A Cubic Millimeter of Your Brain

Are there more connections in a cubic millimeter of your brain than there are stars in the Milky Way? We are going to answer that question in a moment, but first take a look at this image of hippocampal neurons in a mouse's brain. It is an actual color image from a transgenic mouse in which fluorescent protein variations are expressed quasi-randomly in different neurons. This kind of image is known as a brainbow and is aesthetically awesome further it may be one way to empirically examine a cubic millimeter of the brain (neuron tomography).

In reality mapping even an entire cubic millimeter of the brain is an extremely daunting task, but we can still answer my original question. First, I know that there are different kinds of neurons that vary in size and that some neurons can have a soma (the big part that has the nucleus from which the dendrites extend) spanning a millimeter in size. Thus if you picked a random cubic millimeter of brain you could run right into the heart of a neuron and you would find very few connections. Given this fact, we can very easily answer this question with a resounding no, however, this seems like an unsatisfactory trite approach. So I looked up some numbers on how many neurons are in the brain, how many connections are in the brain, and how many stars are in the Milky Way. Lets answer the question using the 'average' number of connections per cubic millimeter.

How many neurons and connections there are in the brain? This is kind of a tricky question and I am not a nuerobiologist so I have gone to several resources for the answer. Professor of Computational Neuroscience at MIT Sebastung Seung says in a TED talk

your brain contains 100 billion neurons and 10,000 times as many connections

Professor of Molecular Cellular Physiology at Stanford Stephen Smith says in a press release on brain imaging that

In a human, there are more than 125 trillion synapses just in the cerebral cortex alone

René Marois from the Center for Integrative and Cognitive Neurosciences at Vanderbilt Vision Research Center states in a recent paper ^[1]

The human brain is heralded for its staggering complexity and processing capacity: its hundred billion neurons and several hundred trillion synaptic connections can process and exchange prodigious amounts of information over a distributed neural network in the matter of milliseconds.

I have enough expert sources now to confidently say these experiments agree that the human brain has some 100 billion neurons (10¹¹). The number of connections seems less precise, but it is at least several 100 trillion connections (10¹⁴) as judged by Marios and Smith and as much as 10¹⁵ as judged by Seung.

The number of connections in the brain is tricky to define. We may define a synaptic connection as each place the neuron touches another neuron and a synapse is present. It doesn't seem to make sense to simply count incidental contact. Further, there is the question of whether we should count redundant contacts between neurons. We can obtain an upper bound on the number of connections in the brain by considering the case in which every neuron is connected to every other neuron. Coincidentally the operation of connecting every node in a network with every other node is a process I am familiar with from cross correlating radio signals. Anyways, the equation we are looking for is N(N-1)/2 where N is the number of nodes in the network. Thus, for our N=10¹¹ neurons the maximum number of non-redundant connections is about 10²². This maximum bound is huge! But how huge is it really? Hilariously, while searching for an answer to my original question I found a message board pondering the grand statement

There are more connections in the brain than atoms in the Universe.

A really clever person pointed out that

Theoretically, if we took all the atoms in the universe; wouldn't that include the atoms within the brain?

People have this feeling that the number of connections between items can be much larger than the number of actual items in the collection and while this intuition is true the idea that there are more connections in the brain than there are atoms in the universe is absurd. Lets put it in perspective that a few grams of any substance, like water, is measured units of moles. A mole is standard unit of measurement corresponding to the absolute 6.02 x 10²³. Thus even a drop of water contains more atoms than there are connections in the brain.

Now we need to know how many neurons and connections are in an average cubic millimeter of the brain. How big is the brain? John S. Allen of the Department of Neurology at University of Iowa stated in a recent paper that^[2]

The mean total brain volumes found here (1,273.6 cc for men, and 1,131.1 cc for women) are very comparable to the results from other high-resolution MRI-volumetric studies.

We can take the volume of the brain as 1000cc as a low estimate (which will only over estimate the density of connections).

The final thing we need to know to answer the question at hand is the number of stars in the Milky Way. Like every other number we have been working with it is rather uncertain. Even if we define a star as only those spheres of gas which are large enough to fuse hydrogen at some point in their lifetime we don't know the answer because we can't see the multitudes of dim stars. There are probably at least 500 billion star like objects in the Milky Way. Lets take 100 billion as the number to be conservative.

Finally, lets bring all the numbers together. One cubic millimeter is 1/1000 of a cubic centimeter and 1/1000000 (10^-6) of the entire volume of the brain. We can scale the total number of connections in the brain (using the high estimate of 10¹⁵ connections in the brain) then we find that there are 10⁹ connections in a cubic millimeter of the brain. The 10⁹ connections in a cubic millimeter of the brain is two orders of magnitude smaller than a low estimate of the number of stars in the Milky Way. No, on average there are not more connections in a cubic millimeter of your brain than there are stars in the Milky Way. 

