Advertisement

Transition Path Sampling and Other Advanced Simulation Techniques for Rare Events

  • Christoph DellagoEmail author
  • Peter G. BolhuisEmail author
Chapter
Part of the Advances in Polymer Science book series (POLYMER, volume 221)

Computer simulations of molecular processes such as nucleation in first-order phase transitions or the folding of a protein are often complicated by widely disparate time scales related to important but rare events. Here, we will review sev eral recently developed computational methods designed to address the rare-events problem. In doing so, we will focus on the transition path sampling methodology.

Keywords

Computer simulation Rare events Rate constant Reaction mechanism Transition pathway 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hansen J-P, Löwen H (2002) In: Nielaba P, Mareschal M, Ciccotti G (eds) Bridging time scales: molecular simulations for the next decade. Springer, BerlinGoogle Scholar
  2. 2.
    Abrams C, Delle Site L, Kremer K (2002) In: Nielaba P, Mareschal M, Ciccotti G (eds) Bridging time scales: molecular simulations for the next decade. Springer, BerlinGoogle Scholar
  3. 3.
    Likos CN (2001) Phys Rep 348:267Google Scholar
  4. 4.
    Louis AA, Bolhuis PG, Hansen J-P, Meijer EJ (2000) Phys Rev Lett 85:2522Google Scholar
  5. 5.
    Jaffe C, Ross SD, Lo MW, Marsden J, Farrelly D, Uzer T (2002) Phys Rev Lett 89:011101Google Scholar
  6. 6.
    Eigen M, De Maeyer L (1959) Z Elektrochem 59:986Google Scholar
  7. 7.
    Geissler PL, Dellago C, Chandler D, Hutter J, Parrinello M (2001) Science 291:2121Google Scholar
  8. 8.
    Wilding N, Landau DP (2002) In: Nielaba P, Mareschal M, Ciccotti G (eds) Bridging time scales: molecular simulations for the next decade. Springer, BerlinGoogle Scholar
  9. 9.
    Frenkel D, Smit B (2002) Understanding molecular simulation, 2nd edn. Academic, San DiegoGoogle Scholar
  10. 10.
    Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Science 220:671Google Scholar
  11. 11.
    Gottwald D, Kahl G, Likos CN (2005) J Chem Phys 122:204503Google Scholar
  12. 12.
    Oganov AR, Glass CW (2006) J Chem Phys 124:244704Google Scholar
  13. 13.
    Meerbach E, Dittmer E, Horenko I, Schütte C (2006) In: Ferrario M, Ciccotti G, Binder K (eds) Computer simulations in condensed matter: from materials to chemical biology. Springer, BerlinGoogle Scholar
  14. 14.
    Deuflhard P, Huisinga W, Fischer A, Schütte C (2000) Lin Alg Appl 315:39Google Scholar
  15. 15.
    Wales DJ, Doye JPK (1997) J Chem Phys A 101:5111Google Scholar
  16. 16.
    Henkelman G, Johannesson G, Jónsson H (2000) In: Schwartz SD (ed) Progress on theoretical chemistry and physics. Kluwer, DordrechtGoogle Scholar
  17. 17.
    Elber R, Ghosh A, Cárdenas A, Stern H (2004) Adv Chem Phys 126:123Google Scholar
  18. 18.
    Hummer G, Kevrikidis IG (2003) J Chem Phys 118:10762Google Scholar
  19. 19.
    Laio A, Parrinello M (2002) Proc Natl Acad Sci USA 99:12562Google Scholar
  20. 20.
    Voter AF, Montalenti F, Germann TC (2002) Annu Rev Mater Res 32:321Google Scholar
  21. 21.
    Dellago C, Bolhuis PG, Csajka FS, Chandler D (1998) J Chem Phys 108:1964Google Scholar
  22. 22.
    Dellago C, Bolhuis PG, Chandler D (1998) J Chem Phys 108:9263Google Scholar
  23. 23.
    Bolhuis PG, Dellago C, Chandler D (1998) Faraday Discuss 110:421Google Scholar
  24. 24.
    Pratt LR (1986) J Chem Phys 85:5045Google Scholar
  25. 25.
    Geissler PL, Dellago C, Chandler D (1999) J Phys Chem B 103:3706Google Scholar
  26. 26.
    Bolhuis PG, Dellago C, Chandler D (2000) Proc Natl Acad Sci USA 97:5877Google Scholar
  27. 27.
    Hummer G (2004) J Chem Phys 120:516Google Scholar
  28. 28.
    Ma A, Dinner AR (2005) J Phys Chem B 109:6769Google Scholar
  29. 29.
