent, an iterative density modification method has been proposed [55"]. Both a real-space restraint and a reciprocal restraint were applied, the former consisting of the requirement that the protein density should remain unchanged, the latter of the requirement that observed and calculated structure factors should agree as closely as possible. Fluctuations in solvent density in cubic insulin crystals were observed, demonstrating non-random arrangements of water molecules extending several layers from the first solvation shell of the protein [55"]. The significance of these results will need further investigation, i.e. a criterium is needed which determines whether better agreement with the diffraction data results from a better physical model or from fitting noise in the data.
Thermal motion and disorder The flexibility of macromolecular structure results in thermal motion and disorder which should be accounted for during refinement [56]. The customary use of isotropic temperature factors and a single molecular conformation provides only a poor description of these phenomena. As a result of the adverse atomic parameter to observed diffraction data ratio typical for macromolecular diffraction data, anisotropic temperature factors normally cannot be used. Thus, the goal is to describe thermal motion and disorder with a minimal set of additional parameters. Conformational disorder has been studied by including two independent structures ('twins') with isotropic temperature factors in least-squares optimization [57"]. For several test cases, a reduction in the R factor was achieved by slight displacement of the corresponding atoms in the twin structure. The displacements were correlated with anisotropic temperature factors in the case of crambin for which 0.945-~, resolution data allows anisotropic temperature factor refinement. By repeated application, the method automatically identified side chains with two or more altemative conformations. Another approach makes use of a minimal set of low-fiequency normal modes for modelling temperature factors [58",59"]. Refinement is carried out for amplitude coefficients for the normal modes [58"] or, more generally, for the variances and covariances o f the normal modes [59"]. In the case of a crystal structure of bovine pancreatic trypsin inhibitor, the refinement of the amplitude coefficients for only 19 normal modes derived from a molecular mechanics computation [60] was sulficient to reproduce a description obtained from 892 isotropic temperature factors. The rigid-body translational and vibrational modes provided the largest contribution to the temperature factors. This is corroborated by the observation that
the simple TIS model [61] for the oscillations of rigid protein molecules is successful in qualitatively reproducing the isotropic temperature factor profiles of a wide range of proteins ranging in size from lysozyme to influenza virus hemagglutinin [62.]. For more detailed descriptions, a segmented TLS refinement has been suggested [63]. Although these results are encouraging, a clear rule has yet to be established that determines what type of temperature-factor refinement is most optimal for a given resolution range. Diffraction data reflect a time average over many possible conformations of the crystallized molecule. It has been suggested that a time-average restraint or memory term should be used that incorporates previously encountered conformations of the macromolecule during a long molecular dynamics simulation [64.]. This method yields an ensemble of structures in which many possible thermal motions are allowed, including anisotropic and anharmonic motions. These calculations involve the introduction of a large and uncertain number of additional free parameters and, thus, the significance of the reduction in R factor achievable by this method needs to be studied further.
Crystal packing Conformational flexibility of macromolecules and crystal packing may account for differences between solution and crystal conformations. A particularly striking example has recently been obtained in a mutant T4 lysozyme crystal that crystallized with four independent molecules in the crystal [65.]. The four molecules have distinctly different hinge-bending angles between the two lysozyme domains (between 9.1 and 32.3°).
Conclusion The availability of high-performance computers has opened the door for 'experimentation' based on new mathematical techniques, such as maximum entropy [6,7] or SA [41]. The applications of these modern techniques are not restricted to crystalline substances. For example, the maximum entropy principle has been successfully applied to powder diffraction data [66.]. Diffraction data of a S i O 4 glasS were described by a Monte Carlo simulation [67"]. There are few other examples in physical chemistry where experiment and computation are so tightly interwoven as in crystallography. We can expect that, with the availability of ever increasing computing power and the application of novel mathematical algorithms, significant advances in solving the crystallographic phase problem and important insights into the physical character of crystallized macromolecules will be achieved in the near future.
10z0 Biophysicalmethods References and recommended reading
This paper describes how a metropolis-like algorithm can improve the method of density modification.
Papers of special interest, published within the annual period of review, have been highlighted as: • of~ interest •• of outstanding interest
16
SAYRED: The Squaring Method: a New Method for Phase Determination. Acta Crystallogr 1952, 5:60--65.
17.
PODJARNYAn, BHATTN, ZWICKM: Imptovinlg CrystallogtaDhic Macromolecular Images. The Real-space Approach. Annu Rev BR~Ws B~0hys Chem 1987, 16:351-373.
1.
2.
WYCKOFFHW, HIRS CHW, TIMASHEFF SN (EDS): Diffraction Methods for Biological Macromolecules, Part A. Methods Enzymo/ 1985, 114. WYCKOFFHW, Hms CHW, TIMASHEFF SN (El)S): Diffraction Methods for Biological Macromolecules. Methods Enzymol 1985, 115.
3. BRONGERAT: Simulated Annealing in Crystallography. Annu • Rev Phys O0em 1991, 42:197-223. A comprehensive review of recent applications of SA to phasing and refinement procedures.
18. •
SUBB~ S: Low-resolution Real-space Envelopes: an Approach to the ab initto Macromolecular Phase Problem. Science 1991, 252:128--133. A novel approach to low-resolution phasing. A random gas of hardsphere point scatterers is allowed to condense under the constraint of solvent fraction and the restraint of the observed Fourier amplitude data. 19.
WANGB-C: Resolution of Phase Ambiguity in Maeromolecular Crystallograph),. Methods Enzymol 1985, 115:90--112.
20.
HOPPE W: Die Faltmoleldilmethode - - eine Neue Methode zur Bestimmung der Kristallstruktur bei Ganz oder Teilwelse Bekannter Moleldilstn~tur. Acta Crystallogr 1957, 10:750-751.
21.
ROSSMANMG, BLOW DM: The Detection of Sub-units Within the Crystallographic Asymmetric Unit. Acta Crystallogr [,4] 1962, 15:24-31.
4.
HAUPTMANHA: The Phase Problem o f X-ray Crystallography. Phys Today 1989, 42:24-29.
5.
KARIE J, HAUPTMAN H: The Phases and Magnitudes of the Structure Factors. Acta Crystallogr 1950, 3:181-187.
6.
BmCOGNEG: Maximum Entropy and the Foundation of Direct Methods. Acta Crystallogr [A] 1984, 40:410-445.
22.
COLLINSDM: Electron Density Images from Imperfect Data by Iterative Entropy Maximization. Nature 1982, 298:49-51.
HUBER~ Die Automatisierte Faltmolekiiimethode. Acta Cry~ tal~gr [,4] 1965, 19:353-356.
23.
ROSSMANMG (ED): The Molecular Replacement MethO~ International Science Review No. 13 [book]. New York: Gordon and Breach, 1972.
