Theoretical Chemistry Accounts

, 131:1076 | Cite as

Toward ab initio refinement of protein X-ray crystal structures: interpreting and correlating structural fluctuations

  • Olle Falklöf
  • Charles A. Collyer
  • Jeffrey R. ReimersEmail author
Regular Article
Part of the following topical collections:
  1. 50th Anniversary Collection


The refinement of protein crystal structures currently involves the use of empirical restraints and force fields that are known to work well in many situations but nevertheless yield structural models with some features that are inconsistent with detailed chemical analysis and therefore warrant further improvement. Ab initio electronic structure computational methods have now advanced to the point at which they can deliver reliable results for macromolecules in realistic times using linear-scaling algorithms. The replacement of empirical force fields with ab initio methods in a final refinement stage could allow new structural features to be identified in complex structures, reduce errors and remove computational bias from structural models. In contrast to empirical approaches, ab initio refinements can only be performed on models that obey basic qualitative chemical rules, imposing constraints on the parameter space of existing refinements, and this in turn inhibits the inclusion of unlikely structural features. Here, we focus on methods for determining an appropriate ensemble of initial structural models for an ab initio X-ray refinement, modeling as an example the high-resolution single-crystal X-ray diffraction data reported for the structure of lysozyme (PDB entry “2VB1”). The AMBER force field is used in a Monte Carlo calculation to determine an ensemble of 8 structures that together embody all of the partial atomic occupancies noted in the original refinement, correlating these variations into a set of feasible chemical structures while simultaneously retaining consistency with the X-ray diffraction data. Subsequent analysis of these results strongly suggests that the occupancies in the empirically refined model are inconsistent with protein energetic considerations, thus depicting the 2VB1 structure as a deep-lying minimum in its optimized parameter space that actually embodies chemically unreasonable features. Indeed, density-functional theory calculations for one specific nitrate ion with an occupancy of 62% indicate that water replaces this ion 38% of the time, a result confirmed by subsequent crystallographic analysis. It is foreseeable that any subsequent ab initio refinement of the whole structure would need to locate a globally improved structure involving significant changes to 2VB1 which correct these identified local structural inconsistencies.


Monte Carlo Density-functional theory Protein refinement Ensemble refinement Lysozyme 



We thank Aaron McGrath for technical advice, Zbigniew Dauter from the Argonne National Laboratory Biosciences Division for providing detailed information regarding the refinement of 2VB1, the Australian Research Council for funding this research, and National Computer Infrastructure (NCI) and Australian Centre for Advanced Computing and Communications (AC3) for computing resources.


  1. 1.
    Engh RA, Huber R (1991) Acta Crystallogr A 47:392–400CrossRefGoogle Scholar
  2. 2.
    Sheldrick G, Schneider T (1997) SHELXL: high-resolution refinement. Methods Enzymol 277:319–343CrossRefGoogle Scholar
  3. 3.
    Murshudov GN, Vagin AA, Dobson EJ (1997) Refinement of macromolecular structures by the maximum-likelihood method. Acta Crystallogr D Biol Crystallogr 53:240–255CrossRefGoogle Scholar
  4. 4.
    Canfield P, Dahlbom MG, Hush N, Reimers JR (2006) Density-functional geometry optimization of the 150000-atom photosystem-I trimer. J Chem Phys 124:024301CrossRefGoogle Scholar
  5. 5.
    Kleywegt GJ (1999) Experimental assessment of differences between related protein crystal structures. Acta Crystallogr D Biol Crystallogr 55:1878–1884CrossRefGoogle Scholar
  6. 6.
    Cruickshank DWJ (1999) Remarks about protein structure precision. Acta Crystallogr D Biol Crystallogr 55:583–601CrossRefGoogle Scholar
  7. 7.
    DePristo MA, De Bakker PIW, Blundell TL (2004) Heterogeneity and inaccuracy in protein structures solved by X-ray crystallography. Structure 12:831–838CrossRefGoogle Scholar
  8. 8.
    Jaskolski M, Gilski M, Dauter Z, Wlodawer A (2007) Stereochemical restraints revisited: how accurate are refinement targets and how much should protein structures be allowed to deviate from them? Acta Crystallogr D Biol Crystallogr 63:611–620CrossRefGoogle Scholar
  9. 9.
