Nonlinear lattice models for biopolymers: Dynamical coupling to a ionic cloud and application to actin filaments

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October 2011, 4(5): 1147-1166. doi: 10.3934/dcdss.2011.4.1147

Nonlinear lattice models for biopolymers: Dynamical coupling to a ionic cloud and application to actin filaments

Cynthia Ferreira ^1, , Guillaume James ^2, and Michel Peyrard ^3,

1.	Institut de Mathématiques de Toulouse (UMR 5219), Département de Mathématiques, INSA-Toulouse, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France
2.	Laboratoire Jean Kuntzmann, Université de Grenoble and CNRS, BP 53, 38041 Grenoble Cedex 9, France
3.	Laboratoire de Physique, Ecole Normale Supérieure de Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07, France

Received September 2009 Revised January 2010 Published December 2010

Abstract
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This paper is a first attempt to derive a qualitatively simple model coupling the dynamics of a charged biopolymer and its diffuse cloud of counterions. We consider here the case of a single actin filament. A zig-zag chain model introduced by Zolotaryuk et al [28] is used to represent the actin helix, and calibrated using experimental data on the stiffness constant of actin. Starting from the continuum drift-diffusion model describing counterion dynamics, we derive a discrete damped diffusion equation for the quantity of ionic charges in a one-dimensional grid along actin. The actin and ionic cloud models are coupled via electrostatic effects. Numerical simulations of the coupled system show that mechanical waves propagating along the polymer can generate charge density waves with intensities in the $pA$ range, in agreement with experimental measurements of ionic currents along actin.

Keywords: soliton propagation, nonlinear lattice, Biopolymer dynamics, actin filament, drift-diffusion model, charge density waves., dynamical coupling to condensed ions.

Mathematics Subject Classification: Primary: 92B99, 74J30, 70F99, 35Q99; Secondary: 74J3.

Citation: Cynthia Ferreira, Guillaume James, Michel Peyrard. Nonlinear lattice models for biopolymers: Dynamical coupling to a ionic cloud and application to actin filaments. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1147-1166. doi: 10.3934/dcdss.2011.4.1147

References:

