### 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.

### The flying Wallendas of the synapse

Is anything similar to the flying Wallendas ( https://en.wikipedia.org/wiki/The_Flying_Wallendas) going on in the synapse? The first electron micrographs of the synaptic cleft back in the day showed a clear space about 400 Angstroms (40 nanoMeters) thick.  Well we now know that there are tons of proteins occupying this space — a copy of a previous post

## The bouillabaisse of the synaptic cleft

appears after the **** at the end of this post.  It shows just how many proteins occupy that clear space. Could a presynaptic protein directly bond to a postsynaptic protein across the cleft (perhaps with the help of a third or fourth Wallenda protein between the two?  A nice review [ Neuron vol. 96 pp. 680 – 696 ’17b ] http://www.cell.com/neuron/fulltext/S0896-6273(17)30935-2 sets out what is known.

We know that neurexins (presynaptic) bind to neuroligins (postsynaptic) across the cleft.  This is the best studied pair, and most of the earlier post discusses what is known about them.

Figure 1c p. 682 is particularly fascinating as it shows that there are many more molecules which shake hands across the cleft.  Even more interesting is the fact that just where they are relative to the center/periphery of  the synapses isn’t shown for the neurexin/neuroligin pair and the LAR/Strk pair (e.g. one of the best studied pairs) because apparently this isn’t known.   The ephrins/ephrin pair and the syncam pair are in the center, while N-cadherin is shown at the edge.

One of the crucial elements of the post-synaptic membrane, the AMPAR receptor for glutamic protrudes its amino terminal domain 1/3 of the way across the cleft (assuming it is 40 nanoMeters thick).

Postsynaptic receptors are said to be clustered in nanoDomains 80 – 100 nanoMeters in diameter, Similarly, presynaptic RIM nanoClusters are the same size and are said to be aligned with postSynaptic nanoClusters of PSD95 as measured by 3D-STORM, the current most cutting edge technique we have for visualizing these things [ Nature vol. 536 pp. 210 – 214 ’17 ].

So, all in all, the paper is fascinating and shows how much more there is to know.

Unfortunately the paper contains one statement which raises my chemical hackles;  “A consistent prediction across models is that the glutamate concentration profile reaches a very high peak (over 1 milliMolar), but only for a brief time period (100 microSeconds) and over a small distance (100 nanoMeters).” Glutamate is the major excitatory neurotransmitter in brain and is what binds to AMPAR.

Models are lovely, but how many molecules of glutamic acid are they talking about?  It’s easy (but tedious) to figure this out.

We know the volume they are talking about: a cylinder 100 nanoMeters in diameter and 40 nanoMeters tall (the width of the synaptic cleft).   So it contains pi * 100 * 40 = 12,566 cubic nanometers –round this down to 10^4 cubic nanoMeters. A liter is a cube .1 meters (10 centimeters) on a side. So 10 centimeters is 10^8 nanoMeters, meaning that a liter contains (10^8)^3 = 10^24 cubic nanoMeters.

A 1 molar solution of anything contains 6 * 10^23 molecules per liter (Avogadro’s number), so a 1 milliMolar solution (of glutamate in this case) contains 6 * 10^20 molecules/liter or  6 * 10^-4 molecules per cubic nanoMeter. Multiply this by the volume of the cylinder and you get a grand total of 6 molecules of glutamic acid in the cylinder.

If I’ve done the calculations correctly (and I think I have), “a very high peak (over 1 milliMolar)” is basically scientific garbage, the concept of concentration being stretched far beyond its range of meaningful applicability.

I’d love to stand corrected if my calculations are incorrect. Just make a comment.

Addendum 12 Dec — well my calculation is wrong. Here’s the dialog

APAJ — “We know the volume they are talking about: a cylinder 100 nanoMeters in diameter and 40 nanoMeters tall (the width of the synaptic cleft). So it contains pi * 100 * 40 = 12,566 cubic nanometers –round this down to 10^4 cubic nanoMeters.”
Just one err0r in the maths: the volume is r^2*pi*h so it’s closer to 3^5 cubic nm. This leads to ~188 glumate molecules following your further calculations. A more significant number, but I agree concentrations should not be used in these kind of volumes.

APAJ — Thanks — you’re correct and I’m embarrassed — pi * diameter is circumference not volume. so its pi * 50^2 * 40 = 314,259 cubic microns == 25 x more than 12,566 bringing the number of glutamic acids up to 150 (when 12,566 is rounded down to 10^4).

