×

Geometrical effects on nonlinear electrodiffusion in cell physiology. (English) Zbl 1379.35063

Summary: We report here new electrical laws, derived from nonlinear electrodiffusion theory, about the effect of the local geometrical structure, such as curvature, on the electrical properties of a cell. We adopt the Poisson-Nernst-Planck equations for charge concentration and electric potential as a model of electrodiffusion. In the case at hand, the entire boundary is impermeable to ions and the electric field satisfies the compatibility condition of Poisson’s equation. We construct an asymptotic approximation for certain singular limits to the steady-state solution in a ball with an attached cusp-shaped funnel on its surface. As the number of charge increases, they concentrate at the end of cusp-shaped funnel. These results can be used in the design of nanopipettes and help to understand the local voltage changes inside dendrites and axons with heterogeneous local geometry.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J66 Nonlinear boundary value problems for nonlinear elliptic equations
92C05 Biophysics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bezanilla, F.: How membrane proteins sense voltage. Nat. Rev. Mol. Cell Biol. 9, 323-332 (2008) · Zbl 1197.34080
[2] Cartailler, J., Schuss, Z., Holcman, D.: Analysis of the Poisson-Nernst-Planck equation in a ball for modeling the voltage-current relation in neurobiological microdomains. Phys. D Nonlinear Phenom. 339, 39-48 (2016) · Zbl 1376.92012 · doi:10.1016/j.physd.2016.09.001
[3] Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 2. Wiley Interscience, New York (1989) · Zbl 0729.00007 · doi:10.1002/9783527617210
[4] Debye, P., Hückel, E.: Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen. Phys. Z. 24(9), 185-206 (1923) · JFM 49.0587.11
[5] Delgado, M., Coombs, D.: Conditional mean first passage times to small traps in a 3-D domain with a sticky boundary. SIAM J. Multiscale Anal. Simul. 13(4), 1224-1258 (2015) · Zbl 1339.92009 · doi:10.1137/140978314
[6] Eisenberg, R.S.: From structure to function in open ionic channels. J. Membr. Biol. 171, 1-24 (1998a) · doi:10.1007/s002329900554
[7] Eisenberg, R.S.: Ionic channels in biological membranes. Electrostatic analysis of a natural nanotube. Contemp. Phys. 39(6), 447-466 (1998b) · doi:10.1080/001075198181775
[8] Eisenberg, R.S., Klosek, M.M., Schuss, Z.: Diffusion as a chemical reaction: stochastic trajectories between fixed concentrations. J. Chem. Phys. 102(4), 1767-1780 (1995) · doi:10.1063/1.468704
[9] Hille, B.: Ion Channels of Excitable Membranes, 3rd edn. Sinauer Associates, Sunderland (2001) · Zbl 1376.92012
[10] Holcman, D., Schuss, Z.: Brownian motion in dire straits. SIAM J. Multiscale Model. Simul. 10(4), 1204-1231 (2012a) · Zbl 1266.60144 · doi:10.1137/110857519
[11] Holcman, D., Schuss, Z.: Brownian motion in dire straits. Multiscale Model. Simul. 10(4), 1204-1231 (2012b) · Zbl 1266.60144 · doi:10.1137/110857519
[12] Holcman, D., Schuss, Z.: Stochastic Narrow Escape in Molecular an Cellular Biology. Analysis and Applications. Springer, New York (2015) · Zbl 1321.92008 · doi:10.1007/978-1-4939-3103-3
[13] Holcman, D., Yuste, R.: The new nanophysiology: regulation of ionic flow in neuronal subcompartments. Nat. Rev. Neurosci. 16, 685-692 (2015) · doi:10.1038/nrn4022
[14] Holcman, D., Hoze, N., Schuss, Z.: Narrow escape through a funnel and effective diffusion on a crowded membrane. Phys. Rev. E 84, 021906 (2011). Erratum. Phys. Rev. E 85, 039903 (2012) · doi:10.1103/PhysRevE.85.039903
[15] Horn, R., Roux, B., Aqvist, J.: Permeation redux: thermodynamics and kinetics of ion movement through potassium channels. Biophys J. 106(9), 1859-1863 (2014) · doi:10.1016/j.bpj.2014.03.039
[16] Lindsay, A.E., Bernoff, A.J., Ward, M.J.: First passage statistics for the capture of a Brownian particle by a structured spherical target with multiple surface traps. SIAM J. Multiscale Model. Simul. 15(1), 74-109 (2016) · Zbl 1372.35021
[17] Mamonov, A., Coalson, R., Nitzan, A., Kurnikova, M.: The role of the dielectric barrier in narrow biological channels: a novel composite approach to modeling single channel currents. Biophys. J. 84, 3646-3661 (2003) · doi:10.1016/S0006-3495(03)75095-4
[18] Perry, D., Momotenko, D., Lazenby, R.A., Kang, M., Unwin, P.R.: Characterization of nanopipettes. Anal. Chem. 88(10), 5523-5530 (2016) · doi:10.1021/acs.analchem.6b01095
[19] Pillay, S., Peirce, A., Kolokolnikov, T., Ward, M.: An asymptotic analysis of the mean first passage time for narrow escape problems: part I: two-dimensional domains. SIAM Multiscale Model. Simul. 8(3), 803-835 (2010) · Zbl 1203.35023 · doi:10.1137/090752511
[20] Qian, N., Sejnowski, T.J.: An electro-diffusion model for computing membrane potentials and ionic concentrations in branching dendrites, spines and axons. Biol. Cybern. 62, 1-15 (1989) · Zbl 0683.92004 · doi:10.1007/BF00217656
[21] Rall, W.; Koch, C. (ed.); Segev, I. (ed.), Cable theory for dendritic neurons, 9-63 (1989), Cambridge
[22] Roux, B., Karplus, M.: Ion transport in the gramicidin channel: free energy of the solvated right-handed dimer in a model membrane. J. Am. Chem. Soc. 115, 3250-3262 (1993) · doi:10.1021/ja00061a025
[23] Ruiz, F.J.G., Godoy, A., Gamiz, F., Sampedro, C., Donetti, L.: A comprehensive study of the corner effects in Pi-gate MOSFETs including quantum effects. IEEE Trans. Electron Devices 54(12), 3369-3377 (2007) · doi:10.1109/TED.2007.909206
[24] Savtchenko, L.P., Kulahin, N., Korogod, S.M., Rusakov, D.A.: Electric fields of synaptic currents could influence diffusion of charged neurotransmitter molecules. Synapse 51(4), 270-278 (2004) · doi:10.1002/syn.10300
[25] Schuss, Z., Nadler, B., Eisenberg, R.S.: Derivation of Poisson and Nernst-Planck equations in a bath and channel from a molecular model. Phys. Rev. E 64, 036116 (2001) · doi:10.1103/PhysRevE.64.036116
[26] Singer, A., Norbury, J.: A Poisson-Nernst-Planck model for biological ion channels-an asymptotic analysis in a three-dimensional narrow funnel. SIAM Appl. Math. 7(3), 949-968 (2009) · Zbl 1197.34080 · doi:10.1137/070687037
[27] Sparreboom, W., van den Berg, A., Eijkel, J.C.T.: Principles and applications of nanofluidic transport. Nat. Nanotechnol 4, 713-720 (2009) · doi:10.1038/nnano.2009.332
[28] Yuste, R.: Dendritic Spines. The MIT Press, Cambridge (2010) · doi:10.7551/mitpress/9780262013505.001.0001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.