The above videos should be helpful in understanding the structure and function of the basic type of potassium channel, the "inward rectifier". My lecture and Notes describe the basic features of selective permeation, especially the narrowest part of the open pore the "selectivity filter". This is lined with 20 (5X4) carbonyl groups, all pointing to the interior. These are part of the backbone of the polypeptides, and the corresponding amino acid side chains point outward and interact with the rest of the protein, so the carbonyls are held rigidly, and cannot move inward to better contact sodium ions. Therefore these ions cannot lower their energy enough to compensate for that gained by losing H20 "hydration" molecules as they squeeze into the narrow pore. But K+ ions can lower their energy and enter the narrow part of the pore, and then easily move from site to site (e.g s1 to s2 or s3 to s2), depending on whether adjacent sites are already occupied by a K ion. Because the narrow part of the pore (the "filter") always has 2 ions (otherwise the complex structure collapses, eg at very low K cocentration) the only movements we need consider are between the 2 possible states 1,3 or 2,4. Both can occur, but the relative numbers depend on factors such as K concentration on both sides of the membrane, Vm and the chemical energies of the ions in different states. While I diagrammed a simplified "chemical energy" diagram, which showed only equally low energy wells, with the ion pair are located at the lowest energy positions, of course the ions are moving and the total chemical energy depends at any time depends on state and position. It's the complete energy profile for all possible combinations of state (though to a good approximation only movements between the 1,3 and 2/4 states are important and position that actually determine the movements of ions and thus the unidirectional and net potassium currents (more about this in a later lecture).
Crucially the membrane potential affects the energies at all positions. A reasonable approximation is to assume the field is constant at all positions (this "constant field" assumption is used in deriving the GHK membrane equation referred to in the lecture on membrane potential), i.e. that the voltage changes linearly with position (though perhaps most of the potential drop occurs along the narrow filter, so we would first consider the situation at zero Vm, then at other Vms. (I will post diagrams for this)
Notice one crucial point: whatever the current at various Vms and K external/internal concentrations we know that the net K current at the K Nernst potential must be zero, and our quantative analysis must yield this result.
THe main point of this discussion that embodying our recent understanding of molecular structural details can take us far beyond the simple pictures we used before (either Ohm's law or electrodiffusion across a uniform membrane as in the GHK picture). However it can get complicated , and these more detailed models don't really affect the basic conclusions we reached previously: qualitaively Ik = Gk(Vm-Ek). They do offer a more complete picture of ion channel function, and you should know the gist of the ideas involved.
SUMMARY: the movement of K ions through the selectivity filter can be modeled as cycling between 1,3 and 2/4 states. Inic current occurs when there are move cycles in 1 direction than the other. When the pore moves form the first state to the second, an ion has moved inward, and vice versa. If the concentration difference on either side is zero, and Vm is too, then clearly there cannot be net cycling either way. If Kout/Kin is > 1, with Vm = 0, there will be net inward cycling; if K out/Kin = 1 bit Vm is < 0, there will also be net inward cycling. If Vm = Ek (the Nernst potential) there is no net cycling though neither Vm = 0 nor Ko/Ki = 1. In general knowing the energy profile for various pore occupancy states allows one to calculate Ik under any conditions.
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