Very roughly how many water molecules would you guess there are in a single cell? Within a couple of orders of magnitude? I got it quite wrong.
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So call it 30μ, i.e. 3*10^-5. Then the ratio of linear dimensions is 10^5, or volumetrically 10^15. But water isn’t the whole of the cell, so less than that. Maybe 10^14. This is larger than most numbers I found on the web, but not way off.
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Several tweeps did estimates based on total mass of human body divided by the number of cells, and came up with similar numbers. I wouldn’t have thought of that approach!
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I think my original guess was dumb in an interesting way. 10^9 is the cube of only 10^3. Would it be reasonable for a cell to be 10^3 water molecules across? If you think about all the stuff that’s crammed into a cell, made of proteins, and proteins are WAY bigger—no, not 10^3.
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The lesson for me is that cubing numbers makes them grow faster than I realized at a gut level. “Polynomial algorithms are fast!” is not so true for O(n^3).
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Apparently there are on the order of 10^8 protein molecules per cell, which sounds like an awful lot until you take the cube root, to get a few hundred, which seems plausible if you think of all the organelles, made of protein, that have to fit in there.
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I went for 100μm, from vague memories of onion cells in school biology class. But probably we were using onion cells because they were unusually big.
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10 μm is actually just right if you're talking about blood cells. An RBC is 7 μm on average, whites are slightly bigger. Other human cells are even bigger though. Frog egg cells are 1mm.
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