    Chen J, Brooks CL (2007) Can molecular dynamics simulations provide high-resolution refinement of protein structure? Proteins Struct Funct Bioinform 67:922–930CrossRefGoogle Scholar
  10. 10.
    Karplus PA, Shapovalov MV, Dunbrack RL, Berkholz DS (2008) A forward-looking suggestion for resolving the stereochemical restraints debate: ideal geometry functions. Acta Crystallogr D Biol Crystallogr 64:335–336CrossRefGoogle Scholar
  11. 11.
    Rashin AA, Rashin AHL, Jernigan RL (2009) Protein flexibility: coordinate uncertainties and interpretation of structural differences. Acta Crystallogr D Biol Crystallogr 65:1140–1161CrossRefGoogle Scholar
  12. 12.
    Jaskolski M (2010) From atomic resolution to molecular giants: an overview of crystallographic studies of biological macromolecules with synchrotron radiation. Acta Physica Polonica A 117:257–263Google Scholar
  13. 13.
    Eyal E, Gerzon S, Potapov V, Edelman M, Sobolev V (2005) The limit of accuracy of protein modeling: influence of crystal packing on protein structure. J Mol Biol 351:431–442CrossRefGoogle Scholar
  14. 14.
    Konnert JH (1976) A restrained parameter structure-factor least-squares refinement procedure for large asymmetric units. Acta Crystallogr A 32:614–617CrossRefGoogle Scholar
  15. 15.
    Hendrickson WA, Konnert JH (1979) Stereochemically restrained crystallographic least-squares refinement of macromolecule structures. In: Srinivasan R (ed) Biomolecular structure, conformation, function, and evolution, vol 1. Pergamon Press, Oxford, pp 43–57Google Scholar
  16. 16.
    Konnert JH, Hendrickson WA (1980) A restrained-parameter thermal-factor refinement procedure. Acta Crystallogr A 36:344–350CrossRefGoogle Scholar
  17. 17.
    Hendrickson WA (1985) Stereochemically restrained refinement of macromolecular structures. Methods Enzymol 115:252–270CrossRefGoogle Scholar
  18. 18.
    Jack A, Levitt M (1978) Refinement of large structures by simultaneous minimization of energy and R factor. Acta Crystallogr A 34:931–935CrossRefGoogle Scholar
  19. 19.
    Brunger AT, Kuriyan J, Karplus M (1987) Crystallographic R factor refinement by molecular dynamics. Science 235:458–460CrossRefGoogle Scholar
  20. 20.
    Ohta K, Yoshioka Y, Morokuma K, Kitaura K (1983) The effective fragment potential method. An approximate ab initio mo method for large molecules. Chem Phys Lett 101:12–17CrossRefGoogle Scholar
  21. 21.
    Stewart JJP (1996) Application of localized molecular orbitals to the solution of semiempirical self-consistent field equations. Int J Quantum Chem 58:133–146CrossRefGoogle Scholar
  22. 22.
    White CA, Johnson BG, Gill PMW, Head-Gordon M (1996) Linear scaling density functional calculations via the continuous fast multipole method. Chem Phys Lett 253:268–278CrossRefGoogle Scholar
  23. 23.
    Stewart JJP (1997) Calculation of the geometry of a small protein using semiempirical methods. J Mol Struct Theochem 401:195–205CrossRefGoogle Scholar
  24. 24.
    Lee TS, Lewis JP, Yang W (1998) Linear-scaling quantum mechanical calculations of biological molecules: the divide-and-conquer approach. Comput Mater Sci 12:259–277CrossRefGoogle Scholar
  25. 25.
    Van Alsenoy C, Yu CH, Peeters A, Martin JML, Schäfer L (1998) Ab initio geometry determinations of proteins. 1. Crambin. J Phys Chem A 102:2246–2251CrossRefGoogle Scholar
  26. 26.
    Artacho E, Sánchez-Portal D, Ordejón P, García A, Soler JM (1999) Linear-scaling ab initio calculations for large and complex systems. Phys Status Solidi B 215:809–817CrossRefGoogle Scholar
  27. 27.