    Peters B, Trout BL (2006) J Chem Phys 125:054108Google Scholar
  30. 30.
    Weinan E, Ren W, Vanden-Eijnden E (2002) Phys Rev B 66:052301Google Scholar
  31. 31.
    Weinan E, Ren W, Vanden-Eijnden E (2005) Chem Phys Lett 413:242Google Scholar
  32. 32.
    Maragliano L, Fischer A, Vanden-Eijnden E, Ciccotti G (2006) J Chem Phys 125:024106Google Scholar
  33. 33.
    Weinan E, Vanden-Eijnden E (2006) J Stat Phys 123:503Google Scholar
  34. 34.
    Stillinger FH (1976) J Chem Phys 65:3968Google Scholar
  35. 35.
    Louis AA, Bolhuis PG, Hansen J-P (2000) Phys Rev E 62:7961Google Scholar
  36. 36.
    Flory PJ (1953) Principles of polymer chemistry. Cornell University Press, IthacaGoogle Scholar
  37. 37.
    Wales DJ (2003) Energy landscapes. Cambridge University Press, CambridgeGoogle Scholar
  38. 38.
    Stillinger FH, Weber TA (1983) Phys Rev A 28:2408Google Scholar
  39. 39.
    Swendsen RH, Wang J-S (1986) Phys Rev Lett 57:2607Google Scholar
  40. 40.
    Sugita Y, Okamoto Y (1999) Chem Phys Lett 314:141Google Scholar
  41. 41.
    Earl DJ, Deem MW (2005) Phys Chem Chem Phys 7:3910Google Scholar
  42. 42.
    Torrie GM, Valleau JP (1977) J Comput Phys 23:187Google Scholar
  43. 43.
    Kirkwood J (1935) J Chem Phys 3:300Google Scholar
  44. 44.
    Carter EA, Ciccotti G, Hynes JT, Kapral R (1989) Chem Phys Lett 156:472Google Scholar
  45. 45.
    Darve E, Pohorille AJ (2001) J Chem Phys 115:9169Google Scholar
  46. 46.
    Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in C. Cam bridge University Press, CambridgeGoogle Scholar
  47. 47.
    Cerjan CJ, Miller WH (1981) J Chem Phys 75:2800Google Scholar
  48. 48.
    Doye JPK, Wales DJ (1997) Z Phys D 40:194Google Scholar
  49. 49.
    Munro LJ, Wales DJ (1999) Phys Rev B 59:3969Google Scholar
  50. 50.
    Mousseau N, Barkema G (1996) Phys Rev Lett 77:4358Google Scholar
  51. 51.
    Mousseau N, Barkema G (1998) Phys Rev E 57:2419Google Scholar
  52. 52.
    Vocks H, Chubynsky MV, Barkema GT, Mousseau N (2005) J Chem Phys 123:244707Google Scholar
  53. 53.
    Henkelman G, Jónsson H (1999) J Chem Phys 111:7010Google Scholar
  54. 54.
    Tanase-Nicola S, Kurchan J (2003) Phys Rev Lett 91:188302Google Scholar
  55. 55.
    Tailleur J, Kurchan J (2007) Nat Phys 3:203Google Scholar
  56. 56.
    Elber R, Karplus M (1987) Chem Phys Lett 139:375Google Scholar
  57. 57.
    Czerminski R, Elber R (1990) Int J Quantum Chem 24:167Google Scholar
  58. 58.
    Ulitzky A, Elber R (1990) J Chem Phys 92:1519Google Scholar
  59. 59.
    Fischer S, Karplus M (1992) Chem Phys Lett 194:252Google Scholar
  60. 60.
    Sevick EM, Bell AT, Theodorou DN (1993) J Chem Phys 98:3196Google Scholar
  61. 61.
    Gillilan RE, Wilson KR (1992) J Chem Phys 97:1757Google Scholar
  62. 62.
    Jonssón H, Mills G, Jacobsen KW (1998) In: Berne BJ, Ciccotti G, Coker DF (eds) Classical and quantum dynamics in condensed phase simulations. World Scientific, Singapore, p. 385Google Scholar
  63. 63.
    Ionova IV, Carter EA (1993) J Chem Phys 98:6377Google Scholar
  64. 64.
    Dewar MJS, Healy EF, Stewart JJP (1984) J Chem Soc, Farad Trans II 80:227Google Scholar
  65. 65.
    Henkelman G, Jónsson H (2000) J Chem Phys 113:9978Google Scholar
  66. 66.
    Chu J-W, Trout BL, Brooks BR (2003) J Chem Phys 119:12708Google Scholar
  67. 67.
