6: Some real numbers

Before we get into the psychedelic equations, let’s do some simpler maths with ions and channels.

A typical cell volume is about 10-10 litres, and the concentration of K+ ions is 140 mM (millimoles per litre). So, about how many K+ ions are in a typical cell?

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Eight trillion — let’s see, about 1000 times the population of the earth…

Now let’s see how many of those 8 trillion ions could be leaving the cell at any given time:

A typical cell has 10,000 K+ channels, and each channel can let 100,000 ions through per second that it is open. However, the typical channel is only open for 1 millisecond out of every second. So, how many K+ can leave in one second?

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A million per second — if ions were people, that would be the population of Adelaide every second — sounds like a stampede to me!

What PERCENTAGE of the cells K+ ions can typically leave in 1 second?

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water flowing over a damSo on the one hand, ions are rushing out at the pace of a million a second… on the other hand, that’s only about ten-thousandths of a per cent of the number of ions present! This is a very tiny percentage!! And the movement of that tiny percentage is what causes the tenth-of-a-volt membrane potential.

Here’s another way to think about it: water can gush over a dam at a rate of hundreds or thousands of litres a minute, yet the level of the water above and below the dam doesn’t change perceptibly — because there are millions of litres of water involved. And, despite the fact that the flow is only a small percentage of the total water, it can still do a significant amount of work as it falls.

Finally, let’s imagine for a moment that all of the K+ ions could continue to leak out of the cell at the rate of 1 million per second.

How long would it take to completely empty the cell of K+ ions at this rate?

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This last calculation is a fantasy, because in fact the cell will not continue to empty out at the same rate. Remember the two opposing gradients? As diffusion moves ions out of the cell, a voltage gradient builds up which pushes them back in. Eventually the two forces even out and … voila, equilibrium.