Abstract
A method is described that allows experimental \(S^2\) order parameters to be enforced as a time-averaged quantity in molecular dynamics simulations. The two parameters that characterize time-averaged restraining, the memory relaxation time and the weight of the restraining potential energy term in the potential energy function used in the simulation, are systematically investigated based on two model systems, a vector with one end restrained in space and a pentapeptide. For the latter it is shown that the backbone N–H order parameter of individual residues can be enforced such that the spatial fluctuations of quantities depending on atomic coordinates are not significantly perturbed. The applicability to realistic systems is illustrated for the B3 domain of protein G in aqueous solution.
Similar content being viewed by others
References
Barker JA, Watts RO (1973) Monte Carlo studies of the dielectric properties of water-like models. Mol Phys 26:789–792
Beauchamp KA, Lin YS, Das R, Pande VS (2012) Are protein force fields getting better? A systematic benchmark on 524 diverse NMR measurements. J Chem Theory Comput 8:1409–1414
Berendsen HJC (1985) Treatment of long-range forces in molecular dynamics. In: Hermans J (ed) Molecular dynamics and protein structure. Polycrystal Book Service, Western Springs, pp 18–22
Berendsen HJC, Postma JPM, van Gunsteren WF, Hermans J (1981) Interaction models for water in relation to protein hydration. In: Pullmann B (ed) Intermolecular forces. Reidel, Dordrecht, pp 331–342
Berendsen HJC, Postma JPM, van Gunsteren WF, DiNola A, Haak JR (1984) Molecular dynamics with coupling to an external bath. J Chem Phys 81:3684–3690
Best RB, Vendruscolo M (2004) Determination of protein structures consistent with NMR order parameters. J Am Chem Soc 126:8090–8091
Brainard JR, Szabo A (1981) Theory for nuclear magnetic relaxation of probes in anisotropic systems: application to cholesterol in phospholipid vesicles. Biochemistry 20:4618–4628
Braun W, Bösch C, Brown LR, Gō N, Wüthrich K (1981) Combined use of proton–proton Overhauser enhancements and a distance geometry algorithm for determination of polypeptide conformations. Application to micelle-bound glucagon. Biochim Biophys Acta 667:377–396
Brüschweiler R, Wright PE (1994) NMR order parameters of biomolecules: a new analytical representation and application to the Gaussian axial fluctuation model. J Am Chem Soc 116:8426–8427
Brüschweiler R, Roux B, Blackledge M, Griesinger C, Karplus M, Ernst RR (1992) Influence of rapid intramolecular motion on NMR cross relaxation rates. A molecular dynamics study of antamanide in solution. J Am Chem Soc 114:2289–2302
Buck M, Bouguet-Bonnet S, Pastor RW, MacKerell AD (2006) Importance of the CMAP correction to the CHARMM22 protein force field: dynamics of hen lysozyme. Biophys J 90:L36–L38
Cavalli A, Camilloni C, Vendruscolo M (2013) Molecular dynamics simulations with replica-averaged structural restraints generate structural ensembles according to the maximum entropy principle. J Chem Phys 138:094112
Chandrasekhar I, Clore GM, Szabo A, Gronenborn AM, Brooks BR (1992) A 500 ps molecular dynamics simulation study of interleukin-1\(\beta \) in water. Correlation with nuclear magnetic resonance spectroscopy and crystallography. J Mol Biol 226:239–250
Christen M, Hünenberger PH, Bakowies D, Baron R, Bürgi R, Geerke DP, Heinz TN, Kastenholz MA, Kräutler V, Oostenbrink C, Peter C, Trzesniak D, van Gunsteren WF (2005) The GROMOS software for biomolecular simulation: GROMOS05. J Comput Chem 26:1719–1751