    Henkelman G, Uberaga BP, Jónsson H (2000) J Chem Phys 113:9901Google Scholar
  68. 68.
    Mills G, Jónsson H (1994) Phys Rev Lett 72:1124Google Scholar
  69. 69.
    Villarba M, Jónsson H (1995) Surf Sci 324:35Google Scholar
  70. 70.
    Eichler A, Hafner J (1999) Surf Sci 435:58Google Scholar
  71. 71.
    Ciobica IM, van Santen RA (2002) J Phys Chem B 106:6200Google Scholar
  72. 72.
    Sorensen MR, Mishin Y, Voter AF (2000) Phys Rev B 62:3658Google Scholar
  73. 73.
    Mathews DH, Case DA (2006) J Mol Biol 357:1683Google Scholar
  74. 74.
    Goldstein H (1980) Classical mechanics. Addison-Wesley, ReadingGoogle Scholar
  75. 75.
    Landau LD, Lifshitz EM (1984) Mechanics. Pergamon, OxfordGoogle Scholar
  76. 76.
    Elber R (2006) In: Ferrario M, Ciccotti G, K. Binder (eds) Computer simulations in condensed matter: from materials to chemical biology. Springer, BerlinGoogle Scholar
  77. 77.
    Whittaker ET (1964) Analytical dynamics, 4th edn. Cambridge University Press, CambridgeGoogle Scholar
  78. 78.
    Passerone D, Parrinello M (2001) Phys Rev Lett 87:108302Google Scholar
  79. 79.
    Passerone D, Ceccarelli M, Parrinello M (2003) J Chem Phys 118:2025Google Scholar
  80. 80.
    Cho AE, Doll JD, Freeman DL (1994) Chem Phys Lett 229:218Google Scholar
  81. 81.
    Olender R, Elber R (1996) J Chem Phys 105:9299Google Scholar
  82. 82.
    Elber R, Ghosh A, Cárdenas A (2002) In: Nielaba P, Mareschal M, Ciccotti G (eds) Bridging time scales: molecular simulations for the next decade. Springer, BerlinGoogle Scholar
  83. 83.
    Elber R, Ghosh A, Cárdenas A (2002) Acc Chem Res 35:396Google Scholar
  84. 84.
    Onsager L, Machlup S (1953) Phys Rev 91:1505Google Scholar
  85. 85.
    Eastman P, Gronbech-Jensen N, Doniach S (2001) J Chem Phys 114:3823Google Scholar
  86. 86.
    Zaloj V, Elber R (2000) Comput Phys Commun 128:118Google Scholar
  87. 87.
    Uitdehaag JCM, van der Veen BA, Dijkhuizen L, Elber R, Dijkstra BW (2001) Prot Struct Funct Gen 43:327Google Scholar
  88. 88.
    Siva K, Elber R (2003) Prot Struct Funct Gen 50:63Google Scholar
  89. 89.
    Ghosh A, Elber R, Scheraga H (2002) Proc Natl Acad Sci USA 99:10394Google Scholar
  90. 90.
    Cárdenas A, Elber R (2003) Proteins 51:245Google Scholar
  91. 91.
    Cárdenas A, Elber R (2003) Biophys J 85:2919Google Scholar
  92. 92.
    Kevrikidis IG, Gear CW, Hyman JM, Kevrekidis PG, Runborg O, Theodoropoulos K (2003) Comm Math Sci 14:715Google Scholar
  93. 93.
    Amait MA, Kevrekidis IG, Maroudas D (2007) Appl Phys Lett 90:171910Google Scholar
  94. 94.
    Sriraman S, Kevrekidis IG, Hummer G (2005) Phys Rev Lett 95:130603Google Scholar
  95. 95.
    Iannuzzi M, Laio A, Parrinello M (2003) Phys Rev Lett 90:238302Google Scholar
  96. 96.
    Laio A, Parrinello M (2006) In: Ferrario M, Ciccotti G, K. Binder (eds) Computer simulations in condensed matter: from materials to chemical biology. Springer, BerlinGoogle Scholar
  97. 97.
    Grubmüller H (1995) Phys Rev E 52:2893Google Scholar
  98. 98.
    Schulze BG, Grubmüller H, Evanseck JD (2000) J Am Chem Soc 122:8700Google Scholar
  99. 99.
    Müller EM, de Meijere A, Grubmüller H (2002) J Chem Phys 116:897Google Scholar
  100. 100.
    Huber T, Torda A, van Gunsteren W (1994) J Comput Aided Mol Des 8:695Google Scholar
  101. 101.
    Wang F, Landau D (2001) Phys Rev Lett 86:2050Google Scholar
  102. 102.
