## Numerology – I

It’s time to put some numbers on the formulas of statistical mechanics to bring home just how fantastic the goings on inside our cells actually are.

To start — we live at temperatures of 300 Kelvin (27 Centigrade, 80 Fahrenheit). If you’ve studied statistical mechanics you know that the kinetic energy of a molecule is 3/2 k * T — where k is the Boltzmann constant and T is the temperature in Kelvin. The Boltzmann constant is the gas constant R divided by Avogadro’s number. R is to be found in the perfect gas law familiar from elementary PChem or physics — PV – nRT, where P is Pressure, V is volume, and n is the number of moles.

If you’re a bit foggy on this look at https://luysii.wordpress.com/2016/01/10/the-road-to-the-boltzmann-constant/ where you’ll find an explanation of why the dimensional units of R are energy divided by temperature times the number of moles.

This is all very nice but how fast are things moving at room temperature? We need to choose some units and stick to them. We’ve got Kelvin already. We can get from k (the Boltzmann constant) to R (the gas constant) easily by multiplying k by Avogadro’s number.

So now we have kinetic energy per mole (not molecule) is 3/2 R * T

You now need a choice of units for expressing the gas constant. The first part of every course in grad school was consumed with units. Don Voet used to say he preferred the hand stone fortnight system, but that isn’t used much anymore. We’ll use the MKS (Meter KiloGram Second) system. This gives kinetic energy in Joules.

A Joule is the kinetic energy of a mass of 1 kiloGram moving at a velocity of 1 meter/second — or in units — kilogram (meter/second)^2.

Now we’re getting somewhere. The next step is to get molar mass in kiloGrams. Chemists use the Dalton, where the mass of 1 mole of hydrogen is 1 Dalton (1 gram — not kilogram).

Kinetic energy = 1/2 *mass * velocity^2 = mass * (meter/second)^2 == 3/2 R*T

So velocity (in meters/second) = Sqrt ( 3 * R * T / molar mass in kilograms).

To keep things simple I’m going to assume that we’re dealing with hydrogen atoms — so its molar mass is 1 gram (10^-3 kiloGrams)

Putting it all together — the velocity of a hydrogen atom at 300 Kelvin is Sqrt ( 3 *8.314 * 300 / 10^-3 ) == 2,735 meters second

Pretty fast. To convert this to kilometers per hour multiple by 3600 and divide by 1000 == 9,846 Kilometers/hour

In Miles per hour this is 9846 (miles/kilometer) = 6,113 miles per hour.

Recall the number 2735. All you have to do to find out how fast ANY molecular species is moving at room temperature is divide this by the square root of the molecules mass (in Daltons not kiloGrams). So that of water is 2735/ sqrt (18) = 644 meters/second.

I never could be sure that some of the energy of a molecule wasn’t sucked up in vibrations and conformation change. Multiple attempts at understanding the equipartition of energy theorem didn’t help. Finally one of authors of one of 3 biophysics books I’m reading said that “the speed just depends on mass. That’s the translational part. Other degrees of Freedom (like vibrations) can absorb potential energy. But it doesn’t affect velocity.

The velocity formula works even for something as large as RNA polymerase II (500 kilodaltons). To make things really easy lets work with a molecular complex of mass 1,000,000 daltons (1 megaDalton) — there are plenty of such protein complexes of this size (and more) in the cell. A 1 megaDalton mass has a velocity of 2.7 meters a second.

Cells are small. The 3 polymerases transcribed DNA into RNA have masses in the megaDalton range. So how long should it take them to traverse a nucleus 10 microns (10^-5 meters) in diameter. It’s going at 2.7 meters/second so it will traverse 270,000 in a second or one every 4 microSeconds.

Clearly I’ve left something out — nothing in the cell moves in a straight line. It is very crowded, so that even though things are moving very quickly their trajectory isn’t straight (although the numbers I’ve given are correct for the total length of the trajectory when straightened out.  I’ll be writing about diffusion constants etc. etc. in the future, but here’s one more numerological example.

Consider pure water. How many moles of water are in a liter (1 kilogram) of water. 1000/18 – 55.5 moles. How many molecules is that

55.5 * 6.023 * 10^23. How big is water — I found a source that water can be considered a squashed sphere of maximum diameter 2.82 Angstroms. Now Angstroms are something chemist’s deal with — the hydrogen atom is about 1 Angstrom in diameter, and the carbon carbon single bond is 1.54 Angstroms.

So what is the volume of a water molecule — its (4/3) * pi * (2.82/2)^3 == 11.7 cubic Angstroms.

What is the volume of a liter in cubic Angstroms? An Angstrom is 10^-10 meters and a liter is a cube .1 meter on a side — so there are 10^27 cubic Angstroms in a liter. How many cubic Angstroms do the 55.5 moles of water in a liter take up

11.7 * 55.5 * 6.023 ^ 10^23  == 3.9 * 10^26 cubic Angstroms — 40% of the volume of a liter. So water molecule1 is likely to hit another one in 2.5 * 2.28 Angstroms or in about 7 Angstroms. How long will that take ? It’s moving at 6.44 * 10^2 meters/second and 7 Angstroms is a distance of 7 * 10^-10 meters, so it’s like to meet another water in (roughly) 10^-12 seconds (1 picoSecond).

There’s all sorts of hell breaking loose with the water inside our cells. That’s enough for now.

• John F  On February 29, 2016 at 3:05 am

My goodness. It’s like a right royal demolition derby going on in there, but one where the field is chock full of all sorts of vehicles moving in every possible way, and in three dimensions rather than two.

One typo, though: “A Joule is the kidney energy…” Kinetic, yes?

John

• Rick Bryan  On February 29, 2016 at 9:17 am

Oops! A liter is not a cubic meter, it’s a cubic decimeter, 1,000 cc’s. Also, the dalton and angstrom arithmetic could be sanity-checked against the density of water, 1 g/cc.

• luysii  On February 29, 2016 at 10:10 am

Agree ! — This is actually what I said — ” a liter is a cube .1 meter on a side”

• profhurst  On February 29, 2016 at 12:31 pm

Why do Americans not use leading zeros I wonder? (0.1 instead of .1)

I once read somewhere it was a great way to get an overdose of medication.
Ten times your regular dose of opiate painkiller coming right up!