Category Archives: Neurology & Psychiatry

The staggering implications of one axon synapsing on another

It isn’t often that a single paper can change the way we think the brain works.  But such is the case for the paper described in the previous post (full copy below *** ) if the implications I draw from it are correct.

Unfortunately this post requires a deep dive into neuroanatomy, neurophysiology, neuropharmacology and cellular molecular biology.  I hope to put in enough background to make some of it comprehensible, but it is really written for the cognoscenti in these fields.

I’m pretty sure these thoughts are both original and unique

Briefly, the paper provided excellent evidence for one axon causing another to fire an impulse (an action potential).   The fireror was from a neuron using acetyl choline as a neurotransmitter, and the fireree was a dopamine axon going to the striatum.

Dopamine axons are special.  They go all over the brain. The cell body of the parent neuron of the axon to be synapsed on uses dopamine as a neurotransmitter.  It sits in the pars compacta of the substantia nigra a fair piece away from the target they studied. “Individual neurons of the pars compact are calculated to give rise to 4.5 meters of axons once all the branches are summed”  — [ Neuron vol. 96 p. 651 ’17 ].”  These axons release dopamine all over the brain.  There aren’t many dopamine neurons to begin with just 80,000 which is 1 millionth of the current (probably unreliable) estimate of the number of neurons in the brain 80,000,000,000.

Now synapses between neurons are easy to spot using electron microscopy.  The presynaptic terminal contains a bunch of small vesicles and is closely apposed (300 Angstroms — way below anything the our eyes can see) to the post synaptic neuron which also looks different, usually having a density just under the membrane (called, logically enough, post-synaptic density).  Embedded in the postsynaptic membrane are proteins which conduct ions such as Na+, K+, Cl- into the postsynaptic neuron triggering an action potential.

But the dopamine axons going all over the brain have a lot of presynaptic specialization, but the post-synaptic neuron and its postsynaptic density is nowhere to be found.  This is called volume neurotransmission.

The story doesn’t end with dopamine.  There are 3 other similar systems of small numbers of neurons collected into nuclei, using different neurotransmitters, but whose axons branch and branch so they go all over the brain.

These are the locus coeruleus which uses norepinephrine as a neurotransmitter, the dorsal raphe nucleus which uses serotonin and the basal nucleus of Meynert which uses acetyl choline.

What is so remarkable about the paper, that it allows the receiving neurons to (partially) control what dopamine input it gets.

But dopamine doesn’t work at the synapse, and the 5 receptors for it (called G Protein Coupled Receptors — GPCRs) aren’t found there. None of the GPCRs conduct ions or trigger action potentials (immediately anyway).  Instead, they produce their effects much more slowly and change the metabolism of the interior of the cell.

Neither does norepinephrine all of whose receptors are GPCRs.  Serotonin does have one of its 16 or so receptors which conducts ions, but the rest are GPCRs.

Acetyl choline does have one class of receptors (nicotinic) which conducts ions, and which the paper shows is what is triggering the axon on axon synapse.  The other class (muscarinic) of acetyl choline receptor is a GPCR.

We do know that the norepinephrine and serotonin axons work by volume neurotransmission (not sure about those of the basal nucleus of Meynert).

Now the paper tested axon to axon firing in one of the four systems (dopamine) in one of the places its axons goes (the striatum).  There is no question that the axons of all 4 systems ramify widely.

Suppose axon to axon firing is general, so a given region can control in someway how much dopamine/serotonin/norepinephrine/acetyl choline it is getting.

Does this remind you of any system you are familiar with?  Maybe, because my wife went to architecture school, it reminds me of an old apartment building, with separate systems to distribute electricity, plumbing, steam heat and water to each apartment, which controls how much of each it gets.

Perhaps these four systems are basically neurological utilities, necessary for  the function of the brain, but possibly irrelevant to the computations it is carrying out, like a mother heating a bottle for her baby in water on a gas stove on a cold winter night.  The nature of steam heat, electricity, water and gas tell you very little about what is going on in her apartment.

The paper is so new (the Neuron issue of 21 September) that more implications are sure to present themselves.

Quibbles are sure to arise.  One is that fact that the gray matter of our brain doesn’t contain much in the way of neurons using acetyl choline as a neurotransmitter.  What it does have is lots of neurons using GABA which we know can act on axons, inhibiting axon potential generation.  This has been well worked out with synapses where the axon emerges from the neuron cell body (the initial segment).

The work was done in living animals, so no microscopy is available showing the synapse. Such work is sure to be done.  No classical presynaptic apparatus may be present, just two naked axons touching each other and interacting by ephaptic transmission.

