Breather-mediated energy transfer in proteins

Previous Article
Continuation and bifurcations of breathers in a finite discrete NLS equation
DCDS-S Home
This Issue
Next Article
Persistent mobile lattice excitations in a crystalline insulator

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

Abstract
Related Papers

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:

[1]	F. Abdullaev, O. Bang and M. P. Sorensen (eds.), "Nonlinearity and Disorder: Theory and Applications,", vol. \textbf{45}, 45 (2001).
[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.
[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.
[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.
[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).
[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.
[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.
[8]	F. Columbus (ed.), "Soft Condensed Matter. New Research,", Nova Science Publishers, (2005).
[9]	T. Dauxois, R. Khomeriki, F. Piazza and S. Ruffo, The anti-FPU problem,, Chaos, 15 (2005). doi: 10.1063/1.1854273.
[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).
[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.
[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.
[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.
[14]	J. J. Falke, Enzymology: A moving story,, Science, 295 (2002), 1480. doi: 10.1126/science.1069823.
[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.
[16]	S. Flach and C. R. Willis, Discrete breathers,, Physics Reports, 295 (1998), 181. doi: 10.1016/S0370-1573(97)00068-9.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	D. M. Leitner, Vibrational energy transfer in helices,, Phys. Rev. Lett., 87 (2001). doi: 10.1103/PhysRevLett.87.188102.
[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.
[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.
[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.
[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.
[29]	M. Peyrard, "Nonlinear Excitations in Biomolecules,", Springer, (1995).
[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.
[31]	F. Piazza and Y.-H. Sanejouand, Discrete breathers in protein structures,, Phys. Biol, 5 (2008). doi: 10.1088/1478-3975/5/2/026001.
[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.
[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.
[34]	J. Ross, Energy transfer from adenosine triphosphate,, The Journal of Physical Chemistry B, 110 (2006), 6987. doi: 10.1021/jp0556862.
[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.
[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.
[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.
[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.
[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.
[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.
[41]	A. Scott, Davydov's soliton,, Physics Reports, 217 (1992), 1. doi: 10.1016/0370-1573(92)90093-F.
[42]	E. Segré (ed.), "Collected Papers of Enrico Fermi,", University of Chicago Press, (1965).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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).
[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.
[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.
[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.
[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).
[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.
[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.
[8]	F. Columbus (ed.), "Soft Condensed Matter. New Research,", Nova Science Publishers, (2005).
[9]	T. Dauxois, R. Khomeriki, F. Piazza and S. Ruffo, The anti-FPU problem,, Chaos, 15 (2005). doi: 10.1063/1.1854273.
[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).
[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.
[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.
[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.
[14]	J. J. Falke, Enzymology: A moving story,, Science, 295 (2002), 1480. doi: 10.1126/science.1069823.
[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.
[16]	S. Flach and C. R. Willis, Discrete breathers,, Physics Reports, 295 (1998), 181. doi: 10.1016/S0370-1573(97)00068-9.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	D. M. Leitner, Vibrational energy transfer in helices,, Phys. Rev. Lett., 87 (2001). doi: 10.1103/PhysRevLett.87.188102.
[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.
[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.
[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.
[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.
[29]	M. Peyrard, "Nonlinear Excitations in Biomolecules,", Springer, (1995).
[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.
[31]	F. Piazza and Y.-H. Sanejouand, Discrete breathers in protein structures,, Phys. Biol, 5 (2008). doi: 10.1088/1478-3975/5/2/026001.
[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.
[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.
[34]	J. Ross, Energy transfer from adenosine triphosphate,, The Journal of Physical Chemistry B, 110 (2006), 6987. doi: 10.1021/jp0556862.
[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.
[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.
[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.
[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.
[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.
[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.
[41]	A. Scott, Davydov's soliton,, Physics Reports, 217 (1992), 1. doi: 10.1016/0370-1573(92)90093-F.
[42]	E. Segré (ed.), "Collected Papers of Enrico Fermi,", University of Chicago Press, (1965).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