Christen M, Keller B, van Gunsteren WF (2007) Biomolecular structure refinement based on adaptive restraints using local-elevation simulation. J Biomol NMR 39:265–273
Daura X, Mark AE, van Gunsteren WF (1999) Peptide folding simulations: No solvent required? Comput Phys Commun 123:97–102
d’Auvergne EJ, Gooley PR (2003) The use of model selection in the model-free analysis of protein dynamics. J Biomol NMR 25:25–39
Dolenc J, Missimer JH, Steinmetz MO, van Gunsteren WF (2010) Methods of NMR structure refinement: molecular dynamics simulations improve the agreement with measured NMR data of a C-terminal peptide of GCN4-p1. J Biomol NMR 47:221–235
Duan Y, Wu C, Chowdhury S, Lee MC, Xiong G, Zhang W, Yang R, Cieplak P, Luo R, Lee T, Caldwell J, Wang J, Kollman P (2003) A point-charge force field for molecular mechanics simulations of proteins based on condensed-phase quantum mechanical calculations. J Comput Chem 24:1999–2012
Eichenberger AP, Allison JR, Dolenc J, Geerke DP, Horta BAC, Meier K, Oostenbrink C, Schmid N, Steiner D, Wang D, van Gunsteren WF (2011) The GROMOS++ software for the analysis of biomolecular simulation trajectories. J Chem Theory Comput 7:3379–3390
Evenäs J, Forsén S, Malmendal A, Akke M (1999) Backbone dynamics and energetics of a calmodulin domain mutant exchanging between closed and open conformations. J Mol Biol 289:603–617
Feenstra KA, Peter C, Scheek RM, van Gunsteren WF, Mark AE (2002) A comparison of methods for calculating NMR cross-relaxation rates (NOESY and ROESY intensities) in small peptides. J Biomol NMR 23:181–194
Fennen J, Torda AE, van Gunsteren WF (1995) Structure refinement with molecular dynamics and a Boltzmann-weighted ensemble. J Biomol NMR 6:163–170
Fukunishi H, Watanabe O, Takada S (2002) On the Hamiltonian replica exchange method for efficient sampling of biomolecular systems: application to protein structure prediction. J Chem Phys 116:9058–9067
Gapsys V, de Groot BL (2013) Optimal superpositioning of flexible molecule ensembles. Biophys J 104:196–207
Gáspári Z, Perczel A (2010) Protein dynamics as reported by NMR. Annu Rep NMR Spectrosc 71:35–75
Gattin Z, Schwartz J, Mathad RI, Jaun B, van Gunsteren WF (2009) Interpreting experimental data by using molecular simulation instead of model building. Chem Eur J 15:6389–6398
Gniewek P, Kolinski A, Jernigan RL, Kloczkowski A (2012) How noise in force fields can affect the structural refinement of protein models. Proteins Struct Funct Bioinf 80:335–341
Gros P, van Gunsteren WF (1993) Crystallographic refinement and structure-factor time-averaging by molecular dynamics in the absence of a physical force field. Mol Sim 10:377–395
Hall JB, Fushman D (2003) Characterization of the overall and local dynamics of a protein with intermediate rotational anisotropy: differentiating between conformational exchange and anisotropic diffusion in the B3 domain of protein G. J Biomol NMR 27:261–275
Harvey TS, van Gunsteren WF (1993) The application of chemical shift calculation to protein structure determination by NMR. In: Angeletti RH (ed) Tech Protein Chem, vol 4. Academic Press, New York, pp 615–622
Heinz TN, van Gunsteren WF, Hünenberger PH (2001) Comparison of four methods to compute the dielectric permittivity of liquids from molecular dynamics simulations. J Chem Phys 115:1125–1136
Henry ER, Szabo A (1985) Influence of vibrational motion on solid state line shapes and NMR relaxation. J Chem Phys 82:4753–4761