My first response to this question was bullshit! This question (or rather statement) is made by David Eagleman here at a TEDx talk and here on the Colbert Report. Colbert also called out Eagleman when he dropped this factoid, but it didn't stop the interview. I have also contacted some actual neuroscientists to see what they thought of this statement and they agree with me that it is not true. Maybe there is special part of the brain particularly more dense in connections than the brain on average, but that would be misleading like saying the density of the Milky Way is that of water because, you know, certain parts of the Milky Way are water. The better statement would be to say that there are are more connections in the brain than there are stars in the Milky Way. As Colbert would say, I am putting you on notice Eagleman.

While we are on the subject I want to mention my favorite talk about the brain which mixes just the right amount of wonder and fact. It is the TED talk I mentioned earlier by Sebastian Seung on what he calls the connectome - the network of connections in your brain between neurons which physically dictates how you think. In the video he discusses another volume tomography technique in the brain using a cube of mouse brain tissue just 6 microns on a side. It is another great visualization for what is actually in a cubic millimeter of your brain.

References

[1] Marois, R., & Ivanoff, J. (2005). Capacity limits of information processing in the brain Trends in Cognitive Sciences, 9 (6), 296-305 DOI: 10.1016/j.tics.2005.04.010

[2] Allen, J., Damasio, H., & Grabowski, T. (2002). Normal neuroanatomical variation in the human brain: An MRI-volumetric study American Journal of Physical Anthropology, 118 (4), 341-358 DOI: 10.1002/ajpa.10092

The Universe and Life is asymmetric: Chirality

The shadow of symmetry haunts physics. Symmetry is invoked to understand nature concisely, but broken symmetry is invoked to understand nature completely. Physics is filled with examples of shattered symmetries: there is more matter than antimatter, neutrinos only come in the left handed spin flavor, and quantum processes break symmetries constantly, but nature also violates symmetry in chemistry and biology in a very clever manner. Chemistry and biology are subjects I do no normally touch upon, but I am intrigued by the curious circumstance of life on Earth: many molecules are not superimposable upon their mirror images, a property called chirality, and life on Earth has a preference for these chiral mirror configurations. Physics and life is inherently asymmetric.

That something is not identical to its mirror image is a property known as chirality. Hands (etymologically the word chirality is derived from the Greek word for hand), spiral galaxies, and the DNA helix are all examples of chiral objects. In particle physics chirality is a more abstract notion defined by transformations of a particle with respect to a left right of left handed representation in the Poincaré group. In chemistry chirality is well described by analogy to your hands wherein left and right hands cannot be superposed on each other even though the fingers are the same and match up.

This article is an exploration of chirality in biochemistry. I want to ask what makes life chiral, why is life chiral, and how did life become chiral. In order to supplement my limited knowledge of the subject I interviewed a world expert and author of over twenty papers on the subject, Robert Compton, who I must give a deep thanks to for being willing to answer my silly questions.

It is important to accept that the concept of symmetry is tinted by the human notion of harmonious or aesthetically pleasing forms, but the strict mathematical interpretation of symmetry relies upon metrics of geometry. To this end many seemingly symmetric forms in the living natural world are actually examples of broken symmetries: spiral tree trunks, the human form, and sea shells (which generally only coil in one particular direction according to species). The remarkable thing is that this macro asymmetry can be traced back to a micro asymmetry in the chemistry of life. The arrangement of atoms in a molecule defines the function of that molecule, but even molecules with the same chemical configuration can behave differently as in the case of chiral molecules which are like mirror images of the same molecule that come in 'left' and 'right' handed forms. The great asymmetry of life is that all living organisms on Earth almost exclusively utilize the left handed (or levorotatory) configuration for amino acids and the right handed (or dextrorotatory) configuration for sugars belonging to DNA or RNA.

Perhaps it is trivial or obvious that life is chiral when looking at the nautilus, but this obvious chirality is a macroscopic feature which belies the fine arrangement of atoms which defines the chirality of biomolecules.

Different structural forms of compounds with the same molecular formula are known as isomers to chemists. A stereoisomer is an isomeric molecule which has the identical constitution and sequence of bonded atoms, but has a different three-dimensional geometry in space. An enantiomer is one specific steroisomer of the two possible mirror images that are non-superposable. The dominance of the left handed chiral enantiomer in biology is a massive blow to the idea that nature is perfectly symmetric and is an unsolved mystery as to why nature is this way.

Many molecules are chiral, however because molecules are constantly vibrating the instantaneous structure of a molecule may lack the exact structure or symmetry seen in an ideal model.  Regardless, enantiomers have identical chemical properties except when they react with other molecules which are also enantiomeric in which case chiral forces yield a difference in behavior. Further, and perhaps more important for biology, particularly astrobiology, is that enantiomers have identical physical properties except with respect to the way they interact with plane-polarized or circularly polarized light or other chiral compounds. A pure enantiomer compound will rotate the plane of a monochromatic plane-polarized light by a certain angle in one direction, say clockwise, while the other enantiomer form of the compound will rotate the light by an equal amount in the opposite direction. Things that rotate light are said to be optically active.  Measurements with a polarimeter allow chemists to determine if a compound is chiral or not. Polarization of light by organic compounds was discovered in 1815 by the French physicist and chemist Jean-Baptiste Biot. He found that organically produced chemical solutions consistently rotated plane polarized light, but laboratory synthesized chemicals did not reproduce the rotation. Beyond conjectures he had no explanation for the phenomena.Years later Louis Pasteur preformed a similar experiment with tartaric acid produced from grapes and tartaric acid synthesized in the lab. Pasteur went further and somehow used tweezers and a microscope (I do not conceive to understand how) to separate the tartaric acid crystals which he produced in the laboratory into piles of left and right handed molecules. He found that polarized light was rotated by the left handed molecules that he had selected in the same way the polarized light was rotated by the organic tartaric acid. He concluded that chiral molecules are responsible for the rotation of polarized light.

