Slow Collective Variables of Macromolecular Systems

Part of the Scientific Computation book series (SCIENTCOMP)


This chapter aims to discuss different methods used to determine the frequency spectrum of the motions in a macromolecular system, namely the normal modes, principal components analysis, and the time-lagged auto-encoder machine learning approach.


  1. Aalten, D., Amadei, A., Linssen, A., Eijsink, V., Vriend, G., Berendsen, H.: The essential dynamics of thermolysin-conformation of the hinge-bending motion and comparison of simulations in vacuum and water. Proteins 22, 45–54 (1993)CrossRefGoogle Scholar
  2. Albers, J., Deutch, J.M., Oppenheim, I.: Generalized Langevin equations. J. Chem. Phys. 54(8), 3541–3546 (1971)CrossRefADSGoogle Scholar
  3. Amadei, A., Linssen, A.B.M., Berendsen, H.J.C.: Essential dynamics of proteins. Proteins Struct. Funct. Genet. 17, 412–425 (1993)CrossRefGoogle Scholar
  4. Amadei, A., de Groot, B., Ceruso, M., Paci, M., Di Nola, A., Berendsen, H.: A kinetic model for the internal motions of proteins: diffusion between multiple harmonic wells. Proteins 35(3), 283–292 (1999a)CrossRefGoogle Scholar
  5. Amadei, A., de Groot, B.L., Ceruso, M.A., Paci, M., Di Nola, A., Berendsen, H.J.C.: A kinetic model for the internal motions of proteins: diffusion between multiple harmonic wells. Proteins Struct. Funct. Genet. 35, 283 (1999b)CrossRefGoogle Scholar
  6. Andricioaei, I., Straub, J.E.: Finding the needle in the haystack: algorithms for conformational optimisation. Comput. Phys. 10, 449–454 (1996b)CrossRefADSGoogle Scholar
  7. Andricioaei, I., Straub, J.E.: Global optimisation using bad derivatives: derivative-free method for molecular energy minimisation. J. Comput. Chem. 19(13), 1445–1455 (1998)CrossRefGoogle Scholar
  8. Balucani, U., Zoppi, M.: Dynamics of the Liquid State. Clarendon, Oxford (1994)Google Scholar
  9. Benguria, R., Kac, M.: Quantum langevin equation. Phys. Rev. Lett. 46, 1 (1981)CrossRefADSMathSciNetGoogle Scholar
  10. Brooks, B.B., Janezic, D., Karplus, M.: Hamonic-analysis of large systems. 1. Methodology. J. Comput. Chem. 16, 1522–1542 (1995)Google Scholar
  11. Chandrasekhar, S.: Brownian motion, dynamical friction, and stellar dynamics. Mod. Mod. Phys. 21, 383 (1949)CrossRefADSMathSciNetzbMATHGoogle Scholar
  12. Chen, R., Liu, X., Jin, S., Lin, J., Liu, J.: Machine learning for drug-target interaction prediction. Molecules 23, 2208–2215 (2018)CrossRefGoogle Scholar
  13. Coffey, W.T., Kalmykov, Y.P., Waldron, J.T.: The Langevin Equation, vol. 14. World Scientific Series in Contemporary Chemical Physics, 2nd Edition, World Scientific, (1996)Google Scholar
  14. Collins, C.R., Gordon, G.J., von Lilienfeld, O.A., Yaron, D.J.: Constant size descriptors for accurate machine learning models of molecular properties. J. Chem. Phys. 148, 241718–241711 (2018)CrossRefADSGoogle Scholar
  15. Decherchi, S., Berteotti, A., Bottegoni, G., Rocchia, W., Cavalli, A.: The ligand binding mechanism to purine nucleoside phosphorylase elucidated via molecular dynamics and machine learning. Nat. Commun. 6(6155), 1–10 (2015)Google Scholar
  16. Einstein, A.: Investigations on the Theory of the Brownian Movement. (Edited by Fürth), Methuen and Co. Ltd., London (1926)Google Scholar
  17. Ford, G.W., Kac, M.: On the quantum Langevin equation. J. Stat. Phys. 46, 803–810 (1987)CrossRefADSMathSciNetzbMATHGoogle Scholar