So a lot of work should be done, the first of which should be replication. As the late Carl Sagan said “extraordinary claims require extraordinary evidence”.


As Mark Twain said ” There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.”


Synapses on Axons !

Every now and then a paper comes along which shows how little we really know about the brain and how it works.  Even better, it demands a major rethink of what we thought we knew.  Such a paper is —

which I doubt you can get unless you are a subscriber to Neuron.    What [ Neuron vol. 110 pp. 2889 – 2890 ’22 ] does is pretty much prove that an axon from one neuron can synapse on an axon of another neuron.  When one neuron is stimulated the axon of another neuron fires an impulse (an action potential) as measured by patch clamping the second axon.  This happens way too fast after stimulation to be explained by volume neurotransmission (about which more later).  Such synapses are well known on the initial segment of the axon as it leaves the cell body (the soma) of the neuron.

But these synapses occur very near to the end of the axon in the part of the brain (the striatum) the parent neuron (a midbrain dopamine neuron) innervates (the striatum).   The neurotransmitter involved is acetylcholine and the striatum has lots of neurons using acetylcholine as a neurotransmitter.  There are two basic types of acetylcholine receptor in the brain — muscarinic and nicotinic.  Muscarinic receptors are slow acting and change the internal chemistry of the neuron.  This takes time.  Nicotinic receptors are ion channels, and when they open, an action potential is nearly immediate.  Also using a drug to block the nicotinic acetyl choline receptor, blocks action potential formation after stimulation.

Why is this work so radical? (which of course means that it must be repeated by others).  It implies that all sorts of computations in the brain can occur locally at the end of an axon, far away from the neuron cell body which is supposed to be in total control of it.  The computations could occur without any input from the cell body, and spontaneous activity of the axons they studied occur without an impulse from the cell body.   If replicated, we’re going to have to rethink our models of how the brain actually works.  The authors note that they have just studied one system, but other workers are certain to study others, to find out how general this.

Neuropil, is an old term for areas of the brain with few neuron or glial cell bodies, but lots of neural and glial processes.  It never was much studied, and our brain has lots of it.  Perhaps it is actually performing computations, in which case it must be added to the 80 billion neurons we are thought to have.

Now for a bit more detail

The cell body of the parent neuron of the axon to be synapsed on uses dopamine as a neurotransmitter.  It sits in the pars compacta of the substantia nigra a fair piece away from the target they studied. “Individual neurons of the pars compact are calculated to give rise to 4.5 meters of axons once all the branches are summed”  — [ Neuron vol. 96 p. 651 ’17 ].”  These axons release dopamine all over the brain, and not necessarily synapsing with a neuron.  So when that single neuron fires, dopamine is likely to bathe every neuron in the brain.This is called volume neurotransmission which is important because the following neurotransmitters use it — dopamine, serotonin, acetyl choline and norepinephrine. Each has only a small number of cells using them as a transmitter.  The ramification of these neurons is incredible.

So now you see why massive release of any of the 4 neurotransmitters mentioned (norepinephrine, serotonin, dopamine, acetyl choline) would have profound effects on brain states.  The four are vitally involved in emotional state and psychiatric disease. The SSRIs treat depression, they prevent reuptake of released serotonin.  Cocaine has similar effects on dopamine.  The list goes on and on and on.

Axons synapsing on other axons is yet another reason to modify our rather tattered wiring diagram of the brain —

FDA Amylyx approval 7 September implies Simufilam will be FDA approved this year

On 7 September an FDA advisory board reversed itself and recommended approval for a drug for ALS —  The head of the FDA Office of Neuroscience (Billy Dunn) gave a verbal endorsement, making it likely that Amylyx’s drug would be approved.

What does this have to do with the approval of Simufilam this year? Amylyx did a post-hoc, retrospective “responder analysis” that showed patients who did respond to drug (vs placebo) had “an usually strong response”, i.e., a bunch of non-responders in the general population masked the beneficial effects of the drug. This, after the same committee in March turned the drug down due to lack of efficacy in the studied cohort as a whole.

You may recall that I thought Cassava’s results with Simufilam were better than they realized after they released the data on the first 50 patients in the open trail reaching the 9 month endpoint. The full post published 25 August 2021 can be found below the &&&&&. 5/50 had a greater than 50% improvement in their ADAS-Cog11 score (by more than 10 points).  Data like this in Alzheimer’s has never been seen before in any study, or in my clinical experience.  So the data can not be explained by Cherry-picking.  The only other explanations are (1) Fraud (2) incompetent ADAS-Cog11 measurement (3) people without Alzheimer’s entering the study for the money, all of which I think are remote.  Also, the average decline at one year in ADAS-Cog in Alzheimer patients is 5 points.