    Bussi G, Laio A, Parrinello M (2006) Phys Rev Lett 96:090601Google Scholar
  103. 103.
    Micheletti C, Laio A, Parrinello M (2004) Phys Rev Lett 92:170601Google Scholar
  104. 104.
    Car R, Parrinello M (1985) Phys Rev Lett 55:2471Google Scholar
  105. 105.
    Laio A, Rodriguez-Fortea A, Gervasio FL, Ceccarelli M, Parrinello M (2005) J Phys Chem B 109:6714Google Scholar
  106. 106.
    Piana S, Laio A (2007) J Phys Chem B 111:4553Google Scholar
  107. 107.
    Bussi G, Gervasio FL, Laio A, Parrinello M (2006) J Am Chem Soc 128:13435Google Scholar
  108. 108.
    Martoňák R, Laio A, Parrinello M (2003) Phys Rev Lett 90:75503Google Scholar
  109. 109.
    Raitieri P, Martoňák R, Parrinello M (2005) Angew Chem Int Ed 44:3769Google Scholar
  110. 110.
    Ceccarelli M, Mercuri F, Passerone D, Parrinello M (2005) J Phys Chem B 109:17094Google Scholar
  111. 111.
    Stirling A, Iannuzzi M, Laio A, Parrinello M (2004) Chem Phys Chem 5:1558Google Scholar
  112. 112.
    Ensing B, de Vivo M, Liu Z, Moore P, Klein ML (2006) Acc Chem Res 39:73Google Scholar
  113. 113.
    Ceccarelli M, Danelon C, Laio A, Parrinello M (2004) Biophys J 87:58Google Scholar
  114. 114.
    Gervasio FL, Laio A, Parrinello M (2005) J Am Chem Soc 127:2600Google Scholar
  115. 115.
    Hanggi P, Talkner P, Borkovec M (1990) Rev Mod Phys 62:2551Google Scholar
  116. 116.
    Pechukas P (1981) Ann Rev Phys Chem 32:15Google Scholar
  117. 117.
    Anderson JB (1995) Adv Chem Phys 91:381Google Scholar
  118. 118.
    Vanden-Eijnden E, Tal FA (2005) J Chem Phys 123:184103Google Scholar
  119. 119.
    Marcelin A (1995) Ann Chim Phys 3:158Google Scholar
  120. 120.
    Eyring H (1935) J Chem Phys 3:107Google Scholar
  121. 121.
    Horiuti J (1938) Bull Chem Soc Jpn 13:210Google Scholar
  122. 122.
    Eyring H, Polanyi M (1931) Z Phys Chem B 12:279Google Scholar
  123. 123.
    Wigner E (1938) Trans Faraday Soc 34:29Google Scholar
  124. 124.
    Dellago C, Bolhuis PG, Geissler PL (2006) In: Ferrario M, Ciccotti G, Binder K (eds) Com puter simulations in condensed matter: from materials to chemical biology. Springer Lecture Notes in Physics. Springer, Berlin, p. 349Google Scholar
  125. 125.
    Chandler D (1978) J Chem Phys 68:2959Google Scholar
  126. 126.
    Anderson JB (1973) J Chem Phys 58:4684Google Scholar
  127. 127.
    Kassel LS (1928) J Phys Chem 32:225; Rice OK, Ramsperger HC (1927) J Am Chem Soc 49:1617; Rice OK, Ramsperger HC (1927) J Am Chem Soc 50:617; Marcus R, Rice OK (1951) J Phys Colloid Chem 55:894Google Scholar
  128. 128.
    Geissler PL, Dellago C, Chandler D (1999) Phys Chem Chem Phys 1:1317Google Scholar
  129. 129.
    Truhlar DG, Garrett B (1984) Ann Rev Phys Chem 35:159Google Scholar
  130. 130.
    Truhlar DG, Garrett BC, Klippenstein SJ (1996) J Phys Chem 100:31Google Scholar
  131. 131.
    Tucker SC (1995) In: Talkner P, Hnggi P (eds) New trends in Kramers' reaction rate theory. Kluwer, DordrechtGoogle Scholar
  132. 132.
    Johannesson G, Jónsson H (2001) J Chem Phys 115:9644Google Scholar
  133. 133.
    Fernández-Ramos A, Miller JA, Klippenstein SJ, Truhlar DG (2006) Chem Rev 106:4518Google Scholar
  134. 134.
    Yamamoto T (1960) J Chem Phys 33:281Google Scholar
  135. 135.
    Keck JC (1962) Discuss Faraday Soc 33:173Google Scholar
  136. 136.