  18. Ford, G.W., Lewis, J.T., O’Connell, R.F.: Quantum Langevin equation. Phys. Rev. A 37, 4419 (1988)CrossRefADSMathSciNetGoogle Scholar
  19. Frenkel, D., Smit, B.: Understanding Molecular Simulation from Algorithms to Applications. Academic, San Diego (2001). ISBN 9780122673511Google Scholar
  20. Gallo, P., Rovere, M., Ricci, M.A., Hartnig, C., Spohr, E.: Non-exponential kinetic behaviour of confined water. Europhys. Lett. 49(2), 183 (2000)CrossRefADSGoogle Scholar
  21. Garcia, A.E.: Large-amplitude nonlinear motions in proteins. Phys. Rev. Lett. 68, 2696 (1992)CrossRefADSGoogle Scholar
  22. Gastegger, M., Schwiedrzik, L., Bittermann, M., Berzsenyi, F., Marquetand, P.: wACSF-Weighted atom-centered symmetry functions as descriptors in machine learning potentials. J. Chem. Phys. 148, 241709–241711 (2018)CrossRefADSGoogle Scholar
  23. Go, N.: A theorem on amplitudes of thermal atomic fluctuations in large molecules assuming specific conformations calculated by normal mode analysis. Biophys. Chem. 35, 105–112 (1990)CrossRefGoogle Scholar
  24. Goh, G.B., Siegel, C., Vishnu, A., Hodas, N., Baker, N.: How much chemistry does a deep neural network need to know to make accurate predictions? In: 2018 IEEE Winter Conference on Applications of Computer Vision (WACV), pp. 1340–1349 (2018)Google Scholar
  25. Goldstein, H.: Classical Mechanics, 2nd edn. Addison-Wesley, San Francisco (2002)zbMATHGoogle Scholar
  26. Grubmüller, H.: Predicting slow structural transitions in macromolecular systems: conformational flooding. Phys. Rev. E 52, 2893 (1995)CrossRefADSGoogle Scholar
  27. Hänggi, P., Ingold, G.-L.: Fundamental aspects of quantum Brownian motion. Chaos 15, 026105–1 (2005)CrossRefADSMathSciNetzbMATHGoogle Scholar
  28. Herr, J.E., Yao, K., McIntyre, R., Toth, D.W., Parkhill, J.: Metadynamics for training neural network model chemistries: a competitive assessment. J. Chem. Phys. 148, 241710–9 (2018)CrossRefADSGoogle Scholar
  29. Ichiye, T., Karplus, M.: Collective motions in proteins: a covariance analysis of atomic fluctuations in molecular-dynamics and normal mode simulations. Proteins 11, 205–217 (1991)CrossRefGoogle Scholar
  30. Islam, M.A.: Einstein - Smoluchowski diffusion equation: A discussion. Physica Scripta. 70, 120 (2004)CrossRefADSzbMATHGoogle Scholar
  31. Janezic, D., Brooks, B.B.: Harmonic analysis of large systems: II. Comparison of different protein models. J. Comput. Chem. 16, 1543–1553 (1995)Google Scholar
  32. Kamath, A., Vargas-Hernández, R.A., Krems, R.V., Carrington, T. Jr., Manzhos, S.: Neural networks vs Gaussian process regression for representing potential energy surface: a comparative study of fit quality and vibrational spectrum accuracy. J. Chem. Phys. 148, 241702–7 (2018)CrossRefADSGoogle Scholar
  33. Kamberaj, H.: A theoretical model for the collective motion of proteins by means of principal component analysis. Cent. Eur. J. Phys. 9(1), 96–109 (2011)Google Scholar
  34. Kamberaj, H.: Sampling Convergence of collective motions in proteins. J. Appl. Phys. Sci. Int. 8(3), 101–112 (2017)Google Scholar
  35. Karhunen, K.: Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fenn. Ser. A1, 37, 1–79 (1947)MathSciNetzbMATHGoogle Scholar
  36. Karplus, M., Jushick, J.N.: Method for estimating the configurational entropy of macromolecules. Macromolecules 14, 325–332 (1981)CrossRefADSGoogle Scholar
