A theoretical model for the collective motion of proteins by means of principal component analysis

Hiqmet Kamberaj 1
  • 1 Faculty of Technical Sciences, International Balkan University, Avtokomanda Tashko Karaxha BB, Skopje, R. of Macedonia

Abstract

A coarse grained model in the frame work of principal component analysis is presented. We used a bath of harmonic oscillators approach, based on classical mechanics, to derive the generalized Langevin equations of motion for the collective coordinates. The dynamics of the protein collective coordinates derived from molecular dynamics simulations have been studied for the Bovine Pancreatic Trypsin Inhibitor. We analyzed the stability of the method by studying structural fluctuations of the Ca atoms obtained from a 20 ns molecular dynamics simulation. Subsequently, the dynamics of the collective coordinates of protein were characterized by calculating the dynamical friction coefficient and diffusion coefficients along with time-dependent correlation functions of collective coordinates. A dual diffusion behavior was observed with a fast relaxation time of short diffusion regime 0.2–0.4 ps and slow relaxation time of long diffusion about 1–2 ps. In addition, we observed a power law decay of dynamical friction coefficient with exponent for the first five collective coordinates varying from −0.746 to −0.938 for the real part and from −0.528 to −0.665 for its magnitude. It was found that only the first ten collective coordinates are responsible for configuration transitions occurring on time scale longer than 50 ps.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] M. Karplus, J.A. McCammon, Nat. Struct. Biol. 9, 646 (2002) http://dx.doi.org/10.1038/nsb0902-646

  • [2] M. Karplus, J.A. McCammon, Nat. Struct. Biol. 9, 788 (2002) http://dx.doi.org/10.1038/nsb0902-646

  • [3] W.F. van Gunsteren et al., Angew. Chem. Int. Edit. 9, 4064 (2006)

  • [4] G.M. Torrie, J.P. Valleau, J. Comput. Phys. 23, 187 (1977) http://dx.doi.org/10.1016/0021-9991(77)90121-8

  • [5] H. Grubmüller, Phys. Rev. E 52, 2893 (1995) http://dx.doi.org/10.1103/PhysRevE.52.2893

  • [6] Y. Sugita, Y. Okamoto, Chem. Phys. Lett. 314, 141 (1999) http://dx.doi.org/10.1016/S0009-2614(99)01123-9

  • [7] A. Laio, M. Parrinello, Proc. Nat. Acad. Sci. U.S.A. 99, 12562 (2002) http://dx.doi.org/10.1073/pnas.202427399

  • [8] P. Minary, G.J. Martyna, M.E. Tuckerman, Phys. Rev. Lett. 93, 150201 (2004) http://dx.doi.org/10.1103/PhysRevLett.93.150201

  • [9] H. Kamberaj, A. van der Vaart, J. Chem. Phys. 127, 234102 (2007) http://dx.doi.org/10.1063/1.2806930

  • [10] H. Kamberaj, A. van der Vaart, J. Chem. Phys. 130, 074906 (2009) http://dx.doi.org/10.1063/1.3077857

  • [11] R.G. Palmer, Adv. Phys. 32, 669 (1982) http://dx.doi.org/10.1080/00018738200101438

  • [12] Y. Ueeda, H. Taketomi, N. Gō Biopolymers 17, 1531 (1978) http://dx.doi.org/10.1002/bip.1978.360170612

  • [13] J.A. McCammon, S.H. Northrup, M. Karplus, R.M. Levy, Biopolymers 19, 2033 (1984) http://dx.doi.org/10.1002/bip.1980.360191108

  • [14] I. Bahar, J. Mol. Biol. 266, 195 (1997) http://dx.doi.org/10.1006/jmbi.1996.0758

  • [15] A. Irbäck, F. Sjunnesson, S. Wallin, Proc. Nat. Acad. Sci. U.S.A. 97, 13614 (2000) http://dx.doi.org/10.1073/pnas.240245297

  • [16] A.V. Smith, C.K. Hall, Proteins 44, 344 (2001) http://dx.doi.org/10.1002/prot.1100

  • [17] A.V. Smith, C.K. Hall, Proteins 44, 376 (2001) http://dx.doi.org/10.1002/prot.1103

  • [18] A. Liwo et al., J. Phys. Chem. B 108, 16918 (2004) http://dx.doi.org/10.1021/jp040327c

  • [19] S. Oldziej, A. Liwo, C. Czaplewski, J. Pillardy, H.A. Scheraga, J. Phys. Chem. B 108, 16934 (2004) http://dx.doi.org/10.1021/jp0403285