So Cassava has data similar to Amylyx’s on the first 50 of the 200 in the open label study.  The last of the 200 will complete their full year on the drug by the end of 2022, at which point data will be released.  If the results on the 200 patients are similar to those on the first 50 (say 20/200 having significant (greater than 50% change for the better in ADAS-Cog) improvement, Cassava will have a (strong in my opinion) argument for Simufilam approval.

Clinicians know that patients always respond variably to any sort of therapy. We now know why.  Given that the human genome contains 3,200,000,000 positions.  Full genome sequencing of well over 100,000 people has shown that any two people will differ at one position in a thousand — that’s 3,200,000 differences  — source


Gentlemen start your engines


Cassava Sciences 9 month data is probably better than they realize

My own analysis of the Cassava Sciences 9 month data shows that it is probably even better than they realize.

Here is a link to what they released — keep it handy

I was unable to listen to Lindsay Burn’s presentation at the Alzheimer Association International Conference in July as I wasn’t signed up.  I have been unable to find either a video or a transcript, so perhaps Lindsay did realize what I’m about to say.

Apparently today 25 August there was another bear attack on the company and its data.  I’ve not read it or even seen what the stock did.  In what follows I am assuming that everything they’ve said about their data is true and that their data is what they say it is.

So the other day I had a look at what Cassava released at the time of Lindsay’s talk.

First some background on their study.  It is a report on the first 50 patients who had received Simulfilam for 9 months.  It is very important to understand how they were measuring cognition.  It is something called ADAS-Cog11

Here it is and how it is scored and my source —

The original version of the ADAS-Cog consists of 11 items, including:1

1. Word Recall Task: You are given three chances to recall as many words as possible from a list of 10 words that you were shown. This tests short-term memory.

2. Naming Objects and Fingers: Several real objects are shown to you, such as a flower, pencil and a comb, and you are asked to name them. You then have to state the name of each of the fingers on the hand, such as pinky, thumb, etc. This is similar to the Boston Naming Test in that it tests for naming ability, although the BNT uses pictures instead of real objects, to prompt a reply.

3. Following Commands: You are asked to follow a series of simple but sometimes multi-step directions, such as, “Make a fist” and “Place the pencil on top of the card.”

4. Constructional Praxis: This task involves showing you four different shapes, progressively more difficult such as overlapping rectangles, and then you will be asked to draw each one. Visuospatial abilities become impaired as dementia progresses and this task can help measure these skills.

5. Ideational Praxis: In this section, the test administrator asks you to pretend you have written a letter to yourself, fold it, place it in the envelope, seal the envelope, address it and demonstrate where to place the stamp. (While this task is still appropriate now, this could become less relevant as people write and send fewer letters through the mail.)

6. Orientation: Your orientation is measured by asking you what your first and last name are, the day of the week, date, month, year, season, time of day, and location. This will determine whether you are oriented x 1, 2, 3 or 4.

7. Word Recognition Task: In this section, you are asked to read and try to remember a list of twelve words. You are then presented with those words along with several other words and asked if each word is one that you saw earlier or not. This task is similar to the first task, with the exception that it measures your ability to recognize information, instead of recall it.

8. Remembering Test Directions: Your ability to remember directions without reminders or with a limited amount of reminders is assessed.

9. Spoken Language: The ability to use language to make yourself understood is evaluated throughout the duration of the test.

10. Comprehension: Your ability to understand the meaning of words and language over the course of the test is assessed by the test administrator.

11. Word-Finding Difficulty: Throughout the test, the test administrator assesses your word-finding ability throughout spontaneous conversation.

What the ADAS-Cog Assesses

The ADAS-Cog helps evaluate cognition and differentiates between normal cognitive functioning and impaired cognitive functioning. It is especially useful for determining the extent of cognitive decline and can help evaluate which stage of Alzheimer’s disease a person is in, based on his answers and score. The ADAS-Cog is often used in clinical trials because it can determine incremental improvements or declines in cognitive functioning.2


The test administrator adds up points for the errors in each task of the ADAS-Cog for a total score ranging from 0 to 70. The greater the dysfunction, the greater the score. A score of 70 represents the most severe impairment and 0 represents the least impairment.

The average score of the 50 individuals entering was 17 with a standard deviation of 8, meaning that about 2/3 of the group entering had scores of 9 to 25 and that 96% had scores of 1 to 32 (but I doubt that anyone would have entered the study with a score of 1 — so I’m assuming that the lowest score on entry was 9 and the highest was 25).  Cassava Sciences has this data but I don’t know what it is.