  37. Kitao, A., Hirata, F., Go, N.: The effect of solvent on the conformation and the collective motions of protein: normal mode analysis and molecular dynamics simulations of Melittin in water and in vacuum. Chem. Phys. 158, 447–472 (1991)CrossRefGoogle Scholar
  38. Lange, O.F., Grubmüller, H.: Can principal components yield a dimension reduced description of protein dynamics on long time scales? J. Phys. Chem. B 110, 22842–22852 (2006)CrossRefGoogle Scholar
  39. Lange, O.F., Schäfer, L.V., Grubmüller, H.: Flooding in GROMACS: accelerated barrier crossing in molecular dynamics. J. Comput. Chem. 27(14), 1693–1702 (2006)CrossRefGoogle Scholar
  40. Levitt, M., Sander, C., Stern, P.S.: Protein normal-mode dynamics: trypsin inhibitor, cram bin, ribonuclease and lysozyme. J. Mol. Biol. 181, 423–447 (1985)CrossRefGoogle Scholar
  41. Lifson, S., Warshel, A.: Consistent force field for calculations of conformations, vibrational spectra, and enthalpies of cycloalkane and n-alkane molecules. J. Chem. Phys. 49, 5116–5129 (1968)CrossRefADSGoogle Scholar
  42. Lubbers, N., Smith, J.S., Barros, K.: Hirarchical modeling of molecular energies using a deep neural network. J. Chem. Phys. 148, 241715–8 (2018)CrossRefADSGoogle Scholar
  43. McCulloch, W.S., Pitts, W.H.: A logical calculus of the ideas immanent in neural nets. Bull. Math. Biophys. 5, 115–133 (1943)CrossRefMathSciNetzbMATHGoogle Scholar
  44. Mokshin, A.V., Yulmetyev, R.M., Hänggi, P.: Diffusion processes and memory effects. New J. Phys., 7:9 (2005)CrossRefGoogle Scholar
  45. Paass, G.: Assessing and improving neural network predictions by the bootstrap algorithm. In: Hanson, S.J., Cowan, J.D., Giles, C.L. (eds.) Advances in Neural Information Processing Systems, vol. 5, pp. 196–203. Morgan-Kaufmann, San Francisco (1993)Google Scholar
  46. Park, S., Schulten, K.: Calculating potentials of mean force from steered molecular dynamics simulations. J. Chem. Phys. 120(13), 5946 (2004)CrossRefADSGoogle Scholar
  47. Qian, N.: On the momentum term in gradient descent learning algorithms. Neural Netw. 12, 145–151 (1999)CrossRefGoogle Scholar
  48. Schneider, E., Dai, L., Topper, R.Q., Drechsel-Grau, C., Tuckerman, M.E.: Stochastic neural network approach for learning high-dimensional free energy surfaces. Phys. Rev. Lett. 119, 150601 (2017)CrossRefADSMathSciNetGoogle Scholar
  49. Srivastava, N., Hinton, G.E., Krizhevsky, A., Sutskever, I., Salakhutdinov, R.: A simple way to prevent neural networks from overfitting. J. Mach. Learn. Res. 15, 1929–1958 (2014)MathSciNetzbMATHGoogle Scholar
  50. Stepanova, M.: Dynamics of essential collective motions in proteins: theory. Phys. Rev. E 76(5), 051918 (2007)CrossRefADSMathSciNetGoogle Scholar
  51. Wainwright, T., Alder, B.J., Gass, D.M.: Decay time correlations in two dimensions. Phys. Rev. A 4, 233 (1971)CrossRefADSGoogle Scholar
  52. Wehmeyer, C., Noé, F.: Time-lagged autoencoders: deep learning of slow collective variables for molecular kinetics. J. Chem. Phys. 148, 241703–9 (2018)CrossRefADSGoogle Scholar
  53. Yulmetyev, R.M., Mokshin, A.V., Hänggi, P.: Diffusion time-scale invariance, randomization processes, and memory effects in lennard-jones liquids. Phys. Rev. E 68, 051201 (2003)CrossRefADSGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Computer EngineeringInternational Balkan UniversitySkopjeNorth Macedonia
  2. 2.Advanced Computing Research CenterUniversity of New York TiranaTiranaAlbania

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