    Sato F, Yoshihiro T, Era M, Kashiwagi H (2001) Calculation of all-electron wavefunction of hemoprotein cytochrome c by density functional theory. Chem Phys Lett 341:645–651CrossRefGoogle Scholar
  28. 28.
    Inaba T, Tahara S, Nisikawa N, Kashiwagi H, Sato F (2005) All-electron density functional calculation on insulin with quasi-canonical localized orbitals. J Comput Chem 26:987–993CrossRefGoogle Scholar
  29. 29.
    Wada M, Sakurai M (2005) A quantum chemical method for rapid optimization of protein structures. J Comput Chem 26:160–168CrossRefGoogle Scholar
  30. 30.
    Li S, Shen J, Li W, Jiang Y (2006) An efficient implementation of the “cluster-in-molecule” approach for local electron correlation calculations. J Chem Phys 125:074109CrossRefGoogle Scholar
  31. 31.
    Sale P, Høst S, Thøgersen L, Jørgensen P, Manninen P, Olsen J, Jansik B, Reine S, Pawlowski F, Tellgren E, Helgaker T, Coriani S (2007) Linear-scaling implementation of molecular electronic self-consistent field theory. J Chem Phys 126:114110CrossRefGoogle Scholar
  32. 32.
    Cankurtaran BO, Gale JD, Ford MJ (2008) First principles calculations using density matrix divide-and-conquer within the SIESTA methodology. J Phys Condens Matter 20:294208CrossRefGoogle Scholar
  33. 33.
    Stewart JJP (2009) Application of the PM6 method to modeling proteins. J Mol Model 15:765–805CrossRefGoogle Scholar
  34. 34.
    Gordon MS, Mullin JM, Pruitt SR, Roskop LB, Slipchenko LV, Boatz JA (2009) Accurate methods for large molecular systems. J Phys Chem B 113:9646–9663CrossRefGoogle Scholar
  35. 35.
    Fedorov DG, Alexeev Y, Kitaura K (2010) Geometry optimization of the active site of a large system with the fragment molecular orbital method. J Phys Chem Lett 2:282–288CrossRefGoogle Scholar
  36. 36.
    Kobayashi M, Kunisada T, Akama T, Sakura D, Nakai H (2010) Reconsidering an analytical gradient expression within a divide-and-conquer self-consistent field approach: exact formula and its approximate treatment. J Chem Phys 134:034105CrossRefGoogle Scholar
  37. 37.
    Mayhall NJ, Raghavachari K (2010) Molecules-in-molecules: an extrapolated fragment-based approach for accurate calculations on large molecules and materials. J Chem Theory Comput 7:1336–1343CrossRefGoogle Scholar
  38. 38.
    Nagata T, Brorsen K, Fedorov DG, Kitaura K, Gordon MS (2010) Fully analytic energy gradient in the fragment molecular orbital method. J Chem Phys 134:124115CrossRefGoogle Scholar
  39. 39.
    Reine S, Krapp A, Iozzi MF, Bakken V, Helgaker T, Pawowski F, Saek P (2010) An efficient density-functional-theory force evaluation for large molecular systems. J Chem Phys 133:044102CrossRefGoogle Scholar
  40. 40.
    Bylaska E, Tsemekhman K, Govind N, Valiev M (2011) Large-scale plane-wave-based density-functional theory: formalism, parallelization, and applications. In: Reimers JR (ed) Computational methods for large systems: electronic structure approaches for biotechnology and nanotechnology. Wiley, Hoboken, pp 77–116CrossRefGoogle Scholar
  41. 41.
    Gale JD (2011) SIESTA: a linear-scaling method for density functional calculations. In: Reimers JR (ed) Computational methods for large systems: electronic structure approaches for biotechnology and nanotechnology. Wiley, Hoboken, pp 45–74CrossRefGoogle Scholar
  42. 42.
    Li W, Hua W, Fang T, Li S (2011) The energy-based fragmentation approach for computing total energies, structures, and molecular properties of large systems at the ab initio levels. In: Reimers JR (ed) Computational methods for large systems: electronic structure approaches for biotechnology and nanotechnology. Wiley, Hoboken, pp 227–258Google Scholar
  43. 43.
