October  2012, 17(7): 2465-2482. doi: 10.3934/dcdsb.2012.17.2465

Nonlinear conformation response in the finite channel: Existence of a unique solution for the dynamic PNP model

1. 

Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, United States

Received  September 2011 Revised  February 2012 Published  July 2012

The standard PNP model for ion transport in channels in cell membranes has been widely studied during the previous two decades; there is a substantial literature for both the dynamic and steady models. What is currently lacking is a generally accepted gating model, which is linked to the observed conformation changes on the protein molecule. In [SIAM J. Appl. Math. 61 (2000), no.3, 792–802], C.W. Gardner, the author, and R.S. Eisen- berg suggested a model for the net charge density in the infinite channel, which has connections to stochastic dynamical systems, and which predicted rectan- gular current pulses. The finite channel was analyzed by these authors in [J. Theoret. Biol. 219 (2002), no. 3, 291–299]. The finite channel cannot, in general, be analyzed by a traveling wave approach. In this paper, a rigorous study of the initial-boundary value problem is carried out for the deterministic version of the finite channel; an existence/uniqueness result, with a weak maximum principle, is derived on the space-time domain under assumptions on the inital and boundary data which confine the channel to certain states. Significant open problems remain and are discussed
Citation: Joseph W. Jerome. Nonlinear conformation response in the finite channel: Existence of a unique solution for the dynamic PNP model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2465-2482. doi: 10.3934/dcdsb.2012.17.2465
References:
[1]

F. Alabau, On the existence of multiple steady-state solutions in the theory of electrodiffusion. I: The nonelectroneutral case. II: A constructive method for the electroneutral case, Trans. Amer. Math. Soc., 350 (1998), 4709-4756.

[2]

J.-P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044.

[3]

V. Barcilon, D.-P. Chen, J. W. Jerome and R. S. Eisenberg, Qualitative properties of solutions of steady-state Poisson-Nernst-Planck systems: Perturbation and simulation study, SIAM J. Appl. Math., 57 (1997), 631-648.

[4]

R. S. Eisenberg, M. M. Klosek and Z. Schuss, Diffusion as a chemical reaction: Stochastic trajectories between fixed concentrations, J. Chemical Physics, 102 (1995), 1767-1780. doi: 10.1063/1.468704.

[5]

B. Fiedler, S. Liebscher and J. C. Alexander, General Hopf bifurcation from lines of equilibria without parameters. I. Theory, J. Differential Equations, 167 (2000), 16-35.

[6]

B. Fiedler, S. Liebscher and J. C. Alexander, Generic Hopf bifurcation from lines of equilibria without parameters. III. Binary oscillations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1613-1621.

[7]

C. L. Gardner, private communication, December, 2011.

[8]

C. L. Gardner, J. W. Jerome and R. S. Eisenberg, Electrodiffusion model of rectangular current pulses in ionic channels of cellular membranes, SIAM J. Appl. Math., 61 (2000), 792-802.

[9]

C. L. Gardner, J. W. Jerome and R. S. Eisenberg, Electrodiffusion model simulation of rectangular current pulses ina voltage-biased biological channel, J. Theoret. Biol., 219 (2002), 291-299.

[10]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[11]

I. Halperin, "Introduction to the Theory of Distributions. Based on the Lectures Given by Laurent Schwartz,'' University ofToronto Press, Toronto, 1952.

[12]

B. Hille, "Ionic Channels of Excitable Membranes,'' Second edition, Sinauer, Sunderland, MA, 1992.

[13]

J. W. Jerome, "Approximation of Nonlinear Evolution Systems,'' Mathematics in Science and Engineering, 164, Academic Press, Inc., Orlando, FL, 1983.

[14]

J. W. Jerome, Consistency of semiconductor modeling: An existence/stability analysis for the stationary Van Roosbroeck system, SIAM J. Appl. Math., 45 (1985), 565-590.

[15]

J. W. Jerome, Evolution systems in semiconductor device modeling: A cyclic uncoupled line analysis for the Gummel map, Math. Methods Appl. Sci., 9 (1987), 455-492.

[16]

J. W. Jerome, A trapping principle and convergence results for finite element approximate solutions of steady reaction/diffusion systems, Numer. Math., 109 (2008), 121-142.

[17]

M. Krupa, Robust heteroclinic cycles, J. Nonlinear Sci., 7 (1997), 129-176.

[18]

W. Liu and B. Wang, Poisson-Nernst-Planck systems for narrow tubular-like membrane channels, J. Dynam. Differential Equations, 22 (2010), 413-437.

[19]

E. Neher and B. Sackmann, Single-channel currents recorded from membrane of denervated frog muscle-fibers, Nature, 260 (1976), 799-802.