    Bennett CH (1977) In: Christofferson R (ed) Algorithms for Chemical Computations, ACS Symposium Series No. 46. American Chemical Society, Washington DCGoogle Scholar
  137. 137.
    Ruiz-Montero MJ, Frenkel D, Brey JJ (1997) Mol Phys 90:925Google Scholar
  138. 138.
    van Erp TS, Bolhuis PG (2005) J Comput Phys 205:157Google Scholar
  139. 139.
    Waalkens H, Burbanks A, Wiggins S (2005) J Phys A Math Gen 38:L759Google Scholar
  140. 140.
    Waalkens H, Burbanks A, Wiggins S (2004) J Phys A 37:L257Google Scholar
  141. 141.
    Waalkens H, Burbanks A, Wiggins S (2004) J Chem Phys 121:6207Google Scholar
  142. 142.
    Tal FA, Vanden-Eijnden E (2006) Nonlinearity 19:501Google Scholar
  143. 143.
    Bortz AB, Kalos MH, Lebowitz JL (1975) J Comput Phys 17:10Google Scholar
  144. 144.
    Gillespie DT (1977) J Comput Phys 22:2340Google Scholar
  145. 145.
    Henkelman G, Jónsson H (2001) J Chem Phys 115:9657Google Scholar
  146. 146.
    Shirts MR, Pande VS (2001) Phys Rev Lett 86:4983Google Scholar
  147. 147.
    Voter AF (1998) Phys Rev B 57:13985Google Scholar
  148. 148.
    Pande VS, Baker I, Chapman J, Elmer SP, Khaliq S, Larson SM, Rhee YM, Shirts MR, Snow CD, Sorin EJ, Zagrovic B (2003) Biopolymers 68:91Google Scholar
  149. 149.
    Kum O, Dickson BM, Stuart SJ, Uberuaga BP, Voter AF (2004) J Chem Phys 121:9808Google Scholar
  150. 150.
    Birner S, Kim J, Richie DA, Wilkins JW, Voter AF, Lenosky T (2001) Solid State Commun 120:279Google Scholar
  151. 151.
    Uberuaga BP, Voter AF, Sieber KK, Sholl DS (2003) Phys Rev Lett 91:105901Google Scholar
  152. 152.
    Uberuaga BP, Stuart SJ, Voter AF (2007) Phys Rev B 75:014301Google Scholar
  153. 153.
    Voter AF (1997) J Chem Phys 106:4665Google Scholar
  154. 154.
    Voter AF (1997) Phys Rev Lett 78:3908Google Scholar
  155. 155.
    Duan XM, Sun DY, Gong XG (2001) Comput Mater Sci 20:151Google Scholar
  156. 156.
    Hamelberg D, McCammon JA (2005) J Am Chem Soc 127:13778Google Scholar
  157. 157.
    Becker KE, Fichthorn KA (2006) J Chem Phys 125:184706Google Scholar
  158. 158.
    Voter AF, Chen SP (1987) Mater Res Soc Symp Proc 82:175Google Scholar
  159. 159.
    Sørensen MR, Voter AF (2000) J Chem Phys 112:9599Google Scholar
  160. 160.
    Montalenti F, Voter AF, Ferrando R (2002) Phys Rev B 66:205404Google Scholar
  161. 161.
    Montalenti F, Sørensen MR, Voter AF (2001) Phys Rev Lett 87:126101Google Scholar
  162. 162.
    Cogoni M, Mattoni A, Uberuaga BP, Voter AF, Colombo L (2005) Appl Phys Lett 87:191912Google Scholar
  163. 163.
    Sprague JA, Montalenti F, Uberuaga BP, Kress JD, Voter AF (2002) Phys Rev B 66:205415Google Scholar
  164. 164.
    Bolhuis PG, Chandler D, Dellago C, Geissler PL (2002) Ann Rev Phys Chem 53:291Google Scholar
  165. 165.
    Dellago C, Bolhuis PG, Geissler PL (2002) Adv Chem Phys 123:1Google Scholar
  166. 166.
    Dellago C, Chandler D (2002) In: Nielaba P, Mareschal M, Ciccotti G (eds) Molecular simu lation for the next decade. Springer, Berlin, p. 321Google Scholar
  167. 167.
    Dellago C (2005) In: Yip S (ed) Handbook of materials modeling. Springer, Berlin, p. 1585Google Scholar
  168. 168.
    Dellago C (2007) In: Pohorille A, Chipot C (eds) Free energy calculations: theory and appli cations in chemistry and biology. Springer, BerlinGoogle Scholar
  169. 169.
    Dellago C, Bolhuis PG (2007) In: Reiher M (ed) Topics in Current Chemistry, vol. 268. Springer, Berlin, p. 291Google Scholar
  170. 170.