24.
LATrMANEE: Use of the Rotation and Translation Function. Methods Enzymol 1985, 115:55-77.
7.
8. ,
BPaCOGNEG, GILMORECJ: A Multiisolation Method of Phase Determination by Combined Maximization of Entropy and Likelihood. I. Theory, Al~gorithms and Strategy. Acta Crystallogr [A] 1990, 46:284-297. An organized search for combinations of phase associated with a basis set of reflections that have maximum likelihood is described. Likelihood is defined as the conditional probability of the observed amplitude belonging to reflections outside the basis set. 9. VENTKATESANR: The Phase Problem and its Relation to the • Spin-glass Problem Acta Crystallogr [A] 1991, 47:400-404. An analogy between the phase problem and the spin-glass problem of condensed matter physics is discussed. The entropy surface in the space of unknown phases is studied.
SJOLINL, PRINCEE, SVENSSONLA, GILLIIANDGL: Ab tnitto Phase Determination for X-ray Diffraction Data from Crystals of a Native Protein. Acta Crystallogr [A] 1991, 47:216-223. A systematic procedure for testing the signs of centric reflections, using the total entropy of the map as a figure of merit, is used to produce a low-resolution map of recombinant bovine chymosin. The phases of acentric and additional centric reflectiofis are then chosen by adding them to the map with various possible phases and computing the total entropy. This is the first successful application of maximum entropy phasing to a molecule of this size. 10. •
11.
KIRKPATRICKS, GEIATr CD JR, VECCHI MP: Optimization by Simulated Annealing. Science 1983, 220:671q~80.
12. •
SHELDRICKGM: Phase Annealing in SHELX-90: Direct Methotis for Larger Structures. Acta Crystallogr [A] 1990, 46:467-473, This paper describes the first application of simulated annealing to direct methods. 13.
COCHRANW, WOOLFSON MM: The Theory of Sign Relations Between Structure Factors. Acta CrystaUogr 1955, 8:1-12.
14.
STOUTGH, JENSENLH: X-ray Structure Determination, A Practical Guide. New York: John Wiley and Sons, 1989.
15. •
BHATTN: A Metropolis-like Algorithm to Improve Phases: some Preliminary Results. Acta Crystallogr [A] 1990, 46:739-742.
25. •
BRI~NGERAT: Extension of Molecular Replacement: a New Search Strategy Based on Patterson Correlation Refinement. Acta Crystallogr [A] 1990, 46:46-57. This paper describes the methodology of generalized molecular replacement through PC refinement. 26. •
YEATESTO, RINI JM: Intensity-based Domain Refinement of Orientated but Unpositioned Molecular Refinement of Molecular Replacement Models. Acta Crystallogr [A] 1990, 46:352-359. These authors independently discovered the usefulness of PC-refinement to improve the translation function. 27. •
BRf3NGERAT: Solution of a Fab (26-10)/Di4goxin Complex by Generalized Molecular Replacement. Acta Crystallogr [,4] 1991, 47:195-204. Another application of generalized molecular replacement through PC refinement. Two molecules in the asymmetric unit and an elbow-angle difference of ~ 10° between the search model and the final structure made the solution dil/icult. 28. •
BRfdNGERAT, MILBURN MV, TONG I, DE VOS AM, JANCARIKJ, YAMAIZUMI Z, NISHIMURAS, OHTSUKAE, KIM S-H: Crystal Structure of an Active Form of Ras Protein, a Complex of GTP Analog and c-H-Ras P21 Catalytic Domain. Proc Naa Acad Sci USA 1990, 87:4849-4853. One of the first applications of generalized molecular replacement through PC refinement for a structure other than a Fab. The challenge in this case consists of an approximate fourfold non-crystallographic symmetry which reduces the significance of conventional rotation functions. Here, PC refinement is used to refine the overall orientation of a single monomer. 29. .
BRONGERAT, LEAHYDJ, HYNES TR, FOX RO: The 2.9A Re.solution Structure o f an Anti-dinitrophenyl-spin-label Monoclonal Antibody Fab Fragment with Bound Hapten. J Mo/ BR~/1991, 221:239-256. One of the first applications of generalized molecular replacement through PC refinement. This technique is essential in obtaining the correct orientation and position of the Fab. The elbow-angle changed by 8 ° between the initial search model and the final refined structure.
Crystallographic phasing and refinement of macromolecules TONGL, ROSSMANNMG: The Locked Rotation Function. Acta Crystallogr [A] 1990, 46:783-792. A new rotation function is presented where all non-crystallographic symmetry operators are searched simultaneously. 30.
•
31.
PRESSWH, FLANNERYBP, TEUKOLOSKYSA: Numerical Rec~es [book]. Cambridge: Cambridge University Press, 1986.
32.
BLOW DM, CRICK FHC: The T r e a t m e n t of Errors in the Isomorphous Replacement Method. Acta CrystaUogr 1959, 12:794-802.
33.
BRANDENC-l, JONES TA: Between Objectivity and Subjectivity. Nature 1990, 343:687Jo89. An overview of the current state of crystallographic structure determination.
•
34.
BRUNGERAT: Crystallographic Refinement by Simulated Annealing: Application to a 2.8A Resolution Structure of Aspartate Aminotransferase. J Mol Biol 1988, 203:803-816.
35.
JENSEN LH: Overview of Refinement in Macromolecular Structure Analysis. Methods Enzymol 1985, 115:227-234.
36.
SUSSMANJL, HOLBROOKSR, CHURCH GM, KIM SH: A Structurefactor Least-squares Refinement Procedure for Macromolecular Structure using Constrained and Restrained Parameters. Acta Crystallogr [,4] 1977, 33:800-804.
37.
KONNERT JH, HENDRICKSON WA~ A Restrained-parameter Thermal-factor Refinement Procedure. Acta Crystallogr [A] 1980, 36:344-349.
38.
HENDRICKSON WA: Stereochemically Restrained Refinement of Macromolecular Structures. Methods Enzymol 1985, 11:252-270.
Brfinger
Bond-length and bond-angle parameters are derived from a statistical survey of X-ray structures from the Cambridge Structural Database. 47.
KURIYANJ, BRONGER AT, KARPLUSM, HENDR1CKSONWA: X-ray Refinement of Protein Structures by Simulated Annealing: Test of the Method o n Myohemerythrin. Acta Crystallogr [A] 1989, 45:396-409.
48. •
JONES TA, ZOU J-Y, COWAN S&V, KJELDGAARD M: Improved Methods for Building Protein Models in Electron Density Maps and the Location of Errors in These Models. Acta Crystallogr [,4] 1991, 47:110-119. Strategies and computational tools are described that help to detect errors in crystal structures. Of particular interest is the definition of a real space R factor. 49. READ RJ: Structure Factor Probabilities for Related Struc• tures. Acta Crystallogr [,4] 1990, 46:900-912. A revised theory of probability relationships between structure factors is presented that has implications for computing more accurate phrases and refining structures. 50.
