Inverse problems related to ion channel selectivity

SIAM Journal on Applied Mathematics

Volumn 67, Issue 4, 2007, Pages 960-989

  Burger, Martin ^a,b   Eisenberg, Robert S ^c   Engl, Heinz W ^a  

^a JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

^b UNIVERSITY OF MÜNSTER   (Germany)

^c RUSH MEDICAL COLLEGE   (United States)

Author keywords

Current voltage relations; Identification; Ion channels; Optimal design; Permanent charge; Poisson nernst planck equations

Indexed keywords

FUNCTION EVALUATION; INVERSE PROBLEMS; IONIZATION; MATHEMATICAL MODELS;

CURRENT-VOLTAGE RELATIONS; ION CHANNELS; PERMANENT CHARGES; POISSON-NERNST-PLANCK EQUATIONS;

BIOACTIVITY;

EID: 34548270391     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/060664689     Document Type: Article

References (64)

1. B. ALBERTS, D. BRAY, J. LEWIS, M. RAFF, K. ROBERTS, AND J. D. WATSON, Molecular Biology of the Cell, Garland, New York, 1994.
2. F. M. ASHCROFT, Ion Channels and Disease, Academic Press, New York, 1999.
3. D. ATTWELL, Problems in the interpretation of membrane current-voltage relations, in Membrane Transport Processes, Vol. 3, C. F. Stevens and R. W. Tsien, eds., Raven Press, New York, 1979, pp. 21-41.
4. D. ATTWELL AND J. J. B. JACK, The interpretation of membrane current voltage relations: A Nernst-Planck analysis, Prog. Biophys. Mol. Biol., 34 (1978), pp. 81-107.
5. F. BEZANILLA AND E. STEFANI, Gating currents, Methods Enzymol., 293 (1998), pp. 331-352.
6. D. BODA, M. VALISKO, B. EISENBERG, W. NONNER, D. HENDERSON, AND D. GILLESPIE, Effect of protein dielectric coefficient on the ionic selectivity of a calcium channel, J. Chem. Phys., 125 (2006), 034901.
7. M. BURGER, B. EISENBERG, AND H. W. ENGL, Mathematical Design of Ion Channel Selectivity via Inverse Problems Technology, U.S. Patent Application, Serial Number 60/791, 185.
8. M. BURGER, H. W. ENGL, A. LEITAO, AND P. MARKOWICH, On inverse problems for semiconductor equations, Milan J. Math., 72 (2004), pp. 273-314.
9. M. BURGER, H. W. ENGL, AND P. MARKOWICH, Inverse doping problems for semiconductor devices, in Recent Progress in Computational and Applied PDEs, T. F. Chan, Y. Huang, T. Tang, J. A. Xu, and L. A. Ying, eds., Kluwer, Boston, Dordrecht, London, 2002, pp. 39-54.
10. M. BURGER, H. W. ENGL, P. MARKOWICH, AND P. PIETRA, Identification of doping profiles in semiconductor devices, Inverse Problems, 17 (2001), pp. 1765-1795.
11. M. BURGER AND W. MÜHLHUBER, Numerical approximation of an SQP-type method for parameter identification, SIAM J. Numer. Anal., 40 (2002), pp. 1775-1797.
12. M. BURGER AND R. PINNAU, Fast optimal design of semiconductor devices, SIAM J. Appl. Math., 64 (2003), pp. 108-126.
13. M. BURGER AND M. T. WOLFRAM, Symmetric Discretization of PNP-Systems, manuscript.
14. D. P. CHEN AND R. S. EISENBERG, Charges, currents and potentials in ionic channels of one conformation, Biophys. J., 64 (1993), pp. 1405-1421.
15. D. COLTON, R. EWING, AND W. RUNDELL, Inverse Problems in Partial Differential Equations, SIAM, Philadelphia, 1990.
16. K. DEIMLING, Nonlinear Functional Analysis, Springer, Berlin, 1985.
17. R. S. EISENBERG, Channels as enzymes, J. Memb. Biol., 115 (1990), pp. 1-12.
18. R. S. EISENBERG, Ionic channels in biological membranes. Electrostatic analysis of a natural nanotube, Contemp. Phys., 39 (1998), pp. 447-466.
19. H. ENGL, Integralgleichungen, Springer, Wien, 1997.
20. H. W. ENGL, M. HANKE, AND A. NEUBAUER, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 1996.
