Skip to main content
SpringerLink
Log in

Unraveling the Beautiful Complexity of Simple Lattice Model Polymers and Proteins Using Wang-Landau Sampling

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We describe a class of “bare bones” models of homopolymers which undergo coil-globule collapse and proteins which fold into their native states in free space or into denatured states when captured by an attractive substrate as the temperature is lowered. We then show how, with the use of a properly chosen trial move set, Wang-Landau Monte Carlo sampling can be used to study the rough free energy landscape and ground (native) states of these intriguingly simple systems and thus elucidate their thermodynamic complexity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anfinsen, C.B.: Principles that govern the folding of protein chains. Science 181, 223–230 (1973)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  3. Bachmann, M., Janke, W.: Thermodynamics of lattice heteropolymers. J. Chem. Phys. 120, 6779–6791 (2004)

    Article  ADS  Google Scholar 

  4. Bachmann, M., Janke, W.: Conformational transitions of nongrafted polymers near an absorbing substrate. Phys. Rev. Lett. 95, 058102 (2005)

    Article  ADS  Google Scholar 

  5. Bachmann, M., Janke, W.: Substrate adhesion of a nongrafted flexible polymer in a cavity. Phys. Rev. E 73, 041802 (2006)

    Article  ADS  Google Scholar 

  6. Bachmann, M., Janke, W.: Substrate specificity of peptide adsorption: A model study. Phys. Rev. E 73, 020901(R) (2006)

    ADS  Google Scholar 

  7. Backofen, R., Will, S.: A constraint-based approach to fast and exact structure prediction in three-dimensional protein models. Constraints 11, 5–30 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bastolla, U., Frauenkron, H., Gerstner, E., Grassberger, P., Nadler, W.: Testing a new Monte Carlo algorithm for protein folding. Proteins 32, 52–66 (1998)

    Article  Google Scholar 

  9. Berg, B.A., Neuhaus, T.: Multicanonical algorithms for first order phase transitions. Phys. Lett. B 267, 249–253 (1991)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  11. Berger, B., Leighton, T.: Protein folding in the hydrophobic-hydrophilic (HP) model is NP-complete. J. Comput. Biol. 5, 27–40 (1998)

    Article  Google Scholar 

  12. Binder, K., Paul, W.: Recent developments in Monte Carlo simulations of lattice models for polymer systems. Macromolecules 41, 4537–4550 (2008)

    Article  ADS  Google Scholar 

  13. Bonaccini, R., Seno, F.: Simple model to study insertion of a protein into a membrane. Phys. Rev. E 60, 7290–7298 (1999)

    Article  ADS  Google Scholar 

  14. Bryngelson, J.D., Onuchic, J.N., Socci, N.D., Wolynes, P.G.: Funnels, pathways and the energy landscape of protein folding: a synthesis. Proteins 21, 167–195 (1995)

    Article  Google Scholar 

  15. Castells, V., Yang, S., Van Tassel, P.R.: Surface-induced conformational changes in lattice model proteins by Monte Carlo simulation. Phys. Rev. E 65, 031912 (2002)

    Article  ADS  Google Scholar 

  16. Cellmer, T., Bratko, D., Prausnitz, J.M., Blanch, H.: Protein-folding landscapes in multichain systems. Proc. Natl. Acad. Sci. USA 102, 11692–11697 (2005)

    Article  ADS  Google Scholar 

  17. Chikenji, G., Kikuchi, M., Iba, Y.: Multi-self-overlap ensemble for protein folding: ground state search and thermodynamics. Phys. Rev. Lett. 83, 1886–1889 (1999)

    Article  ADS  Google Scholar 

  18. Crescenzi, P., Goldman, D., Papadimitriou, C., Piccolboni, A., Yannakakis, M.: On the complexity of protein folding. J. Comput. Biol. 5, 423–465 (1998)

    Article  Google Scholar 

  19. Deutsch, J.M.: Long range moves for high density polymer simulations. J. Chem. Phys. 106, 8849–8854 (1997)

    Article  ADS  Google Scholar 

  20. Dill, K.A.: Theory for the folding and stability of globular proteins. Biochemistry 24, 1501–1509 (1985)

    Article  Google Scholar 

  21. Dill, K.A.: The meaning of hydrophobicity. Science 250, 297–298 (1990)

    Article  ADS  Google Scholar 

  22. Dill, K.A.: Polymer principles and protein folding. Protein Sci. 8, 1166–1180 (1999)

    Article  Google Scholar 

  23. Dill, K.A., Bromberg, S., Yue, K., Fiebig, K.M., Yee, D.P., Thomas, P.D., Chan, H.S.: Principles of protein folding—a perspective from simple exact models. Protein Sci. 4, 561–602 (1995)

