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.

The road to the Boltzmann constant

If you’re going to think seriously about cellular biophysics, you’ve really got to clearly understand the Boltzmann constant and what it means.

The road to it starts well outside the cell, in the perfect gas law, part of Chem 101. This seems rather paradoxic. Cells (particularly neurons) do use gases (carbon monoxide, hydrogen sulfide, nitric oxide, and of course oxygen and CO2) as they function, but they are far from all gas.

Get out your colored pencils with separate colors for pressure, energy, work, force, area, acceleration, volume. All of them are combinations of just 3 things mass, distance and time for which you don’t need a colored pen,

The perfect gas law is Pressure * Volume = n R Temperature — the units don’t matter at this point. R is the gas constant, and n is the number of moles (chem 101 again).

Pressure = Force / Area
Force = Mass * Acceleration
Acceleration = distance / (time * time )
Area = Distance * distance

Volume = Distance * distance * distance

So Pressure * Volume = { Mass * distance / (time * time * distance * distance ) } * { distance * distance * distance }

= mass * distance * distance / ( time * time )

This looks suspiciously like kinetic energy (because it is )

Since work is defined as force * distance == mass * acceleration * distance

This also comes out to mass * distance * distance / ( time * time )

So Pressure * Volume has the units of work or kinetic energy

Back to the perfect gas law P * V = n * R * T

It’s time to bring in the units we actually use to  measure energy and work.

Energy and work are measured in Joules. Temperature in degrees above absolute zero (e.g. degrees Kelvin) — 300 is close to body temperature at 81.

Assume we have one mole of gas. Then the gas constant (R) is just PV/T or Joules/degree kelvin == energy/degree kelvin.

Statistical mechanics thinks about molecules not moles (6.022 * 10^23 molecules).

So the Boltzmann constant is just the Gas constant (R) divided by (the number of molecules in a mole * one degree Kelvin ) — it’s basically the energy each molecule posses divided by the current temperature — it is called k and equals 8.31441 Joules/ (mole * degree kelvin)

Biophysicists are far more interested in how much energy a given molecule has at body temperature — to find this multiply k by T (which is why you see kT all over the place.

At 300 Kelvin

kT is
4 picoNewton * nanoMNeters — work
23 milliElectron volts
.6 kiloCalories/mole
4.1 * 10^-21 joules/molecule — energy

Now we’re ready to start thinking about the molecular world.

I should do it, but hopefully someone out there can use this information to find how fast a sodium ion is moving around in our cells. Perhaps I’ll do this in a future post if no one does — it’s probably out there on the net.

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?