Rugged Free Energy Landscapes pp 369-407 | Cite as
Generalized-Ensemble Algorithms for Protein Folding Simulations
Abstract
Conventional simulations of complex systems in the canonical ensemble suffer from the quasi-ergodicity problem. A simulation in generalized ensemble overcomes this difficulty by performing a random walk in potential energy space and other parameter space. 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 the generalized-ensemble algorithms. 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 further extensions of the above three methods.
Keywords
Random Walk Simulated Tempering Weight Factor Monte Carlo Replica ExchangePreview
Unable to display preview. Download preview PDF.
References
- 1.U. H. E. Hansmann, Y. Okamoto: Generalized-ensemble approach for protein folding simulations. In Annual Reviews of Computational Physics VI, edited by D. Stauffer (World Scientific, Singapore, 1999), pp. 129–157Google Scholar
- 2.A. Mitsutake, Y. Sugita, Y. Okamoto: Biopolymers (Peptide Science) 60, 96–123 (2001)CrossRefGoogle Scholar
- 3.Y. Sugita, Y. Okamoto: Free-energy calculations in protein folding by generalized-ensemble algorithms. In Lecture Notes in Computational Science and Engineering, edited by T. Schlick, H. H. Gan (Springer-Verlag, Berlin, 2002), pp. 304–332 [cond-mat/0102296]Google Scholar
- 4.B. A. Berg: Generalized Ensemble Simulations for Complex Systems, Comput. Phys. Commun. 147, 52. 52–57 (2002)MATHADSGoogle Scholar
- 5.Y. Okamoto: J. Mol. Graphics Mod. 22, 425–439 (2004) [cond-mat/0308360]CrossRefGoogle Scholar
- 6.H. Kokubo, Y. Okamoto: Mol. Sim. 32, 791–801 (2006)MATHCrossRefGoogle Scholar
- 7.S. G. Itoh, H. Okumura, Y. Okamoto: Mol. Sim. 33, 47–56 (2007)MATHCrossRefGoogle Scholar
- 8.A. M. Ferrenberg, R. H. Swendsen: Phys. Rev. Lett. 61, 2635–2638 (1988); A. M. Ferrenberg, R. H. Swendsen: Phys. Rev. Lett. 63, 1658 (1989)ADSCrossRefGoogle Scholar
- 9.A. M. Ferrenberg, R. H. Swendsen: Phys. Rev. Lett. 63, 1195–1198 (1989)ADSCrossRefGoogle Scholar
- 10.S. Kumar, D. Bouzida, R. H. Swendsen, P. A. Kollman, J. M. Rosenberg: J. Comput. Chem. 13, 1011–1021 (1992)CrossRefGoogle Scholar
- 11.R. H. Swendsen, J. S. Wang: Phys. Rev. Lett. 58, 86–88 (1987)ADSCrossRefGoogle Scholar
- 12.U. Wolff: Phys. Rev. Lett. 62, 361–364 (1989)ADSCrossRefGoogle Scholar
- 13.H. G. Evertz, G. Lana, M. Marcu: Phys. Rev. Lett. 70, 875–879 (1993).ADSCrossRefGoogle Scholar
- 14.B. A. Berg, T. Neuhaus: Phys. Lett. B 267, 249–253 (1991)ADSCrossRefGoogle Scholar
- 15.B. A. Berg, T. Neuhaus: Phys. Rev. Lett. 68, 9–12 (1992)ADSCrossRefGoogle Scholar
- 16.B. A. Berg: Fields Inst. Commun. 26, 1–24 (2000) [cond-mat/9909236]Google Scholar
- 17.W. Janke: Physica A 254, 164–178 (1998)CrossRefGoogle Scholar
