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Transition Networks: A Unifying Theme for Molecular Simulation and Computer Science

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Mathematical Modeling of Biological Systems, Volume I

Summary

A transition network (TN) is a graph-theoretical concept describing the transitions between (meta)stable states of dynamical systems. Here, we review methods to generate and analyze a TN for molecular systems. The appropriate identification of states and transitions from the potential energy surface of the molecule is discussed. We describe a formalism transforming a TN on a static energy surface into a time-dependent dynamic TN that yields the population probabilities for each system state and the interstate transition rates. Three analysis methods that help in understanding the dynamics of the molecular system based on the TN are discussed: (1) Disconnectivity graphs allow important features of the energy surface captured in a static TN to be visualized, (2) graph-theoretical methods enable the computation of the best transition paths between two predefined states of the TN, and (3) statistical methods from complex network analysis identify important features of the TN topology. A broad review of the literature is given, and some open research directions are discussed.

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References

  1. Stillinger, F.H., Weber, T.H.: Hidden structure in liquids. Phys. Rev. A, 25, 978–989 (1982).

    Article  Google Scholar 

  2. Elber, R., Karplus, M.: Multiple conformational states of proteins: A molecular dynamics study of myoglobin. Science, 235, 318–321 (1987).

    Article  Google Scholar 

  3. Miller, M.A., Doye, J.P.K., Wales, D.J.: Energy landscapes of model polyalanines. J. Chem. Phys., 117, 1363–1376 (2002).

    Article  Google Scholar 

  4. Stillinger, F.H., Weber, T.A.: Packing structures and transitions in liquids and solids. Science, 228, 983–989 (1984).

    Article  Google Scholar 

  5. Stillinger, F.H.: A topographic view of supercooled liquids and glass formation. Science, 267, 1935–1939 (1995).

    Article  Google Scholar 

  6. Berry, R.S., Breitengraser-Kunz, R.: Topography and dynamics of multidimensional interatomic potential surfaces. Phys. Rev. Lett., 74, 3951–3954 (1995).

    Article  Google Scholar 

  7. Brooks III, C.L., Onuchic, J.N., Wales, D.J.: Taking a walk on a landscape. Science, 293, 612–613, (2001).

    Article  Google Scholar 

  8. Schütte, C., Huisinga, W.: Biomolecular conformations can be identified as metastable sets of molecular dynamics. In: Ciaret, P.G., Lions, J.L. (eds) Handbook of Numerical Analysis. Computational Chemistry, 699–744, North-Holland, Amsterdam, (2003).

    Google Scholar 

  9. Schlick, T., Barth, E., Mandziuk, M.: Biomolecular dynamics at long timesteps: Bridging the timescale gap between simulation and experimentation Annu. Rev. Biophys. Biomol. Struct., 26, 181–222 (1997).

    Article  Google Scholar 

  10. Frenkel, D.: Introduction to monte carlo methods. In: Attig, N., Binder, K., Grubmuller, H., Kremer, K. (eds) Computational Soft Matter: From Synthetic Polymers to Proteins. 23, 29–59, NIC Series, Jülich, Germany, (2004).

    Google Scholar 

  11. Schlick, T., Skeel, R.D., Brunger, A.T., Kalé, L.V., Board Jr., J.A., Hermans, J., Schulten, K.: Algorithmic challenges in computational molecular biophysics. J. Comp. Phys., 151, 9–48 (1999).

    Article  MATH  Google Scholar 

  12. Czerminski, R., Elber, R.: Reaction path study of conformational transitions and helix formation in a tetrapeptide. Proc. Nat. Acad. Sci. USA, 86, 6963–6967 (1989).

    Article  Google Scholar 

  13. Czerminski, R., Elber, R.: Reaction path study of conformational transitions in flexible systems: Application to peptides. J. Chem. Phys., 92, 5580–5601 (1990).

    Article  Google Scholar 

  14. Wales, D.J.: Structure, dynamics, and thermodynamics of clusters: Tales from topographic potential surfaces. Science, 271, 925–933 (1996).

    Article  Google Scholar 

  15. Ball, K.D., Berry, R.S., Kunz, R.E., Li, F.Y., Proykova, A., Wales, D.J.: From topographies to dynamics on multidimensional potential energy surfaces of atomic clusters. Science, 271, 963–967 (1996).

    Article  Google Scholar 

  16. Becker, O.M., Karplus, M.: The topology of multidimensional potential energy surfaces: Theory and application to peptide structure and kinetics. J. Chem. Phys., 106, 1495–1517 (1996).

    Article  Google Scholar 

  17. Levy, Y., Becker, O.M.: Effect of conformational constraints on the topography of complex potential energy surfaces. Phys. Rev. Lett., 81, 1126–1132 (1998).