    Clark T, Stewart JJP (2011) MNDO-like semiempirical molecular orbital theory and its application to large systems. In: Reimers JR (ed) Computational methods for large systems: electronic structure approaches for biotechnology and nanotechnology. Wiley, Hoboken, pp 259–286CrossRefGoogle Scholar
  44. 44.
    Elstner M, Gaus M (2011) The self-consistent-charge density-functional tight-binding (SCC-DFTB) method: an efficient approximation of density functional theory. In: Reimers JR (ed) Computational methods for large systems: electronic structure approaches for biotechnology and nanotechnology. Wiley, Hoboken, pp 287–308CrossRefGoogle Scholar
  45. 45.
    Zimmerli U, Parrinello M, Koumoutsakos P (2004) Dispersion corrections to density functionals for water aromatic interactions. J Chem Phys 120:2693–2699CrossRefGoogle Scholar
  46. 46.
    Antony J, Grimme S (2006) Density functional theory including dispersion corrections for intermolecular interactions in a large benchmark set of biologically relevant molecules. Phys Chem Chem Phys 8:5287–5293CrossRefGoogle Scholar
  47. 47.
    Grimme S, Antony J, Schwabe T, Mück-Lichtenfeld C (2007) Density functional theory with dispersion corrections for supramolecular structures, aggregates, and complexes of (bio)organic molecules. Org Biomol Chem 5:741–758CrossRefGoogle Scholar
  48. 48.
    Zhao Y, Truhlar DG (2007) Density functionals for noncovalent interaction energies of biological importance. J Chem Theory Comput 3:289–300CrossRefGoogle Scholar
  49. 49.
    Murdachaew G, De Gironcoli S, Scoles G (2008) Toward an accurate and efficient theory of physisorption. I. Development of an augmented density-functional theory model. J Phys Chem A 112:9993–10005CrossRefGoogle Scholar
  50. 50.
    DiLabio GA (2008) Accurate treatment of van der Waals interactions using standard density functional theory methods with effective core-type potentials: application to carbon-containing dimers. Chem Phys Lett 455:348–353CrossRefGoogle Scholar
  51. 51.
    Gräfenstein J, Cremer D (2009) An efficient algorithm for the density-functional theory treatment of dispersion interactions. J Chem Phys 130:124105CrossRefGoogle Scholar
  52. 52.
    Liu Y, Goddard WA (2009) A universal damping function for empirical dispersion correction on density functional theory. Mater Trans 50:1664–1670CrossRefGoogle Scholar
  53. 53.
    Sato T, Nakai H (2009) Density functional method including weak interactions: dispersion coefficients based on the local response approximation. J Chem Phys 131:224104CrossRefGoogle Scholar
  54. 54.
    Foster ME, Sohlberg K (2010) Empirically corrected DFT and semi-empirical methods for non-bonding interactions. Phys Chem Chem Phys 12:307–322CrossRefGoogle Scholar
  55. 55.
    Grimme S, Antony J, Ehrlich S, Krieg H (2010) A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J Chem Phys 132:154104CrossRefGoogle Scholar
  56. 56.
    Riley KE, Pitončák M, Jurecčka P, Hobza P (2010) Stabilization and structure calculations for noncovalent interactions in extended molecular systems based on wave function and density functional theories. Chem Rev 110:5023–5063CrossRefGoogle Scholar
  57. 57.
    MacKie ID, Dilabio GA (2010) Accurate dispersion interactions from standard density-functional theory methods with small basis sets. Phys Chem Chem Phys 12:6092–6098CrossRefGoogle Scholar
  58. 58.
    Goerigk L, Grimme S (2011) A thorough benchmark of density functional methods for general main group thermochemistry, kinetics, and noncovalent interactions. Phys Chem Chem Phys 13:6670–6688CrossRefGoogle Scholar
  59. 59.
    Grimme S, Ehrlich S, Goerigk L (2011) Effect of the damping function in dispersion corrected density functional theory. J Comput Chem 32:1456–1465CrossRefGoogle Scholar
  60. 60.
    Steinmann SN, Corminboeuf C (2011) A density dependent dispersion correction. Chimia 65:240–244CrossRefGoogle Scholar
  61. 61.
