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I'm reading Medical Physiology by Boron and Boulpaep (a really terrific book). In the chapter Electrophysiology of the Cell Membrane, section Membrane Potential Is Generated by Ion Gradients, Not Directly by Ion Pumps, the text reads:

It may seem that the inside negative Vm originates from the continuous pumping of positive charges out of the cell by the electrogenic Na-K pump. The resting potential of large cells -- whose surface-to-volume ratio is so large that ion gradients run down slowly -- is maintained for a long time even when metabolic poisons block ATP-dependent energy metabolism. This finding implies that an ATP-dependent pump is not the immediate energy source underlying the membrane potential.

I totally get the main takeaway, that Vm results from the net accumulation of ion gradients, rather than the immediate consequences of the ion pumps. But I'm unclear on the phrase large cells -- whose surface-to-volume is so large that ion gradients run down slowly.

Presumably a cell with a large surface-to-volume ratio, like a long thin neuron, would have many ion channels which leak constitutively. So I'd expect the net ion conductance to be high, and gradients would run down quickly. But that contradicts the book's run down slowly point, so I'm confused.

Does passive diffusion play a role here? A long, thin cell would have slow passive ion diffusion, so is that why Vm would run down slowly?

Am I overthinking this?

What is the authors' point here?

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In a typical neuron at rest, potassium is high inside the cell and low outside, with the opposite true for sodium. The membrane is mostly permeable to potassium. Let's ignore the other ions.

The resting potential in this situation will be something like -70 mV. Rest means that the net current flow is zero; however, there is still current: potassium is flowing out of the cell and sodium is flowing in. Therefore, if we turn off the sodium-potassium pump, over time, the concentration gradients will slowly equalize.

The authors are making an assumption that the current is roughly a function of membrane surface area: more membrane = more current. However, the "reservoir" of concentration imbalance is a function of volume: a bigger cell has more total potassium ions than a smaller cell.

Therefore, in a large cell, if the membrane potential is mostly driven by the concentration gradient, and it takes a long time for the concentration inside to equalize with the outside because the current is small relative to the volume, the membrane potential will only change slowly if you use a toxin to stop the sodium-potassium pump.

I don't think the authors are really intending to say anything special about large cells versus small ones, they are just setting up some assumptions under which their argument about the resting potential is going to be most evident. This is a bit like in a physics textbook where you read something like "assume a uniform spherical baseball."

The claim they are making: "the membrane potential is because of the concentration gradient, not the action of the pump itself" is true for all cellular membranes, not just in large cells. This particular piece of evidence they are referring to, however, is going to be most obvious in a large cell. That's all.

If you tried to do this experiment in a very small cell by turning off the pumps and watching the membrane potential, you might erroneously conclude that the pump was responsible for the membrane potential, but this would be a false conclusion caused by the concentration gradient equalizing too quickly. Instead you should test it in a big cell to see the effect more clearly.

In my personal opinion, this is not the best evidence to explain this feature, but it's a hard concept for many people to grasp and I suspect the authors of your text approach it in several different ways to find one that sticks.

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    $\begingroup$ I also thought about a large spherical cell, and had the authors stopped there, the point would have been clear. Given that spherical area grows at $O(r^2)$ and volume grows at $O(r^3)$, the resting dynamics of a large cell would be rate-limited by the small number of open membrane channels compared the huge number of intracellular ions. And if that's correct, they probably meant to write, "whose surface-to-volume ratio is so small"... Might just be a typo. $\endgroup$ – user50080 Feb 25 at 19:29
  • $\begingroup$ @user50080 Oops, I hadn't even read the quoted passage clearly. Yes there is definitely a typo, they mean to say "whose surface-to-volume ratio is so small" or conversely "whose volume-to-surface ratio is so large", I think they caught up with the cell being large and the ordinary way to arrange these things is "surface-to-volume ratio" and made a mistake. The meaning is more clear from the rest of the text. $\endgroup$ – Bryan Krause Feb 25 at 19:46

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