- 18.J. Lee: Phys. Rev. Lett. 71, 211–214 (1993); J. Lee: Phys. Rev. Lett. 71, 2353(E) (1993)Google Scholar
- 19.M. Mezei: J. Comput. Phys. 68, 237–248 (1987)MATHADSMathSciNetCrossRefGoogle Scholar
- 20.C. Bartels, M. Karplus: J. Phys. Chem. B 102, 865–880 (1998)CrossRefGoogle Scholar
- 21.G. M. Torrie, J. P. Valleau: J. Comput. Phys. 23, 187–199 (1977)ADSCrossRefGoogle Scholar
- 22.J. S. Wang, R. H. Swendsen: J. Stat. Phys. 106, 245–285 (2002)MATHMathSciNetCrossRefGoogle Scholar
- 23.F. Wang, D. P. Landau: Phys. Rev. Lett. 86, 2050–2053 (2001)ADSCrossRefGoogle Scholar
- 24.F. Wang, D. P. Landau: Phys. Rev. E 64, 056101 (2001)ADSCrossRefGoogle Scholar
- 25.Q. Yan, R. Faller, J. J. de Pablo: J. Chem. Phys. 116, 8745–8749 (2002)ADSCrossRefGoogle Scholar
- 26.S. Trebst, D. A. Huse, M. Troyer: Phys. Rev. E 70, 046701 (2004)ADSCrossRefGoogle Scholar
- 27.B. A. Berg, T. Celik: Phys. Rev. Lett. 69, 2292–2295 (1992)ADSCrossRefGoogle Scholar
- 28.B. A. Berg, U. H. E. Hansmann, T. Neuhaus: Phys. Rev. B 47, 497–500 (1993)ADSCrossRefGoogle Scholar
- 29.W. Janke, S. Kappler: Phys. Rev. Lett. 74, 212–215 (1995)ADSCrossRefGoogle Scholar
- 30.B. A. Berg, W. Janke: Phys. Rev. Lett. 80, 4771–4774 (1998)ADSCrossRefGoogle Scholar
- 31.N. Hatano, J. E. Gubernatis: Prog. Theor. Phys. (Suppl.) 138, 442–447 (2000)ADSGoogle Scholar
- 32.B. A. Berg, A. Billoire, W. Janke: Phys. Rev. B 61, 12143–12150 (2000)ADSCrossRefGoogle Scholar
- 33.U. H. E. Hansmann, Y. Okamoto: J. Comput. Chem. 14, 1333–1338 (1993)CrossRefGoogle Scholar
- 34.U. H. E. Hansmann, Y. Okamoto: Physica A 212, 415–437 (1994)ADSCrossRefGoogle Scholar
- 35.M. H. Hao, H. A. Scheraga: J. Phys. Chem. 98, 4940–4948 (1994)CrossRefGoogle Scholar
- 36.Y. Okamoto, U. H. E. Hansmann: J. Phys. Chem. 99, 11276–11287 (1995)CrossRefGoogle Scholar
- 37.N. B. Wilding: Phys. Rev. E 52, 602–611 (1995)ADSCrossRefGoogle Scholar
- 38.A. Kolinski, W. Galazka, J. Skolnick: Proteins 26, 271–287 (1996)CrossRefGoogle Scholar
- 39.N. Urakami, M. Takasu: J. Phys. Soc. Jpn. 65, 2694–2699 (1996)ADSCrossRefGoogle Scholar
- 40.S. Kumar, P. Payne, M. Vásquez: J. Comput. Chem. 17, 1269–1275 (1996)CrossRefGoogle Scholar
- 41.U. H. E. Hansmann, Y. Okamoto, F. Eisenmenger: Chem. Phys. Lett. 259, 321–330 (1996)CrossRefADSGoogle Scholar
- 42.U. H. E. Hansmann, Y. Okamoto: Phys. Rev. E 54, 5863–5865 (1996)ADSCrossRefGoogle Scholar
- 43.U. H. E. Hansmann, Y. Okamoto: J. Comput. Chem. 18, 920–933 (1997)CrossRefGoogle Scholar
- 44.N. Nakajima, H. Nakamura, A. Kidera: J. Phys. Chem. B 101, 817–824 (1997)CrossRefGoogle Scholar
- 45.C. Bartels, M. Karplus: J. Comput. Chem. 18, 1450–1462 (1997)CrossRefGoogle Scholar
- 46.J. Higo, N. Nakajima, H. Shirai, A. Kidera, H. Nakamura: J. Comput. Chem. 18, 2086–2092 (1997)CrossRefGoogle Scholar