[20]

H. L. Royden, "Real Analysis,'' Third edition, Macmillan Publishing Company, New York, 1988.

[21]

T. Seidman and G. M. Troianiello, Time-dependent solutions of a nonlinear system arising in semiconductor theory, Nonlinear Analysis, 9 (1985), 1137-1157.

[22]

Z. S. Siwy, M. R. Powell, A. Petrov, E. Kalman, C. Trautmann and R. S. Eisenberg, Calcium induced voltage gating in single conical nanopores, Nano Letters, 6 (2006), 1729-1734.

[23]

E. Stone and P. Holmes, Random perturbations of heteroclinic attractors, SIAM J. Appl. Math., 50 (1990), 726-743.

[24]

P. Szmolyan, Traveling waves in GaAs-semiconductors, Physica D, 39 (1989), 393-404.

show all references

References:
[1]

F. Alabau, On the existence of multiple steady-state solutions in the theory of electrodiffusion. I: The nonelectroneutral case. II: A constructive method for the electroneutral case, Trans. Amer. Math. Soc., 350 (1998), 4709-4756.

[2]

J.-P. Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042-5044.

[3]

V. Barcilon, D.-P. Chen, J. W. Jerome and R. S. Eisenberg, Qualitative properties of solutions of steady-state Poisson-Nernst-Planck systems: Perturbation and simulation study, SIAM J. Appl. Math., 57 (1997), 631-648.

[4]

R. S. Eisenberg, M. M. Klosek and Z. Schuss, Diffusion as a chemical reaction: Stochastic trajectories between fixed concentrations, J. Chemical Physics, 102 (1995), 1767-1780. doi: 10.1063/1.468704.

[5]

B. Fiedler, S. Liebscher and J. C. Alexander, General Hopf bifurcation from lines of equilibria without parameters. I. Theory, J. Differential Equations, 167 (2000), 16-35.

[6]

B. Fiedler, S. Liebscher and J. C. Alexander, Generic Hopf bifurcation from lines of equilibria without parameters. III. Binary oscillations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1613-1621.

[7]

C. L. Gardner, private communication, December, 2011.

[8]

C. L. Gardner, J. W. Jerome and R. S. Eisenberg, Electrodiffusion model of rectangular current pulses in ionic channels of cellular membranes, SIAM J. Appl. Math., 61 (2000), 792-802.

[9]

C. L. Gardner, J. W. Jerome and R. S. Eisenberg, Electrodiffusion model simulation of rectangular current pulses ina voltage-biased biological channel, J. Theoret. Biol., 219 (2002), 291-299.

[10]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[11]

I. Halperin, "Introduction to the Theory of Distributions. Based on the Lectures Given by Laurent Schwartz,'' University ofToronto Press, Toronto, 1952.

[12]

B. Hille, "Ionic Channels of Excitable Membranes,'' Second edition, Sinauer, Sunderland, MA, 1992.

[13]

J. W. Jerome, "Approximation of Nonlinear Evolution Systems,'' Mathematics in Science and Engineering, 164, Academic Press, Inc., Orlando, FL, 1983.

[14]

J. W. Jerome, Consistency of semiconductor modeling: An existence/stability analysis for the stationary Van Roosbroeck system, SIAM J. Appl. Math., 45 (1985), 565-590.

[15]

J. W. Jerome, Evolution systems in semiconductor device modeling: A cyclic uncoupled line analysis for the Gummel map, Math. Methods Appl. Sci., 9 (1987), 455-492.

[16]

J. W. Jerome, A trapping principle and convergence results for finite element approximate solutions of steady reaction/diffusion systems, Numer. Math., 109 (2008), 121-142.

[17]

M. Krupa, Robust heteroclinic cycles, J. Nonlinear Sci., 7 (1997), 129-176.

[18]

W. Liu and B. Wang, Poisson-Nernst-Planck systems for narrow tubular-like membrane channels, J. Dynam. Differential Equations, 22 (2010), 413-437.

[19]

E. Neher and B. Sackmann, Single-channel currents recorded from membrane of denervated frog muscle-fibers, Nature, 260 (1976), 799-802.

[20]

H. L. Royden, "Real Analysis,'' Third edition, Macmillan Publishing Company, New York, 1988.

[21]

T. Seidman and G. M. Troianiello, Time-dependent solutions of a nonlinear system arising in semiconductor theory, Nonlinear Analysis, 9 (1985), 1137-1157.

[22]

Z. S. Siwy, M. R. Powell, A. Petrov, E. Kalman, C. Trautmann and R. S. Eisenberg, Calcium induced voltage gating in single conical nanopores, Nano Letters, 6 (2006), 1729-1734.

[23]

E. Stone and P. Holmes, Random perturbations of heteroclinic attractors, SIAM J. Appl. Math., 50 (1990), 726-743.

[24]

P. Szmolyan, Traveling waves in GaAs-semiconductors, Physica D, 39 (1989), 393-404.

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