MATrHEWSBW: Solvent C o n t e n t of Protein Crystals. J .'viol Biol 1968, 33:491-497.
51.
FRASERRDB, MACRAE TP, SUZUKI E: An Improved Method for Calculating the Contribution of Solvent to the X-ray Diffraction Pattern of Biological Molecules. Appl Co,stallogr 1978, 11:693-694.
52.
MOEWSPC, KRETS1NGERRH: Refinement of the Structure of the Carp Muscle Calcium-binding Parvalbumin by Model Building and Difference Fourier Analysis. J Mol Biol 1975, 91:201-228.
53.
PHILLIPSSEV: Structure and Refinement of Oxymyoglobin at 1.6A Resolution. J Mol Biol 1980, 142:531-554.
54.
CHENGX, SCHOENBORNBP: Hydration in Protein Crystals. A
•
Neutron Diffraction Analysis of Carbonmonoxymyoglobin.
Acta Crystallogr [B] 1990, 46:195-208. 39.
JACK A, LEVrlT M: Refinement of Large Structures by Simultan e o u s Minimization of Energy and R Factor. Acta Crystallogr [A] 1978, 34:931-935.
40.
TRONRUD DE, TEN EYCK LF, MATFHEWS BW: An Efficient G e n e r a l - p u ~ Least-squares Refinement Program for Macromolecular Structures. Acta Crystallogr [A] 1987, 43:489-500.
41.
BRONGER AT, KUPaYAN J, KARPLUS M: Crystallographic R Factor Refinement by Molecular Dynamics. Science 1987, 235:458-460.
42.
FUJINAGAM, GROS P, VAN GUNSTERENWF: Testing t h e Method of Crystallographic Refinement Using Molecular Dynamics. J Appl Crystallogr 1989, 22:1-8.
43. .
CHANG XZ-B, CHANG C-H, SCHIFFER M: Testing t h e Procedure of Simulated Annealing by Refining Homologous Immunoglobulin Light-chain Dimers. Protein Eng 1990, 3:583-589. Using SA, the structure of one immunoglobulin light-chain dimer is refined against 2.8A diffraction data collected for a homologous lightchain dimer. 44. •
BRONGERAT, KRUKOWSKIA, ERICKSONJ: Slow-cooling Protocols for Crystallographic Refinement by Simulated Annealin_g. A a a Crystallogr [A] 1990, 46:585-593. This paper describes experiences with SA refinement. The best protocol appears to be of the slow-cooling type. 45.
46. •
BERENDSEN HJC, POSTMAJPM, VAN GUNSTERENXVF, DI-NOLAA, JI~ Molecular Dynamics w i t h Coupling to an External Bath. J Chem Phys 1984, 81:3684-3690. ENGH RA, HUBER R: Accurate Bond and Angle Parameters for X-ray Structure Refinement. Acta Crystallogr [A] 1991, 47:392-400.
The contribution of solvent to the low-order reflections is evaluated by dividing the solvent volume into shells extending outward from the surface of the protein. The inclusion of the solvent contribution allows the localization of 87 bound water molecules. 55.
BADGERJ, CASPAR DLD: Wa~er Structure in Cubic Insulin Crystals. Proc Natl Acad Sci USA 1991, 88:622~626. The electron-density distribution of the solvent in the cubic insulin crys tal structure, which occupies 65% of the volume, is mapped from 1.TA reg)lution diffraction data by an iterative difference Fourier and density modification method, keeping the previously determined protein structure as a refinement restraint. 56.
KURIYANJ, PETSKO GA, LEVY RM, KARPLUS M: Effect of Anisotropy and Anharmonicity on Protein Crystallographic Refinement. J Mol Biol 1986, 190:227-254.
57.
KURIYANJ, OSAPAY K, BURLEWSK, BR(INGER AT, HENDRICKSON WA, KARPLUSM: Exploration of Disorder in Protein Structure by X-ray Restrained Molecular Dynamics. Proteins 1991, 10:340-358. Conformational disorder in crystal structures of rilx)nuclease A and crambin is studied by including two independent structures in leastsquares optimization against X-ray data. •
58. •
DIAMONDR: O n the Use of Normal Modes in Thermal Parameter Refinement: Theory and Application to the Bovine Pancreatic Trypsin Inhibitor. Acta Crystallogr [.4] 1990, 46:425435. The description of molecular motion furnished by 892 isotropic temperature factors of bovine pancreatic trypsin inhibitor is largely reproduced using only 19 parameters that represent the amplitude coefficients of molecular normal modes of vibration. 59. •
KIDERAA, NOBUHIRO G: Refinement of Protein Dynamic Structure: Normal Mode Refinement. Proc Natl Acad Sci USA 1990, 87:317-322. The temperature factor is expanded in terms of effective normal modes. The variances and covariances of these effective normal modes are treated as parameters.
1021
1022
Biophysical methods 60.
ILcVlTFM, SANDERC, STERN PS: Protein Normal-mode Dynamics: TrypsIn Inhibitor, Crambin, Ribonuclease and Lysozyme. j Mo/Bto/1985, 181:423---447.
61.
SCHOMAKERV, TRUEBLOODKN: On the Rigid-body Motion of Molecules in Crystals. Acta Crystallogr [B] 1968, 24:63-76.
62. •
KORIYANJ, WEIS WI: Rigid ProteIn Motion as a Model for Crystallographic Temperature Factors. Proc Natl Acad Sci USA 1991, 88:2773-2777. The simple 10-parameter TLS model is shown to qualitatively reproduce the maxima and minima of isotropic backbone mean-square displacements. For the dimers, better results are obtained by treating the monomers as independent rigid bodies. 63.
65. .
FABER HR, MATrHEWS BW: A Mutant T4 Lysozyme Displays Five Different Crystal Conformations. Nature 1990, 348:263-266. A striking example of molecular flexibility and the effect of crystal packing on protein conformation. 66. DAVIDWIF: Extending the Power of Powder Diffraction for • Structure Determination. Nature 1990, 346:731-734. A novel application of maximum entropy methods to powder diffraction. The algorithm evaluates, from first principles, the intensities of overlapping reflections in powder diffraction profiles, 67. •
KEENDA, MCGEEVYRL: Structural Modelling o f Glasses Using Reverse Carlo Simulation. Nature 1990, 344:423-425.
HOWL~ B, MOSS DS, HARMS GW: Segmented Anisotropic Refinement o f Bovine Ribonuclease A by the Application of the Rigid-body TLS ModeL Acta Crystallogr [A] 1989, 45:851-861.