    Geissler PL, Dellago C (2004) J Phys Chem B 108:6667Google Scholar
  171. 171.
    Chandrasekhar S (1943) Rev Mod Phys 15:1Google Scholar
  172. 172.
    Zwanzig R (2001) Nonequilibrium statistical mechanics. Oxford University Press, OxfordGoogle Scholar
  173. 173.
    Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) J Chem Phys 21:1087Google Scholar
  174. 174.
    Dellago C, Bolhuis PG, Csajka FS, Chandler D (1999) J Chem Phys 110:6617Google Scholar
  175. 175.
    Geyer CJ, Thompson EA (1995) J Am Stat Soc 90:909Google Scholar
  176. 176.
    Vlugt TJH, Smit B (2001) Phys Chem Comm 2:1Google Scholar
  177. 177.
    Bolhuis PG (2003) J Phys Cond Matter 15:S113Google Scholar
  178. 178.
    Grünwald M, Geissler PL, Dellago C (2008), J Chem Phys (in print)Google Scholar
  179. 179.
    Hu J, Ma A, Dinner AR (2006) J Chem Phys 125:114101Google Scholar
  180. 180.
    Grünwald M, Rabani E, Dellago C (2006) Phys Rev Lett 96:255701Google Scholar
  181. 181.
    Grünwald M, Geissler PL, Dellago C (2007) J Chem Phys 127:154718Google Scholar
  182. 182.
    Dellago C, Bolhuis PG (2004) Mol Sim 30:795Google Scholar
  183. 183.
    van Erp TS, Moroni D, Bolhuis PG (2003) J Chem Phys 118:7762Google Scholar
  184. 184.
    Juraszek J, Bolhuis PG (2006) Proc Natl Acad Sci USA 103:15859Google Scholar
  185. 185.
    Moroni D, Bolhuis PG, van Erp TS (2004) J Chem Phys 120:4055Google Scholar
  186. 186.
    van Erp TS (2007) Phys Rev Lett 98:268301Google Scholar
  187. 187.
    Bolhuis PG (2008) J. Chem. 129, 114108Google Scholar
  188. 188.
    Moroni D, van Erp TS, Bolhuis PG (2005) Phys Rev E 71:056709Google Scholar
  189. 189.
    Allen RJ, Warren PB, ten Wolde PR (2005) Phys Rev Lett 94:018104Google Scholar
  190. 190.
    Allen RJ, Frenkel D, ten Wolde PR (2006) J Chem Phys 124:024102Google Scholar
  191. 191.
    Allen RJ, Frenkel D, ten Wolde PR (2006) J Chem Phys 124:194111Google Scholar
  192. 192.
    Valeriani C, Allen RJ, Morelli MJ, Frenkel D, ten Wolde PR (2007) J Chem Phys 127:114109Google Scholar
  193. 193.
    Borrero EE, Escobedo FA (2006) J Chem Phys 125:164904Google Scholar
  194. 194.
    Faradjian T, Elber R (2004) J Chem Phys 120:10882Google Scholar
  195. 195.
    West AMA, Elber R, Shalloway D (2007) J Chem Phys 126:145104Google Scholar
  196. 196.
    Elber R (2007) Biophys J 92:L85Google Scholar
  197. 197.
    Auer S, Frenkel D (2005) Adv Poly Sci 173:149Google Scholar
  198. 198.
    Pan AC, Chandler D (2004) J Phys Chem B 108:19681Google Scholar
  199. 199.
    Moroni D, ten Wolde PR, Bolhuis PG (2005) Phys Rev Lett 94:235703Google Scholar
  200. 200.
    Zahn D (2007) J Phys Chem B 111:5249Google Scholar
  201. 201.
    Beckham GT, Peters B, Starbuck C, Variankaval N, Trout BL (2007) J Am Chem Soc 129:4714Google Scholar
  202. 202.
    Zahn D (2006) J Phys Chem B 110:19601Google Scholar
  203. 203.
    Zahn D, Grin Y, Leoni S (2005) Phys Rev B 72:064110Google Scholar
  204. 204.
    Zahn D (2004) Phys Rev Lett 93:227801Google Scholar
  205. 205.
    Zahn D, Leoni S (2006) J Phys Chem B 110:10873Google Scholar
  206. 206.
    Zahn D, Hochrein O, Leoni S (2005) Phys Rev B 72(9):094106Google Scholar
  207. 207.
    Zahn D (2004) J Solid State Chem 177:3590Google Scholar
  208. 208.
    Leoni S, Zahn D (2004) Z Kristall 219:339Google Scholar
  209. 209.
