American Institute of Mathematical Sciences

Advanced
October  2011, 4(5): 1247-1266. doi: 10.3934/dcdss.2011.4.1247

Breather-mediated energy transfer in proteins

Francesco Piazza 1, and  Yves-Henri Sanejouand 2,

1. 

Centre de Biophysique Moleculaire (CBM-CNRS), University of Orleans, Rue Charles Sadron, 45071 Orleans, France
2. 

Laboratoire Biotechnologie, Biocatalyse et Biorégulation, UMR 6204 du CNRS, Faculté des Sciences et des Techniques, 2, rue de la Houssinière, 44322 Nantes Cedex 3, France

Received  September 2009 Revised  December 2009 Published  December 2010

In this paper we investigate how energy is redistributed across protein structures, following localized kicks, within the framework of a nonlinear network model. We show that energy is directed most of the times to a few specific sites, systematically within the stiffest regions. This effect is sharpened as the energy of the kicks is increased, with fractions of transferred energy as high as 70% already for kicks above $20$ kcal/mol. Remarkably, we show that such site-selective, high-yield transfers mark the spontaneous formation of spatially localized, time-periodic vibrations at the target sites, acting as efficient energy-collecting centers. A comparison of our simulations with a previously developed theory reveals that such energy-pinning modes are discrete breathers, able to carry energy across the structure in an quasi-coherent fashion by jumping from site to site.
Keywords: Discrete Breathers, Energy transfer., Nonlinear dynamical systems, Network models, Enzymes, Proteins.
Mathematics Subject Classification: Primary: 70-XX, 37-XX, 92-XX; Secondary: 70K75, 34C15, 37N25, 92C4.
Citation: Francesco Piazza, Yves-Henri Sanejouand. Breather-mediated energy transfer in proteins. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1247-1266. doi: 10.3934/dcdss.2011.4.1247
References:
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[2]

J. F. R. Archilla, Yu. B. Gaididei, P. L. Christiansen and J. Cuevas, Stationary and moving breathers in a simplified model of curved alpha-helix proteins,, Journal of Physics A: Mathematical and General, 35 (2002), 8885.  doi: 10.1088/0305-4470/35/42/301.  Google Scholar

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S. Aubry, Discrete breathers: Localization and transfer of energy in discrete hamiltonian nonlinear systems,, Physica D: Nonlinear Phenomena, 216 (2006), 1.  doi: 10.1016/j.physd.2005.12.020.  Google Scholar

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I. Bahar, A. R. Atilgan and B. Erman, Direct evaluation of thermal fluctuations in proteins using a single-parameter harmonic potential,, Fold. Des., 2 (1997), 173.  doi: 10.1016/S1359-0278(97)00024-2.  Google Scholar

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I. Bahar and Q. Cui (eds.), "Normal Mode Analysis: Theory and Applications to Biological and Chemical Systems,", C&H/CRC Mathematical & Computational Biology Series, 9 (2005).   Google Scholar

[6]

B. R. Brooks and M. Karplus, Harmonic dynamics of proteins: Normal modes and fluctuations in bovine pancreatic trypsin inhibitor,, Proc. Natl. Acad. Sci. USA, 80 (1983), 6571.  doi: 10.1073/pnas.80.21.6571.  Google Scholar

[7]

V. M. Burlakov, S. A. Kiselev and V. N. Pyrkov, Computer-simulation of intrinsic localized modes in one-dimensional and 2-dimensional anharmonic lattices,, Physical Review B, 42 (1990), 4921.  doi: 10.1103/PhysRevB.42.4921.  Google Scholar

[8]

F. Columbus (ed.), "Soft Condensed Matter. New Research,", Nova Science Publishers, (2005).   Google Scholar

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T. Dauxois, R. Khomeriki, F. Piazza and S. Ruffo, The anti-FPU problem,, Chaos, 15 (2005).  doi: 10.1063/1.1854273.  Google Scholar

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T. Dauxois, A. Litvak-Hinenzon, R. MacKay and A. Spanoudaki (eds.), "Energy Localisation and Transfer in Crystals, Biomolecules and Josephson Arrays,", Advanced Series in Nonlinear Dynamics, 22 (2004).   Google Scholar