[1]	S. Aubry, G. Kopidakis, V. Kadelburg. Variational proof for hard Discrete breathers in some classes of Hamiltonian dynamical systems. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 271-298. doi: 10.3934/dcdsb.2001.1.271
[2]	Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 91-107. doi: 10.3934/dcds.2011.29.91
[3]	Hideo Takaoka. Energy transfer model for the derivative nonlinear Schrödinger equations on the torus. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5819-5841. doi: 10.3934/dcds.2017253
[4]	Tanka Nath Dhamala. A survey on models and algorithms for discrete evacuation planning network problems. Journal of Industrial & Management Optimization, 2015, 11 (1) : 265-289. doi: 10.3934/jimo.2015.11.265
[5]	Panayotis Panayotaros. Continuation and bifurcations of breathers in a finite discrete NLS equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1227-1245. doi: 10.3934/dcdss.2011.4.1227
[6]	Jesús Cuevas, Bernardo Sánchez-Rey, J. C. Eilbeck, Francis Michael Russell. Interaction of moving discrete breathers with interstitial defects. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1057-1067. doi: 10.3934/dcdss.2011.4.1057
[7]	Michael Kastner, Jacques-Alexandre Sepulchre. Effective Hamiltonian for traveling discrete breathers in the FPU chain. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 719-734. doi: 10.3934/dcdsb.2005.5.719
[8]	Jean-Pierre Eckmann, C. Eugene Wayne. Breathers as metastable states for the discrete NLS equation. Discrete & Continuous Dynamical Systems - A, 2018, 0 (0) : 1-13. doi: 10.3934/dcds.2018136
[9]	B. Coll, A. Gasull, R. Prohens. On a criterium of global attraction for discrete dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (3) : 537-550. doi: 10.3934/cpaa.2006.5.537
[10]	Jean-Luc Chabert, Ai-Hua Fan, Youssef Fares. Minimal dynamical systems on a discrete valuation domain. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 777-795. doi: 10.3934/dcds.2009.25.777
[11]	Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187
[12]	Mostafa Abounouh, H. Al Moatassime, J. P. Chehab, S. Dumont, Olivier Goubet. Discrete Schrödinger equations and dissipative dynamical systems. Communications on Pure & Applied Analysis, 2008, 7 (2) : 211-227. doi: 10.3934/cpaa.2008.7.211
[13]	Adina Luminiţa Sasu, Bogdan Sasu. Discrete admissibility and exponential trichotomy of dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2929-2962. doi: 10.3934/dcds.2014.34.2929
[14]	Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039
[15]	Aleksandar Zatezalo, Dušan M. Stipanović. Control of dynamical systems with discrete and uncertain observations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4665-4681. doi: 10.3934/dcds.2015.35.4665
[16]	Karl P. Hadeler. Quiescent phases and stability in discrete time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 129-152. doi: 10.3934/dcdsb.2015.20.129
[17]	Robert Skiba, Nils Waterstraat. The index bundle and multiparameter bifurcation for discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5603-5629. doi: 10.3934/dcds.2017243
[18]	Jacobo Pejsachowicz, Robert Skiba. Topology and homoclinic trajectories of discrete dynamical systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1077-1094. doi: 10.3934/dcdss.2013.6.1077
[19]	K. L. Mak, J. G. Peng, Z. B. Xu, K. F. C. Yiu. A novel neural network for associative memory via dynamical systems. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 573-590. doi: 10.3934/dcdsb.2006.6.573
[20]	Quoc T. Luu, Paul DuChateau. The relative biologic effectiveness versus linear energy transfer curve as an output-input relation for linear cellular systems. Mathematical Biosciences & Engineering, 2009, 6 (3) : 591-602. doi: 10.3934/mbe.2009.6.591

2016 Impact Factor: 0.781

American Institute of Mathematical Sciences