  • [20] S. Oldziej et al., J. Phys. Chem. B. 108, 16950 (2004) http://dx.doi.org/10.1021/jp040329x

  • [21] V. Tozzini, Curr. Opin. Struc. Biol. 15, 144 (2005) http://dx.doi.org/10.1016/j.sbi.2005.02.005

  • [22] V. Tozzini, J.A. McCammon, Chem. Phys. Lett. 413, 123 (2005) http://dx.doi.org/10.1016/j.cplett.2005.07.075

  • [23] V. Tozzini, W. Rocchia, J.A. McCammon, J. Chem. Theory Comput. 2, 667 (2006) http://dx.doi.org/10.1021/ct050294k

  • [24] V. Tozzini, J. Trylska, C.-E Chang, J.A. McCammon, J. Struct. Biol. 157, 606 (2007) http://dx.doi.org/10.1016/j.jsb.2006.08.005

  • [25] J. Trylska, V. Tozzini, C.-E Chang, J.A. McCammon, Biophys. J. 92, 4179 (2007) http://dx.doi.org/10.1529/biophysj.106.100560

  • [26] K. Karhunen, Ann. Acad. Sci. Fenn. A1 37, 3 (1947)

  • [27] B.B. Brooks, D. Janezic, M. Karplus, J. Comput. Chem. 16, 1522 (1995) http://dx.doi.org/10.1002/jcc.540161209

  • [28] D. Janezic, B.B. Brooks, J. Comput. Chem. 16, 1543 (1995) http://dx.doi.org/10.1002/jcc.540161210

  • [29] M. Karplus, J.N. Jushick, Macromolecules 14, 325 (1981) http://dx.doi.org/10.1021/ma50003a019

  • [30] T. Ichiye, M. Karplus, Proteins 11, 205 (1991) http://dx.doi.org/10.1002/prot.340110305

  • [31] T. Horiuchi, N. Gō, Proteins 10, 106 (1990) http://dx.doi.org/10.1002/prot.340100204

  • [32] A. Kitao, F. Hirata, N. Gō, Chem. Phys. 158, 447 (1991) http://dx.doi.org/10.1016/0301-0104(91)87082-7

  • [33] A. Amadei, A.B.M. Linssen, H.J.C. Berendsen, Proteins: Structure, Function and Genetics 17, 412 (1993) http://dx.doi.org/10.1002/prot.340170408

  • [34] D. Aalten et al., Proteins: Structure, Function, and Genetics 22, 45 (1995) http://dx.doi.org/10.1002/prot.340220107

  • [35] N. Gō, Biophys. Chem. 35, 105 (1990) http://dx.doi.org/10.1016/0301-4622(90)80065-F

  • [36] A.E. Garcia, Phys. Rev. Lett. 68, 2696 (1992) http://dx.doi.org/10.1103/PhysRevLett.68.2696

  • [37] H. Grubmüller, P. Tavan, J. Chem. Phys. 101, 5047 (1994) http://dx.doi.org/10.1063/1.467427

  • [38] M.A. Balsera, W. Wriggers, Y. Oono, K. Schulten, J. Phys. Chem. B 100, 2567 (1996) http://dx.doi.org/10.1021/jp9536920

  • [39] O.F. Lange, H. Grubmüller, J. Chem. Phys. 124, 214903 (2006) http://dx.doi.org/10.1063/1.2199530

  • [40] M. Stepanova, Phys. Rev. E 76, 051918 (2007) http://dx.doi.org/10.1103/PhysRevE.76.051918

  • [41] R. Benguria, M. Kac, Phys. Rev. Lett. 46, 1 (1981) http://dx.doi.org/10.1103/PhysRevLett.46.1

  • [42] G.W. Ford, M. Kac, J. Stat. Phys. 46, 803 (1981) http://dx.doi.org/10.1007/BF01011142

  • [43] G.W. Ford, J.T. Lewis, R.F. O’Connell, Phys. Rev. A 37, 4419 (1988) http://dx.doi.org/10.1103/PhysRevA.37.4419

  • [44] P. Hänggi, G.-L. Ingold, Chaos 15, 026105 (2005) http://dx.doi.org/10.1063/1.1853631

  • [45] O.F. Lange, H. Grubmüller, J. Phys. Chem. B 110, 22842 (2006) http://dx.doi.org/10.1021/jp062548j

  • [46] B. Hess, Phys. Rev. E 65, 031910 (2002) http://dx.doi.org/10.1103/PhysRevE.65.031910

