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Enhanced Sampling Algorithms

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Book cover Biomolecular Simulations

Part of the book series: Methods in Molecular Biology ((MIMB,volume 924))

Abstract

In biomolecular systems (especially all-atom models) with many degrees of freedom such as proteins and nucleic acids, there exist an astronomically large number of local-minimum-energy states. Conventional simulations in the canonical ensemble are of little use, because they tend to get trapped in states of these energy local minima. Enhanced conformational sampling techniques are thus in great demand. A simulation in generalized ensemble performs a random walk in potential energy space and can overcome this difficulty. From only one simulation run, one can obtain canonical-ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review uses of the generalized-ensemble algorithms in biomolecular systems. Three well-known methods, namely, multicanonical algorithm, simulated tempering, and replica-exchange method, are described first. Both Monte Carlo and molecular dynamics versions of the algorithms are given. We then present various extensions of these three generalized-ensemble algorithms. The effectiveness of the methods is tested with short peptide and protein systems.

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References

  1. Kirkpatrick S, Gelatt CD Jr, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–680

    Article  PubMed  CAS  Google Scholar 

  2. Nilges M, Clore GM, Gronenborn AM (1988) Determination of three-dimensional structures of proteins from interproton distance data by hybrid distance geometry-dynamical simulated annealing calculations. FEBS Lett 229:317–324

    Article  PubMed  CAS  Google Scholar 

  3. Brünger AT (1988) Crystallographic refinement by simulated annealing. Application to a 2.8 Å resolution structure of aspartate aminotransferase. J Mol Biol 203:803–816

    Article  PubMed  Google Scholar 

  4. Wilson SR, Cui W, Moskowitz JW, Schmidt KE (1988) Conformational analysis of flexible molecules—location of the global minimum energy conformation by the simulated annealing method. Tetrahedron Lett 29:4373–4376

    Article  CAS  Google Scholar 

  5. Kawai H, Kikuchi T, Okamoto Y (1989) A prediction of tertiary structures of peptide by the Monte Carlo simulated annealing method. Protein Eng 3:85–94

    Article  PubMed  CAS  Google Scholar 

  6. Wilson C, Doniach S (1989) A computer model to dynamically simulate protein folding: studies with crambin. Proteins 6:193–209

    Article  PubMed  CAS  Google Scholar 

  7. Kawai H, Okamoto Y, Fukugita M, Nakazawa T, Kikuchi T (1991) Prediction of α-helix folding of isolated C-peptide of ribonuclease A by Monte Calro simulated annealing. Chem Lett 1991:213–216

    Article  Google Scholar 

  8. Okamoto Y, Fukugita M, Nakazawa T, Kawai H (1991) α-helix folding by Monte Carlo simulated annealing in isolated C-peptide of ribonuclease A. Protein Eng 4:639–647

    Article  PubMed  CAS  Google Scholar 

  9. Hansmann UHE, Okamoto Y (1999) Generalized-ensemble approach for protein folding simulations. In: Stauffer D (ed) Annual Reviews of Computational Physics VI. World Scientific, Singapore, pp 129–157

    Chapter  Google Scholar 

  10. Mitsutake A, Sugita Y, Okamoto Y (2001) Generalized-ensemble algorithms for molecular simulations of biopolymers. Biopolymers 60:96–123

    Article  PubMed  CAS  Google Scholar 

  11. Sugita Y, Okamoto Y (2002) Free-energy calculations in protein folding by generalized-ensemble algorithms. In: Schlick T, Gan HH (eds) Lecture notes in computational science and engineering. Springer, Berlin, pp 304–332. e-print: cond-mat/0102296

    Google Scholar 

  12. Okamoto Y (2004) Generalized-ensemble algorithms: enhanced sampling techniques for Monte Carlo and molecular dynamics simulations. J Mol Graphics Mod 22:425–439. e-print: cond-mat/0308360

    Google Scholar 

  13. Kokubo H, Okamoto Y (2006) Replica-exchange methods and predictions of helix configurations of membrane proteins. Mol Sim 32:791–801

    Article  CAS  Google Scholar 

  14. Itoh SG, Okumura H, Okamoto Y (2007) Generalized-ensemble algorithms for molecular dynamics simulations. Mol Sim 33:47–56

    Article  CAS  Google Scholar 

  15. Sugita Y, Mitsutake A, Okamoto Y (2008) Generalized-ensemble algorithms for protein folding simulations. In: Janke W (ed) Lecture notes in physics. Rugged free energy landscapes: common computational approaches in spin glasses, structural glasses and biological macromolecules. Springer, Berlin, pp 369–407. e-print: arXiv:0707.3382v1[cond-mat.stat-mech]

    Google Scholar 

  16. Okamoto Y (2009) Generalized-ensemble algorithms for studying protein folding. In: Kuwajima K, Goto Y, Hirata F, Kataoka M, Terazima M (eds) Water and Biomolecules. Springer, Berlin, pp 61–95

    Chapter  Google Scholar 

  17. Ferrenberg AM, Swendsen RH (1988) New Monte Carlo technique for studying phase transitions. Phys Rev Lett 61:2635–2638

    Google Scholar 

  18. Ferrenberg AM, Swendsen RH (1989) New Monte Carlo technique for studying phase transitions errata. Phys Rev Lett 63:1658