- 47.Y. Iba, G. Chikenji, M. Kikuchi: J. Phys. Soc. Jpn. 67, 3327–3330 (1998)ADSCrossRefGoogle Scholar
- 48.A. Mitsutake, U. H. E. Hansmann, Y. Okamoto: J. Mol. Graph. Mod. 16, 226–238; 262–263 (1998)Google Scholar
- 49.U. H. E. Hansmann, Y. Okamoto: J. Phys. Chem. B 103, 1595–1604 (1999)CrossRefGoogle Scholar
- 50.H. Shimizu, K. Uehara, K. Yamamoto, Y. Hiwatari: Mol. Sim. 22, 285–301 (1999)CrossRefGoogle Scholar
- 51.S. Ono, N. Nakajima, J. Higo, H. Nakamura: Chem. Phys. Lett. 312, 247–254 (1999)CrossRefADSGoogle Scholar
- 52.A. Mitsutake, Y. Okamoto: J. Chem. Phys. 112, 10638–10647 (2000)ADSCrossRefGoogle Scholar
- 53.K. Sayano, H. Kono, M. M. Gromiha, A. Sarai: J. Comput. Chem. 21, 954–962 (2000)CrossRefGoogle Scholar
- 54.F. Yasar, T. Celik, B. A. Berg, H. Meirovitch: J. Comput. Chem. 21, 1251–1261 (2000)CrossRefGoogle Scholar
- 55.A. Mitsutake, M. Kinoshita, Y. Okamoto, F. Hirata: Chem. Phys. Lett. 329, 295–303 (2000)CrossRefADSGoogle Scholar
- 56.M. S. Cheung, A. E. Garcia, J. N. Onuchic: Proc. Natl. Acad. Sci. USA 99, 685–690 (2002)ADSCrossRefGoogle Scholar
- 57.N. Kamiya, J. Higo, H. Nakamura: Protein Sci. 11, 2297–2307 (2002)CrossRefGoogle Scholar
- 58.S. W. Jang, Y. Pak, S. M. Shin: J. Chem. Phys. 116, 4782–4786 (2002)ADSCrossRefGoogle Scholar
- 59.J. G. Kim, Y. Fukunishi, H. Nakamura: Phys. Rev. E 67, 011105 (2003)ADSCrossRefGoogle Scholar
- 60.N. Rathore, T. A. Knotts, IV, J. J. de Pablo: J. Chem. Phys. 118, 4285–4290 (2003)ADSCrossRefGoogle Scholar
- 61.T. Terada, Y. Matsuo, A. Kidera: J. Chem. Phys. 118, 4306–4311 (2003)ADSCrossRefGoogle Scholar
- 62.B. A. Berg, H. Noguchi, Y. Okamoto: Phys. Rev. E 68, 036126 (2003)ADSCrossRefGoogle Scholar
- 63.M. Bachmann, W. Janke: Phys. Rev. Lett. 91, 208105 (2003)ADSCrossRefGoogle Scholar
- 64.H. Okumura, Y. Okamoto: Chem. Phys. Lett. 383, 391–396 (2004)CrossRefADSGoogle Scholar
- 65.T. Munakata, S. Oyama: Phys. Rev. E 54, 4394–4398 (1996)ADSCrossRefGoogle Scholar
- 66.A. P. Lyubartsev, A. A. Martinovski, S. V. Shevkunov, P. N. Vorontsov-Velyaminov: J. Chem. Phys. 96, 1776–1783 (1992)ADSCrossRefGoogle Scholar
- 67.E. Marinari, G. Parisi: Europhys. Lett. 19, 451–458 (1992)ADSCrossRefGoogle Scholar
- 68.E. Marinari, G. Parisi, J. J. Ruiz-Lorenzo: In Spin Glasses and Random Fields, edited by A. P. Young (World Scientific, Singapore, 1998), pp. 59–98Google Scholar
- 69.A. Irbäck, F. Potthast: J. Chem. Phys. 103, 10298–10305 (1995)ADSCrossRefGoogle Scholar
- 70.A. Irbäck, E. Sandelin: J. Chem. Phys. 110, 12256–12262 (1999)ADSCrossRefGoogle Scholar
- 71.G. R. Smith, A. D. Bruce: Phys. Rev. E 53, 6530–6543 (1996)ADSCrossRefGoogle Scholar
- 72.U. H. E. Hansmann: Phys. Rev. E 56, 6200–6203 (1997)ADSCrossRefGoogle Scholar
