Abstract
We present initial results regarding the existence, stability and interactionof linear and nonlinear vibrational modes in a system of two coupled, onedimensional lattices with unequal numbers of masses. The effects on thesenonlinear modes of coupling a near continuum system to a discrete systemusing a nonlinear coupling are examined. This numerical model is a firststep towards investigating the dynamical behavior of a flexible sheetcoupled nonlinearly to a semi-rigid support, a system which couldconceivably represent a biological cell membrane with a supporting proteinnetwork. General implications for the dynamical behavior of continuumsystems coupled nonlinearly to discrete systems are introduced.
Keywords: Breather modes, discrete systems, intrinsic localized modes, membrane dynamics, nonlinear systems
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References
- 1.Alberts B., Bray D., Lewis J., Raff M., Roberts K., Watson J.D. Molecular Biology of the Cell. New York: Garland Publishing, Inc; 1994. [Google Scholar]
- 2.Bloom M., Evans E., Mouritsen O.G. Physical Properties of the Fluid Lipid Bilayer Component of Cell Membranes: A Perspective. Quart. Rev. Biophys. 1991;24(3):293–397. doi: 10.1017/s0033583500003735. [DOI] [PubMed] [Google Scholar]
- 3.Tobochnik J., Gould H. Lattice Simulations of Biological Membranes. Computers in Physics. 1996;10(6):542–547. [Google Scholar]
- 4.Boal D.H. Computer Simulation of a Model Network for the Erythrocyte Cytoskeleton. Biophys. J. 1994;67(2):521–529. doi: 10.1016/S0006-3495(94)80511-9. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 5.Boey S.K., Boal D.H., Discher D.E. Simulations of the Erythrocyte Cytoskeleton at Large Deformation. I. Microscopic Models. Biophys. J. 1998;75(3):1573–1583. doi: 10.1016/S0006-3495(98)74075-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 6.Hansen J.C., Skalak R., Chien S., Hoger A. An Elastic Network Model Based on the Structure of the Red Blood Cell Membrane Skeleton. Biophys. J. 1996;70(1):146–166. doi: 10.1016/S0006-3495(96)79556-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 7.Krishnaswamy S. A Csserat-Type Model of the Red Blood CellWall. Int. J. Engng. Sci. 1996;34(8):873–899. [Google Scholar]
- 8.Brody J.P., Han Y., Austin R.H., Bitensky M. Deformation and Flow of Red Blood Cells in a Sythetic Lattice: Evidence for an Active Cytoskeleton. Biophys. J. 1995;68(6):2224–2232. doi: 10.1016/S0006-3495(95)80443-1. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 9.Waugh R.E. Elastic Energy Curvature Driven Bump Formation on Red Blood Cell Membrane. Biophys. J. 1996;70(2):1027–1035. doi: 10.1016/S0006-3495(96)79648-0. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 10.Stout A.L., Axelrod D. Reversible Binding Kinetics of a Cytoskeleal Protein at the Erythrocyte Submembrane. Biophys. J. 1994;67(3):1324–1334. doi: 10.1016/S0006-3495(94)80604-6. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 11.Waugh R.E., Bauserman R.G. Physical Measurements of Bilayer Skeletal Separation Forces. Ann. Biomed. Eng. 1995;23:308–321. doi: 10.1007/BF02584431. [DOI] [PubMed] [Google Scholar]
- 12.Levin S., Korenstein R. Membrane Fluctuations in Erythrocytes are Linked to MgATP Dependent Dynamic Assembly of the Membrane Skeleton. Biophys. J. 1991;60(3):733. doi: 10.1016/S0006-3495(91)82104-X. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 13.Peyrard M., Bishop A.R. Statistical Mechanics of a Nonlinear Model for DNA Denaturation. Phys. Rev. Lett. 1989;62:2755. doi: 10.1103/PhysRevLett.62.2755. [DOI] [PubMed] [Google Scholar]
- 14.Forinash K., Bishop A.R., Lomdahl P. Nonlinear Dynamics in a Double Chain Model of DNA. Phys. Rev. B. 1991;43(13):10743–10750. doi: 10.1103/physrevb.43.10743. [DOI] [PubMed] [Google Scholar]
- 15.Forinash K., Cretegny T., Peyrard M. Local Modes and Localization in a Multi-Component Nonlinear Lattice. Phys. Rev. E. 1997;55:4740–4759. [Google Scholar]
- 16.Scott, A.: Nonlinear Science: Emergence and Dynamics of Coherent Structures, Oxford University Press, 1999.
- 17.Press W.H., Flannery B.P., Teukolsky S.A., Vetterling W.T. Numerical Recipes. Cambridge: Cambridge University Press; 1989. [Google Scholar]
- 18.Johnson, R.M.:Membrane Stress Increases Cation Permeability in Red Cells, Biophys. J.67(5) (1994), 1876–1881. [DOI] [PMC free article] [PubMed]
- 19.Flach S., Willis C.R. Physics Reports. 1998;295:181. [Google Scholar]
- 20.Bang O., Peyrard M. Generation of High-Energy Localized Vibrational Modes in Nonlinear Klein-Gordon Lattices. Phys. Rev. E. 1996;53:4143. doi: 10.1103/physreve.53.4143. [DOI] [PubMed] [Google Scholar]
- 21.Oster, G.F. and Moore, H.H.: The Budding of Membranes, In A. Goldbeter (ed.), Cell to Cell Signalling: From Experiments to Theoretical Models, Academic Press, 1989, pp. 171–187.
- 22.MacKay R.S., Aubry S. Proof of Existence of Breathers for Time-Reversible or Hamiltonian Networks of Weakly Coupled Oscillators. Nonlinearity. 1994;7:1623–1643. [Google Scholar]
- 23.Vanossi A., Rasmussen K.Ø., Bishop A.R., Malomed B.A., Bortolani V. Spontaneous Pattern Formation in Driven Nonlinear Lattices. Phys. Rev. E. 2000;62:7353–7357. doi: 10.1103/physreve.62.7353. [DOI] [PubMed] [Google Scholar]