    Zhao Y, Truhlar DG (2011) Density functional theory for reaction energies: test of meta and hybrid meta functionals, range-separated functionals, and other high-performance functionals. J Chem Theory Comput 7:669–676CrossRefGoogle Scholar
  62. 62.
    Brüning J, Alig E, Van De Streek J, Schmidt MU (2011) The use of dispersion-corrected DFT calculations to prevent an incorrect structure determination from powder data: The case of acetolone, C 11H11N3O3. Z Kristallogr 226:476–482CrossRefGoogle Scholar
  63. 63.
    Ryde U, Olsen L, Nilsson K (2002) Quantum chemical geometry optimizations in proteins using crystallographic raw data. J Comput Chem 23:1058–1070CrossRefGoogle Scholar
  64. 64.
    Ryde U, Nilsson K (2003) Quantum chemistry can locally improve protein crystal structures. J Am Chem Soc 125:14232–14233CrossRefGoogle Scholar
  65. 65.
    Ryde U (2007) Accurate metal-site structures in proteins obtained by combining experimental data and quantum chemistry. Dalton Trans 607–625Google Scholar
  66. 66.
    Ryde U, Greco C, De Gioia L (2010) Quantum refinement of [FeFe] hydrogenase indicates a dithiomethylamine ligand. J Am Chem Soc 132:4512–4513CrossRefGoogle Scholar
  67. 67.
    Yu N, Yennawar HP, Merz KM Jr (2005) Refinement of protein crystal structures using energy restraints derived from linear-scaling quantum mechanics. Acta Crystallogr D Biol Crystallogr 61:322–332CrossRefGoogle Scholar
  68. 68.
    Yu N, Li X, Cui G, Hayik SA, Merz KM Jr (2006) Critical assessment of quantum mechanics based energy restraints in protein crystal structure refinement. Protein Sci 15:2773–2784CrossRefGoogle Scholar
  69. 69.
    Yu N, Hayik SA, Wang B, Liao N, Reynolds CH, Merz KM Jr (2006) Assigning the protonation states of the key aspartates in beta-secretase using QM/MM X-ray structure refinement. J Chem Theory Comput 2:1057–1069CrossRefGoogle Scholar
  70. 70.
    Van Der Vaart A, Suárez D, Merz KM Jr (2000) Critical assessment of the performance of the semiempirical divide and conquer method for single point calculations and geometry optimizations of large chemical systems. J Chem Phys 113:10512–10523CrossRefGoogle Scholar
  71. 71.
    Van Der Vaart A, Gogonea V, Dixon SL, Merz KM Jr (2000) Linear scaling molecular orbital calculations of biological systems using the semiempirical divide and conquer method. J Comput Chem 21:1494–1504CrossRefGoogle Scholar
  72. 72.
    Dixon SL, Merz KM Jr (1997) Fast, accurate semiempirical molecular orbital calculations for macromolecules. J Chem Phys 107:879–893CrossRefGoogle Scholar
  73. 73.
    Dixon SL, Merz KM Jr (1996) Semiempirical molecular orbital calculations with linear system size scaling. J Chem Phys 104:6643–6649CrossRefGoogle Scholar
  74. 74.
    Pellegrini M, Grønbech-Jensen N, Kelly JA, Pfluegl GMU, Yeates TO (1997) Highly constrained multiple-copy refinement of protein crystal structures. Proteins Struct Funct Bioinform 29:426–432CrossRefGoogle Scholar
  75. 75.
    Levin EJ, Kondrashov DA, Wesenberg GE, Phillips GN Jr (2007) Ensemble refinement of protein crystal structures: validation and application. Structure 15:1040–1052CrossRefGoogle Scholar
  76. 76.
    Terwilliger TC, Grosse-Kunstleve RW, Afonine PV, Adams PD, Moriarty NW, Zwart P, Read RJ, Turk D, Hung LW (2007) Interpretation of ensembles created by multiple iterative rebuilding of macromolecular models. Acta Crystallogr D Biol Crystallogr 63:597–610CrossRefGoogle Scholar
  77. 77.
    Stewart KA, Robinson DA, Lapthorn AJ (2008) Type II dehydroquinase: molecular replacement with many copies. Acta Crystallogr D Biol Crystallogr 64:108–118CrossRefGoogle Scholar
  78. 78.