An interesting application of Monte Carlo optimization to model the diffraction data of #asses,
GROS P, VAN GUNSTERENNX/F,HOL WGJ: Inclusion of Thermal Motion in Crystallographic Structures by Restrained Molecular Dynamics. Science 1990, 249:1149--1152. A molecular dynamics method is described which makes use of a timeaveraged crystallographic restraint. The method produces an ensemble of structures in which all possible thermal motions are allowed.
AT Briinger, The Howard Hughes Medical Institute and Department of Molecular Biophysics and Biochemistry, Yale University, New Haven, Connecticut 06511, USA.
64. •
Introduction X-ray crystallography (for reviews, see [1,2]) is becoming an increasingly important tool for understanding the structure, function and control of biological macromolecules. Developments in genetics, data collection and computer hardware have produced an unprecedented growth in macromolecular crystallographic studies. X-ray crystallography produces large amounts of diffraction data whose interpretation is entirely dependent upon the availability of powerful computers and sophisticated gorithms. After crystallization and data collection, a number of computational procedures are required in order to solve and refine the structure. These procedures include methods of phasing, density modification, chain tracing, refinement and correction of errors. Many of these computational procedures can be formulated as non-linear optimization problems, i.e. one is trying to optimize a target function, usually the discrepancy between observed and computed diffraction data, as a function of certain parameters, such as phases, scale factors between structure factors, or parameters of an atomic model (for a review, see [3 o]). Current research has two primary goals: to increase the efficiency and extend the applicability of phasing and refinement procedures; and, to improve the modelling of crystal structures in order to obtain a better understanding of thermal motion, disorder of macromolecules and bulk solvent. The past year has seen a number of important developments in both areas.
reflections: yet phases are required to compute electrondensity maps by Fourier transformation of the structure factor, which is described by a set of complex numbers for each reflection. Although the phase problem has been solved in the case of small molecules (up to a few hundred atoms in the unit cell) by the use of so-called direct methods [5], the application of these methods to macromolecules has not yet succeeded and one has to resort to time-consuming and sometimes unsuccessful experimental methods. Current efforts focus on new mathematical approaches to ab initio phasing and to knowledge-based phasing. (Recent developments in experimental phasing techniques are not discussed in this review.)
Ab initio
phasing
Phasing
The past year has seen a number of interesting results in the area of maximum entropy methods [6,7]. A new multi-solution method for direct phase determination by combined maximization of entropy and likelihood has been proposed [8.] and successfully applied in the solution of hard-to-solve small-molecule crystal structures. The relationship between entropy maximization and the spin-glass problem has been discussed and it has been shown that the entropy 'landscape' in phase space contains many local minima which may correspond to fragments of a crystal structure [9°]. Very encouraging is the successful test of a maximum entropy approach to the phasing of crystals of recombinant bovine chymosin [10o], the structure of which was previously solved by molecular replacement. It remains to be seen whether or not this result represents a breakthrough that is generally applicable.
The phase problem [4] arises from the fact that a monochromatic diffraction experiment on single crystals provides only the amplitudes, but not the phases, of the
The direct-methods approach to the phase problem has recently, been formulated as an optimization problem of a minimal function which essentially consists of the
Abbreviation SA~-simulated annealing.
1016
(~) Current Biology Lid ISSN 0959-440X
Crystallographic phasing and refinement of macromolecules Bringer difference between the actual and expected values of certain structure invariants (DY Guo, HA Hauptman: abstract T02, American Crystallographic Association, 40th Anniversary Meeting, April 8-13, 1990, New Orleans, Louisiana, USA, 18:59). Clearly, powerful optimization algorithms will be needed to solve this problem for macromolecular structures. In fact, one such algorithm, simulated annealing (SA) [11], has already been applied to a number of phasing techniques during the past year. SA has been successfully used for ab initio phasing of small molecules by direct methods [12o]. The optimized target for direct methods consists of Cochran and Woolfson's [13] formula for centrosymmetric structures or the tangent formula [14] for non-centrosymmetric cases. SA was implemented as a Monte Carlo simulation in phase space. Examples showed that SA extends the applicability of direct methods to several hundred atoms as long as atomic resolution data to at least 1.1A are available [12.]. An SA-like procedure was applied to phase improvement through density modification [15"]. The objective of phase improvement is to modify a given set of phases until the electron density computed with this set of phases and observed amplitudes satisfies the nonnegativity rule [5,16]. A cycle of conventional phase improvement [17] consists of the computation of an electron-density map using the current phases and obsewed structure factor amplitudes, truncation of the negative portions of the map, and Fourier transformation of the truncated map. This yields somewhat modified phases which are inputed into the next cycle. This fixedpoint procedure converges to a certain set of phases but is subject to the multiple minimum problem. The new SA procedure includes random phase shifts as an intervening step in the iterative phase-improvement procedure in order to overcome barriers between local minima [15"]. Approximate molecular envelopes for macromolecules using low-resolution data (typically > 8 A.) have been obtained by Monte Carlo optimization [18.]. The strategy consisted of carrying out a Monte Carlo simulation of a hard-sphere gas of point scatterers where the target is given by the Patterson correlation coefficient. The number and size of the hard-sphere atoms is derived empirically from the number of atoms of the macromolecule. In the few cases studied, phase separation occurred, i.e. regions of higher and lower density emerged. It was ob served that regions of higher density corresponded to bulk-solvent portions or, altematively, the macromolecular portion of the unit cell. The method appears promising in being able to define an approximate low-resolution molecular envelope. This is a powerful constraint which can improve the phases through solvent flattening [19] or can significantly reduce the number of orientations and positions that need to be checked in Patterson searches. It should be noted that the separation of point scatterers provides no phase information p e r se, i.e. phases computed from the distribution of scatterers are random (AT Brfinger, unpublished data).
