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When looking at the action potential generation I do not understand why potassium is needed. I know that potassium is used to repolarize the cell. But why are sodium channels and/or sodium pumps not used to push the cell back into its resting potential?

For example in this video at minute 2:00, sodium is pumped in, and potassium out of the cell. But both are positive ions. You get a less negative membrane potential but wouldn't it go faster if the potassium ions stayed in the cell?

Has potassium an extra property compared to sodium that makes it especially suitable for the repolarization step?

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The action potential is mediated by the opening of Na+ channels, which happens in the order of milliseconds (e.g., Aldrich & Stevens (1987)). Repolarization occurs through the opening of K+channels. Their opening is delayed by a few milliseconds (Purves et al., 2001) to allow Na+ to pour into the cell through Na+ channels. After that delay, the opening of K+ channels occurs also in the order of milliseconds. After these two processes are over, the membrane enters a refractory period, again on a millisecond scale. Then the membrane is ready for another action potential.

The Na+,K+-ATPase, on the other hand, has nothing to do in the action potential process as such. It operates much more sluggishly and it can by no means keep up with the rapid channel gating during action potentials. Instead, the Na+,K+-ATPase activity operates at the background to maintain the overall ion homeostasis. As an analogy, the power adapter of a laptop PC can never keep up with the processor speeds; instead, it functions in the background keeping the battery charged.

Then why two different ions? Because two different species of ions are necessary because they fulfill opposite roles in terms of polarization and reploarization.

References
- Aldrich & Stevens, J Neurosci (1987); 7(2): 418-31
- Purves et al., Reconstruction of the Action Potential. In: Neuroscience, 2nd ed. Sunderland (MA): Sinauer Associates (2001)

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