Now follow the link to Individual Patient Changes in ADAS-Cog (N = 50) and you will see 50 dots, some red, some yellow, some green.

Look at the 5 individuals who fall between -10 and – 15 and think about what this means.  -10 means that an individual made 10 fewer errors at 9 months than on entry into the study.  Again, I have no idea what the scores of the 5 were on entry.

So assume the worst and that the 5 all had scores of 25 on entry.  The group still showed a 50% improvement from baseline as they look like they either made 12, 13, or 14 fewer errors.  If you assume that the 5 had the average impairment of 17 on entry, they were nearly normal after 9 months of treatment.  That doesn’t happen in Alzheimer’s and is a tremendous result.   Lindsay may have pointed this out in her talk, but I don’t know although I’ve tried to find out.

Is there another neurologic disease with responses like this.  Yes there is, and I’ve seen it.

I was one of the first neurologists in the USA to use L-DOPA for Parkinsonism.  All patients improved, and I actually saw one or two wheelchair bound Parkinsonians walk again (without going to Lourdes).  They were far from normal, but ever so much better.

However,  treated mildly impaired Parkinsonians became indistinguishable from normal, to the extent that I wondered if I’d misdiagnosed them.

12 to 14 fewer errors is a big deal, an average decrease of 3 errors, not so much, but still unprecedented in Alzheimer’s disease.   Whether this is clinically meaningful is hard to tell.  However, 12 month data on the 50 will be available in the fourth quarter of ’21, and if the group as a whole continues to improve over baseline it will be a very big deal as it will tell us a lot about Alzheimer’s.

Cassava Sciences has all sorts of data we’ve not seen (not that they are hiding it).  Each of the 50 has 4 data points (entry, 3, 6 and 9 months) and it would be interesting to see the actual scores rather than the changes between them in all 50.  Were the 5 patients with the 12 – 14 fewer errors more impaired (high ADAS-Cog11 score in entry) or less.

Was the marked improvement in the 5 slow and steady or sudden?   Ditto for the ones who deteriorated or who got much worse or who slightly improved.

Even if such dramatic improvement is confined to 10% of those receiving therapy it is worth a shot to give it to all.  Immune checkpoint blockade has dramatically helped some patients with cancer  (far from all), yet it is tried in many.

Disclaimer:  My wife and I have known Lindsay since she was a teenager and we were friendly with her parents.  However, everything in this post is on the basis of public information available to anyone (and of course my decades of experience as a clinical neurologist)


4 diseases explained at one blow said the protein chemist — part 2 — TDP43

A brilliant paper [ Science vol. 377 eabn5582 pp. 1 –> 20 ’22 ] explains how changing a single amino acid (proline) to another  can cause 4 different diseases, depending on the particular protein it is found in (and which proline of many is changed).

There is so much in this paper that it will take several posts to go over it all.  The chemistry in the paper is particularly fine.  So it’s back to Biochemistry 101 and the alpha helix and the beta sheet.

A lot of the paper concerns TDP43, a protein familiar to neurologists because it is involved in FTD-ALS (FrontoTemporal Dementia — Amyotrophic Lateral Sclerosis) and ALS itself.

I actually saw a case early in training.  I had been taught that ALS patients remained cognitively intact until the end (certainly true in my experience — think of Stephen Hawking), so here was this ALS case who was mildly demented.  My education, deficient at that time, so I’d never heard of FTD-ALS, had me writing in the chart “we’re missing something here”.  These were calmer times in the medical malpractice world.

TDP43 is a protein with a lot of different parts in its 414 amino acids.  There are two regions which bind to RNA (Rna Recognition Motifs { RRMs } ), and a glycine rich low complexity domain at the carboxy terminal end.

TDP43 proteins are found in the neuronal inclusions of ALS (interestingly, these weren’t recognized when I was in training).  The low complexity domain of TDP43 aggregates and form fibers.  Some 50 different mutations have been found here in patients.

Just this year the cryoEM structure of TDP43 aggregates from two patients with FTD-ALS were described [ Nature vol. 601 pp. 29 – 30, 139 – 143 ’22 ].  It appears to be a typical amyloid structure with all 79 amino acids (from # 282 Glycine to #360  Glutamine) in a single plane.  Here’s a link to the actual paper —  It is likely behind a paywall, but if you can get it, look at figure 2 p. 140, which has the structure.  Who would have ever thought that a protein could flatten out this much.

Both structures were from TDP-43 with none of the 24 mutations known to cause FTD-ALS.