- 73.B. A. Berg: Nucl. Phys. B (Proc. Suppl.) 63A-C, 982–984 (1998)ADSCrossRefGoogle Scholar
- 74.W. Janke: Histograms and all that. In Computer Simulations of Surfaces and Interfaces V, NATO Science Series, II Mathematics, Physics and Chemistry Vol. 114, Proceedings of the NATO Advanced Study Institute, edited by B. Dünweg, D. P. Landau, A. I. Milchev (Kluwer, Dordrecht, 2003), pp. 137–157Google Scholar
- 75.K. Hukushima, K. Nemoto: J. Phys. Soc. Jpn. 65, 1604–1608 (1996)ADSCrossRefGoogle Scholar
- 76.K. Hukushima, H. Takayama, K. Nemoto: Int. J. Mod. Phys. C 7, 337–344 (1996)ADSCrossRefGoogle Scholar
- 77.C. J. Geyer: In Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, edited by E. M. Keramidas (Interface Foundation, Fairfax Station, 1991), pp. 156–163Google Scholar
- 78.R. H. Swendsen, J.-S. Wang: Phys. Rev. Lett. 57, 2607–2609 (1986)ADSMathSciNetCrossRefGoogle Scholar
- 79.K. Kimura, K. Taki: In Proceedings of the 13th IMACS World Congress on Computation and Applied Mathematics (IMACS ’91), edited by R. Vichnevetsky, J. J. H. Miller, Vol. 2, pp. 827–828Google Scholar
- 80.D. D. Frantz, D. L. Freeman, J. D. Doll: J. Chem. Phys. 93, 2769–2784 (1990)ADSCrossRefGoogle Scholar
- 81.M. C. Tesi, E. J. J. van Rensburg, E. Orlandini, S. G. Whittington: J. Stat. Phys. 82, 155–181 (1996)MATHCrossRefADSGoogle Scholar
- 82.Y. Iba: Int. J. Mod. Phys. C 12, 623–656 (2001)ADSCrossRefGoogle Scholar
- 83.U. H. E. Hansmann: Chem. Phys. Lett. 281, 140–150 (1997)CrossRefADSGoogle Scholar
- 84.Y. Sugita, Y. Okamoto: Chem. Phys. Lett. 314, 141–151 (1999)CrossRefADSGoogle Scholar
- 85.M. G. Wu, M. W. Deem: Mol. Phys. 97, 559–580 (1999)ADSCrossRefGoogle Scholar
- 86.Y. Sugita, A. Kitao, Y. Okamoto: J. Chem. Phys. 113, 6042–6051 (2000)ADSCrossRefGoogle Scholar
- 87.C. J. Woods, J. W. Essex, M. A. King: J. Phys. Chem. B 107, 13703–13710 (2003)CrossRefGoogle Scholar
- 88.Y. Sugita, Y. Okamoto: Chem. Phys. Lett. 329, 261–270 (2000)CrossRefADSGoogle Scholar
- 89.A. Mitsutake, Y. Okamoto: Chem. Phys. Lett. 332, 131–138 (2000)CrossRefADSGoogle Scholar
- 90.D. Gront, A. Kolinski, J. Skolnick: J. Chem. Phys. 113, 5065–5071 (2000).ADSCrossRefGoogle Scholar
- 91.G. M. Verkhivker, P. A. Rejto, D. Bouzida, S. Arthurs, A. B. Colson, S. T. Freer, D. K. Gehlhaar, V. Larson, B. A. Luty, T. Marrone, P. W. Rose: Chem. Phys. Lett. 337, 181–189 (2001)CrossRefADSGoogle Scholar
- 92.H. Fukunishi, O. Watanabe, S. Takada: J. Chem. Phys. 116, 9058–9067 (2002)ADSCrossRefGoogle Scholar
- 93.A. Mitsutake, Y. Sugita, Y. Okamoto: J. Chem. Phys. 118, 6664–6675 (2003)ADSCrossRefGoogle Scholar
- 94.A. Mitsutake, Y. Sugita, Y. Okamoto: J. Chem. Phys. 118, 6676–6688 (2003)ADSCrossRefGoogle Scholar
- 95.A. Sikorski, P. Romiszowski: Biopolymers 69, 391–398 (2003)CrossRefGoogle Scholar