Knowledge-based phasing (molecular replacement) By Fourier transformation of the observed diffraction intensities, one obtains the autocorrelation function of the electron density, called a Patterson map. Although direct visual interpretation of the Patterson map is very difficult, the initial determination of phases by Patterson searches [20-24] is often attempted if the structure of a similar or homologous molecule ('search model') is known. Patterson searches, which are also referred to as molecular replacement, involve the placement (i.e. rotation and translation) of the search model in the unit cell of the target crystal in order to obtain the best agreement between calculated and observed diffraction data. The optimally placed search model is used to obtain initial phases for crystallographic structure fitting and refinement. The method has been applied to both small molecules and macromolecules. It was recently proposed that an intervening step might be introduced between the rotation function and the translation function [23]. The purpose of this step would be to: improve the discrimination between correct and incorrect orientations for the rotation function; and, improve the accuracy of the search model before the translation function is applied. Firstly, a conventional rotation function is evaluated. A large number of orientations corresponding to the highest peaks of the rotation function are selected. Here, the a d hoc assumption is made that the correct orientation is included within this selected subset. Secondly, a small number of parameters that describe the most dominant differences that are expected to occur between the crystal structure and the search model are introduced. Thirdly, and most importantly, refinements are made to the l~arameters against the neg ative correlation, PC, between normalized structure-factor intensities. A necessary condition for the correct solution of the crystal structure is that PC assumes a maximum. Thus, PC-refinement is used as a filter for the highest rotation-function peaks. The final step of the strategy consists of translation functions based on the PC-refined search models for this subset of orientations. The net result of the strategy is to reduce the number of possible orientations to be checked by subsequent translation functions [25"] and to improve the accuracy of the search model before the translation function is applied [25",26"]. Phases for several new crystal structures have been obtained by the generalized Patterson search strategy [27"-29"]. The new method was particularly successful for Fab fragments of antibodies. In retrospect, what "all of these cases have in common is that the inaccuracy of the model, in combination with the presence of non-crystallographic symmetry or the high crystal symmetry of the space group, made it difficult to identify the correct orientation of the molecule(s) or single domains by conventional rotation functions, whereas the new method clearly identified the correct orientation. It was recently proposed that the new method enhances the power of a rotation function by searching for all non-crystallographic syrmnetry operations simultaneously [30°]. In the case of
1017
1018
Biophysicalmethods the crystal structure of the 26-10 Fab, however, which contained a non-crystallographic twofold operation, this would not have produced the correct orientation [27°].
Crystallographic refinements The high dimensionality of the parameter space of the atomic model (typically three times the number of atoms) introduces many local minima into the target function; thus, gradient-descent methods, such as conjugate gradient minimization or least-squares methods [31], do not normally achieve shifts of atomic positions large enough to fully refine the structure. In a sense, the difficulty of refinement arises from the crystallographic phase problem. Phases for new crystal structures are usually obtained from experimental methods such as multiple isomorphous replacement [32], and such phases are, in most cases, rather imprecise. Electron-density maps computed by a combination of native crystal amplitudes and multiple isomorphous replacement phases are then sometimes insufficient to allow a complete and unambiguous tracing of the macromolecule. Furthermore, electron-density maps for macromolecules are usually obtained at lower than atomic resolution and are thus prone to human errors when interpreted [33°]. Thus, initial atomic models are likely to contain (partially) incorrect regions and require refinement with a large radius of convergence. The average phase difference between the initial and a refined atomic model can be as large as 54* [34]. Up to now, even the best protein crystal structures produce relatively large final R factors of 10-20% [35] compared with the noise in the data which is typically ,,, 5%. This is probably the result of an inadequate description of thermal motion, disorder and bulk solvent.
able to Langevin dynamics in the context of SA as the constant friction term of the latter slows the atomic motions. Slow-cooling protocols (typically a decrease in temperature of 25 K every 25 fs) produce lower R factors than faster-cooling protocols. So far, slow cooling appears to be the best compromise for crystallographic refinement in terms of generality and efficiency [44.]. The result of crystallographic refinement is not very sensitive to the accuracy of the geometric or empirical energy function employed. For instance, the root mean square deviation for backbone atoms between a structure of crambin refined by PROLSQ [37] using a geometric energy function, and a structure refined by conjugate gradient minimization at 2A resolution using X-PLOR [41] was 0.05A [41]. However, the geometry statistics (deviations of bond lengths and bond angles from ideality) are clearly dependent of the parameters of the energy function. Recently, it was suggested that small-molecule structural databases might be used to derive equilibrium geometry and force constants based on average values and their observed variability [46°].
Improved convergence
SA has greatly improved the efficiency of crystallographic refinement. However, automated algorithms alone are still insufficient to fully refine a crystal structure at present [34,43°,47]. Thus, crystallographic refinement of macromolecules frequently proceeds in a series of steps, each of which consists of a round of SA or minimization followed by re-fitting of the model structure to difference electron-density maps using interactive computer graphics. Improved graphics programs for automated building of protein models in electron-density maps and the location of errors through a real space R factor are important recent developments [48°]. Methods to improve phases and minimize model bias are necessary when interpreting electron-density maps. As was recently shown, weighting schemes based on probability relationships between structure factors can play an important role in this regard [49°]. The automatic detection and correction of errors in crystal structures is an area that will need further investigation.
Several algorithms for the refinement of macromolecular crystal structures have been developed over the past 20 years [14]. These algorithms can be generally classified into constrained or restrained least-squares optimization [36-38], conjugate gradient minimization [39,40] and SA refinement [41]. The two implementations of SA refinement reported make use of molecular dynamics [41,42]. The first implementation [ 41 ] involved heating and cooling the system whereas the second implementation [42] maintained a constant temperature while modifying the relative weighting between diffraction data and chemical restraints. Both methods can be viewed as spedal cases of more general annealing schedules [43°]. The influence of the temperature-variation method, weighting of the energy terms, cooling rate, and duration of the heating stage on the success of SA refinement has been studied [44°]. It was concluded that a temperature-coupling method [45] is preferable to velocity scaling as the latter sometimes causes large temperature fluctuations during high-temperature molecular dynamics. It is also prefer-
Water constitutes a large portion of the volume in water-soluble macromolecule crystals [50]. The macromolecule is surrounded by a tighdy bound layer of water molecules; the remaining bulk water is disordered, and its contribution in refinement procedures is usually neglected or approximated by a 'flat' solvent model. Structure factors of the flat solvent model have been computed using Babinet's principle [51,52] or by construction of a solvent mask [53]. This produces discrepancies between observed and calculated structure factors, especially at low and medium resolution. Recently, a more detailed description of the solvent has been achieved by dividing the solvent volume into shells extending outward from the surface of the protein and by the refinement of two parameters for each shell, the solvent-scattering density and an isotropic temperature factor [54°]. The method is implemented on a finely spaced lattice of scat-
Bulk solvent
Crystallographic phasing and refinement of macromolecules BrOnger 1019 tering centers and produces a radial.distribution function of water shells around protein crystals. Two hydration layers in myoglobin crystals were observed [54"]. In order to obtain more detailed information about the threedimensional structure of bulk solvent, an iterative density modification method has been proposed [55"]. Both a real-space restraint and a reciprocal restraint were applied, the former consisting of the requirement that the protein density should remain unchanged, the latter of the requirement that observed and calculated structure factors should agree as closely as possible. Fluctuations in solvent density in cubic insulin crystals were observed, demonstrating non-random arrangements of water molecules extending several layers from the first solvation shell of the protein [55"]. The significance of these results will need further investigation, i.e. a criterium is needed which determines whether better agreement with the diffraction data results from a better physical model or from fitting noise in the data.