    Article  Google Scholar 

  24. Ferrenberg, A.M., Swendsen, R.H.: New Monte Carlo technique for studying phase transitions. Phys. Rev. Lett. 61, 2635–2638 (1988)

    Article  ADS  Google Scholar 

  25. Ferrenberg, A.M., Swendsen, R.H.: Optimized Monte Carlo data analysis. Phys. Rev. Lett. 63, 1195–1198 (1989)

    Article  ADS  Google Scholar 

  26. Fraser, R., Glasgow, J.I.: A demonstration of clustering in protein contact maps for alpha helix pairs. In: Proc. of ICANNGA 2007, vol. 1, pp. 758–766

  27. Frauenkron, H., Bastolla, U., Gerstner, E., Grassberger, P., Nadler, W.: New Monte Carlo algorithm for protein folding. Phys. Rev. Lett. 80, 3149–3152 (1998)

    Article  ADS  Google Scholar 

  28. Grassberger, P.: Pruned-enriched Rosenbluth method: Simulations of θ polymers of chain length up to 1 000 000. Phys. Rev. E 56, 3682–3693 (1997)

    Article  MathSciNet  ADS  Google Scholar 

  29. Hansmann, U., Okamoto, Y.: New Monte Carlo algorithms for protein folding. Curr. Opin. Struct. Biol. 9, 177–183 (1999)

    Article  Google Scholar 

  30. Harrison, P.M., Chan, H.S., Prusiner, S.B., Cohen, F.E.: Thermodynamics of model prions and its implications for the problem of prion protein folding. J. Mol. Biol. 286, 593–606 (1999)

    Article  Google Scholar 

  31. Hsu, H.-P., Mehra, V., Nadler, W., Grassberger, P.: Growth-based optimization algorithm for lattice heteropolymers. Phys. Rev. E 68, 021113 (2003)

    Article  ADS  Google Scholar 

  32. Hsu, H.-P., Mehra, V., Nadler, W., Grassberger, P.: Growth algorithms for lattice heteropolymers at low temperatures. J. Chem. Phys. 118, 444–451 (2003)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  34. Janke, W.: Multicanonical Monte Carlo simulations. Physica A 254, 164–178 (1998)

    Article  Google Scholar 

  35. Kauzmann, W.: Some factors in the interpretation of protein denaturation. Adv. Protein Chem. 14, 1–63 (1959)

    Article  Google Scholar 

  36. Kolinski, A., Skolnick, J.: Reduced models of proteins and their applications. Polymer 45, 511–524 (2004)

    Article  Google Scholar 

  37. Kou, S.C., Oh, J., Wong, W.H.: A study of density of states and ground states in hydrophobic-hydrophilic protein folding models by equi-energy sampling. J. Chem. Phys. 124, 244903 (2006)

    Article  ADS  Google Scholar 

  38. Landau, D.P., Binder, K.: A Guide to Monte Carlo Methods in Statistical Physics. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  39. Landau, D.P., Tsai, S.-H., Exler, M.: A new approach to Monte Carlo simulation: Wang-Landau sampling. Am. J. Phys. 72, 1294–1301 (2004)

    Article  ADS  Google Scholar 

  40. Lau, K.F., Dill, K.A.: A lattice statistical mechanics model of the conformational and sequence spaces of proteins. Macromolecules 22, 3986–3997 (1989)

    Article  ADS  Google Scholar 

  41. Lesh, N., Mitzenmacher, M., Whitesides, S.: A complete and effective move set for simplified protein folding. In: RECOMB 2003, pp. 188–195

  42. Li, Y.W., Wüst, T., Landau, D.P.: Monte Carlo simulations of the HP model (the “Ising model” of protein folding). Comput. Phys. Commun. 182, 1896–1899 (2011)

    Article  ADS  Google Scholar 

  43. Liang, F.: A generalized Wang-Landau algorithm for Monte Carlo computation. J. Am. Stat. Assoc. 100, 1311 (2005)

    Article  MATH  Google Scholar 

  44. Luettmer-Strathmann, J., Rampf, F., Paul, W., Binder, K.: Transitions of tethered polymer chains: a simulation study with the bond fluctuation lattice model. J. Chem. Phys. 128, 064903 (2008)

    Article  ADS  Google Scholar 

  45. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.M., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953)

    Article  ADS  Google Scholar 

  46. Onuchic, J.N., Luthey-Schulten, Z., Wolynes, P.G.: Theory of protein folding: the energy landscape perspective. Annu. Rev. Phys. Chem. 48, 545–600 (1997)

    Article  ADS  Google Scholar 

  47. Parsons, D.F., Williams, D.R.M.: An off-lattice Wang-Landau study of the coil-globule and melting transitions of a flexible homopolymer. J. Chem. Phys. 124, 221103 (2006)