- 96.C. Y. Lin, C. K. Hu, U. H. E. Hansmann: Proteins 52, 436–445 (2003)CrossRefGoogle Scholar
- 97.G. La Penna, A. Mitsutake, M. Masuya, Y. Okamoto: Chem. Phys. Lett. 380, 609–619 (2003)CrossRefGoogle Scholar
- 98.M. Falcioni, M. W. Deem: J. Chem. Phys. 110, 1754–1766 (1999)ADSCrossRefGoogle Scholar
- 99.Q. Yan, J. J. de Pablo: J. Chem. Phys. 111, 9509–9516 (1999)ADSCrossRefGoogle Scholar
- 100.T. Nishikawa, H. Ohtsuka, Y. Sugita, M. Mikami, Y. Okamoto: Prog. Theor. Phys. (Suppl.) 138, 270–271 (2000)ADSGoogle Scholar
- 101.R. Yamamoto, W. Kob: Phys. Rev. E 61, 5473–5476 (2000)ADSCrossRefGoogle Scholar
- 102.D. A. Kofke: J. Chem. Phys. 117, 6911–6914 (2002)ADSCrossRefGoogle Scholar
- 103.T. Okabe, M. Kawata, Y. Okamoto, M. Mikami: Chem. Phys. Lett. 335, 435–439 (2001)CrossRefADSGoogle Scholar
- 104.Y. Ishikawa, Y. Sugita, T. Nishikawa, Y. Okamoto: Chem. Phys. Lett. 333, 199–206 (2001)CrossRefADSGoogle Scholar
- 105.A. E. Garcia, K. Y. Sanbonmatsu: Proteins 42, 345–354 (2001)CrossRefGoogle Scholar
- 106.R. H. Zhou, B. J. Berne, R. Germain: Proc. Natl. Acad. Sci. USA 98, 14931–14936 (2001)ADSCrossRefGoogle Scholar
- 107.A. E. Garcia, K. Y. Sanbonmatsu: Proc. Natl. Acad. Sci. USA 99, 2782–2787 (2002)ADSCrossRefGoogle Scholar
- 108.R. H. Zhou, B. J. Berne: Proc. Natl. Acad. Sci. USA 99, 12777–12782 (2002)ADSCrossRefGoogle Scholar
- 109.M. Feig, A. D. MacKerell, C. L. Brooks, III: J. Phys. Chem. B 107, 2831–2836 (2003)CrossRefGoogle Scholar
- 110.Y. M. Rhee, V. S. Pande: Biophys. J. 84, 775–786 (2003)CrossRefADSGoogle Scholar
- 111.J. W. Pitera, W. Swope: Proc. Natl. Acad. Sci. USA 100, 7587–7592 (2003)ADSCrossRefGoogle Scholar
- 112.M. K. Fenwick, F. A. Escobedo: Biopolymers 68, 160–177 (2003)CrossRefGoogle Scholar
- 113.H. F. Xu, B. J. Berne: J. Chem. Phys. 112, 2701–2708 (2000)ADSCrossRefGoogle Scholar
- 114.R. Faller, Q. Yan, J. J. de Pablo: J. Chem. Phys. 116, 5419–5423 (2002)ADSCrossRefGoogle Scholar
- 115.A. Mitsutake, Y. Okamoto: J. Chem. Phys. 121, 2491–2504 (2004)ADSCrossRefGoogle Scholar
- 116.M. K. Fenwick, F. A. Escobedo: J. Chem. Phys. 119, 11998–12010 (2003)ADSCrossRefGoogle Scholar
- 117.K. Hukushima: Phys. Rev. E 60, 3606–3614 (1999)ADSCrossRefGoogle Scholar
- 118.T. W. Whitfield, L. Bu, J. E. Straub: Physica A 305, 157–171 (2002)MATHADSCrossRefGoogle Scholar
- 119.W. Kwak, U. H. E. Hansmann: Phys. Rev. Lett. 95, 138102 (2005)Google Scholar
- 120.N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller: J. Chem. Phys. 21, 1087–1092 (1953)ADSCrossRefGoogle Scholar
- 121.S. Nosé: Mol. Phys. 52, 255–268 (1984)ADSCrossRefGoogle Scholar
- 122.S. Nosé: J. Chem. Phys. 81, 511–519 (1984)ADSCrossRefGoogle Scholar