    Article  Google Scholar 

  18. Miller, M.A., Wales, D.J.: Energy landscape of a model protein. J. Chem. Phys., 111, 6610– 6616 (1999).

    Article  Google Scholar 

  19. Mortenson, P.N., Wales, D.J.: Energy landscapes, global optimization and dynamics of the polyalanine Ac(ala)8NHMe. Journal of Chemical Physics, 114, 6443–6453 (2001).

    Article  Google Scholar 

  20. Levy, Y., Becker, O.M.: Energy landscapes of conformationally constrained peptides. J. Chem. Phys., 114, 993–1009 (2001).

    Article  Google Scholar 

  21. Levy, Y., Jortner, J., Becker, O.M.: Dynamics of hierarchical folding on energy landscapes of hexapeptides. J. Chem. Phys., 115, 10533–10547 (2001).

    Article  Google Scholar 

  22. Levy, Y., Jortner, J., Becker, O.M.: Solvent effects on the energy landscapes and folding kinetics of polyalanines. Proc. Nat. Acad. Sci. USA, 98, 2188–2193 (2001).

    Article  Google Scholar 

  23. Mortenson, P.N., Evans, D.A., Wales, D.J.: Energy landscapes of model polyalanines. J. Chem. Phys., 117, 1363–1376 (2002).

    Article  Google Scholar 

  24. Levy, Y., Becker, O.M.: Conformational polymorphism of wild-type and mutant prion proteins: energy landscape analysis. Proteins, 47, 458–468 (2002).

    Article  Google Scholar 

  25. Evans, D.A., Wales, D.J.: Free energy landscapes of model peptides and proteins. J. Chem. Phys., 118, 3891–3897 (2003).

    Article  Google Scholar 

  26. Evans, D.A., Wales, D.J.: The free energy landscape and dynamics of met-enkephalin. J. Chem. Phys., 119, 9947–9955 (2003).

    Article  Google Scholar 

  27. Wales, D.J., Doye, J.P.K.: Stationary points and dynamics in high-dimensional systems. J. Chem. Phys., 119, 12409–12416 (2003).

    Article  Google Scholar 

  28. Evans, D.A., Wales, D.J.: Folding of the gb1 hairpin peptide from discrete path sampling. J. Chem. Phys., 121, 1080–1090 (2004).

    Article  Google Scholar 

  29. Despa, F., Wales, D.J., Berry, R.S.: Archetypal energy landscapes: Dynamical diagnosis. J. Chem. Phys., 122, 024103 (2005).

    Article  Google Scholar 

  30. Kollmann, P.: Free energy calculations: Applications to chemical and biochemical phenomena. Chem. Rev., 93, 2395–2417 (1993).

    Article  Google Scholar 

  31. Horenko, I., Dittmer, E., Fischer, A., Schutte, C.: Automated model reduction for complex systems exhibiting metastability. SIAM Mult. Model.Sim., 5, 802–827 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  32. Rao, F., Caflisch, A.: The protein folding network. J. Mol. Bio., 342, 299–306 (2004).

    Article  Google Scholar 

  33. Moore, W.J., Hummel, D.O.: Physikalische Chemie, 3rd ed. Walter de Gruyter, Berlin, (1983).

    Google Scholar 

  34. Im, W., Beglow, D., Roux, B.: Continuum solvation model: Electrostatic forces from numerical solutions to the Poisson–Boltzmann equation. Comp. Phys. Comm., 111, 59–75 (1998).

    Article  MATH  Google Scholar 

  35. Schaefer, M., Karplus, M.: A comprehensive analytical treatment of continuum electrostatics. J. Chem. Phys., 100, 1578–1599 (1996).

    Article  Google Scholar 

  36. Bashford, D., Case, D.A.: Generalized Born models of macromolecular solvation effects. Ann. Rev. Phys. Chem., 51, 129–152 (2000).

    Article  Google Scholar 

  37. Im, W., Lee, M.S., Brooks III, C.L.: Generalized Born model with a simple smoothing function. J. Comp. Chem., 24, 1691–1702 (2003).

    Article  Google Scholar 

  38. Noé, F., Krachtus, D., Smith, C.J., Fischer, S.: Transition networks for the comprehensive characterization of complex conformational change in proteins. J. Chem. Theo. Comput., 2, 840–857 (2006).

    Article  Google Scholar 

  39. Henkelman, G., J’ohannesson, G., J’onsson, H.: Progress on Theoretical Chemistry and Physics, ch. Methods for Finding Saddle Points and Minimum Energy Paths, 269–300, Kluwer Academic Publishers, Dordrecht, (2000).