[1]	T. E. Angelini, et al., Counterions between charged polymers exhibit liquid-like organization and dynamics,, Proc. Natl. Acad Sci. USA, 103 (2006), 7962. doi: 10.1073/pnas.0601435103.
[2]	N. Ben Abdallah, F. Méhats and N. Vauchelet, A note on the long time behavior for the drift-diffusion-Poisson system,, C. R. Acad. Sci. Paris Ser. I, 339 (2004), 683. doi: 10.1016/j.crma.2004.09.025.
[3]	D. ben-Avraham and M. Tirion, Dynamic and elastic properties of F-actin: A normal mode analysis,, Biophys. J., 68 (1995), 1231. doi: 10.1016/S0006-3495(95)80299-7.
[4]	L. A. Bulavin, S. N. Volkov, S. Yu. Kutuvy and S. M. Perepelytsya, Observation of the DNA ion-phosphate vibrations,, Proceedings of National Academy of Science of Ukraine, 11 (2007), 69.
[5]	M.-F. Carlier, Actin: Protein structure and filament dynamics,, J. Biol. Chem., 266 (1991), 1.
[6]	P. L. Christiansen, A. V. Savin and A. V. Zolotaryuk, Soliton analysis in complex molecular systems: A zig-zag chain,, Journal of Computational Physics, 134 (1997), 108. doi: 10.1006/jcph.1997.5676.
[7]	J. Edler, R. Pfister, V. Pouthier, C. Falvo and P. Hamm, Direct observation of self-trapped vibrational states in $\alpha$-helices,, Phys. Rev. Lett., 93 (2004). doi: 10.1103/PhysRevLett.93.106405.
[8]	F. Fogolari, P. Zuccato, G. Esposito and P. Viglino, Biomolecular electrostatics with the linearized Poisson-Boltzmann equation,, Biophys. J., 76 (1999), 1. doi: 10.1016/S0006-3495(99)77173-0.
[9]	M. K Gilson, M. E. Davis, B. A. Luty and J. A. McCammon, Computation of electrostatic forces on solvated molecules using the Poisson-Boltzmann equation,, J. Phys. Chem., 97 (1993), 3591. doi: 10.1021/j100116a025.
[10]	K. C. Holmes, et al., Atomic model of the actin filament,, Nature, 347 (1990), 44. doi: 10.1038/347044a0.
[11]	M. Karplus, Y. Q. Gao, J. Ma, A. van der Vaart and W. Yang, Protein structural transitions and their functional role,, Phil. Trans. R. Soc. A, 363 (2005), 331. doi: 10.1098/rsta.2004.1496.
[12]	H. Kojima, A.Ishijima and T.Yanagida, Direct measurements of stiffness of single actin filaments with and without tropomyosin by in vitro nanomanipulation,, Proc. Natl. Acad. Sci. USA, 91 (1994), 12962. doi: 10.1073/pnas.91.26.12962.
[13]	A. Lader, H. Woodward, E. Lin and H. Cantiello, Modeling of ionic waves along actin filaments by discrete electrical transmission lines,, METMBS'00 International Conference, (2000), 77.
[14]	E. Lin and H. Cantiello, A novel method to study the electrodynamic behavior of actin filaments. Evidence for cable-like properties of actin,, Biophys. J., 65 (1993), 1371. doi: 10.1016/S0006-3495(93)81188-3.
[15]	X. Liu and H. Pollack, Mechanics of F-actin characterized with microfabricated cantilevers,, Biophys. J., 83 (2002), 2705. doi: 10.1016/S0006-3495(02)75280-6.
[16]	T. Odijk, Stiff chains and filaments under tension,, Macromolecules, 28 (1995), 7016. doi: 10.1021/ma00124a044.
[17]	A. Orlova and E. H. Egelman, F-actin retains a memory of angular order,, Biophys. J., 78 (2000), 2180. doi: 10.1016/S0006-3495(00)76765-8.
[18]	S. M. Perepelytsya and S. N. Volkov, Counterion vibrations in the DNA low-frequency spectra,, Eur. Phys. J. E., 24 (2007), 261. doi: 10.1140/epje/i2007-10236-x.
[19]	M. Peyrard, Nonlinear dynamics and statistical physics of DNA,, Nonlinearity, 17 (2004). doi: 10.1088/0951-7715/17/2/R01.
[20]	M. Peyrard, S. Cuesta-López and G. James, Nonlinear analysis of the dynamics of DNA breathing,, Journal of Biological Physics, 35 (2009), 73. doi: 10.1007/s10867-009-9127-2.
[21]	S. Portet, C. Hogue, J. A. Tuszyński and J. M. Dixon, Elastic vibrations in seamless microtubules,, European Biophysics Journal, 34 (2005), 912. doi: 10.1007/s00249-005-0461-4.
[22]	A. Priel, A. J. Ramos, J. A. Tuszyński and H. F. Cantiello, A biopolymer transistor: Electrical amplification by microtubules,, Biophys. J., 90 (2006), 4639. doi: 10.1529/biophysj.105.078915.
[23]	A. Priel and J. A. Tuszyński, A nonlinear cable-like model of amplified ionic wave propagation along microtubules,, EPL, 83 (2008).
[24]	M. Salerno and Y. S. Kivshar, DNA promoters and nonlinear dynamics,, Phys. Lett. A, 193 (1994), 263. doi: 10.1016/0375-9601(94)90594-0.
[25]	A. V. Savin, L. I. Manevich, P. L. Christiansen and A. V. Zolotaryuk, Nonlinear dynamics of zigzag molecular chains,, Physics-Uspekhi, 42 (1999), 245. doi: 10.1070/PU1999v042n03ABEH000539.
[26]	J. A. Tuszyński, S. Portet, J. M. Dixon, C. Luxford and H. F. Cantiello, Ionic wave propagation along actin filaments,, Biophys. J., 86 (2004), 1890.
[27]	L. V. Yakushevich, A. V. Savin and L. I. Manevitch, Nonlinear dynamics of topological solitons in DNA,, Phys. Rev. E, 66 (2002). doi: 10.1103/PhysRevE.66.016614.
[28]	A. V. Zolotaryuk, P. L. Christiansen and A. V.Savin, Two-dimensional dynamics of a free molecular chain with a secondary structure,, Phys. Rev. E, 54 (1996), 3881. doi: 10.1103/PhysRevE.54.3881.

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References:

[1]	T. E. Angelini, et al., Counterions between charged polymers exhibit liquid-like organization and dynamics,, Proc. Natl. Acad Sci. USA, 103 (2006), 7962. doi: 10.1073/pnas.0601435103.
[2]	N. Ben Abdallah, F. Méhats and N. Vauchelet, A note on the long time behavior for the drift-diffusion-Poisson system,, C. R. Acad. Sci. Paris Ser. I, 339 (2004), 683. doi: 10.1016/j.crma.2004.09.025.
[3]	D. ben-Avraham and M. Tirion, Dynamic and elastic properties of F-actin: A normal mode analysis,, Biophys. J., 68 (1995), 1231. doi: 10.1016/S0006-3495(95)80299-7.
[4]	L. A. Bulavin, S. N. Volkov, S. Yu. Kutuvy and S. M. Perepelytsya, Observation of the DNA ion-phosphate vibrations,, Proceedings of National Academy of Science of Ukraine, 11 (2007), 69.
[5]	M.-F. Carlier, Actin: Protein structure and filament dynamics,, J. Biol. Chem., 266 (1991), 1.
[6]	P. L. Christiansen, A. V. Savin and A. V. Zolotaryuk, Soliton analysis in complex molecular systems: A zig-zag chain,, Journal of Computational Physics, 134 (1997), 108. doi: 10.1006/jcph.1997.5676.
[7]	J. Edler, R. Pfister, V. Pouthier, C. Falvo and P. Hamm, Direct observation of self-trapped vibrational states in $\alpha$-helices,, Phys. Rev. Lett., 93 (2004). doi: 10.1103/PhysRevLett.93.106405.
[8]	F. Fogolari, P. Zuccato, G. Esposito and P. Viglino, Biomolecular electrostatics with the linearized Poisson-Boltzmann equation,, Biophys. J., 76 (1999), 1. doi: 10.1016/S0006-3495(99)77173-0.
[9]	M. K Gilson, M. E. Davis, B. A. Luty and J. A. McCammon, Computation of electrostatic forces on solvated molecules using the Poisson-Boltzmann equation,, J. Phys. Chem., 97 (1993), 3591. doi: 10.1021/j100116a025.
[10]	K. C. Holmes, et al., Atomic model of the actin filament,, Nature, 347 (1990), 44. doi: 10.1038/347044a0.
[11]	M. Karplus, Y. Q. Gao, J. Ma, A. van der Vaart and W. Yang, Protein structural transitions and their functional role,, Phil. Trans. R. Soc. A, 363 (2005), 331. doi: 10.1098/rsta.2004.1496.
[12]	H. Kojima, A.Ishijima and T.Yanagida, Direct measurements of stiffness of single actin filaments with and without tropomyosin by in vitro nanomanipulation,, Proc. Natl. Acad. Sci. USA, 91 (1994), 12962. doi: 10.1073/pnas.91.26.12962.
[13]	A. Lader, H. Woodward, E. Lin and H. Cantiello, Modeling of ionic waves along actin filaments by discrete electrical transmission lines,, METMBS'00 International Conference, (2000), 77.
[14]	E. Lin and H. Cantiello, A novel method to study the electrodynamic behavior of actin filaments. Evidence for cable-like properties of actin,, Biophys. J., 65 (1993), 1371. doi: 10.1016/S0006-3495(93)81188-3.
[15]	X. Liu and H. Pollack, Mechanics of F-actin characterized with microfabricated cantilevers,, Biophys. J., 83 (2002), 2705. doi: 10.1016/S0006-3495(02)75280-6.
[16]	T. Odijk, Stiff chains and filaments under tension,, Macromolecules, 28 (1995), 7016. doi: 10.1021/ma00124a044.
[17]	A. Orlova and E. H. Egelman, F-actin retains a memory of angular order,, Biophys. J., 78 (2000), 2180. doi: 10.1016/S0006-3495(00)76765-8.
[18]	S. M. Perepelytsya and S. N. Volkov, Counterion vibrations in the DNA low-frequency spectra,, Eur. Phys. J. E., 24 (2007), 261. doi: 10.1140/epje/i2007-10236-x.
[19]	M. Peyrard, Nonlinear dynamics and statistical physics of DNA,, Nonlinearity, 17 (2004). doi: 10.1088/0951-7715/17/2/R01.
[20]	M. Peyrard, S. Cuesta-López and G. James, Nonlinear analysis of the dynamics of DNA breathing,, Journal of Biological Physics, 35 (2009), 73. doi: 10.1007/s10867-009-9127-2.
[21]	S. Portet, C. Hogue, J. A. Tuszyński and J. M. Dixon, Elastic vibrations in seamless microtubules,, European Biophysics Journal, 34 (2005), 912. doi: 10.1007/s00249-005-0461-4.
[22]	A. Priel, A. J. Ramos, J. A. Tuszyński and H. F. Cantiello, A biopolymer transistor: Electrical amplification by microtubules,, Biophys. J., 90 (2006), 4639. doi: 10.1529/biophysj.105.078915.
[23]	A. Priel and J. A. Tuszyński, A nonlinear cable-like model of amplified ionic wave propagation along microtubules,, EPL, 83 (2008).
[24]	M. Salerno and Y. S. Kivshar, DNA promoters and nonlinear dynamics,, Phys. Lett. A, 193 (1994), 263. doi: 10.1016/0375-9601(94)90594-0.
[25]	A. V. Savin, L. I. Manevich, P. L. Christiansen and A. V. Zolotaryuk, Nonlinear dynamics of zigzag molecular chains,, Physics-Uspekhi, 42 (1999), 245. doi: 10.1070/PU1999v042n03ABEH000539.
[26]	J. A. Tuszyński, S. Portet, J. M. Dixon, C. Luxford and H. F. Cantiello, Ionic wave propagation along actin filaments,, Biophys. J., 86 (2004), 1890.
[27]	L. V. Yakushevich, A. V. Savin and L. I. Manevitch, Nonlinear dynamics of topological solitons in DNA,, Phys. Rev. E, 66 (2002). doi: 10.1103/PhysRevE.66.016614.
[28]	A. V. Zolotaryuk, P. L. Christiansen and A. V.Savin, Two-dimensional dynamics of a free molecular chain with a secondary structure,, Phys. Rev. E, 54 (1996), 3881. doi: 10.1103/PhysRevE.54.3881.

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