- 123.B. A. Berg: Markov Chain Monte Carlo Simulations and Their Statistical Analysis (World Scientific, Singapore, 2004), p. 253Google Scholar
- 124.B. A. Berg: Multicanonical simulations step by step, cond-mat/0206333. Comput. Phys. Commun. 153, 397–406 (2003)ADSCrossRefMATHGoogle Scholar
- 125.T. Ooi, M. Oobatake, G. Némethy, H. A. Scheraga: Proc. Natl. Acad. Sci. USA 84, 3086–3090 (1987)ADSCrossRefGoogle Scholar
- 126.H. Kawai, Y. Okamoto, M. Fukugita, T. Nakazawa, T. Kikuchi: Chem. Lett. 1991, 213–216 (1991)CrossRefGoogle Scholar
- 127.Y. Okamoto, M. Fukugita, T. Nakazawa, H. Kawai: Protein Eng. 4, 639–647 (1991)CrossRefGoogle Scholar
- 128.M. Masuya: Manuscript in preparation; see also http://biocomputing.cc/nsol/Google Scholar
- 129.Y. Sugita, Y. Okamoto: UnpublishedGoogle Scholar
- 130.P. A. Kollman, R. Dixon, W. Cornell, T. Fox, C. Chipot, A. Pohorille: In Computer Simulation of Biomolecular Systems, Vol. 3, edited by A. Wilkinson, P. Weiner, W. F. van Gunsteren (Kluwer, Dordrecht, 1997), pp. 83–96Google Scholar
- 131.W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, M. L. Klein: J. Chem. Phys. 79, 926–935 (1983)ADSCrossRefGoogle Scholar
- 132.Y. Sugita, A. Kitao: Proteins 30, 388–400 (1998)CrossRefGoogle Scholar
- 133.A. Kitao, S. Hayward, N. Go: Proteins 33, 496–517 (1998)CrossRefGoogle Scholar
- 134.K. Morikami, T. Nakai, A. Kidera, M. Saito, H. Nakamura: Comput. Chem. 16, 243–248 (1992)CrossRefGoogle Scholar
- 135.Y. Sugita, Y. Okamoto: Biophys. J. 88, 3180–3190 (2005)CrossRefADSGoogle Scholar
- 136.K. R. Shoemaker, P. S. Kim, E. J. York, J. M. Stewart, R. L. Baldwin: Nature 326, 563–567 (1987)ADSCrossRefGoogle Scholar
- 137.K. R. Shoemaker, R. Fairman, D. A. Schultz, A. D. Robertson, E. J. York, J. M. Stewart, R. L. Baldwin: Biopolymers 29, 1–11 (1990)CrossRefGoogle Scholar
- 138.J. Wang, P. Cieplak, P. A. Kollman: J. Comput. Chem. 21, 1049-1074 (2000)Google Scholar
- 139.T. Yoda, Y. Sugita, Y. Okamoto: Chem. Phys. Lett. 386, 460–467 (2004)CrossRefADSGoogle Scholar
- 140.P. J. Kraulis: J. Appl. Crystallogr. 24, 946–950 (1991)CrossRefGoogle Scholar
- 141.E. A. Merritt, D. J. Bacon: Methods Enzymol. 277, 505–524 (1997)Google Scholar
- 142.T. Yoda, Y. Sugita, Y. Okamoto: Proteins 66, 846–859 (2007)CrossRefGoogle Scholar
- 143.T. Nagasima, Y. Sugita, A. Mitsutake, Y. Okamoto: UnpublishedGoogle Scholar
- 144.T. Nagasima, Y. Sugita, A. Mitsutake, Y. Okamoto: Comput. Phys. Commun. 146, 69–76 (2002)MATHADSCrossRefGoogle Scholar
- 145.Y. Okamoto: Metropolis algorithms in generalized ensemble. In The Monte Carlo Method in the Physical Sciences: Celebrating the 50th Anniversary of the Metropolis Algorithm, edited by J. E. Gubernatis (American Institute of Physics, Melville, 2003), pp. 248–260 [cond-mat/0308119]Google Scholar