    Article  CAS  Google Scholar 

  19. Ferrenberg AM, Swendsen RH (1989) Optimized Monte Carlo data analysis. Phys Rev Lett 63:1195–1198

    Article  PubMed  CAS  Google Scholar 

  20. Kumar S, Bouzida D, Swendsen RH, Kollman PA, Rosenberg JM (1992) The weighted histogram analysis method for free-energy calculations on biomolecules. 1. The method. J Comput Chem 13:1011–1021

    Article  CAS  Google Scholar 

  21. Berg BA, Neuhaus T (1991) Multicanonical algorithms for 1st order phase transitions. Phys Lett B267:249–253

    Google Scholar 

  22. Berg BA, Neuhaus T (1992) Multicanonical ensemble: a new approach to simulate first-order phase transitions. Phys Rev Lett 68:9–12

    Article  PubMed  Google Scholar 

  23. Berg BA (2004) Introduction to Monte Carlo simulations and their statistical analysis. World Scientific, Singapore

    Google Scholar 

  24. Janke W (1998) Multicanonical Monte Carlo simulations. Phys A 254:164–178

    Article  CAS  Google Scholar 

  25. Lee J (1993) New Monte Carlo algorithm: entropic sampling. Phys Rev Lett 71:211–214

    Article  PubMed  CAS  Google Scholar 

  26. Lee J (1993) New Monte Carlo algorithm: entropic sampling errata. Phys Rev Lett 71:2353

    Google Scholar 

  27. Hao WH, Scheraga HA (1994) Monte Carlo simulation of a first-order transition for protein folding. J Phys Chem 98:4940–4948

    Article  CAS  Google Scholar 

  28. Mezei M (1987) Adaptive umbrella sampling—self-consistent determination of the non-Boltzmann bias. J Comput Phys 68:237–248

    Article  Google Scholar 

  29. Bartels C, Karplus M (1998) Probability distributions for complex systems: adaptive umbrella sampling of the potential energy. J Phys Chem B 102:865–880

    Article  CAS  Google Scholar 

  30. Torrie GM, Valleau JP (1977) Nonphysical sampling distributions in Monte Carlo free-energy estimation: umbrella sampling. J Comput Phys 23:187–199

    Article  Google Scholar 

  31. Wang F, Landau DP (2001) Efficient, multiple-range random walk algorithm to calculate the density of states. Phys Rev Lett 86:2050–2053

    Article  PubMed  CAS  Google Scholar 

  32. Wang F, Landau DP (2001) Determining the density of states for classical statistical models: a random walk algorithm to produce a flat histogram. Phys Rev E 64:056101

    Article  CAS  Google Scholar 

  33. Yan Q, Faller R, de Pablo JJ (2002) Density-of-states Monte Carlo method for simulation of fluids. J Chem Phys 116:8745–8749

    Article  CAS  Google Scholar 

  34. Laio A, Parrinello M (2002) Escaping free-energy minima. Proc Natl Acad Sci USA 99:12562–12566

    Article  PubMed  CAS  Google Scholar 

  35. Trebst S, Huse DA, Troyer M (2004) Optimizing the ensemble for equilibration in broad-histogram Monte Carlo simulations. Phys Rev E 70:046701

    Article  CAS  Google Scholar 

  36. Berg BA, Celik T (1992) New approach to spin-glass simulations. Phys Rev Lett 69:2292–2295

    Article  PubMed  Google Scholar 

  37. Berg BA, Hansmann UHE, Neuhaus T (1993) Simulation of an ensemble with varying magnetic field: a numerical determination of the order-order interface tension in the D=2 Ising model. Phys Rev B 47:497–500

    Article  Google Scholar 

  38. Janke W, Kappler S (1995) Phys Rev Lett 74:212–215

    Article  PubMed  CAS  Google Scholar 

  39. Berg BA, Janke W (1998) Phys Rev Lett 80:4771–4774

    Article  CAS  Google Scholar 

  40. Hatano N, Gubernatis JE (2000) A multicanonical Monte Carlo study of the 3D +/- J spin glass. Prog Theor Phys (Suppl) 138:442–447

    Article  CAS  Google Scholar 

  41. Berg BA, Billoire A, Janke W (2000) Spin-glass overlap barriers in three and four dimensions. Phys Rev B 61:12143–12150

    Article  CAS  Google Scholar 

  42. Berg BA, Muguruma C, Okamoto Y (2007) Residual entropy of ordinary ice from multicanonical simulations. Phys Rev B 75:092202

    Article  CAS  Google Scholar 

  43. Hansmann UHE, Okamoto Y (1993) Prediction of peptide conformation by multicanonical algorithm—new approach to the multiple-minima problem. J Comput Chem 14:1333–1338

    Article  CAS  Google Scholar 

  44. Hansmann UHE, Okamoto Y (1994) Comparative study of multicanonical and simulated annealing algorithms in the protein folding problem. Physica A 212:415–437

    Article  CAS  Google Scholar 

  45. Okamoto Y, Hansmann UHE (1995) Thermodynamics of helix-coil transitions studied by multicanonical algorithms. J Phys Chem 99:11276–11287

    Article  CAS  Google Scholar 

  46. Wilding NB (1995) Critical-point and coexistence-curve properties of the Lennard–Jones fluid: a finite-size scaling study. Phys Rev E 52:602–611