    Stewart JJP (2008) Application of the PM6 method to modeling the solid state. J Mol Model 14:499–535CrossRefGoogle Scholar
  79. 79.
    Genheden S, Ryde U (2011) A comparison of different initialization protocols to obtain statistically independent molecular dynamics simulations. J Comput Chem 32:187–195CrossRefGoogle Scholar
  80. 80.
    Genheden S, Diehl C, Akke M, Ryde U (2010) Starting-condition dependence of order parameters derived from molecular dynamics simulations. J Chem Theory Comput 6:2176–2190CrossRefGoogle Scholar
  81. 81.
    Delarue M (2007) Dealing with structural variability in molecular replacement and crystallographic refinement through normal-mode analysis. Acta Crystallogr D Biol Crystallogr 64:40–48CrossRefGoogle Scholar
  82. 82.
    Knight JL, Zhou Z, Gallicchio E, Himmel DM, Friesner RA, Arnold E, Levy RM (2008) Exploring structural variability in X-ray crystallographic models using protein local optimization by torsion-angle sampling. Acta Crystallogr D Biol Crystallogr 64:383–396CrossRefGoogle Scholar
  83. 83.
    Sellers BD, Zhu K, Zhao S, Friesner RA, Jacobson MP (2008) Toward better refinement of comparative models: predicting loops in inexact environments. Proteins Struct Funct Genet 72:959–971CrossRefGoogle Scholar
  84. 84.
    Yao P, Dhanik A, Marz N, Propper R, Kou C, Liu G, Van Den Bedem H, Latombe JC, Halperin-Landsberg I, Altman RB (2008) Efficient algorithms to explore conformation spaces of flexible protein loops. IEEE/ACM Trans Comput Biol Bioinform 5:534–545CrossRefGoogle Scholar
  85. 85.
    Lindorff-Larsen K, Ferkinghoff-Borg J (2009) Similarity measures for protein ensembles. PLoS One 4:e4203CrossRefGoogle Scholar
  86. 86.
    Yang L, Song G, Jernigan RL (2009) Comparisons of experimental and computed protein anisotropic temperature factors. Proteins Struct Funct Bioinform 76:164–175CrossRefGoogle Scholar
  87. 87.
    Dhanik A, Van Den Bedem H, Deacon A, Latombe JC (2010) Modeling structural heterogeneity in proteins from X-ray data. Springer Tracts Adv Robot 57:551–566CrossRefGoogle Scholar
  88. 88.
    Schwander P, Fung R, Phillips GN Jr, Ourmazd A (2010) Mapping the conformations of biological assemblies. New J Phys 12:035007CrossRefGoogle Scholar
  89. 89.
    Lang PT, Ng HL, Fraser JS, Corn JE, Echols N, Sales M, Holton JM, Alber T (2010) Automated electron-density sampling reveals widespread conformational polymorphism in proteins. Protein Sci 19:1420–1431CrossRefGoogle Scholar
  90. 90.
    Kohn JE, Afonine PV, Ruscio JZ, Adams PD, Head-Gordon T (2010) Evidence of functional protein dynamics from X-ray crystallographic ensembles. PLoS Comput Biol 6:e1000911CrossRefGoogle Scholar
  91. 91.
    Tyka MD, Keedy DA, André I, Dimaio F, Song Y, Richardson DC, Richardson JS, Baker D (2011) Alternate states of proteins revealed by detailed energy landscape mapping. J Mol Biol 405:607–618CrossRefGoogle Scholar
  92. 92.
    Ramelot TA, Raman S, Kuzin AP, Xiao R, Ma L-C, Acton TB, Hunt JF, Montelione GT, Baker D, Kennedy MA (2009) Improving NMR protein structure quality by Rosetta refinement: a molecular replacement study. Proteins Struct Funct Bioinform 75:147–167CrossRefGoogle Scholar
  93. 93.
    Fleming A (1922) On a remarkable bacteriolytic element found in tissues and secretions. Proc R Soc Ser B 93:306–317CrossRefGoogle Scholar
  94. 94.
    Blake CCF, Fenn RH, North ACT, Phillips DC, Poljak RJ (1962) Structure of lysozyme. Nature 196:1173–1176CrossRefGoogle Scholar
  95. 95.
