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This question arises from the explanation of what the resting potential of a cell membrane is. In the Goldman formula, there is no interaction between different ion types.

If diffusion is caused by random movement of ions and collisions, then how is it that the K+ ions do not seem affected by the very high concentration of Na+ ions outside of the membrane? Shouldn't there be an effect due to the fact that outside, the ion concentration overall is higher than inside? How come they are all independent?

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2 Answers 2

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Most importantly, the whole Goldman-Hodgkin-Katz model is, well, a model. It is a way we would like to find and explain phenomena, given pen and paper.

Often, scientists build models in order to explain data in hand, but even then, they will have to add something from their imagination. For example, early astronomers saw planets moving differently from stars, and came up with the idea that, you know, planets do not circle the earth, but maybe around some points in space, which in turn circle the earth. That didn't work well, so they created the heliocentric model, where planets and the Earth circled the sun. That was still perfectible, and Newton suggested the planets do not move on circles, but on ellipses. Each generation of astronomers had almost the same data in hand, but chose to make some assumptions in their models.

The same goes with the lack of interaction between potassium and sodium ions: it is a theoretical assumption. As models go, the GHK model is quite good, because it fits experimental findings better than any other model. Its assumptions have a great chance of being true, of reflecting the physical world. I guess you really asked what are the physical facts that underlie the assumption of ion independence. The fact is, you are looking for explanations for a fact that may be or may not be - so any response you will receive is, to a degree, speculation.

My thought is that the number of water molecules (55 molar) that may collide with an ion is far greater than the number of cations (up to a few hundred nanomolar, about 100 million rarer). Perhaps the number of ion-ion collisions is negligible. There is one bottleneck, in the actual channels, but there different species do not meet, because each has its own channels.

Another, more factual, is that the model is imperfect, and independence is more of a wish than a complete truth. Quoting from http://books.google.com/books?id=SmJoSwnwSh0C&lpg=PA353&ots=3fvI19Sk5Q&dq=ghk%20ion%20independence%20fails&pg=PA353#v=onepage&q&f=false : "The rich literature on how ion channels fail to obey the independence principle is reviewed in Chapter 14 of Hille (1991), and some specific models will be studied in the following chapter."

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  • $\begingroup$ Thanks for this quick reply, which helps to a certain extent. But now that I thought about it for a while... If one assumes that the interactions between cations are negligible, and that therefore K+ diffuses "as if Na+ wasn't there", wouldn't that also mean that inside the cell K+ would hardly diffuse, because... well they don't interact with each other, thus hardly ever collide and thus don't diffuse? $\endgroup$
    – Moppentapper
    Commented Sep 4, 2014 at 6:07
  • $\begingroup$ In solution, potassium ions collide with each other as rarely as with sodium ions. The issue is net movement. When looking at a semiporous membrane that allows 1/10 of the ions hitting it, there will be no net movement, if 10 Na ions bumble on one side, and 10 Na ions on the other; nor if you have 100 on one side and 100 on the other. But there is predictable movement, if one side holds 10 times more Na compared to the other side, regardless of absolute numbers, and regardless of the number of K ions hanging around. $\endgroup$
    – nvja
    Commented Sep 4, 2014 at 17:52
  • $\begingroup$ Hi Nick, and thanks again for taking the time to help me! I think I finally understand what you mean. Are you saying, that the diffusion is only caused by how many potassium ions hit the channel on either side? That would make sense to me... I had a quick Wikipedia search and found that there are inward and outward potassium channels, so that potassium that hits the membrane from the outside can flow into the cell too. The fact that there are inward and outward channels also helps with all this. Thanks again! $\endgroup$
    – Moppentapper
    Commented Sep 4, 2014 at 22:53
  • $\begingroup$ The number of ions moving one way is dependent on the number of ions of that kind. If one in a thousand hits the hole, like the balls in lottery, it matters how many balls one is juggling. More green balls will pass if more greenballs are bouncing. But we have two movements. Most ionic channels allow equal passage both ways. $\endgroup$
    – nvja
    Commented Sep 6, 2014 at 16:40
  • $\begingroup$ (continued) The GHK model implies no direction selectivity. GHK does not distinguish permeability for moving K in from permeability for moving K out. (In reality, this assumption is not 100% correct. Rectifying channels make a distinction between directions. So here's another reason why GHK is a simplified model.) $\endgroup$
    – nvja
    Commented Sep 6, 2014 at 19:45
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Maybe I'm missing something here (and it's not my area of expertise) but...

the GHK formula includes the parameter Vm which is the transmembrane potential. This parameter is determined by the intracellular and extracellular concentrations of all ions. A high external concentration of Na+ will therefore influence the net movement of K+.

One of the assumptions of the GHK equation is that the ions do not directly interact.

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  • $\begingroup$ Hi Alan, in the GHK formula, the movement of K+ is independent of the other ion concentrations, you can see this by setting the permeability of all other ions to 0. According to the GHK model, if the ion cannot pass, it will not have an effect on the membrane voltage. But have a look at Nick's answer! $\endgroup$
    – Moppentapper
    Commented Sep 4, 2014 at 23:04

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