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A. del Sol, C. J. Tsai, B. Y. Ma and R. Nussinov, The origin of allosteric functional modulation: Multiple pre-existing pathways,, Structure, 17 (2009), 1042.  doi: 10.1016/j.str.2009.06.008.  Google Scholar

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J. J. Falke, Enzymology: A moving story,, Science, 295 (2002), 1480.  doi: 10.1126/science.1069823.  Google Scholar

[15]

S. Flach and G. Mutschke, Slow relaxation and phase-space properties of a conservative system with many degrees of freedom,, Physical Review E, 49 (1994), 5018.  doi: 10.1103/PhysRevE.49.5018.  Google Scholar

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S. Hayward, A. Kitao and N. Go, Harmonicity and anharmonicity in protein dynamics: A normal mode analysis and principal component analysis,, Proteins, 23 (1995), 177.  doi: 10.1002/prot.340230207.  Google Scholar

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K. A. Henzler-Wildman, M. Lei, V. Thai, S. Jordan Kerns, M. Karplus and D. Kern, A hierarchy of timescales in protein dynamics is linked to enzyme catalysis,, Nature, 450 (2007), 913.  doi: 10.1038/nature06407.  Google Scholar

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K. Hinsen, Analysis of domain motions by approximate normal mode calculations,, Proteins, 33 (1998), 417.  doi: 10.1002/(SICI)1097-0134(19981115)33:3<417::AID-PROT10>3.0.CO;2-8.  Google Scholar

[21]

B. Juanico, Y.-H. Sanejouand, F. Piazza and P. De Los Rios, Discrete breathers in nonlinear network models of proteins,, Phys. Rev. Lett., 99 (2007).  doi: 10.1103/PhysRevLett.99.238104.  Google Scholar

[22]

G. Kopidakis, S. Aubry and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems,, Phys. Rev. Lett., 87 (2001).  doi: 10.1103/PhysRevLett.87.165501.  Google Scholar

[23]

D. M. Leitner, Anharmonic decay of vibrational states in helical peptides, coils, and one-dimensional glasses,, Journal of Physical Chemistry A, 106 (2002), 10870.  doi: 10.1021/jp0206119.  Google Scholar

[24]

D. M. Leitner, Vibrational energy transfer in helices,, Phys. Rev. Lett., 87 (2001).  doi: 10.1103/PhysRevLett.87.188102.  Google Scholar

[25]

M. Levitt, C. Sander and P. S. Stern, Normal-mode dynamics of a protein: Bovine pancreatic trypsin inhibitor,, Int. J. Quant. Chem., 10 (1983), 181.   Google Scholar

[26]

R. M. Levy, D. Perahia and M. Karplus, Molecular dynamics of an alpha-helical polypeptide: Temperature dependance and deviation from harmonic behavior,, Proc. Natl. Acad. Sci. USA, 79 (1982), 1346.  doi: 10.1073/pnas.79.4.1346.  Google Scholar

[27]

K. Moritsugu, O. Miyashita and A. Kidera, Vibrational energy transfer in a protein molecule,, Physical Review Letters, 85 (2000), 3970.  doi: 10.1103/PhysRevLett.85.3970.  Google Scholar

[28]

T. Noguti and N. Go, Collective variable description of small-amplitude conformational fluctuations in a globular protein,, Nature, 296 (1982), 776.  doi: 10.1038/296776a0.  Google Scholar

[29]

M. Peyrard, "Nonlinear Excitations in Biomolecules,", Springer, (1995).   Google Scholar

[30]

M. Peyrard, The pathway to energy localization in nonlinear lattices,, Physica D: Nonlinear Phenomena, 119 (1998), 184.  doi: 10.1016/S0167-2789(98)00079-7.  Google Scholar

[31]

F. Piazza and Y.-H. Sanejouand, Discrete breathers in protein structures,, Phys. Biol, 5 (2008).  doi: 10.1088/1478-3975/5/2/026001.  Google Scholar

[32]

F. Piazza and Y.-H. Sanejouand, Long-range energy transfer in proteins,, Physical Biology, 6 (2009).  doi: 10.1088/1478-3975/6/4/046014.  Google Scholar

[33]

K. O. Rasmussen, D. Cai, A. R. Bishop and N. Gronbech-Jensen, Localization in a nonlinear disordered system,, Europhysics Letters, 47 (1999), 421.  doi: 10.1209/epl/i1999-00405-1.  Google Scholar