But that’s far from the end of the story.  The same area of TDP43 can also form liquid droplets (perhaps the precursor of the fibers).  But that’s where the brilliant chemistry of [ Science vol. 377 eabn5582 pp. 1 –> 20 ’22 ] comes in.

That’s for next time.  After that, I should be finished with Needham and will have time to write about 6 or so of the interesting papers I’ve run across in the past 6 months.

4 diseases explained at one blow said the protein chemist — part 1

A brilliant paper [ Science vol. 377 eabn5582 pp. 1 –> 20 ’22 ] explains how changing a single amino acid (proline) to another  can cause 4 different diseases, depending on the particular protein it is found in (and which proline of many is changed).

There is so much in this paper that it will take several posts to go over it all.  The chemistry in the paper is particularly fine.  So it’s back to Biochemistry 101 and the alpha helix and the beta sheet.

Have a look at this

If you can tell me how to get a picture like this into a WordPress post please make a comment.

The important point is that hydrogen bonds between the amide hydrogen of one amino acid and the carbonyl group of another hold the alpha helix and the beta pleated sheet together.

Enter proline : p//  Proline when not embedded in a protein has a hydrogen on the nitrogen atom in the ring.  When proline is joined to another amino acid by a peptide bond in a protein, the hydrogen on the nitrogen is no longer present.  So the hydrogen bond helping to hold the two structures (alpha helix and beta sheet) is no longer present at proline, and alpha helices and beta sheets containing proline are not has stable.  Prolines after the fourth amino acid of the alpha helix (e. g. after the first turn of the helix) produce a kink.  The proline can’t adopt the alpha helical configuration of the backbone and it can’t hydrogen bond.

But it’s even worse than that (and this observation may even be original).  Instead of a hydrogen bonding to the free electrons of the oxygen in the carbonyl group you have the two electrons on the nitrogen jammed up against them.  This costs energy and further destabilizes both structures.

Being a 5 membered ring which contains the alpha carbon of the amino acid, proline in proteins isn’t as flexible as other amino acids.

This is why proline is considered to be a helix breaker, and is used all the time in alpha helices spanning cellular membranes to cause kinks, giving them more flexibility.

There is much more to come — liquid liquid phase separation, prion like domains, low complexity sequences, frontotemporal dementia with ALS, TDP43, amyloid, Charcot Marie Tooth disease and Alzheimer’s disease.

So, for the present stare at the link to the diagram above.

Why Cassava’s 1 year results should allow compassionate use of Simufilam

Cassava reported results on 100 Alzheimer patients in an open label (e.g. no controls) trial of Simufilam for 1 year —  The average results were unimpressive (to the uninitiated) with only a minimal average overall improvement of an ADAS-Cog11 score of 1.5 points.  This is probably why the stock (SAVA) dropped a point yesterday after the news.  Since everything turns on ADAS-Cog11 here is a link to a complete description —  The test takes about 45 minutes placing it out of reach of a busy practicing clinical neurologist.

Why is even the 1.5 point improvement impressive to the initiated (me)?  Over 32 years in clinical neurology, I’d estimate that I saw at least 1 demented patient each week.  Now probably only 300 or so of the 1,664 were followed for a year.  Guess what?  None of them remained stable for a year, and all got worse.  Absolutely none of them  ever got better after a year.  So at least some stabilization of the disease is possible for a year.  The statistics say that Alzheimer patients lose 5 points a year on ADAS-Cog.

But that’s pretty small beer.  Who wants to keep a demented patient around but stable.  Here is the remarkable part of the Cassava results at a year.

63% of the 100 Patients Showed an Improvement in ADAS-Cog11 Scores, and This Group of Patients Improved an Average of 5.6 Points (S.D. ± 3.8). The statistics say that Alzheimer patients lose 5 points a year on ADAS-Cog.

This is unprecedented and is a strong argument for quick approval of Simufilam (or at least compassionate use).

The cynic will say that I’m just looking at the happy part of the Bell curve.  There must have been people who declined to average the improvement in the 63% down to a measly 1.5 points on the ADAS-Cog.

This is where clinical experience comes in.  No drug helps everyone with a given disease.  “Only 20% of cancer patients respond long term to a type of immune checkpoint blockade (of PD-1)” Science vol. 363p. 1377 ’19.  Nonetheless immune checkpoint blockade of several types was approved by the FDA, simply because there was nothing better available.

So if nearly 2/3 of Alzheimer patients will improve at one year on Simufilam, why not  let the FDA offer it to them now under compassionate use.



Amyloid Structure at Last ! 3 The Alzheimer mutations

I am republishing this post from last October, because the excellent paper I’m going to write about has similar thinking.