    Article  ADS  Google Scholar 

  48. Paul, W., Strauch, T., Rampf, F., Binder, K.: Unexpectedly normal phase behavior of single homopolymer chains. Phys. Rev. E 75, 060801(R) (2007)

    ADS  Google Scholar 

  49. Ping, G., Yuan, J.M., Vallieres, M., Dong, H., Sun, Z., Wei, Y., Li, F.Y., Lin, S.H.: Effects of confinement on protein folding and protein stability. J. Chem. Phys. 118, 8042–8048 (2003)

    Article  ADS  Google Scholar 

  50. Prellberg, T., Krawczyk, J.: Flat histogram version of the pruned and enriched Rosenbluth method. Phys. Rev. Lett. 92, 120602 (2004)

    Article  ADS  Google Scholar 

  51. Prellberg, T., Krawczyk, J., Rechnitzer, A.: Polymer simulations with a flat histogram stochastic growth algorithm. In: Landau, D.P., Lewis, S.P., Schüttler, H.-B. (eds.) Computer Simulation Studies in Condensed-Matter Physics XVII, pp. 122–135. Springer, Berlin Heidelberg New York (2006)

    Chapter  Google Scholar 

  52. Rampf, F., Binder, K., Paul, W.: The phase diagram of a single polymer chain: New insights from a new simulation method. J. Polym. Sci., Part B, Polym. Phys. 44, 2542–2555 (2006)

    Article  ADS  Google Scholar 

  53. Seaton, D.T., Wüst, T., Landau, D.P.: A Wang-Landau study of the phase transitions in a flexible homopolymer. Comput. Phys. Commun. 180, 587–589 (2008)

    Article  ADS  Google Scholar 

  54. Seaton, D.T., Wüst, T., Landau, D.P.: Collapse transitions in a flexible homopolymer chain: application of the Wang-Landau algorithm. Phys. Rev. E 81, 011802 (2010)

    Article  ADS  Google Scholar 

  55. Sokal, A.D.: Monte Carlo methods for the self-avoiding walk. In: Binder, K. (ed.) Monte Carlo and Molecular Dynamics Simulations in Polymer Science, pp. 47–124. Oxford University Press, New York (1995)

    Google Scholar 

  56. Swetnam, A.D., Allen, M.P.: Improved simulations of lattice peptide adsorption. Phys. Chem. Chem. Phys. 11, 2046–2055 (2009)

    Article  Google Scholar 

  57. Vogel, T., Bachmann, M., Janke, W.: Freezing and collapse of flexible polymers on regular lattices in three dimensions. Phys. Rev. E 76, 061803 (2007)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  60. Wang, F., Landau, D.P.: Determining the density of states for classical statistical models by a flat-histogram random walk. Comput. Phys. Commun. 147, 674–677 (2002)

    Article  ADS  MATH  Google Scholar 

  61. Wüst, T., Landau, D.P.: The HP model of protein folding: A challenging testing ground for Wang-Landau sampling. Comput. Phys. Commun. 179, 124–127 (2008)

    Article  ADS  MATH  Google Scholar 

  62. Wüst, T., Landau, D.P.: Versatile approach to access the low temperature thermodynamics of lattice polymers and proteins. Phys. Rev. Lett. 102, 178101 (2009)

    Article  ADS  Google Scholar 

  63. Zhang, J., Kou, S.C., Liu, J.S.: Biopolymer structure simulation and optimization via fragment regrowth Monte Carlo. J. Chem. Phys. 126, 225101 (2007)

    Article  ADS  Google Scholar 

  64. Zhang, L., Lu, D., Liu, Z.: How native proteins aggregate in solution: a dynamic Monte Carlo simulation. Biophys. Chem. 133, 71–80 (2008)

    Article  Google Scholar 

  65. Zhou, C., Bhatt, R.N.: Understanding and improving the Wang-Landau algorithm. Phys. Rev. E 72, 025701(R) (2005)

    ADS  Google Scholar 

Download references

Author information

Author notes

  1. T. Wüst

    Present address: Swiss Federal Research Institute WSL, 8903, Birmensdorf, Switzerland

Authors and Affiliations

  1. Center for Simulational Physics, The University of Georgia, Athens, GA, 30602, USA

    T. Wüst, Y. W. Li & D. P. Landau

Authors
  1. T. Wüst
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. W. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. P. Landau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Wüst.

Additional information

The research project is supported by the National Science Foundation under Grant No. DMR-0810223.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wüst, T., Li, Y.W. & Landau, D.P. Unraveling the Beautiful Complexity of Simple Lattice Model Polymers and Proteins Using Wang-Landau Sampling. J Stat Phys 144, 638–651 (2011). https://doi.org/10.1007/s10955-011-0266-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-011-0266-z

Keywords

Search

Navigation