21. H. W. ENGL, K. KUNISCH, AND A. NEUBAUER, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Problems, 5 (1989), pp. 523-540.
22. H. W. ENGL AND W. RUNDELL, EDS., Inverse Problems in Diffusion Processes, SIAM, Philadelphia, 1995.
23. H. W. ENGL AND O. SCHERZER, Convergence rate results for iterative methods for solving nonlinear ill-posed problems, in Surveys on Solution Methods for Inverse Problems, D. Colton, H. W. Engl, A. K. Louis, J. McLaughlin, and W. F. Rundell, eds., Springer, Vienna, New York, 2000, pp. 7-34.
24. D. GILLESPIE AND R. EISENBERG, Physical descriptions of experimental selectivity measurements in ion channels, Eur. Biophys. J., 31 (2002), pp. 454-466.
25. D. GILLESPIE, W. NONNER, AND R. EISENBERG, Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux, J. Phys.: Condens. Matter, 14 (2002), pp. 12129-12145.
26. D. GILLESPIE, W. NONNER, AND R. EISENBERG, Density functional theory of charged, hard-sphere fluids, Phys. Rev. E (3), 68 (2003), 031503.
27. M. B. GILES AND N. A. PIERCE, An introduction to the adjoint approach to design, Flow, Turbulence and Combustion, 65 (2000), pp. 393-415.
28. C. W. GROETSCH, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, London, 1984.
29. S. H. HEINEMANN, H. TERLAU, W. STUHMER, K. IMOTO, AND S. NUMA, Calcium channel characteristics conferred on the sodium channel by single mutations, Nature, 356 (1992), pp. 441-143.
30. B. HILLE, Ionic Channels of Excitable Membranes, Sinauer Associates, Sunderland, MA, 2001.
31. M. HINZE AND R. PINNAU, Optimal control of the drift-diffusion model for semiconductor devices, in Optimal Control of Complex Structures, K. H. Hoffmann, I. Lasiecka, G. Leugering, and J. Sprekels, eds., Birkhäuser, Basel, Berlin, 2002, pp. 95-106.
32. M. HINZE AND R. PINNAU, An optimal control approach to semiconductor design, Math. Models Methods Appl. Sci., 12 (2002), pp. 89-107.
33. W. IM AND S. ROUX, Ion permeation and selectivity of OmpF porin: A theoretical study based on molecular dynamics, Brownian dynamics, and continuum electrodiffusion theory, J. Mol. Biol., 322 (2002), pp. 851-869.
34. W. IM AND S. ROUX, Ions and counterions in a biological channel: A molecular dynamics simulation of OmpF porin from Escherichia coli in an explicit membrane with 1 M KCl aqueous salt solution, J. Mol. Biol., 319 (2002), pp. 1177-1197.
35. V. ISAKOV, Inverse Problems for Partial Differential Equations, 2nd ed., Springer, New York, 2005.
36. C. JACOBONI AND P. LUGLI, The Monte Carlo Method for Semiconductor Device Simulation, Springer, New York, 1989.
37. B. KALTENBACHER, A. NEUBAUER, AND O.SCHERZER, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, manuscript, Johannes Kepler University, Linz, Austria, 2007.
38. P. G. KOSTYUK, S. L. MIRONOV, AND Y. M. SHUBA, TWO ion-selective filters in the calcium channel of the somatic membrane of mollusc neurons, J. Memb. Biol., 76 (1983), pp. 83-93.
39. F. LEHMANN-HORN AND K. JURKAT-ROTT, Channelopathies, Elsevier Science, New York, 2000.
40. A. LEITAO, P. A. MARKOWICH, AND J. ZUBELLI, On the inverse doping profile problems for the voltage-current map, Inverse Problems, 22 (2006), pp. 1071-1088.
41. A. B. MAMONOV, R. D. COALSON, A. NITZAN, AND M. G. KURNIKOVA, The role of the dielectric barrier in narrow biological channels: A novel composite approach to modeling single-channel currents, Biophys. J., 84 (2003), pp. 3646-3661.
42. P. A. MARKOWICH, C. A. RINGHOFER, AND C. SCHMEISER, Semiconductor Equations, Springer, Wien, New York, 1990.
