Schrodinger equation, Maxwell-Bolzmann distribution and a single channel current

The dual nature, wave and particle, of ions in the biological system was considered. A theoretical model, based on the steady state Schrodinger equation and Maxwell-Bolzmann distribution of energy, is proposed to describe passive transport of ions through a biological membrane channel in a time independent field. Constant height (V2) and length(L) of the potential energy barrier and effective mass of ions are used in the model. This model shows that an ion may go through or be reflected from a channel whether its energy is lower or higher than the barrier. This is a departure from classical theory. Based on a published I-Vm (channel current-transmembrane voltage) curve from an activated K+ channel in a human erythrocyte membrane [Palle Christophersen, 1991], calculations with our model show that more than 99% of the channel current is contributed by ions with higher energy than the potential barrier. The current can be amplified 10,000 times while V2 is reduced from 0.45 eV (channel closed) to 0.20 eV (channel opened) at Vm = 0.07 V. In contrast, the current changes only 1.5% while L is narrowed from 90 A to 30 A at Vm = 0.07 V, V2 = 0.20 eV. The energy barrier: V2 = 0.16 + 0.43 Vm (eV), at r = 0.99, for 0.02 V < or = Vm < or = 0.12 V.