[34]

J. Ross, Energy transfer from adenosine triphosphate,, The Journal of Physical Chemistry B, 110 (2006), 6987.  doi: 10.1021/jp0556862.  Google Scholar

[35]

M. Rueda, P. Chacon and M. Orozco, Thorough validation of protein normal mode analysis: A comparative study with essential dynamics,, Structure, 15 (2007), 565.  doi: 10.1016/j.str.2007.03.013.  Google Scholar

[36]

B. Rumpf, Growth and erosion of a discrete breather interacting with rayleigh-jeans distributed phonons,, EPL, 78 (2007).  doi: 10.1209/0295-5075/78/26001.  Google Scholar

[37]

S. Sacquin-Mora, E. Laforet and R. Lavery, Locating the active sites of enzymes using mechanical properties,, Proteins, 67 (2007), 350.  doi: 10.1002/prot.21353.  Google Scholar

[38]

D. E. Sagnella, J. E. Straub and D. Thirumalai, Time scales and pathways for kinetic energy relaxation in solvated proteins: Application to carbonmonoxy myoglobin,, J. Chem. Phys., 113 (2000), 7702.  doi: 10.1063/1.1313554.  Google Scholar

[39]

K. W. Sandusky, J. B. Page and K. E. Schmidt, Stability and motion of intrinsic localized modes in nonlinear periodic lattices,, Physical Review B, 46 (1992), 6161.  doi: 10.1103/PhysRevB.46.6161.  Google Scholar

[40]

M. Sato and A. Sievers, Experimental and numerical exploration of intrinsic localized modes in an atomic lattice,, Journal of Biological Physics, 35 (2009), 57.  doi: 10.1007/s10867-009-9135-2.  Google Scholar

[41]

A. Scott, Davydov's soliton,, Physics Reports, 217 (1992), 1.  doi: 10.1016/0370-1573(92)90093-F.  Google Scholar

[42]

E. Segré (ed.), "Collected Papers of Enrico Fermi,", University of Chicago Press, (1965).   Google Scholar

[43]

F. Tama and Y. H. Sanejouand, Conformational change of proteins arising from normal mode calculations,, Protein Engineering Design and Selection, 14 (2001), 1.  doi: 10.1093/protein/14.1.1.  Google Scholar

[44]

M. M. Tirion, Large amplitude elastic motions in proteins from a single-parameter, atomic analysis,, Physical Review Letters, 77 (1996), 1905.  doi: 10.1103/PhysRevLett.77.1905.  Google Scholar

[45]

C. J. Tsai, A. del Sol and R. Nussinov, Allostery: Absence of a change in shape does not imply that allostery is not at play,, Journal of Molecular Biology, 378 (2008), 1.  doi: 10.1016/j.jmb.2008.02.034.  Google Scholar

[46]

C. J. Tsai, A. Del Sol and R. Nussinov, Protein allostery, signal transmission and dynamics: A classification scheme of allosteric mechanisms,, Molecular Biosystems, 5 (2009), 207.  doi: 10.1039/b819720b.  Google Scholar

[47]

P. C. Whitford, J. N. Onuchic and P. G. Wolynes, Energy landscape along an enzymatic reaction trajectory: Hinges or cracks?,, HFSP Journal, 2 (2008), 61.  doi: 10.2976/1.2894846.  Google Scholar

[48]

S. Woutersen and P. Hamm, Nonlinear two-dimensional vibrational spectroscopy of peptides,, Journal of Physics: Condensed Matter, 14 (2002).  doi: 10.1088/0953-8984/14/39/202.  Google Scholar

[49]

A. Xie, A. F. G. van der Meer and R. H. Austin, Excited-state lifetimes of far-infrared collective modes in proteins,, Physical Review Letters, 88 (2001).  doi: 10.1103/PhysRevLett.88.018102.  Google Scholar

[50]

A. Xie, L. van der Meer, W. Hoff and R. H. Austin, Long-lived amide i vibrational modes in myoglobin,, Physical Review Letters, 84 (2000), 5435.  doi: 10.1103/PhysRevLett.84.5435.  Google Scholar

[51]

L. W. Yang and I. Bahar, Coupling between catalytic site and collective dynamics: A requirement for mechanochemical activity of enzymes,, Structure, 13 (2005), 893.  doi: 10.1016/j.str.2005.03.015.  Google Scholar