Thermal motion and disorder The flexibility of macromolecular structure results in thermal motion and disorder which should be accounted for during refinement [56]. The customary use of isotropic temperature factors and a single molecular conformation provides only a poor description of these phenomena. As a result of the adverse atomic parameter to observed diffraction data ratio typical for macromolecular diffraction data, anisotropic temperature factors normally cannot be used. Thus, the goal is to describe thermal motion and disorder with a minimal set of additional parameters. Conformational disorder has been studied by including two independent structures ('twins') with isotropic temperature factors in least-squares optimization [57"]. For several test cases, a reduction in the R factor was achieved by slight displacement of the corresponding atoms in the twin structure. The displacements were correlated with anisotropic temperature factors in the case of crambin for which 0.945-~, resolution data allows anisotropic temperature factor refinement. By repeated application, the method automatically identified side chains with two or more altemative conformations. Another approach makes use of a minimal set of low-fiequency normal modes for modelling temperature factors [58",59"]. Refinement is carried out for amplitude coefficients for the normal modes [58"] or, more generally, for the variances and covariances o f the normal modes [59"]. In the case of a crystal structure of bovine pancreatic trypsin inhibitor, the refinement of the amplitude coefficients for only 19 normal modes derived from a molecular mechanics computation [60] was sulficient to reproduce a description obtained from 892 isotropic temperature factors. The rigid-body translational and vibrational modes provided the largest contribution to the temperature factors. This is corroborated by the observation that
the simple TIS model [61] for the oscillations of rigid protein molecules is successful in qualitatively reproducing the isotropic temperature factor profiles of a wide range of proteins ranging in size from lysozyme to influenza virus hemagglutinin [62.]. For more detailed descriptions, a segmented TLS refinement has been suggested [63]. Although these results are encouraging, a clear rule has yet to be established that determines what type of temperature-factor refinement is most optimal for a given resolution range. Diffraction data reflect a time average over many possible conformations of the crystallized molecule. It has been suggested that a time-average restraint or memory term should be used that incorporates previously encountered conformations of the macromolecule during a long molecular dynamics simulation [64.]. This method yields an ensemble of structures in which many possible thermal motions are allowed, including anisotropic and anharmonic motions. These calculations involve the introduction of a large and uncertain number of additional free parameters and, thus, the significance of the reduction in R factor achievable by this method needs to be studied further.
Crystal packing Conformational flexibility of macromolecules and crystal packing may account for differences between solution and crystal conformations. A particularly striking example has recently been obtained in a mutant T4 lysozyme crystal that crystallized with four independent molecules in the crystal [65.]. The four molecules have distinctly different hinge-bending angles between the two lysozyme domains (between 9.1 and 32.3°).
Conclusion The availability of high-performance computers has opened the door for 'experimentation' based on new mathematical techniques, such as maximum entropy [6,7] or SA [41]. The applications of these modern techniques are not restricted to crystalline substances. For example, the maximum entropy principle has been successfully applied to powder diffraction data [66.]. Diffraction data of a S i O 4 glasS were described by a Monte Carlo simulation [67"]. There are few other examples in physical chemistry where experiment and computation are so tightly interwoven as in crystallography. We can expect that, with the availability of ever increasing computing power and the application of novel mathematical algorithms, significant advances in solving the crystallographic phase problem and important insights into the physical character of crystallized macromolecules will be achieved in the near future.
10z0 Biophysicalmethods References and recommended reading
This paper describes how a metropolis-like algorithm can improve the method of density modification.
Papers of special interest, published within the annual period of review, have been highlighted as: • of~ interest •• of outstanding interest
16
SAYRED: The Squaring Method: a New Method for Phase Determination. Acta Crystallogr 1952, 5:60--65.
17.
PODJARNYAn, BHATTN, ZWICKM: Imptovinlg CrystallogtaDhic Macromolecular Images. The Real-space Approach. Annu Rev BR~Ws B~0hys Chem 1987, 16:351-373.
1.
2.
WYCKOFFHW, HIRS CHW, TIMASHEFF SN (EDS): Diffraction Methods for Biological Macromolecules, Part A. Methods Enzymo/ 1985, 114. WYCKOFFHW, Hms CHW, TIMASHEFF SN (El)S): Diffraction Methods for Biological Macromolecules. Methods Enzymol 1985, 115.
3. BRONGERAT: Simulated Annealing in Crystallography. Annu • Rev Phys O0em 1991, 42:197-223. A comprehensive review of recent applications of SA to phasing and refinement procedures.
18. •
SUBB~ S: Low-resolution Real-space Envelopes: an Approach to the ab initto Macromolecular Phase Problem. Science 1991, 252:128--133. A novel approach to low-resolution phasing. A random gas of hardsphere point scatterers is allowed to condense under the constraint of solvent fraction and the restraint of the observed Fourier amplitude data. 19.
WANGB-C: Resolution of Phase Ambiguity in Maeromolecular Crystallograph),. Methods Enzymol 1985, 115:90--112.
20.
HOPPE W: Die Faltmoleldilmethode - - eine Neue Methode zur Bestimmung der Kristallstruktur bei Ganz oder Teilwelse Bekannter Moleldilstn~tur. Acta Crystallogr 1957, 10:750-751.
21.
ROSSMANMG, BLOW DM: The Detection of Sub-units Within the Crystallographic Asymmetric Unit. Acta Crystallogr [,4] 1962, 15:24-31.
4.
HAUPTMANHA: The Phase Problem o f X-ray Crystallography. Phys Today 1989, 42:24-29.
5.
KARIE J, HAUPTMAN H: The Phases and Magnitudes of the Structure Factors. Acta Crystallogr 1950, 3:181-187.
6.
BmCOGNEG: Maximum Entropy and the Foundation of Direct Methods. Acta Crystallogr [A] 1984, 40:410-445.
22.
COLLINSDM: Electron Density Images from Imperfect Data by Iterative Entropy Maximization. Nature 1982, 298:49-51.
HUBER~ Die Automatisierte Faltmolekiiimethode. Acta Cry~ tal~gr [,4] 1965, 19:353-356.
23.
ROSSMANMG (ED): The Molecular Replacement MethO~ International Science Review No. 13 [book]. New York: Gordon and Breach, 1972.
24.
LATrMANEE: Use of the Rotation and Translation Function. Methods Enzymol 1985, 115:55-77.
7.
8. ,
BPaCOGNEG, GILMORECJ: A Multiisolation Method of Phase Determination by Combined Maximization of Entropy and Likelihood. I. Theory, Al~gorithms and Strategy. Acta Crystallogr [A] 1990, 46:284-297. An organized search for combinations of phase associated with a basis set of reflections that have maximum likelihood is described. Likelihood is defined as the conditional probability of the observed amplitude belonging to reflections outside the basis set. 9. VENTKATESANR: The Phase Problem and its Relation to the • Spin-glass Problem Acta Crystallogr [A] 1991, 47:400-404. An analogy between the phase problem and the spin-glass problem of condensed matter physics is discussed. The entropy surface in the space of unknown phases is studied.