    Google Scholar 

  40. Fischer, S., Karplus, M.: Conjugate peak refinement: an algorithm for finding reaction paths and accurate transition states in systems with many degrees of freedom. Chem. Phys. Lett., 194, 252–261 (1992).

    Article  Google Scholar 

  41. Czerminski, R., Elber, R.: Self avoiding walk between two fixed points as a tool to calculate reaction paths in large molecular systems. Int. J. Quant. Chem., 24, 167 (1990).

    Article  Google Scholar 

  42. Jóonsson, H., Mills, G., Jacobsen, K.W.: Classical and Quantum Dynamics in Condensed Phase Simulations, ch. Nudged Elastic Band Method for Finding Minimum Energy Paths of Transitions, 385–404, World Scientific, Singapore, (1998).

    Google Scholar 

  43. Weinan, E., Ren, W., Vanden-Eijnden, E.: String method for the study of rare events. Phys. Rev. B, 66, 052301 (2002).

    Article  Google Scholar 

  44. Tournier, A., Smith, J.: Principal components of the protein dynamical transition. Phys Rev. Lett., 91, 208106 (2003).

    Article  Google Scholar 

  45. Elber, R.: Recent Developments in Theoretical Studies of Proteins, ch. Reaction Path Studies of Biological Molecules, World Scientific, Singapore (1996).

    Google Scholar 

  46. Wilson, E., Decius, J., Cross, P.: Molecular Vibrations, McGraw-Hill, New York (1955).

    Google Scholar 

  47. Perahia, D., Mouawad, L.: Computation of low-frequency normal modes in macromolecules: Improvements to the method of diagonalization in a mixed basis and application to hemoglobin. Comput. Chem., 19, 241–246 (1995).

    Article  Google Scholar 

  48. Torrie, G.M., Valleau, J.P.: Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling. J. Comp. Phys., 23, 187–199 (1977).

    Article  Google Scholar 

  49. Rajamani, R., Naidoo, K.J., Gao, J.: Implementation of an adaptive umbrella sampling method for the calculation of multidimensional potential of mean force of chemical reactions in solution. Proteins, 24, 1775–1781 (2003).

    Google Scholar 

  50. Karplus, M.: Aspects of protein reaction dynamics: Deviations from simple behavior. J. Phys. Chem. B, 104, 11–27 (2000).

    Article  Google Scholar 

  51. Hanggi, P., Talkner, P., Borkovec, M.: Reaction rate theory: Fifty years after kramers. Rev. Mod. Phys., 62, 251–342 (1990).

    Article  MathSciNet  Google Scholar 

  52. van Kampen, N.G.: Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, (1992).

    Google Scholar 

  53. Herges, T., Wenzel, W.: Free-energy landscape of the villin headpiece in an all-atom force field. Structure, 13, 1–8 (2005).

    Article  Google Scholar 

  54. Olsen, K., Fischer, S., Karplus, M.: A continuous path for the T \to \quad R allosteric transition of hemoglobin. Biophys. J., 78, 394A (2000).

    Article  Google Scholar 

  55. Fischer, S., Windshuegel, B., Horak, D., Holmes, K.C., Smith, J.C.: Structural mechanism of the recovery stroke in the myosin molecular motor. Proc. Nat. Acad. Sci. USA, 102, 6873–6878 (2005).

    Article  Google Scholar 

  56. Coleman, M.L., Marshall, C.J., Olson, M.F.: Ras and Rho GTPases in G1-phase cell-cycle regulation. Nat. Rev. Mol. Cell Bio., 62, 851–891, (1993).

    Google Scholar 

  57. Vojtek, A.B., Der, C.J.: Increasing complexity of the Ras signaling pathway. J. Biol. Chem., 32, 19925–19928 (1998).

    Article  Google Scholar 

  58. Noé, F., Ille, F., Smith, J.C., Fischer, S.: Automated computation of low-energy pathways for complex rearrangements in proteins: Application to the conformational switch of ras p21. Proteins, 59, 534–544 (2005).

    Article  Google Scholar 

  59. Berkowitz, M., Morgan, J.D., McCammon, J.A., Northrup, S.H.: Diffusion-controlled reactions: A variational formula for the optimum reaction coordinate. J. Chem. Phys., 79, 5563–5565 (1983).

    Article  Google Scholar 

  60. Huo, S., Straub, J.E.: The maxflux algorithm for calculating variationally optimized reaction paths for conformational transitions in many body systems at finite temperature. J. Chem. Phys., 107, 5000–5006 (1997).