The criticism still stands. Concentration is meaningless in such small volumes.

*****

## The bouillabaisse of the synaptic cleft

The synaptic cleft is so small ( under 400 Angstroms — 40 nanoMeters ) that it can’t be seen with the light microscope ( the smallest wavelength of visible light 3,900 Angstroms — 390 nanoMeters).  This led to a bruising battle between Cajal and Golgi a just over a century ago over whether the brain was actually made of cells.  Even though Golgi’s work led to the delineation of single neurons he thought the brain was a continuous network.  They both won the Nobel in 1906.

Semifast forward to the mid 60s when I was in medical school.  We finally had the electron microscope, so we could see synapses. They showed up as a small CLEAR spaces (e.g. electrons passed through it easily leaving it white) between neurons.  Neurotransmitters were being discovered at the same time and the synapse was to be the analogy to vacuum tubes, which could pass electricity in just one direction (yes, the transistor although invented hadn’t been used to make anything resembling a computer — the Intel 4004 wasn’t until the 70s).  Of course now we know that information flows back and forth across the synapse, with endocannabinoids (e. g. natural marihuana) being the major retrograde neurotransmitter.

Since there didn’t seem to be anything in the synaptic cleft, neurotransmitters were thought to freely diffuse across it to being to receptors on the other (postsynaptic) side e.g. a free fly zone.

Fast forward to the present to a marvelous (and grueling to read because of the complexity of the subject not the way it’s written) review of just what is in the synaptic cleft [ Cell vol. 171 pp. 745 – 769 ’17 ] http://www.cell.com/cell/fulltext/S0092-8674(17)31246-1 (It is likely behind a paywall).  There are over 120 references, and rather than being just a catalogue, the single author Thomas Sudhof extensively discusseswhich experimental work is to be believed (not that Sudhof  is saying the work is fraudulent, but that it can’t be used to extrapolate to the living human brain).  The review is a staggering piece of work for one individual.

The stuff in the synaptic cleft is so diverse, and so intimately involved with itself and the membranes on either side what what is needed for comprehension is not a chemist but a sociologist.  Probably most of the molecules to be discussed are present in such small numbers that the law of mass action doesn’t apply, nor do binding constants which rely on large numbers of ligands and receptors. Not only that, but the binding constants haven’t been been determined for many of the players.

Now for some anatomic detail and numbers.  It is remarkably hard to find just how far laterally the synaptic cleft extends.  Molecular Biology of the Cell ed. 5 p. 1149 has a fairly typical picture with a size marker and it looks to be about 2 microns (20,000 Angstroms, 2,000 nanoMeters) — that’s 314,159,265 square Angstroms (3.14 square microns).  So let’s assume each protein takes up a square 50 Angstroms on a side (2,500 square Angstroms).  That’s room for 125,600 proteins on each side assuming extremely dense packing.  However the density of acetyl choline receptors at the neuromuscular junction is 8,700/square micron, a packing also thought to be extremely dense which would give only 26,100 such proteins in a similarly distributed CNS synapse. So the numbers are at least in the right ball park (meaning they’re within an order of magnitude e.g. within a power of 10) of being correct.

What’s the point?

When you see how many different proteins and different varieties of the same protein reside in the cleft, the numbers for  each individual element is likely to be small, meaning that you can’t use statistical mechanics but must use sociology instead.

The review focuses on the neurExins (I capitalize the E  to help me remember that they are prEsynaptic).  Why?  Because they are the best studied of all the players.  What a piece of work they are.  Humans have 3 genes for them. One of the 3 contains 1,477 amino acids, spread over 1,112,187 basepairs (1.1 megaBases) along with 74 exons.  This means that just over 1/10 of a percent of the gene is actually coding for for the amino acids making it up.  I think it takes energy for RNA polymerase II to stitch the ribonucleotides into the 1.1 megabase pre-mRNA, but I couldn’t (quickly) find out how much per ribonucleotide.  It seems quite wasteful of energy, unless there is some other function to the process which we haven’t figured out yet.

Most of the molecule resides in the synaptic cleft.  There are 6 LNS domains with 3 interspersed EGFlike repeats, a cysteine loop domain, a transmembrane region and a cytoplasmic sequence of 55 amino acids. There are 6 sites for alternative splicing, and because there are two promoters for each of the 3 genes, there is a shorter form (beta neurexin) with less extracellular stuff than the long form (alpha-neurexin).  When all is said and done there are over 1,000 possible variants of the 3 genes.