SJOLINL, PRINCEE, SVENSSONLA, GILLIIANDGL: Ab tnitto Phase Determination for X-ray Diffraction Data from Crystals of a Native Protein. Acta Crystallogr [A] 1991, 47:216-223. A systematic procedure for testing the signs of centric reflections, using the total entropy of the map as a figure of merit, is used to produce a low-resolution map of recombinant bovine chymosin. The phases of acentric and additional centric reflectiofis are then chosen by adding them to the map with various possible phases and computing the total entropy. This is the first successful application of maximum entropy phasing to a molecule of this size. 10. •
11.
KIRKPATRICKS, GEIATr CD JR, VECCHI MP: Optimization by Simulated Annealing. Science 1983, 220:671q~80.
12. •
SHELDRICKGM: Phase Annealing in SHELX-90: Direct Methotis for Larger Structures. Acta Crystallogr [A] 1990, 46:467-473, This paper describes the first application of simulated annealing to direct methods. 13.
COCHRANW, WOOLFSON MM: The Theory of Sign Relations Between Structure Factors. Acta CrystaUogr 1955, 8:1-12.
14.
STOUTGH, JENSENLH: X-ray Structure Determination, A Practical Guide. New York: John Wiley and Sons, 1989.
15. •
BHATTN: A Metropolis-like Algorithm to Improve Phases: some Preliminary Results. Acta Crystallogr [A] 1990, 46:739-742.
25. •
BRI~NGERAT: Extension of Molecular Replacement: a New Search Strategy Based on Patterson Correlation Refinement. Acta Crystallogr [A] 1990, 46:46-57. This paper describes the methodology of generalized molecular replacement through PC refinement. 26. •
YEATESTO, RINI JM: Intensity-based Domain Refinement of Orientated but Unpositioned Molecular Refinement of Molecular Replacement Models. Acta Crystallogr [A] 1990, 46:352-359. These authors independently discovered the usefulness of PC-refinement to improve the translation function. 27. •
BRf3NGERAT: Solution of a Fab (26-10)/Di4goxin Complex by Generalized Molecular Replacement. Acta Crystallogr [,4] 1991, 47:195-204. Another application of generalized molecular replacement through PC refinement. Two molecules in the asymmetric unit and an elbow-angle difference of ~ 10° between the search model and the final structure made the solution dil/icult. 28. •
BRfdNGERAT, MILBURN MV, TONG I, DE VOS AM, JANCARIKJ, YAMAIZUMI Z, NISHIMURAS, OHTSUKAE, KIM S-H: Crystal Structure of an Active Form of Ras Protein, a Complex of GTP Analog and c-H-Ras P21 Catalytic Domain. Proc Naa Acad Sci USA 1990, 87:4849-4853. One of the first applications of generalized molecular replacement through PC refinement for a structure other than a Fab. The challenge in this case consists of an approximate fourfold non-crystallographic symmetry which reduces the significance of conventional rotation functions. Here, PC refinement is used to refine the overall orientation of a single monomer. 29. .
BRONGERAT, LEAHYDJ, HYNES TR, FOX RO: The 2.9A Re.solution Structure o f an Anti-dinitrophenyl-spin-label Monoclonal Antibody Fab Fragment with Bound Hapten. J Mo/ BR~/1991, 221:239-256. One of the first applications of generalized molecular replacement through PC refinement. This technique is essential in obtaining the correct orientation and position of the Fab. The elbow-angle changed by 8 ° between the initial search model and the final refined structure.
Crystallographic phasing and refinement of macromolecules TONGL, ROSSMANNMG: The Locked Rotation Function. Acta Crystallogr [A] 1990, 46:783-792. A new rotation function is presented where all non-crystallographic symmetry operators are searched simultaneously. 30.
•
31.
PRESSWH, FLANNERYBP, TEUKOLOSKYSA: Numerical Rec~es [book]. Cambridge: Cambridge University Press, 1986.
32.
BLOW DM, CRICK FHC: The T r e a t m e n t of Errors in the Isomorphous Replacement Method. Acta CrystaUogr 1959, 12:794-802.
33.
BRANDENC-l, JONES TA: Between Objectivity and Subjectivity. Nature 1990, 343:687Jo89. An overview of the current state of crystallographic structure determination.
•
34.
BRUNGERAT: Crystallographic Refinement by Simulated Annealing: Application to a 2.8A Resolution Structure of Aspartate Aminotransferase. J Mol Biol 1988, 203:803-816.
35.
JENSEN LH: Overview of Refinement in Macromolecular Structure Analysis. Methods Enzymol 1985, 115:227-234.
36.
SUSSMANJL, HOLBROOKSR, CHURCH GM, KIM SH: A Structurefactor Least-squares Refinement Procedure for Macromolecular Structure using Constrained and Restrained Parameters. Acta Crystallogr [,4] 1977, 33:800-804.
37.
KONNERT JH, HENDRICKSON WA~ A Restrained-parameter Thermal-factor Refinement Procedure. Acta Crystallogr [A] 1980, 36:344-349.
38.
HENDRICKSON WA: Stereochemically Restrained Refinement of Macromolecular Structures. Methods Enzymol 1985, 11:252-270.
Brfinger
Bond-length and bond-angle parameters are derived from a statistical survey of X-ray structures from the Cambridge Structural Database. 47.
KURIYANJ, BRONGER AT, KARPLUSM, HENDR1CKSONWA: X-ray Refinement of Protein Structures by Simulated Annealing: Test of the Method o n Myohemerythrin. Acta Crystallogr [A] 1989, 45:396-409.
48. •
JONES TA, ZOU J-Y, COWAN S&V, KJELDGAARD M: Improved Methods for Building Protein Models in Electron Density Maps and the Location of Errors in These Models. Acta Crystallogr [,4] 1991, 47:110-119. Strategies and computational tools are described that help to detect errors in crystal structures. Of particular interest is the definition of a real space R factor. 49. READ RJ: Structure Factor Probabilities for Related Struc• tures. Acta Crystallogr [,4] 1990, 46:900-912. A revised theory of probability relationships between structure factors is presented that has implications for computing more accurate phrases and refining structures. 50.
MATrHEWSBW: Solvent C o n t e n t of Protein Crystals. J .'viol Biol 1968, 33:491-497.
51.
FRASERRDB, MACRAE TP, SUZUKI E: An Improved Method for Calculating the Contribution of Solvent to the X-ray Diffraction Pattern of Biological Molecules. Appl Co,stallogr 1978, 11:693-694.
52.
