Fig. 3

Positions of ions and water molecules during the permeation cycle. a A cyclic diagram for K⁺ permeation [36]. Among the permeation events, NC-in and SF-in events followed by relaxation are epochs for characterizing the permeation. After the relaxation, ions and water molecules are immobilized in the SF until the next NC-in (frozen period). In this cyclic diagram, two ions are bared in the SF at the NC-in moment, giving rise to the alternative array mode. Variable single-channel conductances of various potassium channels can be accounted for by the elapsed time of the frozen period governed by the entrance geometry of the NC [74]. b Evolution of the ion and water distributions in the SF during the permeation for [K⁺] = 0.05 (left panel) or 0.15 M (right panel). Upper panels: density distributions of I₀ (red), I₁ (blue), I₂ (green), and I₃ (the ion entering ahead of I₂; magenta) along the z axis. Lower panels: density distributions of the water molecules, shaded in sky blue for the water molecules between I₀ and I₁ (i.e., upstream from I₁), and in light blue for those downstream from I₁. For the ion distribution, the numbers beside the peaks indicate their integrals, yielding the fractional number of the corresponding ion (I₀ − I₃; upper panels). For example, at 0.15 M [K⁺], 0.69 of I₁ is located at S₃ and 0.31 is located at S₂ at the SF-in moment. For the water distribution, the numbers beside the peaks indicate the absolute number of water molecules intercalated between I₀ and I₁ (lower panels) at a relevant site