    Article  CAS  Google Scholar 

  47. Kolinski A, Galazka W, Skolnick J (1996) On the origin of the cooperativity of protein folding: implications from model simulations. Proteins 26:271–287

    Article  PubMed  CAS  Google Scholar 

  48. Urakami N, Takasu M (1996) Multicanonical Monte Carlo simulation of a polymer with stickers. J Phys Soc Jpn 65:2694–2699

    Article  CAS  Google Scholar 

  49. Kumar S, Payne P, Vásquez M (1996) Method for free-energy calculations using iterative techniques. J Comput Chem 17:1269–1275

    Article  CAS  Google Scholar 

  50. Hansmann UHE, Okamoto Y, Eisenmenger F (1996) Molecular dynamics, Langevin and hybrid Monte Carlo simulations in a multicanonical ensemble. Chem Phys Lett 259:321–330

    Article  CAS  Google Scholar 

  51. Hansmann UHE, Okamoto Y (1996) Monte Carlo simulations in generalized ensemble: multicanonical algorithm versus simulated tempering. Phys Rev E 54:5863–5865

    Article  CAS  Google Scholar 

  52. Hansmann UHE, Okamoto Y (1997) Numerical comparisons of three recently proposed algorithms in the protein folding problem. J Comput Chem 18:920–933

    Article  CAS  Google Scholar 

  53. Noguchi H, Yoshikawa K (1997) First-order phase transition in a stiff polymer chain. Chem Phys Lett 278:184–188

    Article  CAS  Google Scholar 

  54. Nakajima N, Nakamura H, Kidera A (1997) Multicanonical ensemble generated by molecular dynamics simulation for enhanced conformational sampling of peptides. J Phys Chem B 101:817–824

    Article  CAS  Google Scholar 

  55. Bartels C, Karplus M (1997) Multidimensional adaptive umbrella sampling: applications to main chain and side chain peptide conformations. J Comput Chem 18:1450–1462

    Article  CAS  Google Scholar 

  56. Higo J, Nakajima N, Shirai H, Kidera A, Nakamura H (1997) Two-component multicanonical Monte Carlo method for effective conformation sampling. J Comput Chem 18:2086–2092

    Article  CAS  Google Scholar 

  57. Iba Y, Chikenji G, Kikuchi M (1998) Simulation of lattice polymers with multi-self-overlap ensemble. J Phys Soc Jpn 67:3327–3330

    Article  Google Scholar 

  58. Mitsutake A, Hansmann UHE, Okamoto Y (1998) Temperature dependence of distributions of conformations of a small peptide. J Mol Graphics Mod 16:226–238; 262–263

    Google Scholar 

  59. Hansmann UHE, Okamoto Y (1999) Effects of side-chain charges on alpha-helix stability in C-peptide of ribonuclease A studied by multicanonical algorithm. J Phys Chem B 103:1595–1604

    Article  CAS  Google Scholar 

  60. Shimizu H, Uehara K, Yamamoto K, Hiwatari Y (1999) Structural phase transition of di-block polyampholyte. Mol Sim 22:285–301

    Article  CAS  Google Scholar 

  61. Ono S, Nakajima N, Higo J, Nakamura H (1999) The multicanonical weighted histogram analysis method for the free-energy landscape along structural transition paths. Chem Phys Lett 312:247–254

    Article  CAS  Google Scholar 

  62. Mitsutake A, Okamoto Y (2000) Helix-coil transitions of amino-acid homo-oligomers in aqueous solution studied by multicanonical simulations. J Chem Phys 112:10638–10647

    Article  CAS  Google Scholar 

  63. Sayano K, Kono H, Gromiha MM, Sarai A (2000) Multicanonical Monte Carlo calculation of the free-energy map of the base-amino acid interaction. J Comput Chem 21:954–962

    Article  CAS  Google Scholar 

  64. Yasar F, Celik T, Berg BA, Meirovitch H (2000) Multicanonical procedure for continuum peptide models. J Comput Chem 21:1251–1261

    Article  CAS  Google Scholar 

  65. Mitsutake A, Kinoshita M, Okamoto Y, Hirata F (2000) Multicanonical algorithm combined with the RISM theory for simulating peptides in aqueous solution. Chem Phys Lett 329:295–303

    Article  CAS  Google Scholar 

  66. Cheung MS, Garcia AE, Onuchic JN (2002) Protein folding mediated by solvation: water expulsion and formation of the hydrophobic core occur after the structural collapse. Proc Natl Acad Sci USA 99:685–690

    Article  PubMed  CAS  Google Scholar 

  67. Kamiya N, Higo J, Nakamura H (2002) Conformational transition states of a beta-hairpin peptide between the ordered and disordered conformations in explicit water. Protein Sci 11:2297–2307

    Article  PubMed  CAS  Google Scholar 

  68. Jang SM, Pak Y, Shin SM (2002) Multicanonical ensemble with Nose–Hoover molecular dynamics simulation. J Chem Phys 116:4782–4786

    Article  CAS  Google Scholar 

  69. Terada T, Matsuo Y, Kidera A (2003) A method for evaluating multicanonical potential function without iterative refinement: application to conformational sampling of a globular protein in water. J Chem Phys 118:4306–4311

    Article  CAS  Google Scholar 

  70. Berg BA, Noguchi H, Okamoto Y (2003) Multioverlap simulations for transitions between reference configurations. Phys Rev E 68:036126

