The cell nucleus and its DNA on a human scale – III

So we’re in the grandstand looking at a sphere 150 feet in diameter, which contains 15,000,000 feet of linguini which is 3/8 of an inch thick.  The sphere is a 10 micron spherical nucleus blown up. The volume of the nucleus is 523 x 10^-18 meters, but a meter has 10^3 liters in it as a liter is 1000 cubic centimeters. The 3/8 of an inch is what 20 Angstroms looks like at this magnification.  Can we see water?  Well water is about 4 Angstroms across, or 1/16 of an inch. We’re not going to see any water from our perch, even with good binoculars.  But there’s an even better reason why.  Try and figure out what it is before reading the next paragraph.

Let’s assume that everything in the sphere is at Superbowl temperature (27 Centigrade, 80 Farenheit — it is New Orleans after all).  How fast is water moving at this temperature?  The average velocity of water (mass 18 Daltons, or 0.18 kiloGrams/mole = M) at 300 Kelvin is 

Sqrt[ 3 * R * T /M ]   in Meters/second

R is the gas constant = 8.314 Joules/mole * Kelvin

T is 300 Kelvin

This is 645 Meters/second.  That’s a lot of times around a 10 micron nucleus.  It’s also why molecular dynamics simulations have trouble computing times longer than 1 microSecond, and why they need to see what’s happening on a nanoScond to picoSecond scale.  Things happen fast at the chemical level. 

But we’ve blown up 10 microns to 150 feet, or increased distances by a factor of  4,500,000, so 645 meters/second  times 4,500,000 is a bit faster than the speed of light (which we know is impossible).  So the water molecules can’t be seen — even if they were quite large, and they’re not.   

We’re going to be dealing with far heavier entities than water, so what is the mean speed of something with a mass of 1,000 Daltons (1 kiloGram/Mole).  It’s 87 meters/second.  10,000 Daltons (10 kiloGrams/Mole) has an average speed of 27 meters/second, 100,000 Daltons (100 kiloGrams/Mole) moves at 9 meters/second, 1,000,000 Daltons (1 megaDalton or 1000 kiloGrams/Mole) clips along 2.7 meters/second (about as fast as you walk) .

 Is it meaningful to even think about something with a molecular mass of 1,000,000,000 Daltons (a gigaDalton)?  Of course it is;  any chromosome has a mass far greater than this, figuring around 1000 Daltons per base pair of the double helix (including the sugars and the phosphates) a gigaDalton is only a megaBase  The velocity is .08 meters/second.  The smallest chromosome contains 47 megabases giving a velocity of .012518 meters/second.  Well, that’s about one centiMeter/second, and our nucleus is 10 microns or 1/1000th of a centimeter.

This means that even something as big (and this long) as a chromosome will be all over the nucleus many times in the course of a second.  We’ll see a writhing 15,000,000 feet of linguini, if we see anything at all.   We’re going to have to slow time down if we want to see anything at all.  That’s for next time. 

How many water molecules can our nucleus hold?  By a previous calculation we know that the volume of our nucleus is 523 * 10^-18 cubic meters.  But there are 10^3 liters in a cubic meter.   A liter is 1,000 cubic centimeters (pretty nearly).  So nuclear volume is 523 * 10^-15 liters.  The concentration of water is 1000/18 or 55.5 molar, and there are 6.023 * 10^23 molecules/mole so a liter of water contains 55.5 * 6.023 x 10^23 molecules, and our nucleus contains 1.7 * 10^13 molecules of water (convert that to dollars and you have something of the order of magnitude of the national debt).

So it’s amazing that DNA holds up against the pounding that it takes.  645 Meters/second is 1465 miles/hour, and if there’s nothing in our nucleus but DNA and water (with the DNA making up only 6% of the volume of nucleus) I shudder to think of how many times a second our DNA is getting hit throughout its extent (perhaps one of you can figure it out).  It seems nothing short of miraculous that DNA holds up for a second, let alone a lifetime. Familiarity does not breed contempt.

But every compound chemists deal with is this strong.  Most solvents have molecular masses under 1,000 daltons, so the solvent molecules are moving faster than 87 meters/second.    Most compounds chemists deal with are at most an order of magnitude greater than the solvent, so they’re getting clobbered by something nearly their size (and surviving).  

Most chemistry books (even the magnificent Clayden) mention solvent, but never show it in any reaction mechanism.  It’s the elephant no one sees.   I don’t know enough about molecular dynamics simulations to talk about how (or whether) they handle solvent.

Here’s a link to the next paper in the series

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  • dna  On April 1, 2010 at 4:23 am

    Everyone of intelligence and good conscience will appreciate that the body’s perfect systems could not arise spontaneously from unconscious atoms.

  • Wavefunction  On April 1, 2010 at 8:36 pm

    Sure you can include water in MD simulations, either explicit water (based on a parametrized water model like TIP3P or SPC) or implicit continuum solvation model like GBSA. It doesn’t work all the time, but it works remarkably well.

  • Wavefunction  On April 1, 2010 at 8:44 pm

    If you are talking about the diameter of a water molecule it’s closer to 2.9 A. Also, one thing which you don’t seem to include in your calculation is charge (especially the phosphates in DNA), which might drastically slow down the speeds which we are talking about here. Plus, charge repulsion might keep molecules from approaching each other. So would charged ions like calcium.

    How much energy do you think is imparted to a DNA molecule by all that fast-moving water? It’s probably still not going to be enough to break any of the covalent bonds. What about hydrogen bonds?

  • luysii  On April 1, 2010 at 10:56 pm

    dna: people can look at the same data and come to wildly different conclusions as nearly 300,000,000 Americans did with very few exceptions during the recent real estate bubble. If you want to relax and have a good nonscientific read, check out “The Big Short” by Michael Lewis.

    However, like you, the more I know about what’s going on inside the cell, or inside any organ, particularly the brain, the more difficulty I have accepting that all this exquisite machinery and function arose purely by chance. The explanations I’ve seen remind me of Kipling’s “Just So” stories. But I don’t have anything better to offer.

    Wavefunction: What I did to get 4 Angstroms was to take a ruler to the diagram of water on p. 39 of Voet’s Biochemistry book (Ed. 3). I measured the van der Waals radius of hydrogen (which they have as 1.2 Angstroms (which came out to 3/4″), then I measured across the widest part of the molecule with the same rule which came out to 2 5/8″ and multiplied things out accordingly.

    Good point about the phosphates, but they probably attract counterions (most likely potassium) which would add 80 daltons/basepair, and change the mass by under 10%.

    Also EMail me so we can get together when you’re up this way.

  • Wavefunction  On April 5, 2010 at 1:42 pm

    Hi Retread, did you get my email? I sent it on Sunday I think from my yahoo account. Based on past experience I think our email clients don’t like each other very much. Can you send me an email at my gmail address which I had used to communicate with you earlier? It’s Just substitute myname with…my name.

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