Unlike olfactory neurons which only express one or two of the nearly 1,000 olfactory receptors, neurons express mutiple isoforms of each, increasing the complexity.

The LNS regions of the neurexins are like immunoglobulins and fill at 60 x 60 x 60 Angstrom box.  Since the synaptic cleft is at most 400 Angstroms long, the alpha -neurexins (if extended) reach all the way across.

Here the neurexins bind to the neuroligins which are always postsynaptic — sorry no mnemonic.  They are simpler in structure, but they are the product of 4 genes, and only about 40 isoforms (due to alternative splicing) are possible. Neuroligns 1, 3 and 4 are found at excitatory synapses, neuroligin 2 is found at inhibitory synapses.  The intracleft part of the neuroligins resembles an important enzyme (acetylcholinesterase) but which is catalytically inactive.  This is where the neurexins.

This is complex enough, but Sudhof notes that the neurexins are hubs interacting with multiple classes of post-synaptic molecules, in addition to the neuroligins — dystroglycan, GABA[A] receptors, calsystenins, latrophilins (of which there are 4).   There are at least 50 post-synaptic cell adhesion molecules — “Few are well understood, although many are described.”

The neurexins have 3 major sites where other things bind, and all sites may be occupied at once.  Just to give you a taste of he complexity involved (before I go on to  larger issues).

The second LNS domain (LNS2)is found only in the alpha-neurexins, and binds to neuroexophilin (of which there are 4) and dystroglycan .

The 6th LNS domain (LNS6) binds to neuroligins, LRRTMs, GABA[A] receptors, cerebellins and latrophilins (of which there are 4)_

The juxtamembrane sequence of the neurexins binds to CA10, CA11 and C1ql.

The cerebellins (of which there are 4) bind to all the neurexins (of a particular splice variety) and interestingly to some postsynaptic glutamic acid receptors.  So there is a direct chain across the synapse from neurexin to cerebellin to ion channel (GLuD1, GLuD2).

There is far more to the review. But here is something I didn’t see there.  People have talked about proton wires — sites on proteins that allow protons to jump from one site to another, and move much faster than they would if they had to bump into everything in solution.  Remember that molecules are moving quite rapidly — water is moving at 590 meters a second at room temperature. Since the synaptic cleft is 40 nanoMeters (40 x 10^-9 meters, it should take only 40 * 10^-9 meters/ 590 meters/second   60 trillionths of a second (60 picoSeconds) to cross, assuming the synapse is a free fly zone — but it isn’t as the review exhaustively shows.

It it possible that the various neurotransmitters at the synapse (glutamic acid, gamma amino butyric acid, etc) bind to the various proteins crossing the cleft to get their target in the postsynaptic membrane (e.g. neurotransmitter wires).  I didn’t see any mention of neurotransmitter binding to  the various proteins in the review.  This may actually be an original idea.

I’d like to put more numbers on many of these things, but they are devilishly hard to find.  Both the neuroligins and neurexins are said to have stalks pushing them out from the membrane, but I can’t find how many amino acids they contain.  It can’t find how much energy it takes to copy the 1.1 megabase neurexin gene in to mRNA (or even how much energy it takes to add one ribonucleotide to an existing mRNA chain).

Another point– proteins have a finite lifetime.  How are they replenished?  We know that there is some synaptic protein synthesis — does the cell body send packages of mRNAs to the synapse to be translated there.  There are at least 50 different proteins mentioned in the review, and don’t forget the thousands of possible isoforms, each of which requires a separate mRNA.

Old Chinese saying — the mountains are high and the emperor is far away. Protein synthesis at the synaptic cleft is probably local.  How what gets made and when is an entirely different problem.

A large part of the review concerns mutations in all these proteins associated with neurologic disease (particularly autism).  This whole area has a long and checkered history.  A high degree of cynicism is needed before believing that any of these mutations are causative.  As a neurologist dealing with epilepsy I saw the whole idea of ion channel mutations causing epilepsy crash and burn — here’s a link — https://luysii.wordpress.com/2011/07/17/we’ve-found-the-mutation-causing-your-disease-not-so-fast-says-this-paper/

Once again, hats off to Dr. Sudhof for what must have been a tremendous amount of work

### 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

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?