Although the chemistry explaining why these mutations are associated with Alzheimer’s disease is exquisite and why they point to ‘the’ cause of Alzheimer’s disease — the amyloid fibril, billions have been spent in attempts to remove amyloid fibrils with no useful therapeutic result (and some harm)

Here’s the old post

The structure of the amyloid fibril formed by the aBeta42 peptide exactly shows why certain mutations are associated with hereditary Alzheimer’s disease.   Here is a picture

Scroll down to the picture above “Bonds that Tie”

If you need some refreshing on the general structure of amyloid, have a look at the first post in the series —

Recall that in amyloid fibrils the peptide backbone is flat as a flounder (well in a box 4.8 Angstroms high) with the amino acid side chains confined to this plane.  The backbone winds around in this plane like a snake.  The area in the leftmost loop is particularly crowded with bulky side chains of glutamic acid (single letter E) at position 22 and aspartic acid (single letter D) at position 23 crowding each other.  If that wasn’t enough, at the physiologic pH of 7 both acids are ionized, hence negatively charged.  Putting two negative charges next to each other costs energy and makes the sheet making up the fibril less stable.

The marvelous paper (the source for much of this) Cell vol. 184 pp. 4857 – 4873 ’21 notes that there are 3 types of amyloid — pathological, artificial, and functional, and that the pathological amyloids are the most stable. The most stable amyloids are the pathological ones.  Why this should be so will be the subject of a future post, but accept it as fact for now

In 2007 there were 7 mutations associated with familial Alzheimer’s disease (10 years later there were 11). Here are 5 of them.

Glutamic Acid at 22 to Glycine (Arctic)

Glutamic Acid at 22 to Glutamine (Dutch)

Glutamic Acid at 22 to Lysine (Italian)

Aspartic Acid at 23 to Asparagine (Iowa)

Alanine at 21 to Glycine (Flemish)

All of them lower the energy of the amyloid fiber.

Here’s why

Glutamic Acid at 22 to Glycine (Arctic) — glycine is the smallest amino acid (side chain hydrogen) so this relieves crowding.  It also removes a negatively charged amino acid next to the aspartic acid.  Both lower the energy

Glutamic Acid at 22 to Glutamine (Dutch) — really no change in crowding, but it removes a negative charge next to the negatively charged Aspartic acid

Glutamic Acid at 22 to Lysine (Italian)– no change in crowding, but the lysine is positively charged at physiologic pH, so we have a positive charge next to the negatively charged Aspartic acid, lowering the energy

Aspartic Acid at 23 to Asparagine (Iowa) –really no change in crowding, but it removes a negative charge next to the negatively charged Glutamic acid next door

Alanine at 21 to Glycine (Flemish) — no change in charge, but a reduction in crowding as alanine has a methyl group and glycine a hydrogen.

As a chemist, I find this immensely satisfying.  The structure explains why the mutations in the 42 amino acid aBeta peptide are where they are, and the chemistry explains why the mutations are what they are.

, , , , , , , . No

A visual proof of the the theorem egregium of Gauss

Nothing better illustrates the difference between the intuitive understanding that something is true and being convinced by logic that something is true  than the visual proof of the theorem egregium of Gauss found in “Visual Differential Geometry and Forms” by Tristan Needham and  the 9 step algebraic proof in  “The Geometry of Spacetime” by Jim Callahan.

Mathematicians attempt to tie down the Gulliver of our powerful appreciation of space with Lilliputian strands of logic.

First: some background on the neurology of vision and our perception of space and why it is so compelling to us.

In the old days, we neurologists figured out what the brain was doing by studying what was lost when parts of the brain were destroyed (usually by strokes, but sometimes by tumors or trauma).  This wasn’t terribly logical, as pulling the plug on a lamp plunges you in darkness, but the plug has nothing to do with how the lightbulb or LED produces light.  Even so,  it was clear that the occipital lobe was important — destroy it on both sides and you are blind — but the occipital lobe accounts for only 10% of the gray matter of the cerebral cortex.

The information flowing into your brain from your eyes is enormous.  The optic nerve connecting the eyeball to the brain has a million fibers, and they can fire ‘up to 500 times a second.  If each firing (nerve impulse) is a bit, then that’s an information flow into your brain of a gigaBit/second.   This information is highly processed by the neurons and receptors in the 10 layers of the retina. Over 30 retinal cell types in our retinas are known, each responding to a different aspect of the visual stimulus.  For instance, there are cells responding to color, to movement in one direction, to a light stimulus turning on, to a light stimulus turning off, etc. etc.

