= Solution
\b[<selective permeability>] sets the driving-<force> term. <Potassium> and <sodium> channels provide narrow coordinating pathways whose <free energy> balance of hydration and chemical binding favours the appropriate ion. <voltage-gated potassium channel> carbonyl coordination differs from the charged filter environment in canonical <sodium> channels, so selective permeation is not a pore-size ranking. Ion substitution shifts the <reversal potential>, and selective blockers or filter mutations alter the corresponding <electric current>. An open channel conducts down the <electrochemical gradient>; <ATP>-driven pumps maintain the gradients rather than pushing each ion through the pore.
\b[Voltage-dependent activation] sets channel availability. Charged residues in the S4-containing <voltage-sensing domains> move in the membrane <electric field> and couple to pore opening. A classical <voltage-gated potassium channel> is a tetramer; a <voltage-gated sodium channel> pore-forming <polypeptide> has four homologous domains. <ion-channel gating current> measurements and sensor-residue mutations connect <electric charge> motion to opening, while structures separate the sensor from the permeation pathway. The historical $n^4$ and $m^3$ variables describe kinetics phenomenologically and need not map one-to-one onto individual resolved structural movements.<Voltage-sensing residues in the S2 and S4 segments of the Shaker K+ channel by S. A. Seoh and others>[Sensor perturbations and measured gating charge]{ref}<Crystal structure of a mammalian voltage-dependent Shaker family K+ channel by Stephen B. Long and others>[Voltage-sensor and pore architecture]{ref}
\b[Time dependence, especially <ion-channel inactivation>], explains the different <electric current> waveforms. <Sodium> activation is rapid, but sustained <depolarization> reduces the availability variable $h$, so the <sodium> <electric current> decays despite maintained driving voltage. Fast <sodium> <ion-channel inactivation> involves intracellular machinery including the interdomain III–IV region. <Potassium> activation is slower, supplying delayed outward <electric current>; the classical delayed rectifier is relatively sustained on this time scale. Some <voltage-gated potassium channel> variants do show rapid N-terminal pore-blocking <ion-channel inactivation> or slower filter-associated <ion-channel inactivation>, which should not be assumed identical to <sodium>'s mechanism. <voltage clamp> steps separate activation from decay; recovery protocols, single-channel recordings and targeted perturbations distinguish gating, <ion-channel inactivation> and ion conduction. Removal of the Shaker N-terminal blocker and restoration by a free peptide are particularly direct evidence for one <voltage-gated potassium channel> mechanism.<Biophysical and molecular mechanisms of Shaker potassium channel inactivation by Toshinori Hoshi and others>[N-terminal perturbations separate rapid inactivation from activation]{ref}<Restoration of inactivation in mutants of Shaker potassium channels by a peptide derived from ShB by William N. Zagotta and others>[Cytoplasmic peptide restores the missing blocking process]{ref}
Together these properties produce $n^4\bar g_K(V-E_K)$ and $m^3h\bar g_{Na}(V-E_{Na})$: selectivity fixes <reversal potential>, maximal <electrical conductance> fixes scale, and activation/inactivation variables supply voltage- and history-dependent open <probability>.
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