[52]

X. Yu and D. M. Leitner, Vibrational energy transfer and heat conduction in a protein,, Journal of Physical Chemistry B, 107 (2003), 1698.  doi: 10.1021/jp026462b.  Google Scholar

show all references

References:
[1]

F. Abdullaev, O. Bang and M. P. Sorensen (eds.), "Nonlinearity and Disorder: Theory and Applications,", vol. \textbf{45}, 45 (2001).   Google Scholar

[2]

J. F. R. Archilla, Yu. B. Gaididei, P. L. Christiansen and J. Cuevas, Stationary and moving breathers in a simplified model of curved alpha-helix proteins,, Journal of Physics A: Mathematical and General, 35 (2002), 8885.  doi: 10.1088/0305-4470/35/42/301.  Google Scholar

[3]

S. Aubry, Discrete breathers: Localization and transfer of energy in discrete hamiltonian nonlinear systems,, Physica D: Nonlinear Phenomena, 216 (2006), 1.  doi: 10.1016/j.physd.2005.12.020.  Google Scholar

[4]

I. Bahar, A. R. Atilgan and B. Erman, Direct evaluation of thermal fluctuations in proteins using a single-parameter harmonic potential,, Fold. Des., 2 (1997), 173.  doi: 10.1016/S1359-0278(97)00024-2.  Google Scholar

[5]

I. Bahar and Q. Cui (eds.), "Normal Mode Analysis: Theory and Applications to Biological and Chemical Systems,", C&H/CRC Mathematical & Computational Biology Series, 9 (2005).   Google Scholar

[6]

B. R. Brooks and M. Karplus, Harmonic dynamics of proteins: Normal modes and fluctuations in bovine pancreatic trypsin inhibitor,, Proc. Natl. Acad. Sci. USA, 80 (1983), 6571.  doi: 10.1073/pnas.80.21.6571.  Google Scholar

[7]

V. M. Burlakov, S. A. Kiselev and V. N. Pyrkov, Computer-simulation of intrinsic localized modes in one-dimensional and 2-dimensional anharmonic lattices,, Physical Review B, 42 (1990), 4921.  doi: 10.1103/PhysRevB.42.4921.  Google Scholar

[8]

F. Columbus (ed.), "Soft Condensed Matter. New Research,", Nova Science Publishers, (2005).   Google Scholar

[9]

T. Dauxois, R. Khomeriki, F. Piazza and S. Ruffo, The anti-FPU problem,, Chaos, 15 (2005).  doi: 10.1063/1.1854273.  Google Scholar

[10]

T. Dauxois, A. Litvak-Hinenzon, R. MacKay and A. Spanoudaki (eds.), "Energy Localisation and Transfer in Crystals, Biomolecules and Josephson Arrays,", Advanced Series in Nonlinear Dynamics, 22 (2004).   Google Scholar

[11]

A. del Sol, C. J. Tsai, B. Y. Ma and R. Nussinov, The origin of allosteric functional modulation: Multiple pre-existing pathways,, Structure, 17 (2009), 1042.  doi: 10.1016/j.str.2009.06.008.  Google Scholar

[12]

F. d'Ovidio, H. G. Bohr and P.-A. Lindgård, Solitons on H-bonds in proteins,, Journal of Physics: Condensed Matter, 15 (2003).  doi: 10.1088/0953-8984/15/18/304.  Google Scholar

[13]

F. d'Ovidio, H. G. Bohr and P.-A. Lindgrd, Analytical tools for solitons and periodic waves corresponding to phonons on lennard-jones lattices in helical proteins,, Physical Review E, 71 (2005), 026606.  doi: 10.1103/PhysRevE.71.026606.  Google Scholar

[14]

J. J. Falke, Enzymology: A moving story,, Science, 295 (2002), 1480.  doi: 10.1126/science.1069823.  Google Scholar

[15]

S. Flach and G. Mutschke, Slow relaxation and phase-space properties of a conservative system with many degrees of freedom,, Physical Review E, 49 (1994), 5018.  doi: 10.1103/PhysRevE.49.5018.  Google Scholar

[16]

S. Flach and C. R. Willis, Discrete breathers,, Physics Reports, 295 (1998), 181.  doi: 10.1016/S0370-1573(97)00068-9.  Google Scholar