So how does the relatively small occipital lobe deal with this? It doesn’t.  At least half of your the brain responds to visual stimuli.  How do we know?   It’s complicated, but something called functional Magnetic Resonance Imaging (fMRI) is able to show us increased neuronal activity primarily by the increase in blood flow it causes.

Given that half of your brain is processing what you see, it makes sense to use it to ‘see’ what’s going on in Mathematics involving space.  This is where Tristan Needham’s books come in.

I’ve written several posts about them.

and Here —



OK, so what is the theorem egregium?  Look at any object (say a banana). You can see how curved it is by just looking at its surface (e.g. how it looks in the 3 dimensional space of our existence).  Gauss showed that you don’t
have to even look at an object in 3 space,  just perform local measurements (using the distance between surface points, e.g. the metric e.g.  the metric tensor) .  Curvature is intrinsic to the surface itself, and you don’t have to get outside of the surface (as we are) to find it.



The idea (and mathematical machinery) has been extended to the 3 dimensional space we live in (something we can’t get outside of).  Is our  universe curved or not? To study the question is to determine its intrinsic curvature by extrapolating the tools Gauss gave us to higher dimensions and comparing the mathematical results with experimental observation. The elephant in the room is general relativity which would be impossible without this (which is why I’m studying the theorem egregium in the first place).


So how does Callahan phrase and prove the theorem egregium? He defines curvature as the ratio of the area on a (small) patch on the surface to the area of another patch on the unit sphere. If you took some vector calculus, you’ll know that the area spanned by two nonCollinear vectors is the numeric value of their cross product.



The vectors Callahan needs for the cross product are the normal vectors to the surface.  Herein beginneth the algebra. Callahan parameterizes the surface in 3 space from a region in the plane, uses the metric of the surface to determine a formula for the normal vector to the surface  at a point (which has 3 components  x , y and z,  each of which is the sum of 4 elements, each of which is the product of a second order derivative with a first order derivative of the metric). Forming the cross product of the normal vectors and writing it out is an algebraic nightmare.  At this point you know you are describing something called curvature, but you have no clear conception of what curvature is.  But you have a clear definition in terms of the ratio of areas, which soon disappears in a massive (but necessary) algebraic fandango.



On pages 258 – 262 Callahan breaks down the proof into 9 steps involving various mathematical functions of the metric and its derivatives such as  Christoffel symbols,  the Riemann curvature tensors etc. etc.  It is logically complete, logically convincing, and shows that all this mathematical machinery arises from the metric (intrinsic to the surface) and its derivatives (some as high as third order).



For this we all owe Callahan a great debt.  But unfortunately, although I believe it,  I don’t see it.  This certainly isn’t to denigrate Callahan, who has helped me through his book, and a guy who I consider a friend as I’ve drunk beer with him and his wife while  listening to Irish music in a dive bar north of Amherst.



Callahan’s proof is the way Gauss himself did it and Callahan told me that Gauss didn’t have the notational tools we have today making the theorem even more outstanding (egregious).


Well now,  onto Needham’s geometrical proof.  Disabuse yourself of the notion that it won’t involve much intellectual work on your part even though it uses the geometric intuition you were born with (the green glasses of Immanuel Kant —


Needham’s definition of curvature uses angular excess of a triangle.  Angles are measured in radians, which is the ratio of the arc subtended by the angle to the radius of the circle (not the circumference as I thought I remembered).  Since the circumference of a circle is 2*pi*radius, radian measure varies from 0 to 2*pi.   So a right angle is pi/2 radians.


Here is a triangle with angular excess.  Start with a sphere of radius R.  Go to the north pol and drop a longitude down to the equator.  It meets the equator at a right angle (pi/2).  Go back to the north pole, form an angle of pi/2 with the first longitude, and drop another longitude at that angle which meets the equator at an angle of pi/2.   The two points on the equator and the north pole form a triangle, with total internal angles of 3*(pi/2).  In plane geometry we know that the total angles of a triangle is 2 (pi/2).  (Interestingly this depends on the parallel postulate. See if you can figure out why).  So the angular excess of our triangle is pi/2.  Nothing complicated to understand (or visualize) here.


Needham defines the curvature of the triangle (and any closed area) as the ratio between the angular excess of the triangle to its area



What is the area of the triangle?  Well, the volume of a sphere is (4/3) pi * r^3, and its area is the integral (4 pi * r^2).  The area of the north hemisphere, is 2 pi *r^2, and the area of the triangle just made is 1/2 * Pi * r^2.



So the curvature of the triangle is (pi/2) / (1/2 * pi * r^2) = 1 / r^2.   More to the point, this is the curvature of a sphere of radius r.



