# American Institute of Mathematical Sciences

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.
 [1] Abdelaaziz Sbai, Youssef El Hadfi, Mohammed Srati, Noureddine Aboutabit. Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces Via Leray-Schauder's nonlinear alternative. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 213-227. doi: 10.3934/dcdss.2021015 [2] Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692 [3] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4927-4962. doi: 10.3934/dcdsb.2020320 [4] Zhihong Xia, Peizheng Yu. A fixed point theorem for twist maps. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4051-4059. doi: 10.3934/dcds.2022045 [5] Carlos E. Kenig. The method of energy channels for nonlinear wave equations. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6979-6993. doi: 10.3934/dcds.2019240 [6] Tiziana Cardinali, Paola Rubbioni. Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1947-1955. doi: 10.3934/dcdss.2020152 [7] Zhongyi Huang. Tailored finite point method for the interface problem. Networks and Heterogeneous Media, 2009, 4 (1) : 91-106. doi: 10.3934/nhm.2009.4.91 [8] Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775 [9] Brandon Seward. Krieger's finite generator theorem for actions of countable groups Ⅱ. Journal of Modern Dynamics, 2019, 15: 1-39. doi: 10.3934/jmd.2019012 [10] Stanisław Migórski, Shengda Zeng. The Rothe method for multi-term time fractional integral diffusion equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 719-735. doi: 10.3934/dcdsb.2018204 [11] Igor Griva, Roman A. Polyak. Proximal point nonlinear rescaling method for convex optimization. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 283-299. doi: 10.3934/naco.2011.1.283 [12] Pavlos Xanthopoulos, Georgios E. Zouraris. A linearly implicit finite difference method for a Klein-Gordon-Schrödinger system modeling electron-ion plasma waves. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 239-263. doi: 10.3934/dcdsb.2008.10.239 [13] Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213 [14] Emeka Chigaemezu Godwin, Adeolu Taiwo, Oluwatosin Temitope Mewomo. Iterative method for solving split common fixed point problem of asymptotically demicontractive mappings in Hilbert spaces. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022005 [15] Liyuan Tian, Yong Wang. Solving tensor complementarity problems with $Z$-tensors via a weighted fixed point method. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022093 [16] Jing Cui, Shu-Ming Sun. Nonlinear Schrödinger equations on a finite interval with point dissipation. Mathematical Control and Related Fields, 2019, 9 (2) : 351-384. doi: 10.3934/mcrf.2019017 [17] Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145. [18] Nicholas Long. Fixed point shifts of inert involutions. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297 [19] Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic and Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59 [20] Hao Wang, Wei Yang, Yunqing Huang. An adaptive edge finite element method for the Maxwell's equations in metamaterials. Electronic Research Archive, 2020, 28 (2) : 961-976. doi: 10.3934/era.2020051

2021 Impact Factor: 1.497