43. H. MIEDEMA, A. METER-ARKEMA, J. WIERENGA, J. TANG, B. EISENBERG, W. NONNER, H. HEKTOR, D. GILLESPIE, AND W. MEIJBERG, Permeation properties of an engineered bacterial OmpF porin containing the EEEE-locus of Ca2+ channels, Biophys. J., 87 (2004), pp. 3137-3147.
44. H. MIEDEMA, M. VROUENRAETS, J. WIERENGA, R. S. EISENBERG, D. GILLESPIE, W. MEIJBERG, AND W. NONNER, Ca2+ selectivity of a chemically modified OmpF with reduced pore volume, Biophys J., 91 (2006), pp. 4392-4400.
45. B. NADLER, U. HOLLERBACH, AND R. S. EISENBERG, Dielectric boundary force and its crucial role in gramicidin, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 68 (2003), 021905.
46. F. NATTERER, Imaging and Inverse Problems of Partial Differential Equations, preprint. University of Münster, Münster, Germany, http:/wwwmath1.uni-muenster.de/num/Preprints/2006/artikel.pdf (2006).
47. W. NONNER, L. CATACUZZENO, AND R. EISENBERG, Binding and selectivity in L-type Ca channels: A mean spherical approximation, Biophys. J., 79 (2000), pp. 1976-1992.
48. W. NONNER, A. PEYSER, D. GILLESPIE, AND B. EISENBERG, Relating microscopic charge movement to macroscopic currents: The Ramo-Shockley theorem applied to ion channels, Biophys. J., 87 (2004), pp. 3716-3722.
49. S. OSHER, M. BURGER, D. GOLDFARB, J. XU, AND W. YIN, An iterative regularization method for total variation-based image restoration, Multiscale Model. Simul., 4 (2005), pp. 460-489.
50. I. RUBINSTEIN, Electro-diffusion of Ions, SIAM, Philadelphia, 1990.
51. W. A. SATHER AND E. W. MCCLESKEY, Permeation and selectivity in calcium channels, Annu. Rev. Physiol., 65 (2003), pp. 133-159.
52. O. SCHERZER, An iterative multi-level algorithm for solving nonlinear ill-posed problems, Numer. Math., 80 (1998), pp. 579-600.
53. Z. SCHUSS, B. NADLER, AND R. S. EISENBERG, Derivation of PNP equations in bath and channel from a molecular model, Phys. Rev. E (3), 64 (2001), 036111.
54. T. I. SEIDMAN AND C. R. VOGEL, Well-posedness and convergence of some regularization methods for nonlinear ill-posed problems, Inverse Problems, 5 (1989), pp. 227-238.
55. S. SELBERHERR, Analysis and Simulation of Semiconductor Devices, Springer, New York, 1984.
56. 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 Lett., 6 (2006), pp. 1729-1734.
57. R. E. SHOWALTER, Nonlinear Partial Differential Equations and Monotone Operators in Banach Space, AMS, Providence, RI, 1996.
58. D. P. TIELEMAN, P. C. BIGGIN, G. R. SMITH, AND M. S. SANSOM, Simulation approaches to ion channel structure-function relationships, Q. Rev. Biophys., 34 (2001), pp. 473-561.
59. W. R. VAN ROOSBROECK, Theory of flow of electrons and holes in germanium and other semiconductors, Bell System Tech. J., 29 (1950), pp. 560-607.
60. Y. WANG, L. XU, D. PASEK, D. GILLESPIE, AND G. MEISSNER, Probing the role of negatively charged amino acid residues in ion permeation of skeletal muscle ryanodine receptor, Biophys. J., 89 (2005), pp. 256-265.
61. H. WEYL, Gesammelte Abhandlungen, Springer, Berlin, 1968.
62. M. T. WOLFRAM, Semiconductor Inverse Dopant Profiling from Transient Measurements, SFB Technical Report 2006-4 (SFB F013), Johannes Kepler University, Linz, Austria, 2006, submitted.
63. M. T. WOLFRAM AND M. BURGER, Inverse dopant profiling for highly doped semiconductor devices, in Proceedings of the 5th MATHMOD Conference, F. Breitenecker and I. Troch, eds., Vienna, Austria, 2006.
64. L. XU, Y. WANG, D. GILLESPIE, AND G. MEISSNER, TWO rings of negative charges in the cytosolic vestibule of type-1 ryanodine receptor modulate ion fluxes, Biophys. J., 90 (2006), pp. 443-453.