    Article  CAS  Google Scholar 

  71. Bachmann M, Janke W (2003) Multicanonical chain-growth algorithm. Phys Rev Lett 91:208105

    Google Scholar 

  72. Okumura H, Okamoto Y (2004) Monte Carlo simulations in multibaric-multithermal ensemble. Chem Phys Lett 383:391–396

    Article  CAS  Google Scholar 

  73. Okumura H, Okamoto Y (2004) Monte Carlo simulations in generalized isobaric-isothermal ensembles. Phys Rev E 70:026702

    Article  CAS  Google Scholar 

  74. Okumura H, Okamoto Y (2004) Molecular dynamics simulations in the multibaric-multithermal ensemble. Chem Phys Lett 391:248–253

    Article  CAS  Google Scholar 

  75. Okumura H, Okamoto Y (2006) Multibaric-multithermal ensemble molecular dynamics simulations. J Comput Chem 27:379–395

    Article  PubMed  CAS  Google Scholar 

  76. Itoh SG, Okamoto Y (2004) Multi-overlap molecular dynamics methods for biomolecular systems. Chem Phys Lett 400:308–313

    Article  CAS  Google Scholar 

  77. Sugita Y, Okamoto Y (2005) Molecular mechanism for stabilizing a short helical peptide studied by generalized-ensemble simulations with explicit solvent. Biophys J 88:3180–3190

    Article  PubMed  CAS  Google Scholar 

  78. Itoh SG, Okamoto Y (2007) Effective sampling in the configurational space of a small peptide by the multicanonical-multioverlap algorithm. Phys Rev E 76:026705

    Article  CAS  Google Scholar 

  79. Munakata T, Oyama S (1996) Adaptation and linear-response theory. Phys Rev E 54:4394–4398

    Article  CAS  Google Scholar 

  80. Lyubartsev AP, Martinovski AA, Shevkunov SV, Vorontsov-Velyaminov PN (1992) New approach to Monte Carlo calculation of the free energy—method of expanded ensemble. J Chem Phys 96:1776–1783

    Article  CAS  Google Scholar 

  81. Marinari E, Parisi G (1992) Simulated tempering—a new Monte Carlo scheme. Europhys Lett 19:451–458

    Article  CAS  Google Scholar 

  82. Marinari E, Parisi G, Ruiz-Lorenzo JJ (1997) Numerical simulations of spin glass systems. In: Young AP (ed) Spin glasses and random fields. World Scientific, Singapore, pp 59–98

    Chapter  Google Scholar 

  83. Escobedo FA, de Pablo JJ (1995) Monte Carlo simulation of the chemical potential of polymers in an expanded ensemble. J Chem Phys 103:2703–2710

    Article  CAS  Google Scholar 

  84. Irbäck A, Potthast F (1995) Studies of an off-lattice model for protein folding—sequence dependence and improved sampling at finite temperature. J Chem Phys 103:10298–10305

    Article  Google Scholar 

  85. Irbäck A, Sandelin E (1999) Monte Carlo study of the phase structure of compact polymer chains. J Chem Phys 110:12256–12262

    Article  Google Scholar 

  86. Mitsutake A, Okamoto Y (2000) Replica-exchange simulated tempering method for simulations of frustrated systems. Chem Phys Lett 332:131–138

    Article  CAS  Google Scholar 

  87. Mitsutake A, Okamoto Y (2004) Replica-exchange extensions of simulated tempering method. J Chem Phys 121:2491–2504

    Article  PubMed  CAS  Google Scholar 

  88. Park S, Pande V (2007) Choosing weights for simulated tempering. Phys Rev E 76:016703

    Article  CAS  Google Scholar 

  89. Zheng L, Chen M, Yang W (2009) Simultaneous escaping of explicit and hidden free energy barriers: application of the orthogonal space random walk strategy in generalized ensemble based conformational sampling. J Chem Phys 130:234105

    Article  PubMed  CAS  Google Scholar 

  90. Zhang C, Ma J (2010) Enhanced sampling and applications in protein folding in explicit solvent. J Chem Phys 132:244101

    Article  PubMed  CAS  Google Scholar 

  91. Kim J, Straub JE (2010) Generalized simulated tempering for exploring strong phase transitions. J Chem Phys 133:154101

    Article  PubMed  CAS  Google Scholar 

  92. Hukushima K, Nemoto K (1996) Exchange Monte Carlo method and application to spin glass simulations. J Phys Soc Jpn 65:1604–1608

    Article  CAS  Google Scholar 

  93. Hukushima K, Takayama H, Nemoto K (1996) Application of an extended ensemble method to spin glasses. Int J Mod Phys C 7:337–344

    Article  Google Scholar 

  94. Geyer CJ (1991) Markov chain Monte Carlo maximum likelihood. In: Keramidas EM (ed) Computing science and statistics: proceedings 23rd symposium on the interface. Interface Foundation, Fairfax Station, pp 156–163

    Google Scholar 

  95. Swendsen RH, Wang J-S (1986) Replica Monte Carlo simulation of spin glasses. Phys Rev Lett 57:2607–2609

    Article  PubMed  Google Scholar 

  96. Kimura K, Taki K (1991) Time-homogeneous parallel annealing algorithm. In: Vichnevetsky R, Miller, JJH (eds) IMACS 91 Proceedings of the 13th World Congress on Computation and Applied Mathematics, vol 2. pp 827–828