Hess B, Scheek RM (2003) Orientation restraints in molecular dynamics simulations using time and ensemble averaging. J Magn Reson 164:19–27
Hockney RW (1970) The potential calculation and some applications. Methods Comput Phys 9:136–211
Hornak V, Abel R, Okur A, Strockbine B, Roitberg A, Simmerling C (2006) Comparison of multiple Amber force fields and development of improved protein backbone parameters. Proteins Struct Funct Bioinf 65:712–725
Huber T, van Gunsteren WF (1998) SWARM-MD: searching conformational space by cooperative molecular dynamics. J Phys Chem A 102:5937–5943
Huber T, Torda AE, van Gunsteren WF (1996) Optimization methods for conformational sampling using a Boltzmann-weighted mean field approach. Biopolymers 39:103–114
Jardetzky O (1980) On the nature of molecular conformations inferred from high-resolution NMR. Biochim Biophys Acta 621:227–232
Jarymowycz VA, Stone MJ (2006) Fast time scale dynamics of protein backbones: NMR relaxation methods, applications, and functional consequences. Chem Rev 106:1624–1671
Johnson E (2012) Separability between overall and internal motion: a protein folding problem. Proteins Struct Funct Bioinf 80:2645–2651
Johnson E, Showalter SA, Brüschweiler R (2008) A multifaceted approach to the interpretation of NMR order parameters: a case study of a dynamic \(\alpha \)-helix. J Phys Chem B 112:6203–6210
Kim DE, Blum B, Bradley P, Baker D (2009) Sampling bottlenecks in de novo protein structure prediction. J Mol Biol 393:249–260
Lipari G, Szabo A (1982) Model-free approach to the interpretation of nuclear magnetic resonance relaxation in macromolecules. 1. Theory and range of validity. J Am Chem Soc 104:4546–4559
Luginbühl P, Wüthrich K (2002) Semi-classical nuclear spin relaxation theory revisited for use with biological macromolecules. Prog Nucl Magn Reson Spectrosc 40:199–247
MacKerell AD (2004) Empirical force fields for biological macromolecules: overview and issues. J Comput Chem 25:1584–1604
MacKerell AD, Feig M, Brooks CL (2004) Improved treatment of the protein backbone in empirical force fields. J Am Chem Soc 126:698–699
Marchand S, Roux B (1998) Molecular dynamics study of calbindin \(\text{D}_{\rm 9k}\) in the apo and singly and doubly calcium-loaded state. Proteins Struct Funct Bioinf 33:265–284
Misura KMS, Baker D (2005) Progress and challenges in high-resolution refinement of protein structure models. Proteins Struct Funct Bioinf 59:15–29
Nanzer AP, van Gunsteren WF, Torda AE (1995) Parametrisation of time-averaged distance restraints in MD simulations. J Biomol NMR 6:313–320
Nanzer AP, Torda AE, Bisang C, Weber C, Robinson JA, van Gunsteren WF (1997) Dynamical studies of peptide motifs in the plasmodium falciparum circumsporozoite surface protein by restrained and unrestrained MD simulations. J Mol Biol 267:1012–1025
Olsson S, Frellsen J, Boomsma W, Mardia KV, Hamelryck T (2013) Inference of structure ensembles of flexible biomolecules from sparse, averaged data. PLoS One 8:e79439
Olsson S, Vögeli BR, Cavalli A, Boomsma W, Ferkinghoff-Borg J, Lindorff-Larsen K, Hamelryck T (2014) Probabilistic determination of native state ensembles of proteins. J Chem Theory Comput 10:3484–3491
Palmer AG III, Williams J, McDermott A (1996) Nuclear magnetic resonance studies of biopolymer dynamics. J Phys Chem 100:13,293–13,310
Pearlman DA (1994a) How is an NMR structure best defined? An analysis of molecular dynamics distance based approaches. J Biomol NMR 4:1–16
Pearlman DA (1994b) How well do time-averaged J-coupling restraints work? J Biomol NMR 4:279–299