[17]

S. Flach and A. V. Gorbach, Discrete breathers - advances in theory and applications,, Physics Reports, 467 (2008), 1.  doi: 10.1016/j.physrep.2008.05.002.  Google Scholar

[18]

S. Hayward, A. Kitao and N. Go, Harmonicity and anharmonicity in protein dynamics: A normal mode analysis and principal component analysis,, Proteins, 23 (1995), 177.  doi: 10.1002/prot.340230207.  Google Scholar

[19]

K. A. Henzler-Wildman, M. Lei, V. Thai, S. Jordan Kerns, M. Karplus and D. Kern, A hierarchy of timescales in protein dynamics is linked to enzyme catalysis,, Nature, 450 (2007), 913.  doi: 10.1038/nature06407.  Google Scholar

[20]

K. Hinsen, Analysis of domain motions by approximate normal mode calculations,, Proteins, 33 (1998), 417.  doi: 10.1002/(SICI)1097-0134(19981115)33:3<417::AID-PROT10>3.0.CO;2-8.  Google Scholar

[21]

B. Juanico, Y.-H. Sanejouand, F. Piazza and P. De Los Rios, Discrete breathers in nonlinear network models of proteins,, Phys. Rev. Lett., 99 (2007).  doi: 10.1103/PhysRevLett.99.238104.  Google Scholar

[22]

G. Kopidakis, S. Aubry and G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems,, Phys. Rev. Lett., 87 (2001).  doi: 10.1103/PhysRevLett.87.165501.  Google Scholar

[23]

D. M. Leitner, Anharmonic decay of vibrational states in helical peptides, coils, and one-dimensional glasses,, Journal of Physical Chemistry A, 106 (2002), 10870.  doi: 10.1021/jp0206119.  Google Scholar

[24]

D. M. Leitner, Vibrational energy transfer in helices,, Phys. Rev. Lett., 87 (2001).  doi: 10.1103/PhysRevLett.87.188102.  Google Scholar

[25]

M. Levitt, C. Sander and P. S. Stern, Normal-mode dynamics of a protein: Bovine pancreatic trypsin inhibitor,, Int. J. Quant. Chem., 10 (1983), 181.   Google Scholar

[26]

R. M. Levy, D. Perahia and M. Karplus, Molecular dynamics of an alpha-helical polypeptide: Temperature dependance and deviation from harmonic behavior,, Proc. Natl. Acad. Sci. USA, 79 (1982), 1346.  doi: 10.1073/pnas.79.4.1346.  Google Scholar

[27]

K. Moritsugu, O. Miyashita and A. Kidera, Vibrational energy transfer in a protein molecule,, Physical Review Letters, 85 (2000), 3970.  doi: 10.1103/PhysRevLett.85.3970.  Google Scholar

[28]

T. Noguti and N. Go, Collective variable description of small-amplitude conformational fluctuations in a globular protein,, Nature, 296 (1982), 776.  doi: 10.1038/296776a0.  Google Scholar

[29]

M. Peyrard, "Nonlinear Excitations in Biomolecules,", Springer, (1995).   Google Scholar

[30]

M. Peyrard, The pathway to energy localization in nonlinear lattices,, Physica D: Nonlinear Phenomena, 119 (1998), 184.  doi: 10.1016/S0167-2789(98)00079-7.  Google Scholar

[31]

F. Piazza and Y.-H. Sanejouand, Discrete breathers in protein structures,, Phys. Biol, 5 (2008).  doi: 10.1088/1478-3975/5/2/026001.  Google Scholar

[32]

F. Piazza and Y.-H. Sanejouand, Long-range energy transfer in proteins,, Physical Biology, 6 (2009).  doi: 10.1088/1478-3975/6/4/046014.  Google Scholar

[33]

K. O. Rasmussen, D. Cai, A. R. Bishop and N. Gronbech-Jensen, Localization in a nonlinear disordered system,, Europhysics Letters, 47 (1999), 421.  doi: 10.1209/epl/i1999-00405-1.  Google Scholar

[34]

J. Ross, Energy transfer from adenosine triphosphate,, The Journal of Physical Chemistry B, 110 (2006), 6987.  doi: 10.1021/jp0556862.  Google Scholar

[35]