    Google Scholar 

  97. Frantz DD, Freeman DL, Doll JD (1990) Reducing quasi-ergodic behavior in Monte Carlo simulations by J-walking—applications to atomic clusters. J Chem Phys 93:2769–2784

    Article  CAS  Google Scholar 

  98. Tesi MC, van Rensburg EJJ, Orlandini E, Whittington SG (1996) Monte Carlo study of the interacting self-avoiding walk model in three dimensions. J Stat Phys 82:155–181

    Article  Google Scholar 

  99. Iba Y (2001) Extended ensemble Monte Carlo. Int J Mod Phys C 12:623–656

    Article  Google Scholar 

  100. Hansmann UHE (1997) Parallel tempering algorithm for conformational studies of biological molecules. Chem Phys Lett 281:140–150

    Article  CAS  Google Scholar 

  101. Sugita Y, Okamoto Y (1999) Replica-exchange molecular dynamics method for protein folding. Chem Phys Lett 314:141–151

    Article  CAS  Google Scholar 

  102. Wu MG, Deem MW (1999) Efficient Monte Carlo methods for cyclic peptides. Mol Phys 97:559–580

    Article  CAS  Google Scholar 

  103. Sugita Y, Kitao A, Okamoto Y (2000) Multidimensional replica-exchange method for free-energy calculations. J Chem Phys 113:6042–6051

    Article  CAS  Google Scholar 

  104. Woods CJ, Essex JW, King MA (2003) The development of replica-exchange-based free-energy methods. J Phys Chem B 107:13703–13710

    Article  CAS  Google Scholar 

  105. Sugita Y, Okamoto Y (2000) Replica-exchange multicanonical algorithm and multicanonical replica-exchange method for simulating systems with rough energy landscape. Chem Phys Lett 329:261–270

    Article  CAS  Google Scholar 

  106. Gront D, Kolinski A, Skolnick J (2000) Comparison of three Monte Carlo conformational search strategies for a proteinlike homopolymer model: folding thermodynamics and identification of low-energy structures. J Chem Phys 113:5065–5071

    Article  CAS  Google Scholar 

  107. Verkhivker GM, Rejto PA, Bouzida D, Arthurs S,Colson AB, Freer ST, Gehlhaar DK, Larson V, Luty BA, Marrone T, Rose PW (2001) Parallel simulated tempering dynamics of ligand-protein binding with ensembles of protein conformations. Chem Phys Lett 337:181–189

    Article  CAS  Google Scholar 

  108. Fukunishi F, 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

    Article  CAS  Google Scholar 

  109. Mitsutake A, Sugita Y, Okamoto Y (2003) Replica-exchange multicanonical and multicanonical replica-exchange Monte Carlo simulations of peptides. I. Formulation and benchmark test. J Chem Phys 118:6664–6675

    CAS  Google Scholar 

  110. Mitsutake A, Sugita Y, Okamoto Y (2003) Replica-exchange multicanonical and multicanonical replica-exchange Monte Carlo simulations of peptides. II. Application to a more complex system. J Chem Phys 118:6676–6688

    CAS  Google Scholar 

  111. Sikorski A, Romiszowski P (2003) Thermodynamical properties of simple models of protein-like heteropolymers. Biopolymers 69:391–398

    Article  PubMed  CAS  Google Scholar 

  112. Lin CY, Hu CK, Hansmann UHE (2003) Parallel tempering simulations of HP-36. Proteins 52:436–445

    Article  PubMed  CAS  Google Scholar 

  113. La Penna G, Mitsutake A, Masuya M, Okamoto Y (2003) Molecular dynamics of C-peptide of ribonuclease A studied by replica-exchange Monte Carlo method and diffusion theory. Chem Phys Lett 380:609–619

    Article  CAS  Google Scholar 

  114. Kokubo H, Okamoto Y (2004) Prediction of membrane protein structures by replica-exchange Monte Carlo simulations: case of two helices. J Chem Phys 120:10837

    Article  PubMed  CAS  Google Scholar 

  115. Kokubo H, Okamoto Y (2009) Analysis of helix-helix interactions of bacteriorhodopsin by replica-exhcange simulations. Biophys J 96:765–776

    Article  CAS  Google Scholar 

  116. Falcioni M, Deem DW (1999) A biased Monte Carlo scheme for zeolite structure solution. J Chem Phys 110:1754–1766

    Article  CAS  Google Scholar 

  117. Yan Q, de Pablo JJ (1999) Hyper-parallel tempering Monte Carlo: application to the Lennard–Jones fluid and the restricted primitive model. J Chem Phys 111:9509–9516

    Article  CAS  Google Scholar 

  118. Nishikawa T, Ohtsuka H, Sugita Y, Mikami M, Okamoto Y (2000) Replica-exchange Monte Carlo method for Ar fluid. Prog Theor Phys (Suppl) 138:270–271

    Article  CAS  Google Scholar 

  119. Kofke DA (2002) On the acceptance probability of replica-exchange Monte Carlo trials. J Chem Phys 117:6911–6914

    Article  CAS  Google Scholar 

  120. Okabe T, Kawata M, Okamoto Y, Mikami M (2001) Replica-exchange Monte Carlo method for the isobaric-isothermal ensemble. Chem Phys Lett 335:435–439

