## Numerology

Every class in grad school seemed to begin with a discussion of units. Eventually, Don Voet got fed up and said he preferred the hand stone fortnight system and was going to stick to it. However, even though we all love quantum mechanics dearly for predicting chemical reactivity and spectra, it tells us almost nothing about the events going on in our cells. It’s a crowded environment with objects large and small bumping into one another frequently and at high speeds. At room temperature, a molecule of nitrogen is moving at 500+ meters a second or over 1100 miles an hour. The water in our cells is moving even faster (28/18 times faster to be exact). It’s way too slow for relativity however.

So it’s back classical mechanics to understand cellular events at a physical level, something that will be increasingly important in drug design (but that’s for another post).

The average thermal energy of a molecule at room temperature is kT.

What’s k? It’s the Boltzmann constant. What’s that? It’s the gas constant divided by Avogadro’s number.

I’m assuming that all good chemists know that Avogadro’s number is the number of molecules in a Mole = 6.02 x 10^23

What does the Gas constant have to do with energy?

It’s back to PChem 101 — The ideal gas law is PV = nRT

P = Pressure
V = Volume
n = number of moles
R = Gas constant
T = Temperature

Pressure is Force / Area

Force is Mass * Acceleration
Acceleration is Distance/ (Time * Time)
Area is Distance * Distance
Volume is Distance * Distance * Distance

So PV == [ Force/Area ] * Volume
== { [ Mass * (Distance / Time * Time) ] /( Distance * Distance ) } * ( Distance * Distance * Distance )
== Mass * (Distance/Time) * ( Distance/Time )
== Mass * Velocity * Velocity == mv^2

So PV has the dimensions of (kinetic) energy

The Gas Constant (R) is PV/nT ( == PV/T ) so it has the dimensions of energy/temperature

Now for some actual units (vs. dimensions, although things are much clearer when you think in terms of dimensions)

Force is measured in Newtons which is the force which will accelerate a 1 kiloGram object by 1 meter/second^2

Temperature is measured in Kelvin from absolute zero. A degree Kelvin is the same as 1 degree Celsius (1.8 degrees Fahrenheit)

Room temperature where most of us live is about 27 Centigrade or very close to 300 Kelvin.

So the Boltzmann constant (k) basically energy/temperature per single molecule, which is really what you want to think about when you think about physical processes in the cell.

At room temperature kT works out to 4.1 x 10^-21 Joules.

What’s a Joule? It’s the energy a force of one Newton produces when it moves an object one meter (or you can look at it as the kinetic energy one kilogram has after a force of one Newton has accelerated it over one Meter’s distance.

So a Joule is one Newton * meter

Well 10^-21 is 10^-12 times 10^-9. So what?

This means that at room temperature the average molecule has a thermal energy of 4.1 picoNewton – nanoMeters.

PicoNewtons just happens to be in the range of the force exerted by our molecular motors ( kinesin, dynein, DNA polymerases ) and nanoMeters the range of distances over which they exert forces (act).

Not a coincidence.

Since there are organisms which live at temperatures 20% higher, it would be interesting to know if their motors exert 20% more force. Does anyone out there know?

More interesting even than that are the organisms living at the mid-Ocean ridges where because the extremely high pressures, the water coming from the vents is a lot hotter. What about their motors?

• John Wayne  On October 1, 2015 at 10:26 am

Interesting. Do enzymes have optimized kinetics based on the temperature they have been evolved to operate at? I’m thinking in terms of the polymerases use in PCR. That data must be out there somewhere.

• Curious Wavefunction  On October 1, 2015 at 3:42 pm
• luysii  On October 1, 2015 at 3:50 pm

Looks interesting. Thanks. Will get to it later. Off to a wedding in the teeth of the first serious hurricane of 2015, and possibly the first to hit the continental US since Katrina, a decade ago.

• John Wayne  On October 1, 2015 at 5:20 pm

Interesting paper Ash. Do you all think it is possible to intelligently separate structural changes an enzyme may make to be stable at a higher temperature vs those it may make to compensate for the energy transmitted by an average collision at a higher temperature? Can this be simplified to thermodynamic stability contrasted with optimization of a collision or rotational rate constant? It seems hard to separate those things at the molecule sizes and temperature we’re considering, but I’m just a synthetic guy who TAed physical organic chemistry.

• julien  On October 3, 2015 at 6:38 pm

Cellular motor work fundamentally by diffusion processes. It only goes one way rather than both ways. A bit like on a ratchet.