At this point you should have a geometric intuition of just what curvature is, and how to find it.  So when you are embroiled in the algebra in higher dimensions trying to describe curvature there, you will have a mental image of what the algebra is attempting to describe, rather than just the symbols and machinations of the algebra itself (the Lilliputian strands of logic tying down the Gulliver of curvature).


The road from here to the Einstein gravitational field equations (p. 326 of Needham) and one I haven’t so far traversed,  presently is about 50 pages.Just to get to this point however,  you have been exposed to comprehensible geometrical expositions, of geodesics, holonomy,  parallel transport and vector fields, and you should have mental images of them all.Interested?  Be prepared to work, and to reorient how you think about these things if you’ve met them before.  The 3 links mentioned about will give you a glimpse of Needham’s style.  You probably should read them next.

What you breathe can get into your brain

It used to be ‘you are what you eat’, now it’s you are what you breathe according to a paper from the world capital of air pollution — China — Proc. Natl. Acad. Sci. vol. 118 e2117083119 ’22.par

They found small particles in the cerebrospinal fluid (CSF) of 8/25 people.  What is small?  Well, a hydrogen atom is 1 Angstrom in diameter, E. Coli is a cylinder 10,000 Angstroms in diameter and 20,000 Angstroms long and a red blood cell is 80,000 Angstroms. One particle they show is 500 Angstroms in diameter, a group of 3 was 2,500 Angstroms in diameter.

The paper reviews all sorts of theories of how particles might get into the brain.  One is through the nose, since nerve fibers go outside the skull to the nasal mucosa.  Others include the gut (your are what you eat after all).

So they experimented on mice having them breathe air containing particles 100 – 300 Angstroms made of titanium dioxide (TiO2) which are found in the air we breathe.  To make sure the nose wasn’t involved, they administered the particles into the trachea.  They were able to show the particles got  into the blood and then into the brain across the blood brain barrier, damaging it in the process.  Convincing electron micrographs of the brain are shown.    Whether such particles are harmful is another matter.  Just because they’re there doesn’t mean they’re causing trouble.  We have nearly 10 times as many bacteria on (and in) us as we have cells and we’re still here —

Finding the particles in human CSF is convincing evidence that they’re in the brain, but (at present) we have no electron micrographs of human brains showing that the particles are present in brain tissue itself.


Another neuropharmacologic surprise.

Our genome contains 826 different genes for G Protein Coupled Receptors (GPCRs) which are targeted by at least 475 FDA approved drugs (Nature vol. 587 p. 553 ’20 ). Yet part of the fascination of reading the current literature is the surprises it brings.

Our basic understanding was that the GPCRs sit on the surface of the cell waiting for ligands outside the cell to bind to it, which produces a conformational change on the cytoplasmic side of the cell membrane, changing the way the GPCR binds to the G protein, triggering all sorts of effects inside the cell.

As far as I recall, we never thought that different GPCRs would bind to each other in the cell membrane, even though a single cell can express ‘up to’ 100 different GPCRs [ Mol. Pharm. vol. 88 pp. 181 – 187 ’15 ].  Neurons express GPCRs and some are thought to be involved in neuropathic pain

But that’s exactly what Proc. Natl. Acad. Sci. vol. 119 e2123511119  ’22  is saying.

First a few definitions, if you’re as rusty about them as I was

A cytokine is an extracellular protein or peptide  helping cells to communicate with each other.  A chemokine is an extracellular protein which attracts cells.

Our genome has over 50 chemokines.  Most are  proteins with about 70 amino acids. The are classified by where the cysteines lie in them.  We have 23 receptors for chemokines, 18 of which are GPCRs.   Binding is promiscuous — a given chemokine binds to multiple receptors, and a given receptor binds to multiple chemokines.

Clearly the chemokines and their receptors are intimately involved in inflammation which always involves cell migration.  Neurons express chemokine receptors GPCRs and some are thought to be involved in neuropathic pain.

We also know that the nervous system is involved in immune function, particularly inflammation.  One prominent neurotransmitter is norepinephrine, and a variety of receptors bind to it.  There are 3 alpha1 norepinephrine receptors (a, b and d), all of which are GPCRs.

What is so shocking is that alpha1 GPCRs bind to chemokine receptors (forming heteromers), and that this binding is required for chemokines to have any effect on cell migration.  Even more interesting is that binding of norepinephrine to the alpha1 component of the heteromer INHIBITs cell migration.

So how many of our 826 GPCRs bind to each other, and what effects do they have?

Reading the literature is like opening presents, you find new fascinating toys to play with, some of which may actually benefit humanity