M. Rueda, P. Chacon and M. Orozco, Thorough validation of protein normal mode analysis: A comparative study with essential dynamics,, Structure, 15 (2007), 565.  doi: 10.1016/j.str.2007.03.013.  Google Scholar

[36]

B. Rumpf, Growth and erosion of a discrete breather interacting with rayleigh-jeans distributed phonons,, EPL, 78 (2007).  doi: 10.1209/0295-5075/78/26001.  Google Scholar

[37]

S. Sacquin-Mora, E. Laforet and R. Lavery, Locating the active sites of enzymes using mechanical properties,, Proteins, 67 (2007), 350.  doi: 10.1002/prot.21353.  Google Scholar

[38]

D. E. Sagnella, J. E. Straub and D. Thirumalai, Time scales and pathways for kinetic energy relaxation in solvated proteins: Application to carbonmonoxy myoglobin,, J. Chem. Phys., 113 (2000), 7702.  doi: 10.1063/1.1313554.  Google Scholar

[39]

K. W. Sandusky, J. B. Page and K. E. Schmidt, Stability and motion of intrinsic localized modes in nonlinear periodic lattices,, Physical Review B, 46 (1992), 6161.  doi: 10.1103/PhysRevB.46.6161.  Google Scholar

[40]

M. Sato and A. Sievers, Experimental and numerical exploration of intrinsic localized modes in an atomic lattice,, Journal of Biological Physics, 35 (2009), 57.  doi: 10.1007/s10867-009-9135-2.  Google Scholar

[41]

A. Scott, Davydov's soliton,, Physics Reports, 217 (1992), 1.  doi: 10.1016/0370-1573(92)90093-F.  Google Scholar

[42]

E. Segré (ed.), "Collected Papers of Enrico Fermi,", University of Chicago Press, (1965).   Google Scholar

[43]

F. Tama and Y. H. Sanejouand, Conformational change of proteins arising from normal mode calculations,, Protein Engineering Design and Selection, 14 (2001), 1.  doi: 10.1093/protein/14.1.1.  Google Scholar

[44]

M. M. Tirion, Large amplitude elastic motions in proteins from a single-parameter, atomic analysis,, Physical Review Letters, 77 (1996), 1905.  doi: 10.1103/PhysRevLett.77.1905.  Google Scholar

[45]

C. J. Tsai, A. del Sol and R. Nussinov, Allostery: Absence of a change in shape does not imply that allostery is not at play,, Journal of Molecular Biology, 378 (2008), 1.  doi: 10.1016/j.jmb.2008.02.034.  Google Scholar

[46]

C. J. Tsai, A. Del Sol and R. Nussinov, Protein allostery, signal transmission and dynamics: A classification scheme of allosteric mechanisms,, Molecular Biosystems, 5 (2009), 207.  doi: 10.1039/b819720b.  Google Scholar

[47]

P. C. Whitford, J. N. Onuchic and P. G. Wolynes, Energy landscape along an enzymatic reaction trajectory: Hinges or cracks?,, HFSP Journal, 2 (2008), 61.  doi: 10.2976/1.2894846.  Google Scholar

[48]

S. Woutersen and P. Hamm, Nonlinear two-dimensional vibrational spectroscopy of peptides,, Journal of Physics: Condensed Matter, 14 (2002).  doi: 10.1088/0953-8984/14/39/202.  Google Scholar

[49]

A. Xie, A. F. G. van der Meer and R. H. Austin, Excited-state lifetimes of far-infrared collective modes in proteins,, Physical Review Letters, 88 (2001).  doi: 10.1103/PhysRevLett.88.018102.  Google Scholar

[50]

A. Xie, L. van der Meer, W. Hoff and R. H. Austin, Long-lived amide i vibrational modes in myoglobin,, Physical Review Letters, 84 (2000), 5435.  doi: 10.1103/PhysRevLett.84.5435.  Google Scholar

[51]

L. W. Yang and I. Bahar, Coupling between catalytic site and collective dynamics: A requirement for mechanochemical activity of enzymes,, Structure, 13 (2005), 893.  doi: 10.1016/j.str.2005.03.015.  Google Scholar

[52]

X. Yu and D. M. Leitner, Vibrational energy transfer and heat conduction in a protein,, Journal of Physical Chemistry B, 107 (2003), 1698.  doi: 10.1021/jp026462b.  Google Scholar

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