    Article  CAS  Google Scholar 

  121. Ishikawa Y, Sugita Y, Nishikawa T, Okamoto Y (2001) Ab initio replica-exchange Monte Carlo method for cluster studies. Chem Phys Lett 333:199–206

    Article  CAS  Google Scholar 

  122. Garcia AE, Sanbonmatsu KY (2001) Exploring the energy landscape of a beta hairpin in explicit solvent. Proteins 42:345–354

    Article  PubMed  CAS  Google Scholar 

  123. Zhou RH, Berne BJ, Germain R (2001) The free energy landscape for beta hairpin folding in explicit water. Proc Natl Acad Sci USA 98:14931–14936

    Article  PubMed  CAS  Google Scholar 

  124. Garcia AE, Sanbonmatsu KY (2002) α-Helical stabilization by side chain shielding of backbone hydrogen bonds. Proc Natl Acad Sci USA 99:2782–2787

    Article  PubMed  CAS  Google Scholar 

  125. Zhou RH, Berne BJ (2002) Proc Natl Acad Sci USA 99:12777–12782

    Article  PubMed  CAS  Google Scholar 

  126. Feig M, MacKerell AD, Brooks CL III (2003) Force field influence on the observation of pi- helical protein structures in molecular dynamics simulations. J Phys Chem B 107:2831–2836

    Article  CAS  Google Scholar 

  127. Rhee YM, Pande VS (2003) Multiplexed-replica exchange molecular dynamics method for protein folding simulation. Biophys J 84:775–786

    Article  PubMed  CAS  Google Scholar 

  128. Paschek D, Garcia AE (2004) Reversible temperature and pressure denaturation of a protein fragment: a replica exchange molecular dynamics simulation study. Phys Rev Lett 93:238105

    Article  PubMed  CAS  Google Scholar 

  129. Paschek D, Gnanakaran S, Garcia AE (2005) Simulations of the pressure and temperature unfolding of an α-helical peptide. Proc Natl Acad Sci USA 102:6765–6770

    Article  PubMed  CAS  Google Scholar 

  130. Pitera JW, Swope W (2003) Understanding folding and design: replica-exchange simulations of “Trp-cage” fly miniproteins. Proc Natl Acad Sci USA 100:7587–7592

    Article  PubMed  CAS  Google Scholar 

  131. Ohkubo YZ, Brooks CL III (2003) Exploring Flory’s isolated-pair hypothesis: Statistical mechanics of helix-coil transitions in polyalanine and the C-peptide from RNase A. Proc Natl Acad Sci USA 100:13916–13921

    Article  PubMed  CAS  Google Scholar 

  132. Fenwick MK, Escobedo FA (2003) Hybrid Monte Carlo with multidimensional replica exchanges: conformational equilibria of the hypervariable reigons of a llamma V-HH antibody domain. Biopolymers 68:160–177

    Article  PubMed  CAS  Google Scholar 

  133. Xu HF, Berne BJ (2000) Multicanonical jump walking annealing: an efficient method for geometric optimization. J Chem Phys 112:2701–2708

    Article  CAS  Google Scholar 

  134. Faller R, Yan Q, de Pablo JJ (2002) Multicanonical parallel tempering. J Chem Phys 116:5419–5423

    Article  CAS  Google Scholar 

  135. Fenwick MK, Escobedo FA (2003) Expanded ensemble and replica exchange methods for simulation of protein-like systems. J Chem Phys 119:11998–12010

    Article  CAS  Google Scholar 

  136. Murata K, Sugita Y, Okamoto Y (2004) Free energy calculations for DNA base stacking by replica-exchange umbrella sampling. Chem Phys Lett 385:1–7

    Article  CAS  Google Scholar 

  137. Felts AK, Harano Y, Gallicchio E, Levy RM (2004) Free energy surfaces of β-hairpin and α-helical peptides generated by replica exchange molecular dynamics with the AGBNP implicit solvent model. Proteins 56:310–321

    Article  PubMed  CAS  Google Scholar 

  138. Mitsutake A, Kinoshita M, Okamoto Y, Hirata F (2004) Combination of the replica-exchange Monte Carlo method and the reference interaction site model theory for simulating a peptide molecule in aqueous solution. J Phys Chem B 108:19002–19012

    Article  CAS  Google Scholar 

  139. Baumketner A, Shea JE (2005) Free energy landscapes for amyloidogenic tetrapeptides dimerization. Biophys J 89:1493–1503

    Article  PubMed  CAS  Google Scholar 

  140. Yoda T, Sugita Y, Okamoto Y (2007) Cooperative folding mechanism of a beta-hairpin peptide studied by a multicanonical replica-exchange molecular dynamics simulation. Proteins 66:846–859

    Article  PubMed  CAS  Google Scholar 

  141. Roitberg AE, Okur A, Simmerling C (2007) Coupling of replica exchange simulations to a non-Boltzmann structure reservoir. J Phys Chem B 111:2415–2418

    Article  PubMed  CAS  Google Scholar 

  142. Rosta E, Buchete N-Y, Hummber G (2009) Thermostat artifacts in replica exchange molecular dynamics simulations. J Chem Theory Comput 5:1393–1399

    Article  PubMed  CAS  Google Scholar 

  143. Yoda T, Sugita Y, Okamoto Y (2010) Hydrophobic core formation and dehydration in protein folding studied by generalized-ensemble simulations. Biophys J 99:1637–1644