    Article  Google Scholar 

  61. Dijkstra, E.: A note on two problems in connexion with graphs. Num. Math., 1, 269–271 (1959).

    Article  MATH  MathSciNet  Google Scholar 

  62. Wales, D.J.: Discrete path sampling. Mol. Phys., 100, 3285–3305 (2002).

    Article  Google Scholar 

  63. Eppstein, D.: Finding the k shortest paths. Proc. 35th Symp. Found. Comp. Sci., 154–165 (1994).

    Google Scholar 

  64. Preis, R., Dellnitz, M., Hessel, M., Schutte, C., Meerbach, E.: Dominant paths between almost invariant sets of dynamical systems. Preprint 154 (2004).

    Google Scholar 

  65. Noé, F., Oswald, M., Reinelt, G., Fischer, S., Smith, J.C.: Computing best transition pathways in high-dimensional dynamical systems. SIAM Mult. Model. Sim., 5, 393–419 (2006).

    Article  MATH  Google Scholar 

  66. Newman, M.E.J.: The Structure and function of complex networks. SIAM Review, 45, 167–256 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  67. Karoński, M., Ruciński, A.: The origins of the theory of random graphs. In: Graham, R.L., Nešetřil, J. (eds.) The mathematics of Paul Erdös vol. 13 of Algorithms and Combinatorics, 311–336, Springer, Berlin, (1997).

    Google Scholar 

  68. Albert, R., Barabasi, A.L., Jeong, H.: Diameter of the world wide web. Nature, 401, 130– 131 (1999).

    Article  Google Scholar 

  69. Faloutsos, P.F.M, Faloutsos, C.: On power-law relationships of the internet topology. Comp. Commun. Rev., 29, 251–262 (1999).

    Article  Google Scholar 

  70. Redner, S.: How popular is your paper? an empirical study of the citation distribution. Eur. Phys.J.B, 4, 131–134 (1998).

    Article  Google Scholar 

  71. Jeong, H., Tomber, B., Albert, R., Oltvai, Z.N., Barabasi, A.-L.: The large-scale organization of metabolic networks. Nature, 407, 651 (2000).

    Article  Google Scholar 

  72. Doye, J.P.K.: Network topology of a potential energy landscape: A static scale-free network. Phys. Rev. Lett., 88, 238701 (2002).

    Article  Google Scholar 

  73. Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘smallworld’ networks. Nature, 393, 440–442 (1998).

    Article  Google Scholar 

  74. Strogatz, S.: Exploring complex networks. Nature, 410, 268–276 (2001).

    Article  Google Scholar 

  75. Miller, M.A., Doye, J.P.K., Wales, D.J.: Structural relaxation in morse clusters: Energy landscapes. J. Chem. Phys., 110, 328–334 (1999).

    Article  Google Scholar 

  76. Ball, K.D., Berry, R.S.: Dynamics on statistical samples of potential energy surfaces. J. Chem. Phys., 111, 2060–2070 (1999).

    Article  Google Scholar 

  77. Levy, Y., Jortner, J., Becker, O.M.: Dynamics of hierarchical folding on energy landscapes of hexapeptides. J. Chem. Phys., 115, 10533–10547 (2001).

    Article  Google Scholar 

  78. Krivov, S.V., Karplus, M.: Hidden complexity of free energy surfaces for peptide (protein) folding. Proc. Nat. Acad. Sci. USA, 101, 14766–14770 (2004).

    Article  Google Scholar 

  79. Amadei, A., Linssen, A.B., Berendsen, H.J.C.: Essential dynamics of proteins. Proteins, 17, 412–225 (1993).

    Article  Google Scholar 

  80. Frauenfelder, H., Sligar, S.G., Wolynes, P.G.: The energy landscapes and motions of proteins.. Science, 254, 1598–1603 (1991).

    Article  Google Scholar 

  81. Sullivan, D.C., Kuntz, I.D.: Conformation spaces of proteins. Proteins, 42, 495–511 (2001).

    Article  Google Scholar 

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  1. Interdisziplinäres Zentrum für wissenschaftliches Rechnen (IWR), Universität Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany

    Frank Noé & Jeremy C. Smith

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  1. Frank Noé
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  1. Center for Information Services and High Performance Computing, Technische Universität Dresden, 01062 Dresden, Germany

    Andreas Deutsch

  2. Center for Information Services and High Performance Computing, Technische Universität Dresden, 01062 Dresden, Germany

    Lutz Brusch

  3. Centre for Mathematical Medicine School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, U.K

    Helen Byrne

  4. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada

    Gerda de Vries

  5. Institute of Theoretical Biology, Humboldt-Universität zu Berlin Invalidenstraße 43, 10115 Berlin, Germany

    Hanspeter Herzel

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Noé, F., Smith, J.C. (2007). Transition Networks: A Unifying Theme for Molecular Simulation and Computer Science. In: Deutsch, A., Brusch, L., Byrne, H., Vries, G.d., Herzel, H. (eds) Mathematical Modeling of Biological Systems, Volume I. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4558-8_11

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