Pearlman DA, Kollman PA (1991) Are time-averaged restraints necessary for NMR refinement? A model study for DNA. J Mol Biol 220:457–479
Pepermans H, Tourwé D, van Binst G, Boelens R, Scheek RM, van Gunsteren WF, Kaptein R (1988) The combined use of NMR, distance geometry, and restrained molecular dynamics for the conformational study of a cyclic somatostatin analogue. Biopolymers 27:323–338
Peter C, Daura X, van Gunsteren WF (2001) Calculation of NMR-relaxation parameters for flexible molecules from molecular dynamics simulations. J Biomol NMR 20:297–310
Pfeiffer S, Fushman D, Cowburn D (2001) Simulated and NMR-derived backbone dynamics of a protein with significant flexibility: a comparison of spectral densities for the \(\beta \)ARK1 PH domain. J Am Chem Soc 123:3021–3036
Pitera JW, Chodera JD (2012) On the use of experimental observations to bias simulated ensembles. J Chem Theory Comput 8:4335–3451
Raval A, Piana S, Eastwood MP, Dror RO, Shaw DE (2012) Refinement of protein structure homology models via long, all-atom molecular dynamics simulations. Proteins 80:2071–2079
Richter B, Gsponer J, Várnai P, Salvatella X, Vendruscolo M (2007) The mumo (minimal under-restraining minimal over-restraining) method for the determination of native state ensembles of proteins. J Biomol NMR 37:117–135
Roux B, Weare J (2013) On the statistical equivalence of restrained-ensemble simulations with the maximum entropy method. J Chem Phys 138:084107
Ryckaert JP, Ciccotti G, Berendsen HJC (1977) Numerical integration of the cartesian equations of motion of a system with constraints: molecular dynamics of \(n\)-alkanes. J Comput Phys 23:327–341
Sapienza PJ, Lee AL (2010) Using NMR to study fast dynamics in proteins: methods and applications. Curr Opin Pharmacol 10:723–730
Scheek RM, Torda AE, Kemmink J, van Gunsteren WF (1991) Structure determination by NMR: the modelling of NMR parameters as ensemble averages. In: Hoch JC, Poulsen FM, Redfield C (eds) Computational aspects of the study of biological macromolecules by nuclear magnetic resonance spectroscopy, NATO ASI Series A225. Plenum Press, New York, pp 209–217
Schiffer CA, van Gunsteren WF (1999) Accessibility and order of water sites in and around proteins: a crystallographic time-averaging study. Proteins 36:501–511
Schiffer CA, Gros P, van Gunsteren WF (1995) Time-averaging crystallographic refinement: possibilities and limitations using alpha-cyclodextrin as a test system. Acta Cryst D51:85–92
Schmid N, Allison JR, Dolenc J, Eichenberger AP, Kunz APE, van Gunsteren WF (2011a) Biomolecular structure refinement using the GROMOS simulation software. J Biomol NMR 51:265–281
Schmid N, Eichenberger A, Choutko A, Riniker S, Winger M, Mark AE, van Gunsteren WF (2011b) Definition and testing of the GROMOS force-field versions: 54A7 and 54B7. Eur Biophys J 40:843–856
Schmid N, Christ CD, Christen M, Eichenberger AP, van Gunsteren WF (2012) Architecture, implementation and parallelisation of the GROMOS software for biomolecular simulation. Comput Phys Commun 183:890–903
Schmitz U, Kumar A, James TL (1992) Dynamic interpretation of NMR data: molecular dynamics with weighted time-averaged restraints and ensemble R-factor. J Am Chem Soc 114:10,654–10,656
Schmitz U, Ulyanov B, Kumar A, James TL (1993) Molecular dynamics with weighted time-averaged restraints for a DNA octamer: dynamic interpretation of NMR data. J Mol Biol 234:373–389
Scott WRP, Mark AE, van Gunsteren WF (1998) On using time-averaging restraints in molecular dynamics simulations. J Biomol NMR 12:501–508