  • [47] D. Frenkel, B. Smit, Understanding Molecular Simulation from Algorithms to Applications (Academic, California, 2002)

  • [48] A. Einstein, Investigations on the theory of the Brownian movement, Edited by Fürth (Methuen and Co. Ltd., London, 1926)

  • [49] S. Chandrasekhar, Rev. Mod. Phys. 21, 383 (1949) http://dx.doi.org/10.1103/RevModPhys.21.383

  • [50] M.A. Islam, Phys. Scripta 70, 120 (2004) http://dx.doi.org/10.1088/0031-8949/70/2-3/008

  • [51] U. Balucani, M. Zoppi, Dynamics of the Liquid State (Clarendon, Oxford, 1994)

  • [52] R.M. Yulmetyev, A.V. Mokshin, P. Hänggi, Phys. Rev. E 68, 051201 (2003) http://dx.doi.org/10.1103/PhysRevE.68.051201

  • [53] A.V. Mokshin, R.M. Yulmetyev, P. Hänggi, New J. Phys. 7, 1 (2005) http://dx.doi.org/10.1088/1367-2630/7/1/009

  • [54] A. Amadei et al., Proteins: Structure, Function and Genetics 35, 283 (1999) http://dx.doi.org/10.1002/(SICI)1097-0134(19990515)35:3<283::AID-PROT2>3.0.CO;2-R

  • [55] S. Parkin, B. Rupp, H. Hope, Acta Crystallogr. D 52, 18 (1996) http://dx.doi.org/10.1107/S0907444995008675

  • [56] D.A. Case et al., Amber 8.0 (University of California, San Francisco, CA, 2004)

  • [57] T.E. Cheatham III, M.A. Young, Biopolymers 56, 232 (2001) http://dx.doi.org/10.1002/1097-0282(2000)56:4<232::AID-BIP10037>3.0.CO;2-H

  • [58] J.W. Ponder, D.A. Case, Adv. Protein Chem. 66, 27 (2003) http://dx.doi.org/10.1016/S0065-3233(03)66002-X

  • [59] U. Essmann et al., J. Chem. Phys. 103, 8577 (1995) http://dx.doi.org/10.1063/1.470117

  • [60] J.P. Ryckaert, G. Ciccotti, J. Comput. Phys. 23, 327 (1977) http://dx.doi.org/10.1016/0021-9991(77)90098-5

  • [61] D. Aalten, B.D. Groot, J. Findlay, H.J.C. Berendsen, A. Amadei, J. Comput. Chem. 18, 169 (1997) http://dx.doi.org/10.1002/(SICI)1096-987X(19970130)18:2<169::AID-JCC3>3.0.CO;2-T

  • [62] J.A. Hartigan, M.A. Wong, Appl. Statist. 28, 100 (1979) http://dx.doi.org/10.2307/2346830

  • [63] D. Holmes, E. Mergen, Qual. Eng. 10, 505 (1998) http://dx.doi.org/10.1080/08982119808919164

  • [64] P.H. Hünenberger, A.E. Mark, W.F. van Gunsteren, J. Mol. Biol. 252, 492 (1995) http://dx.doi.org/10.1006/jmbi.1995.0514

  • [65] M. Karplus, T. Ichiye, J. Mol. Biol. 263, 120 (1996) http://dx.doi.org/10.1006/jmbi.1996.0562

  • [66] L. Meinhold, J.C. Smith, Biophys. J. 88, 2554 (2005) http://dx.doi.org/10.1529/biophysj.104.056101

  • [67] D.M.F. van Aalten et al., Biophys. J. 73, 2891 (1997) http://dx.doi.org/10.1016/S0006-3495(97)78317-6

  • [68] Y. Mu, P.H. Nguyen, G. Stock, Proteins 58, 45 (2005) http://dx.doi.org/10.1002/prot.20310

  • [69] G.G. Maisuradze, D.M. Leitner, Proteins 67, 569 (2007) http://dx.doi.org/10.1002/prot.21344

  • [70] G.G. Maisuradze, A. Liwo, H.A. Scheraga, J. Mol. Biol. 385, 312 (2009) http://dx.doi.org/10.1016/j.jmb.2008.10.018

  • [71] W.T. Coffey, Y.P. Kalmykov, J. Waldron, The Langevin Equation, World Scientific Series in Contemporary Chemical Physics volume 10 (World Scientific, Singapore, 1996)

OPEN ACCESS

Journal + Issues

Search