    Article  PubMed  CAS  Google Scholar 

  144. De Simone A, Derreumaux P (2010) Low molecular weight oligomers of amyloid peptides display β-barrel conformations: a replica exchange molecular dynamics study in explicit solvent. J Chem Phys 132:165103

    Article  PubMed  CAS  Google Scholar 

  145. Hukushima K (1999) Domain-wall free energy of spin-glass models: numerical method and boundary conditions. Phys Rev E 60:3606–3614

    Article  CAS  Google Scholar 

  146. Whitfield TW, Bu L, Straub JE (2002) Generalized parallel sampling. Physica A 305:157–171

    Article  Google Scholar 

  147. Kwak W, Hansmann UHE (2005) Efficient sampling of protein structures by model hopping. Phys Rev Lett 95:138102

    Article  PubMed  CAS  Google Scholar 

  148. Bunker A, Dünweg B (2000) Parallel excluded volume tempering for polymer melts. Phys Rev E 63:016701

    Article  CAS  Google Scholar 

  149. Liu P, Kim B, Friesner RA, Bern BJ (2005) Replica exchange with solute tempering: a method for sampling biological systems in explicit water. Proc Natl Acad Sci USA 102:13749–13754

    Article  PubMed  CAS  Google Scholar 

  150. Affentranger R, Tavernelli I, Di Iorio EE (2006) A novel Hamiltonian replica exchange MD protocol to enhance protein conformational space sampling. J Chem Theory Comput 2:217–228

    Article  CAS  Google Scholar 

  151. Lou H, Cukier RI (2006) Molecular dynamics of apo-adenylate kinase: a distance replica exchange method for the free energy of conformational fluctuations. J Phys Chem B 110:24121–24137

    Article  PubMed  CAS  Google Scholar 

  152. Mu Y (2009) Dissociation aided and side chain sampling enhanced Hamiltonian replica exchange. J Chem Phys 130:164107

    Article  PubMed  CAS  Google Scholar 

  153. Itoh SG, Okumura H, Okamoto Y (2010) Replica-exchange method in van der Waals radius space: overcoming steric restrictions for biomolecules. J Chem Phys 132:134105

    Article  PubMed  CAS  Google Scholar 

  154. Mitsutake A, Okamoto Y (2009) From multidimensional replica-exchange method to multidimensional multicanonical algorithm and simulated tempering. Phys Rev E 79:047701

    Article  CAS  Google Scholar 

  155. Mitsutake A, Okamoto Y (2009) Multidimensional generalized-ensemble algorithms for complex systems. J Chem Phys 130:214105

    Article  PubMed  CAS  Google Scholar 

  156. Mitsutake A (2009) Simulated-tempering replica-exchange method for the multidimensional version. J Chem Phys 131:094105

    Article  PubMed  CAS  Google Scholar 

  157. Mori Y, Okamoto Y (2010) Generalized-ensemble algorithms for the isobaric-isothermal ensemble. J Phys Soc Jpn 79:074003

    Article  CAS  Google Scholar 

  158. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21:1087–1092

    Article  CAS  Google Scholar 

  159. Nosé S (1984) A molecular dynamics method for simulations in the canonical ensemble. Mol Phys 52:255–268

    Article  Google Scholar 

  160. Nosé S (1984) A unified formulation of the constant temperature molecular dynamics methods. J Chem Phys 81:511–519

    Article  Google Scholar 

  161. Shirts MR, Chodera JD (2008) Statistically optimal analysis of samples from multiple equilibrium states. J Chem Phys 129:124105

    Article  PubMed  CAS  Google Scholar 

  162. Berg BA (2003) Multicanonical simulations step by step. Comp Phys Commun 153:397–406

    Article  CAS  Google Scholar 

  163. Mori Y, Okamoto Y (2010) Replica-exchange molecular dynamics simulations for various constant temperature algorithms. J Phys Soc Jpn 79:074001

    Article  CAS  Google Scholar 

  164. Allen MP, Tildesley DJ (1987) Computer Simulation of Liquids. Oxford, New York, p 259

    Google Scholar 

  165. Andersen HG (1980) Molecular dynamics simulations at constant pressure and/or temperature. J Chem Phys 72:2384–2393

    Article  CAS  Google Scholar 

  166. Hoover WG, Ladd AJC, Moran B (1982) High strain rate plastic flow studied via non-equilibrium molecular dynamics. Phys Rev Lett 48:1818–1820

    Article  CAS  Google Scholar 

  167. Evans DJ (1983) Computer experiment for non-linear thermodynamics of couette flow. J Chem Phys 78:3297–3302

    Article  CAS  Google Scholar 

  168. Evans DJ, Morriss GP (1983) The isothermal isobaric molecular dynamics ensemble. Phys Lett A 98:433–436

    Article  Google Scholar 

  169. Hoover WG (1985) Canonical dynamics—equilibrium phase space distributions. Phys Rev A 31:1695–1697

    Article  PubMed  Google Scholar 

  170. Martyna GJ, Klein ML, Tuckerman M (1992) Nosé–Hoover chains—the canonical ensemble via continuous dynamics. J Chem Phys 97:2635–2643

    Article  Google Scholar 

  171. Bond SD, Leimkuhler BJ, Laird BB (1999) The Nosé–Poincaré method for constant temperature molecular dynamics. J Comput Phys 151:114–134