Showalter SA, Brüschweiler R (2007) Validation of molecular dynamics simulations of biomolecules. J Chem Theory Comput 3:961–975
Smith LJ, Mark AE, Dobson CM, van Gunsteren WF (1995a) Comparison of MD simulations and NMR experiments for hen lysozyme: analysis of local fluctuations, cooperative motions and global changes. Biochemistry 34:10918–10931
Smith PE, van Schaik RC, Szyperski T, Wüthrich K, van Gunsteren WF (1995b) Internal mobility of the basic pancreatic trypsin inhibitor in solution: a comparison of NMR spin relaxation measurements and molecular dynamics. J Mol Biol 246:356–365
Stocker U, van Gunsteren WF (2000) Molecular dynamics simulation of hen egg white lysozyme: a test of the GROMOS96 force field against nuclear magnetic resonance data. Proteins Struct Funct Bioinf 40:145–153
Sugita Y, Okamoto Y (1999) Replica-exchange molecular dynamics method for protein folding. Chem Phys Lett 314:141–151
Sugita Y, Kitao A, Okamoto Y (2000) Multidimensional replica-exchange method for free-energy calculations. J Chem Phys 113:6042–6051
Tironi IG, Sperb R, Smith PE, van Gunsteren WF (1995) A generalized reaction field method for molecular dynamics simulations. J Chem Phys 102:5451–5459
Torda AE, Scheek RM, van Gunsteren WF (1989) Time-dependent distance restraints in molecular dynamics simulations. Chem Phys Lett 157:289–294
Torda AE, Brunne RM, Huber T, Kessler H, van Gunsteren WF (1993) Structure refinement using time-averaged J-coupling restraints. J Biomol NMR 3:55–66
Trbovic N, Kim B, Friesner RA, Palmer AG (2008) Structural analysis of protein dynamics by MD simulations and NMR spin-relaxation. Proteins Struct Funct Bioinf 71:684–694
van Gunsteren WF, Berendsen HJC (1988) A leap-frog algorithm for stochastic dynamics. Mol Sim 1:173–185
van Gunsteren WF, Berendsen HJC, Rullmann JAC (1981) Stochastic dynamics for molecules with constraints. Brownian dynamics of n-alkanes. Mol Phys 44:69–95
van Gunsteren WF, Brunne RM, Gros P, van Schaik RC, Schiffer CA, Torda AE (1994) Accounting for molecular mobility in structure determination based on nuclear magnetic resonance spectroscopic and x-ray diffraction data. In: James TL, Oppenheimer NJ (eds) Methods in enzymology: nuclear magnetic resonance, vol 239. Academic Press, New York, pp 619–654
van Gunsteren WF, Dolenc J, Mark AE (2008) Molecular simulation as an aid to experimentalists. Curr Opin Struct Biol 18:149–153
White AD, Voth GA (2014) Efficient and minimal method to bias molecular simulations with experimental data. J Chem Theory Comput 10:3023–3030
Wong V, Case DA (2008) Evaluating rotational diffusion from protein MD simulation. J Phys Chem B 112:6013–6024
Yun-Yu S, Lu W, van Gunsteren WF (1988) On the approximation of solvent effects on the conformation and dynamics of cyclosporin A by stochastic dynamics simulation techniques. Mol Sim 1:369–383
Acknowledgments
This work was financially supported by the National Center of Competence in Research (NCCR) in Structural Biology and by Grant Number 200020-137827 of the Swiss National Science Foundation, and by grant number 228076 of the European Research Council, which is gratefully acknowledged. N.H. thanks the German Research Foundation (DFG) for financial support within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hansen, N., Heller, F., Schmid, N. et al. Time-averaged order parameter restraints in molecular dynamics simulations. J Biomol NMR 60, 169–187 (2014). https://doi.org/10.1007/s10858-014-9866-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10858-014-9866-7