    Zahn D, Leoni S (2004) Z Kristall 219:345Google Scholar
  210. 210.
    Zahn D, Leoni S (2004) Phys Rev Lett 92:250201Google Scholar
  211. 211.
    Zahn D (2004) Phys Rev Lett 92:040801Google Scholar
  212. 212.
    Day TJF, Schmitt UW, Voth GA (2000) J Am Chem Soc 122:12027Google Scholar
  213. 213.
    Park JM, Laio A, Iannuzzi M, Parrinello M (2006) J Am Chem Soc 128:11318Google Scholar
  214. 214.
    Lo CS, Radhakrishnan R, Trout BL (2005) Catalaysis Today 105:93Google Scholar
  215. 215.
    Snee PT, Shanoski J, Harris CB (2005) J Am Chem Soc 127:1286Google Scholar
  216. 216.
    Zahn D (2004) Chem Phys 300:79Google Scholar
  217. 217.
    Zahn D, Seifert G (2004) J Phys Chem B 108:16495Google Scholar
  218. 218.
    Ensing B, Baerends EJ (2002) J Phys Chem A 106:7902Google Scholar
  219. 219.
    Pagliai M, Raugei S, Cardini G, Schettino V (2002) J Chem Phys 117:2199Google Scholar
  220. 220.
    Marti J, Csajka FS, Chandler D (2000) Chem Phys Lett 328:169Google Scholar
  221. 221.
    Laria D, Rodriguez J, Dellago C, Chandler D (2001) J Phys Chem A 105:2646Google Scholar
  222. 222.
    Lee JY (2004) Chem Phys 299:123Google Scholar
  223. 223.
    Csajka FS, Chandler D (1998) J Chem Phys 109:1125Google Scholar
  224. 224.
    Marti J, Csajka FS (2000) J Chem Phys 113:1154Google Scholar
  225. 225.
    Marti J (2001) Mol Sim 27:169Google Scholar
  226. 226.
    Bolhuis PG, Chandler D (2000) J Chem Phys 113:8154Google Scholar
  227. 227.
    Vlugt TJH, Dellago C, Smit B (2000) J Chem Phys 113:8791Google Scholar
  228. 228.
    Pool R, Bolhuis PG (2007) J Chem Phys 126:244703Google Scholar
  229. 229.
    Ramirez J, Laso M (2001) J Chem Phys 115:7285Google Scholar
  230. 230.
    Radhakrishnan R, Schlick T (2004) Proc Natl Acad Sci USA 101:5970Google Scholar
  231. 231.
    Radhakrishnan R, Schlick T (2004) J Chem Phys 121:2436Google Scholar
  232. 232.
    McCormick TA, Chandler D (2003) J Phys Chem B 107:2796Google Scholar
  233. 233.
    Hagan MF, Dinner AR, Chandler D, Chakraborty AK (2003) Proc Natl Acad Sci USA 100:13922Google Scholar
  234. 234.
    Crehuet R, Field MJ (2007) J Phys Chem B 111:5708Google Scholar
  235. 235.
    ten Wolde PR, Chandler D (2002) Proc Natl Acad Sci USA 99:6539Google Scholar
  236. 236.
    Bolhuis PG (2003) Proc Natl Acad Sci USA 100:12129Google Scholar
  237. 237.
    Bolhuis PG (2005) Biophys J 88:50Google Scholar
  238. 238.
    Radhakrishnan R, Yang LJ, Arora K, Schlick T (2004) Biophys J 86:34AGoogle Scholar
  239. 239.
    Radhakrishnan R, Schlick T (2005) J Am Chem Soc 127:13245Google Scholar
  240. 240.
    Basner JE, Schwartz SD (2005) J Am Chem Soc 127:13822Google Scholar
  241. 241.
    Marti J, Csajka FS (2004) Phys Rev E 69:061918Google Scholar
  242. 242.
    Marti J (2004) J Phys Condens Matter 16:5669Google Scholar
  243. 243.
    Ma L, Cui Q (2007) J Am Chem Soc 129:10261Google Scholar
  244. 244.
    Wang YL, Schlick T (2007) BMC Struct Biol 7:7Google Scholar
  245. 245.
    Merolle M, Garrahan JP, Chandler D (2005) Proc Natl Acad Sci USA 102:10837Google Scholar
  246. 246.
    Crooks GE, Chandler D (2001) Phys Rev E 64:026109Google Scholar
  247. 247.
    Geissler PL, Chandler D (2000) J Chem Phys 113:9759Google Scholar
  248. 248.
    Sun SX (2003) J Chem Phys 118:5769Google Scholar
  249. 249.