    Berman HM, Henrick K, Nakamura H (2003) Announcing the world wide protein data bank. Nat Struct Biol 10:980CrossRefGoogle Scholar
  96. 96.
    Vocadlo DJ, Davies GJ, Laine R, Withers SG (2001) Catalysis by hen egg-white lysozyme proceeds via a covalent intermediate. Nature 412:835–838CrossRefGoogle Scholar
  97. 97.
    Bottoni A, Miscione GP, De Vivo M (2005) A theoretical DFT investigation of the lysozyme mechanism: computational evidence for a covalent intermediate pathway. Proteins Struct Funct Genet 59:118–130CrossRefGoogle Scholar
  98. 98.
    Wang J, Dauter M, Alkire R, Joachimiak A, Dauter Z (2007) Triclinic lysozyme at 0.65 a resolution. Acta Crystallogr D Biol Crystallogr 63:1254–1268CrossRefGoogle Scholar
  99. 99.
    Blundell TL, Johnson LN (1976) Protein crystallography. Academic Press, LondonGoogle Scholar
  100. 100.
    Chapman HN, Fromme P, Barty A, White TA, Kirian RA, Aquila A, Hunter MS, Schulz J, DePonte DP, Weierstall U, Doak RB, Maia FRNC, Martin AV, Schlichting I, Lomb L, Coppola N, Shoeman RL, Epp SW, Hartmann R, Rolles D, Rudenko A, Foucar L, Kimmel N, Weidenspointner G, Holl P, Liang M, Barthelmess M, Caleman C, Boutet S, Bogan MJ, Krzywinski J, Bostedt C, Bajt S, Gumprecht L, Rudek B, Erk B, Schmidt C, Homke A, Reich C, Pietschner D, Struder L, Hauser G, Gorke H, Ullrich J, Herrmann S, Schaller G, Schopper F, Soltau H, Kuhnel K-U, Messerschmidt M, Bozek JD, Hau-Riege SP, Frank M, Hampton CY, Sierra RG, Starodub D, Williams GJ, Hajdu J, Timneanu N, Seibert MM, Andreasson J, Rocker A, Jonsson O, Svenda M, Stern S, Nass K, Andritschke R, Schroter C-D, Krasniqi F, Bott M, Schmidt KE, Wang X, Grotjohann I, Holton JM, Barends TRM, Neutze R, Marchesini S, Fromme R, Schorb S, Rupp D, Adolph M, Gorkhover T, Andersson I, Hirsemann H, Potdevin G, Graafsma H, Nilsson B, Spence JCH (2011) Femtosecond X-ray protein nanocrystallography. Nature 470:73–77CrossRefGoogle Scholar
  101. 101.
    Brunger AT (1992) Free R value: a novel statistical quantity for assessing the accuracy of crystal structures. Nature 355:472–475CrossRefGoogle Scholar
  102. 102.
    Badger J (1997) Modeling and refinement of water molecules and disordered solvent. Methods Enzymol 277:344–352CrossRefGoogle Scholar
  103. 103.
    Podjarny AD, Howard EI, Urzhumtsev A, Grigera JR (1997) A multicopy modeling of the water distribution in macromolecular crystals. Proteins Struct Funct Bioinform 28:303–312CrossRefGoogle Scholar
  104. 104.
    Colominas C, Luque FJ, Orozco M (1999) Monte Carlo–MST: new strategy for representation of solvent configurational space in solution. J Comput Chem 20:665–678CrossRefGoogle Scholar
  105. 105.
    Liu Y, Beveridge DL (2002) Exploratory studies of ab initio protein structure prediction: multiple copy simulated annealing, AMBER energy functions, and a generalized born/solvent accessibility solvation model. Proteins Struct Funct Bioinform 46:128–146CrossRefGoogle Scholar
  106. 106.
    Das B, Meirovitch H (2003) Solvation parameters for predicting the structure of surface loops in proteins: transferability and entropic effects. Proteins Struct Funct Bioinform 51:470–483CrossRefGoogle Scholar
  107. 107.
