Calcium ion current from an extracellular electrolyte toward a channel opening in an insulating membrane: quantitative model with rotational symmetry

A mathematical model of three-dimensional (3-D) ion transport is formulated in an approximation assuming rotational symmetry. The model consists of three particle-conservation equations for sodium, calcium, and chlorine ions complemented with the Poisson equation. The numerical method of solution is based on the Gummel-Scharfetter semianalytical approach, the program is written in FORTRAN and the system of discrete equations is solved explicitly in the axial direction and by iterations in the radial direction. The present report deals with calcium flux toward a channel opening in an insulating impermeable membrane, assuming depolarization to zero potential. The initial homogeneous concentrations of sodium, calcium, and chlorine ions are 8.729×10 ¹⁹, 6.02×10 ¹⁷, and 8.849×10 ¹⁹ (cm ^-3), respectively, corresponding to molar concentrations of 145-mM NaCl and 1-mM of CaCl ₂; the calcium concentration in the circle representing the channel entry is set at 0.1 μM, corresponding approximately to the concentration of free calcium ions in the cytoplasm. The calculations were carried out up to 3 μs. The calcium flux caused a perturbation of quasi-neutrality and the formation of a space charge, which reached the maximum value (i.e., maximum in absolute value) of -0.2 Ccm ^-3 at the channel entry; the corresponding maximum of the axial component of the electric field was about 1 kV/cm. The maximum value of the calcium current was 0.362 pA, decreasing to 0.283 pA at 3 μs. A review of several experimental studies of calcium currents yielded the average current values for higher and lower conductance channels (mainly L- and T-type) 0.76 and 0.42 pA, respectively. This implies that, at Ca ⁺⁺ concentrations of 1 mM or lower the calcium ion current may be limited by the ion influx from an extracellular medium.