MOEWSPC, KRETS1NGERRH: Refinement of the Structure of the Carp Muscle Calcium-binding Parvalbumin by Model Building and Difference Fourier Analysis. J Mol Biol 1975, 91:201-228.
53.
PHILLIPSSEV: Structure and Refinement of Oxymyoglobin at 1.6A Resolution. J Mol Biol 1980, 142:531-554.
54.
CHENGX, SCHOENBORNBP: Hydration in Protein Crystals. A
•
Neutron Diffraction Analysis of Carbonmonoxymyoglobin.
Acta Crystallogr [B] 1990, 46:195-208. 39.
JACK A, LEVrlT M: Refinement of Large Structures by Simultan e o u s Minimization of Energy and R Factor. Acta Crystallogr [A] 1978, 34:931-935.
40.
TRONRUD DE, TEN EYCK LF, MATFHEWS BW: An Efficient G e n e r a l - p u ~ Least-squares Refinement Program for Macromolecular Structures. Acta Crystallogr [A] 1987, 43:489-500.
41.
BRONGER AT, KUPaYAN J, KARPLUS M: Crystallographic R Factor Refinement by Molecular Dynamics. Science 1987, 235:458-460.
42.
FUJINAGAM, GROS P, VAN GUNSTERENWF: Testing t h e Method of Crystallographic Refinement Using Molecular Dynamics. J Appl Crystallogr 1989, 22:1-8.
43. .
CHANG XZ-B, CHANG C-H, SCHIFFER M: Testing t h e Procedure of Simulated Annealing by Refining Homologous Immunoglobulin Light-chain Dimers. Protein Eng 1990, 3:583-589. Using SA, the structure of one immunoglobulin light-chain dimer is refined against 2.8A diffraction data collected for a homologous lightchain dimer. 44. •
BRONGERAT, KRUKOWSKIA, ERICKSONJ: Slow-cooling Protocols for Crystallographic Refinement by Simulated Annealin_g. A a a Crystallogr [A] 1990, 46:585-593. This paper describes experiences with SA refinement. The best protocol appears to be of the slow-cooling type. 45.
46. •
BERENDSEN HJC, POSTMAJPM, VAN GUNSTERENXVF, DI-NOLAA, JI~ Molecular Dynamics w i t h Coupling to an External Bath. J Chem Phys 1984, 81:3684-3690. ENGH RA, HUBER R: Accurate Bond and Angle Parameters for X-ray Structure Refinement. Acta Crystallogr [A] 1991, 47:392-400.
The contribution of solvent to the low-order reflections is evaluated by dividing the solvent volume into shells extending outward from the surface of the protein. The inclusion of the solvent contribution allows the localization of 87 bound water molecules. 55.
BADGERJ, CASPAR DLD: Wa~er Structure in Cubic Insulin Crystals. Proc Natl Acad Sci USA 1991, 88:622~626. The electron-density distribution of the solvent in the cubic insulin crys tal structure, which occupies 65% of the volume, is mapped from 1.TA reg)lution diffraction data by an iterative difference Fourier and density modification method, keeping the previously determined protein structure as a refinement restraint. 56.
KURIYANJ, PETSKO GA, LEVY RM, KARPLUS M: Effect of Anisotropy and Anharmonicity on Protein Crystallographic Refinement. J Mol Biol 1986, 190:227-254.
57.
KURIYANJ, OSAPAY K, BURLEWSK, BR(INGER AT, HENDRICKSON WA, KARPLUSM: Exploration of Disorder in Protein Structure by X-ray Restrained Molecular Dynamics. Proteins 1991, 10:340-358. Conformational disorder in crystal structures of rilx)nuclease A and crambin is studied by including two independent structures in leastsquares optimization against X-ray data. •
58. •
DIAMONDR: O n the Use of Normal Modes in Thermal Parameter Refinement: Theory and Application to the Bovine Pancreatic Trypsin Inhibitor. Acta Crystallogr [.4] 1990, 46:425435. The description of molecular motion furnished by 892 isotropic temperature factors of bovine pancreatic trypsin inhibitor is largely reproduced using only 19 parameters that represent the amplitude coefficients of molecular normal modes of vibration. 59. •
KIDERAA, NOBUHIRO G: Refinement of Protein Dynamic Structure: Normal Mode Refinement. Proc Natl Acad Sci USA 1990, 87:317-322. The temperature factor is expanded in terms of effective normal modes. The variances and covariances of these effective normal modes are treated as parameters.
1021
1022
Biophysical methods 60.
ILcVlTFM, SANDERC, STERN PS: Protein Normal-mode Dynamics: TrypsIn Inhibitor, Crambin, Ribonuclease and Lysozyme. j Mo/Bto/1985, 181:423---447.
61.
SCHOMAKERV, TRUEBLOODKN: On the Rigid-body Motion of Molecules in Crystals. Acta Crystallogr [B] 1968, 24:63-76.
62. •
KORIYANJ, WEIS WI: Rigid ProteIn Motion as a Model for Crystallographic Temperature Factors. Proc Natl Acad Sci USA 1991, 88:2773-2777. The simple 10-parameter TLS model is shown to qualitatively reproduce the maxima and minima of isotropic backbone mean-square displacements. For the dimers, better results are obtained by treating the monomers as independent rigid bodies. 63.
65. .
FABER HR, MATrHEWS BW: A Mutant T4 Lysozyme Displays Five Different Crystal Conformations. Nature 1990, 348:263-266. A striking example of molecular flexibility and the effect of crystal packing on protein conformation. 66. DAVIDWIF: Extending the Power of Powder Diffraction for • Structure Determination. Nature 1990, 346:731-734. A novel application of maximum entropy methods to powder diffraction. The algorithm evaluates, from first principles, the intensities of overlapping reflections in powder diffraction profiles, 67. •
KEENDA, MCGEEVYRL: Structural Modelling o f Glasses Using Reverse Carlo Simulation. Nature 1990, 344:423-425.
HOWL~ B, MOSS DS, HARMS GW: Segmented Anisotropic Refinement o f Bovine Ribonuclease A by the Application of the Rigid-body TLS ModeL Acta Crystallogr [A] 1989, 45:851-861.
An interesting application of Monte Carlo optimization to model the diffraction data of #asses,
GROS P, VAN GUNSTERENNX/F,HOL WGJ: Inclusion of Thermal Motion in Crystallographic Structures by Restrained Molecular Dynamics. Science 1990, 249:1149--1152. A molecular dynamics method is described which makes use of a timeaveraged crystallographic restraint. The method produces an ensemble of structures in which all possible thermal motions are allowed.
AT Briinger, The Howard Hughes Medical Institute and Department of Molecular Biophysics and Biochemistry, Yale University, New Haven, Connecticut 06511, USA.
64. •