TY - JOUR
T1 - Singular perturbation analysis of the steady-state Poisson-Nernst-Planck system
T2 - Applications to ion channels
AU - Singer, A.
AU - Gillespie, D.
AU - Norbury, J.
AU - Eisenberg, R. S.
PY - 2008/10
Y1 - 2008/10
N2 - Ion channels are proteins with a narrow hole down their middle that control a wide range of biological function by controlling the flow of spherical ions from one macroscopic region to another. Ion channels do not change their conformation on the biological time scale once they are open, so they can be described by a combination of Poisson and drift-diffusion (Nernst-Planck) equations called PNP in biophysics. We use singular perturbation techniques to analyse the steady-state PNP system for a channel with a general geometry and a piecewise constant permanent charge profile. We construct an outer solution for the case of a constant permanent charge density in three dimensions that is also a valid solution of the one-dimensional system. The asymptotical current-voltage (I-V) characteristic curve of the device (obtained by the singular perturbation analysis) is shown to be a very good approximation of the numerical I-V curve (obtained by solving the system numerically). The physical constraint of non-negative concentrations implies a unique solution, i.e., for each given applied potential there corresponds a unique electric current (relaxing this constraint yields non-physical multiple solutions for sufficiently large voltages).
AB - Ion channels are proteins with a narrow hole down their middle that control a wide range of biological function by controlling the flow of spherical ions from one macroscopic region to another. Ion channels do not change their conformation on the biological time scale once they are open, so they can be described by a combination of Poisson and drift-diffusion (Nernst-Planck) equations called PNP in biophysics. We use singular perturbation techniques to analyse the steady-state PNP system for a channel with a general geometry and a piecewise constant permanent charge profile. We construct an outer solution for the case of a constant permanent charge density in three dimensions that is also a valid solution of the one-dimensional system. The asymptotical current-voltage (I-V) characteristic curve of the device (obtained by the singular perturbation analysis) is shown to be a very good approximation of the numerical I-V curve (obtained by solving the system numerically). The physical constraint of non-negative concentrations implies a unique solution, i.e., for each given applied potential there corresponds a unique electric current (relaxing this constraint yields non-physical multiple solutions for sufficiently large voltages).
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U2 - 10.1017/S0956792508007596
DO - 10.1017/S0956792508007596
M3 - Article
C2 - 19809600
AN - SCOPUS:50849111742
SN - 0956-7925
VL - 19
SP - 541
EP - 560
JO - European Journal of Applied Mathematics
JF - European Journal of Applied Mathematics
IS - 5
ER -