Development of 3-D quantum biomechanical models of a single particle movement through a channel and a single channel current

Quantum biomechanical models, based on three dimensional (3-D) steady state Schrodinger equations in a single particle system and in a time independent field, are developed to describe the movement of a particle (an ion or an ion group) in an approximately cuboidal tube-like channel and an approximately cylindrical tube-like channel and to describe a single channel current. The particle is assumed to travel in a longitudinal direction and to wave in transversal directions in the channels. Concepts of the effective constant height (V₂) and length (L_x) of the potential energy barrier in the longitudinal direction, the effective height (L_y) and width (L_z) of the cuboidal channel and the effective radius (a) of the cylindrical channel in the transversal directions, are used in the models to obtain analytical solutions in mathematics. The models elucidate that: (1) A particle is free and trapped in an infinite deep potential energy well in the transversal directions while it penetrates the channels. The particle´s transversal wave functions obey sinusoidal functions for the cuboidal channel and obey Bessel functions of the first kind with zero order for the cylindrical channel. The particle´s transversal energies are discrete in the two groups of channel. The highest probability, a particle can be found in a cross section within the channels, is on or close to the longitudinal axis. (2) V₂ is mostly determined by the repulsion energy, which is produced by the electric interaction between a particle and a channel, V₂∝1/L_y^α and V₂∝1/L_z^α for the cuboidal channel and V₂∝1/a^α for the cylindrical channel, where, α is a coefficient of the interaction and α=1, 2, 3, 4, 5 or 6 depending on the category of the interaction. As an estimation, a single mini-K⁺ channel current may decrease 10,000 times from its open state to its close state, while the effective constant height of potential energy barrier (V₂) increases only 125% (from 0.2 eV to 0.45 eV) and the effective radius (a) decreases only 22% (α=4)