## The uses and abuses of Molarity

Quick what does a one Molar solution of a protein look like?

Answer: It doesn’t. The average protein mass is 100 kiloDaltons — http://book.bionumbers.org/how-big-is-the-average-protein/. That’s 100,000 grams per mole (100 kilograms).

A mole of any chemical is Avogadro’s number of it — or 6.02 x 10^23.  The molar mass counts 1 gram for each hydrogen it contains, 12 for each carbon etc. etc.

A 1 molar concentration of any chemical is its molecular mass dissolved in 1 liter of water, which is 1,000 cubic centimeters (cc.).  The density of water is pretty much the same between 32 and 212 Fahrenheit (or 1 – 100 centigrade).

What is the molar concentration of water, e.g. how many moles of water are in a liter of water.  The molecular mass of water is 18 so there are 1000/18 = 55.6 moles of water per liter of water.

Well you can’t get 220 pounds of our 100 kiloDalton protein into 2.2 pounds (1 kiloGram) of water.  You could decorate each of the 6.02 x 10^23 protein molecules with 55 waters.

Why belabor the obvious?  Because numbers are infinitely divisible and it is possible to talk about concentrations given in moles which make no chemical sense. Why?  Because matter is not infinitely divisible.  Divisibility for chemists stops at the atom level.

Now let’s do some biology.  Cell size is measured in microns or 10^-6 meters.   A liter is a cube 10 centimeters on a side, so it is 10^-3 cubic meters.  A cubic micron is 10^-18 cubic meters, so there are 10^15 cubic microns in a liter.

Now lets put 1 molecule in our cubic micron and each and every cubic micron in a liter of water.  What is its concentration in moles?  Our liter contains 10^15 molecules of our chemical, so its Molar concentration is 10^15/6.02 *10^23or .16 x 10^-8  or 1.6 x 10^-9 or 1.6 nanoMolar.    So 1 cubic micron is the volume  at which concentration less than 1.6 nanoMolar make no sense.

It should be noted that 1 cubic micron contains plenty of water molecules to dissolve our molecule.  The actual number:

55 x 6.02 x 10^23/10^15 = 331 x 10^8  = 3 x 10^10 of them.

Notice that the mass of the molecule makes no difference.  Molar means moles/liter and liter is just a volume.  The number of molecules is what is crucial.

As the volume goes up 1 molecule/volume makes sense at lower and lower concentrations.

At this point the physicist says ‘consider a spherical cow’.  The biologist doesn’t have to.  We have lymphocytes which are nearly spherical with diameters ranging from 6 to 14 microns.

Call it 10 microns.  Then the volume of our lymphocyte is  4/3 * pi * 5^3 = 524 cubic microns (call it 1,000 cubic microns to make things easier).  Recall that a liter contains 10^15 cubic microns.  So a liter can contain at most 10^12 lymphocytes, or 10^12 of our molecules so their concentration is 10^12/6.02 * 10^23 or 1.6 x 10^-12 molar. or 1.6 picoMolar.   Molar concentrations lower than 1.6 picoMolar make no chemical or biological sense in volumes of 1000 cubic microns.

Are there chemicals in the lymphocyte with concentrations that low?  Sure there are.  Each chromosome is a molecule, so in male lymphocytes there is exactly one X chromosome and one Y.

Next up.  Is a dissociation constant (Kd) in the femtoMolar (10^-15 Molar) range biologically meaningful?   I’m not sure and am still thinking about it, but the answer has some relevance to Alzheimer’s disease.

Post a comment or leave a trackback: Trackback URL.