    Article  CAS  Google Scholar 

  172. McDonald IR (1972) NpT-ensemble Monte Carlo calculations for binary liquid mixtures. Mol Phys 23:41-58

    Article  CAS  Google Scholar 

  173. Myers JK, Pace CN, Scholtz JM (1997) A direct comparison of helix propensity in proteins and peptides. Proc Natl Acad Sci USA 94:2833–2837

    Article  PubMed  CAS  Google Scholar 

  174. Momany FA, McGuire RF, Burgess AW, Scheraga HA (1975) Energy parameters in polypeptides. VII. Geometric parameters, partial atomic charges, nonbonded interactions, hydrogen bond interactions, and intrinsic torsional potentials for the naturally occurring amino acids. J Phys Chem 79:2361–2381

    Article  CAS  Google Scholar 

  175. Némethy G, Pottle MS, Scheraga HA (1983) Energy parameters in polypeptides. 9. Updating of geometrical parameters, nonbonded interactions, and hydrogen bond interactions for the naturally occurring amino acids. J Phys Chem 87:1883–1887

    Article  Google Scholar 

  176. Sippl MJ, Némethy G, Scheraga HA (1984) Intermolecular potentials from crystal data. 6. Determination of empirical potentials for O-H…O=C hydrogen bonds from packing configurations. J Phys Chem 88:6231–6233

    Article  CAS  Google Scholar 

  177. Ooi T, Oobatake M, Némethy G, Scheraga HA (1987) Accessible surface areas as a measure of the thermodynamic parameters of hydration of peptides. Proc Natl Acad Sci USA 84:3086–3090

    Article  PubMed  CAS  Google Scholar 

  178. Masuya M, unpublished; see http://biocomputing.cc/nsol/.

  179. Kitahara R, Akasaka K (2003) Close identity of a pressure-stabilized intermediate with a kinetic intermediate in protein folding. Proc Natl Acad Sci USA 100:3167–3172

    Article  PubMed  CAS  Google Scholar 

  180. Kitahara R, Yokoyama S, Akasaka K (2005) NMR snapshots of a fluctuating protein structure: ubiquitin at 30 bar–3 kbar. J Mol Biol 347:277–285

    Article  PubMed  CAS  Google Scholar 

  181. Quigley D, Probert MIJ (2004) Landevin dynamics in constant pressure extended systems. J Chem Phys 120:11432–11441

    Article  PubMed  CAS  Google Scholar 

  182. Darden T, York D, Pedersen L (1993) Particle mesh Ewald—an Nlog(N) method for Ewald sums in large systems. J Chem Phys 98:10089–10092

    Article  CAS  Google Scholar 

  183. Essmann U, Perera L, Berkowitz ML, Darden T, Lee H, Pedersen LG (1995) A smooth particle mesh Ewald method. J Chem Phys 103:8577–8593

    Article  CAS  Google Scholar 

  184. MacKerell AD Jr, Bashford D, Bellott M, Dunbrack RL Jr, Evanseck JD, Field MJ, Fischer S, Gao J, Guo H, Ha S, Joseph-McCarthy D, Kuchnir L, Kuczera K, Lau FTK, Mattos C, Michnick S, Ngo T, Nguyen DT, Prodhom B, Reiher WE III, Roux B, Schlenkrich M, Smith JC, Stote R, Straub J, Watanabe M, Wiórkiewicz-Kuczera J, Yin D, Karplus M (1998) All-atom empirical potential for molecular modeling and dynamics studies of proteins. J Phys Chem B 102:3586–3616

    Article  CAS  Google Scholar 

  185. MacKerell AD Jr, Feig M, Brooks CL III (2004) Improved treatment of the protein backbone in empirical force fields. J Am Chem Soc 126:698–699

    Article  PubMed  CAS  Google Scholar 

  186. MacKerell AD Jr, Feig M, Brooks CL III (2004) Extending the treatment of backbone energetics in protein force fields: Limitations of gas-phase quantum mechanics in reproducing protein conformational distributions in molecular dynamics simulations. J Comput Chem 25:1400–1415

    Article  PubMed  CAS  Google Scholar 

  187. Jorgensen WL, Chandrasekhar J, Madura JD, Impey RW, Klein ML (1983) Comparison of simple potential functions for simulating liquid water. J Chem Phys 79:926–935

    Article  CAS  Google Scholar 

  188. Phillips JC, Braun R, Wang W, Gumbart J, Tajkhorshid E, Villa E, Chipot C, Skeel RD, Kale L, Schulten K (2005) Scalable molecular dynamics with NAMD. J Comput Chem 26:1781–1802

    Article  PubMed  CAS  Google Scholar 

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Acknowledgments

Some of the results were obtained by the computations on the supercomputersat the Institute for Molecular Science, Okazaki, and the Institute for Solid State Physics, University of Tokyo, Japan. This work was supported, in part, by Grants-in-Aid for Scientific Research on Innovative Areas (“Fluctuations and Biological Functions”) and for the Next-Generation Super Computing Project, Nanoscience Program, and Computational Materials Science Initiative from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

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Mitsutake, A., Mori, Y., Okamoto, Y. (2013). Enhanced Sampling Algorithms. In: Monticelli, L., Salonen, E. (eds) Biomolecular Simulations. Methods in Molecular Biology, vol 924. Humana Press, Totowa, NJ. https://doi.org/10.1007/978-1-62703-017-5_7

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