    Ytreberg FM, Zuckerman DM (2004) J Chem Phys 120:10876Google Scholar
  250. 250.
    Oberhofer H, Dellago C, Geissler PL (2005) J Phys Chem B 69:6902–6915Google Scholar
  251. 251.
    Lechner W, Dellago C (2007) J Stat Mech Theor Exp P04001Google Scholar
  252. 252.
    Athènes M (2004) Eur Phys J B 38:651Google Scholar
  253. 253.
    Oberhofer H, Dellago C (2008) Comput Phys Commun 179:41Google Scholar
  254. 254.
    Imparato A, Peliti L (2007) J Stat Mech L02001Google Scholar
  255. 255.
    Imparato A, Peliti L (2007) Compt Rend Physique 8:556Google Scholar
  256. 256.
    Wales DJ (2002) Mol Phys 100:3285Google Scholar
  257. 257.
    Wales DJ (2005) Phys Biol 2:S86Google Scholar
  258. 258.
    Trygubenko SA, Wales DJ (2006) Mol Phys 104:1497Google Scholar
  259. 259.
    Evans DA, Wales DJ (2004) J Chem Phys 121:1080Google Scholar
  260. 260.
    Best RB, Hummer G (2005) Proc Natl Acad Sci USA 102:6732Google Scholar
  261. 261.
    Onsager L (1938) Phys Rev 54:554Google Scholar
  262. 262.
    Du R, Pande V, Grosberg AY, Tanaka T, Shakhnovich EI (1998) J Chem Phys 108:334Google Scholar
  263. 263.
    Gasser U, Weeks ER, Schofield A, Pusey PN, Weitz DA (2001) Science 292:258Google Scholar
  264. 264.
    Ryter D (1987) Physica A 142:103Google Scholar
  265. 265.
    Berezhovski A, Szabo A (2006) J Chem Phys 125:104902Google Scholar
  266. 266.
    Klosek MM, Matkowsky BJ, Schuss Z (1991) Ber Bunsenges Phys Chem 95:331Google Scholar
  267. 267.
    Pollak E, Berezhkovskii AM, Schuss Z (1994) J Chem Phys 100:334Google Scholar
  268. 268.
    Talkner P (1994) Chem Phys 180:199Google Scholar
  269. 269.
    Berezhovski A, Szabo A (2005) J Chem Phys 122:014503Google Scholar
  270. 270.
    Metzner P, Schütte C, Vanden-Eijnden E (2006) J Chem Phys 125:084110Google Scholar
  271. 271.
    Peters B (2006) J Chem Phys 125:241101Google Scholar
  272. 272.
    Quaytman SL, Schwartz SD (2007) Proc Natl Acad Sci USA 104:12253Google Scholar
  273. 273.
    So S-S, Karplus M (1996) J Med Chem 39:1521Google Scholar
  274. 274.
    Dinner A, So S-S, Karplus M (2002) Adv Chem Phys 120:1Google Scholar
  275. 275.
    Edwards AWF (1972) Likelihood. Cambridge University Press, CambridgeGoogle Scholar
  276. 276.
    Peters B, Beckham GT, Trout BL (2007) J Chem Phys 127:034109Google Scholar
  277. 277.
    Vanden-Eijnden E (2006) In: Ferrario M, Ciccotti G, Binder K (eds) Computer simulations in condensed matter: from materials to chemical biology, Springer Lecture Notes in Physics. Springer, Berlin, p. 453Google Scholar
  278. 278.
    Gardiner CW (1985) Handbook of stochastic methods. Springer, BerlinGoogle Scholar
  279. 279.
    Ren W (2003) Comm Math Sci 1:377Google Scholar
  280. 280.
    Weinan E, Ren W, Vanden-Eijnden E (2007) J Chem Phys 126:164103Google Scholar
  281. 281.
    Weinan E, Ren W, Vanden-Eijnden E (2005) J Phys Chem B 109:6688Google Scholar
  282. 282.
    Ren W, Vanden-Eijnden E, Maragakis P, Weinan E (2005) J Chem Phys 123 134109Google Scholar
  283. 283.
    Miller TF III, Vanden-Eijnden E, Chandler D (2007) Proc Natl Acad Sci USA 104:14559Google Scholar
  284. 284.
    ten Wolde P-R, Chandler D (2002) Proc Natl Acad Sci USA 99:6539Google Scholar
  285. 285.
    Wilson MA, Chandler D (1990) Chem Phys 149:11Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Faculty of PhysicsUniversity of ViennaViennaAustria
  2. 2.van't Hoff Institute for Molecular SciencesUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations