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 140mM (millimoles per litre). So, about how many K+ ions are in a typical cell?
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- millimoles to moles : 140 millimoles = about .14 moles
- moles of potassium ions: (0.14 moles/litre) × ( 10-10 litres / cell) = 0.14 × 10-11 moles/cell
- particles per mole: there are about 6.02 × 1023 particles in a mole
- so ... : (0.14 × 10-11 moles/cell) * (6.02 × 1023 ions/mole) = ...
I think I have the answer: 8.4 × 1012 K+ ions per cell, or about 8 trillion
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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- ions per channel opening: 100,000 × 1/1000 = 100 every time a channel opens
- ... if each channel opens once a second ... : 10,000 channels × 100 each ions per opening = 1 million
I think I have the answer: 1 million ions per second
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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- I need a hint: The proportion of K+ ions that typically leave in 1 second is 1 million out of 8.4 trillion
- ... using scientific notation... : (1* 106) / (8.4 × 1012)
- ... to divide, subtract the exponents... : (1/8.4) × 10-6 = 0.119 × 10-6 = 1.19 × 10-7
- ... don't forget to convert to a percentage... : 1.19 × 10-7 × 100 = 1.19 × 10-5
I think I have the answer: 0.0000119%
So 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 percent 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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- how many seconds?: 8.4 trillion ions / 1 million ions per second = 8.4 × 1012 ions / 106 ions per second = 8.4 × 106 seconds
- how long is that? : 8.4 × 106 seconds * (1 hour/3600 seconds) = 2330 hours
I think I have the answer: 97 days, or almost 14 weeks
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.
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