    Hassan SA, Mehler EL, Zhang D, Weinstein H (2003) Molecular dynamics simulations of peptides and proteins with a continuum electrostatic model based on screened coulomb potentials. Proteins Struct Funct Bioinform 51:109–125CrossRefGoogle Scholar
  108. 108.
    Dechene M, Wink G, Smith M, Swartz P, Mattos C (2009) Multiple solvent crystal structures of ribonuclease A: an assessment of the method. Proteins Struct Funct Bioinform 76:861–881CrossRefGoogle Scholar
  109. 109.
    Kannan S, Zacharias M (2010) Application of biasing-potential replica-exchange simulations for loop modeling and refinement of proteins in explicit solvent. Proteins Struct Funct Bioinform 78:2809–2819CrossRefGoogle Scholar
  110. 110.
    Weiner SJ, Kollman PA, Case DA, Singh UC, Ghio C, Alagona G, Profeta SJ, Weiner P (1984) A new force field for molecular mechanical simulation of nucleic acids and proteins. J Am Chem Soc 106:765–784CrossRefGoogle Scholar
  111. 111.
    Cornell WD, Cieplak P, Bayly CI, Gould IR, Merz KM Jr, Ferguson DM, Spellmeyer DC, Fox T, Caldwell JW, Kollman PA (1995) A second generation force field for the simulation of proteins, nucleic acids, and organic molecules. J Am Chem Soc 117:5179–5197CrossRefGoogle Scholar
  112. 112.
    Becke AD (1993) Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys 98:5648–5652CrossRefGoogle Scholar
  113. 113.
    Hehre WJ, Ditchfield R, Pople JA (1972) Self-consistent molecular orbital methods. XII. Further extensions of gaussian-type basis sets for use in molecular orbital studies of organic molecules. J Chem Phys 56:2257–2261CrossRefGoogle Scholar
  114. 114.
    Frisch MJ, Trucks GW, Schlegel HB et al (2009) Gaussian 09, revision A.02. Gaussian, Inc., PittsburghGoogle Scholar
  115. 115.
    Jorgensen WL, Chandrasekhar J, Madura JD, Impey RW, Klein ML (1983) Comparison of simple potential functions for simulating liquid water. J Chem Phys 79:926–935CrossRefGoogle Scholar
  116. 116.
    Fischer RA (1935) The logic of inductive inference. J R Stat Soc A 98:39–54CrossRefGoogle Scholar
  117. 117.
    Freeman GH, Halton JH (1951) Note on an exact treatment of contingency, goodness of fit and other problems of significance. Biometrika 38:141–149Google Scholar
  118. 118.
    Agresti A (1990) Categorical data analysis. Wiley, New YorkGoogle Scholar
  119. 119.
    Bartoszyński R, Niewiadomska-Bugaj M (1996) Probability and statistical inference. Wiley, New YorkGoogle Scholar
  120. 120.
    Walsh MA, Schneider TR, Sieker LC, Dauter Z, Lamzin VS, Wilson KS (1998) Refinement of triclinic hen egg-white lysozyme at atomic resolution. Acta Crystallogr D Biol Crystallogr 54:522–546CrossRefGoogle Scholar
  121. 121.
    Lide DR (ed) (2005) CRC handbook of chemistry and physics, 86th edn. CRC Press, Boca RatonGoogle Scholar
  122. 122.
    Perdew JP, Wang Y (1992) Accurate and simple analytic representation of the electron-gas correlation energy. Phys Rev B 45:13244–13249CrossRefGoogle Scholar
  123. 123.
    Vitkup D, Ringe D, Karplus M, Petsko GA (2002) Why protein R-factors are so large: a self-consistent analysis. Proteins Struct Funct Genet 46:345–354CrossRefGoogle Scholar

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© Springer-Verlag 2012

Authors and Affiliations

  • Olle Falklöf
    • 1
    • 2
    • 4
  • Charles A. Collyer
    • 3
  • Jeffrey R. Reimers
    • 1
    Email author
  1. 1.School of ChemistryThe University of SydneySydneyAustralia
  2. 2.Department of ChemistryThe University of GothenburgGothenburgSweden
  3. 3.School of Molecular BioscienceThe University of SydneySydneyAustralia
  4. 4.Department of Physics, Chemistry and BiologyLinköping UniversityLinköpingSweden

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