## The cell nucleus and its DNA on a human scale – I

The nucleus is a very crowded place, filled with DNA, proteins packing up DNA, proteins patching up DNA, proteins opening up DNA to transcribe it etc. Statements like this produce no physical intuition of the sizes of the various players (to me at least).  How do you go from the 1 Angstrom hydrogen atom, the 3.4 Angstrom thickness per nucleotide (base) of DNA, the roughly 20 Angstrom diameter of the DNA double helix, to any intuition of what it’s like inside a spherical nucleus with a diameter of 10 microns?

How many bases are in the human genome?  It depends on who you read — but 3 billion (3 * 10^9) is a lowball estimate — Wikipedia has 3.08, some sources have 3.4 billion.  3 billion is a nice round number.  How physically long is the genome?  Put the DNA into the form seen in most textbooks — e.g. the double helix.  Well, an Angstrom is one ten billionth (10^-10) of a meter, and multiplying it out we get

3 * 10^9 (bases/genome) * 3.4 * 10^-10 (meters/base) = 1 (meter).

The diameter of a typical nucleus is 10 microns (10 one millionths of a meter == 10 * 10^-6 = 10^-5 meter.   So we’ve got fit the textbook picture of our genome into something 1/100,000 smaller. We’ll definitely have to bend it like Beckham.

As a chemist I think in Angstroms, as a biologist in microns and millimeters, but as an American I think in feet and inches.  To make this stuff comprehensible, think of driving from New York City to Seattle.  It’s 2840 miles or 14,995,200 feet (according to one source on the internet). Now we’re getting somewhere.  I know what a foot is, and I’ve driven most of those miles at one time or other.  Call it 15 million feet, and pack this length down by a factor of 100,000.  It’s 150 feet, half the size of a (US) football field.

Next, consider how thick DNA is relative to its length.  20 Angstroms is 20 * 10^-10 meters or 2 nanoMeters (2 * 10^-9 meters), so our DNA is 500 million times longer than it is thick.  What is 1/500,000,000 of 15,000,000 feet?  Well, it’s 3% of a foot which is .36  of an inch, very close to 3/8 of an inch.   At least in my refrigerator that’s a pair of cooked linguini twisted around each other (the double helix in edible form).  The twisting is pretty tight, a complete turn of the two strands every 35.36 angstroms, or about 1 complete turn every 1.5 thicknesses, more reminiscent of fusilli than linguini, but fusilli is too thick.  Well, no analogy is perfect.  If it were, it would be a description.   One more thing before moving on.

How thinly should the linguini be sliced to split it apart into the constituent bases?  There are roughly 6 bases/thickness, and since the thickness is 3/8 of an inch, about 1/16 of an inch.  So relative to driving from NYC to Seattle, just throw a base out the window every 1/16th of an inch, and you’ll be up to 3 billion before you know it.

You’ve been so good following to this point that you get tickets for 50 yardline seats in the superdome.  You’re sitting far enough back so that you’re 75 feet above the field, placing you right at the equator of our 150 foot sphere. The north and south poles of the sphere are over the 50 yard line. halfway between the two sides.  You are about to the watch the grounds crew pump 15,000,000 feet of linguini into the sphere. Will it burst?  We know it won’t (or we wouldn’t exist).  But how much of the sphere will the linguini take up?

The volume of any sphere is 4/3 * pi * radius^3.  So the volume of our sphere of 10 microns diameter is 4/3 * 3.14 * 5 * 5 * 5 *  = 523 cubic microns. There are 10^18 cubic microns in a meter.  So our spherical nucleus has a volume of 523 * 10^-18 cubic meters.  What is the volume of the DNA cylinder? Its radius is 10 Angstroms or 1 nanoMeter.  So its volume is 1 meter (length of the stretched out DNA) * pi * 10^-9 * 10^-9 meters 3.14 * 10^-18 cubic meters (or 3.14 cubic microns == 3.14 * 10^-6 * 10^-6 * 10^-6

Even though it’s 15,000,000 feet long, the volume of the linguini is only 3.14/523 of the sphere.  Plenty of room for the grounds crew who begin reeling it in at 60 miles an hour.  Since they have 2840 miles of the stuff to reel in, we’ll have to come back in a few days to watch the show.  While we’re waiting, we might think of how anything can be accurately located in 2840 miles of linguini in a 150 foot sphere.