So chiral molecules rotate light, but actually so does an individual achiral molecule! In an ensemble of achiral molecules each individual molecule may rotate the plane of the polarization, but the net rotation averaged over the ensemble will result in zero rotation. A mixture of two enantiomers in a 1:1 ratio (which is what you get when you create chemicals in the lab) is optically inactive because the rotation results in a zero net polarization rotation. When a reagent or catalyst is optically active the chiral product will also be optically active, or in the presence of chiral forces such as circularly polarized light this may also induce optical activity via enantimoeric excess in the products as well. Generally you can get optically active compounds in two ways 1) The reagent in already optically active. 2) The reaction of achiral but optically inactive precursors in a chiral optically active environment occurs. It takes an optically active molecule or chiral force to produce a product that is optically active.

Most chemical reactions are not enantiomerically selective so that the initial reason for a completely homochiral biology on Earth remains a mystery just as when Biot and Pasteur discovered chirality through optical activity. Of course chirality is simply geometric in nature and thus this geometric asymmetry is what makes life chiral. Any molecule that contains a tetrahedral carbon or other central atom bonded to four different atoms or constituents will exist in enantiomeric forms; given that all biological molecules are at least this complicated, then (almost?) all biological molecules exist in enantiomeric forms. It may be that the chirality of biomolcules is simply a consequence of the emergent complexity of basic physics. The conditions necessary for a solution initially containing near equal number of chiral forms to evolve towards pure chirality has been explored (see Frederick Frank 1953) and is plausible. A tiny initial imbalance has spiraled out of control and now each successive generation of biomolecules on Earth is produced by the previous generation of chiral reagents, thus this is the why life is chiral. None of this explains how life is chiral, but a common answer is that because life can be chiral it is chiral.

From this persepctive this topic is not so interesting, honestly. I have come to the conclusion that chriality is as it must be given that each generation of life is spawned from the previous generation under conditions which do have enantiomeric selection forces present. The question is why was left handed chirality chosen for life on our Earth?

Now actually, the chirality of biomolecules is not just a philosophical diversion; it a serious issue of biochemistry in balance. There are over 530 synthetic chiral drugs worldwide today. It is technically and economically prohibitive to make enantiomerically pure drugs in all cases. This results in drugs that may have strange, null, or fatal interactions with human subjects. In some cases the difference between two enantiomeric forms is simple, as in the case of the olfactory exciting chemical carvone; in one configuration it smells like spearmint and the other configuration like caraway seeds. So, "Perhaps," as Alice said to her cat in Lewis Carroll's Though the Looking Glass, "looking-glass milk isn't good to drink".

I digress. Before we hear what Dr. Compton has to say on this matter lets look at what fundamental physics would bias the chirality of life on Earth and how homochirality was selected on Earth.  There are three distinct mechanisms that seem plausible.

• The weak force. Of the fundamental forces, nuclear, electroweak and gravitational, only the weak force can distinguish between left and right parity particles. The weak force it turns out does not conserve parity (although it does conserve CPT symmetry) during some interactions such as the radioactive beta decay corresponding to the emission of an electron with intrinsic spin 1/2 hbar. Also, the weak force induces a parity-violating energy difference, PVED, between molecules or the interactions of left-handed electrons emitted during beta decay with molecules. So the weak force could preselect a handedness in nature through either beta decay or PVED. The idea is that if one chiral configuration is a lower energy state then nature will prefer that configuration (in fact the exact scaling from thermodynamics is that the reaction rate for the oppositely chiral molecules is proportional to the canonical partition function in physics going as e^-PVED/kT where e is the Euler's number, k is Boltzmann's constant, T is the temperature in Kelvin). The difference in energy from the PVED can be theoretically calculated from a Hamiltonian operator that is scaled by the Fermi electroweak coupling constant. The energy difference between chiral configurations is predicted to be small, around 10^-14 Joules per mol which means that not only is this energy difference believed to be minimally important to early life's synthesis, but it is also out of reach of current experimental techniques. However, it turns out that the PVED predicts that the left handed chiral states would be of lower energy, just as they are found dominantly in life on Earth. The weakforce influences chemical reactions because during beta decay, spin polarized electrons produce a an abundance of left-circularly polarized gamma-rays which, if present during the synthesis of biomolecules would tend to create an enantiomeric excess of left handed molecules. However, laboratory experiments have not shown conclusively that this effect is strong enough to matter either. There is some debate as to the exact nature of the PVED which will depend on further experimental measurements. The weakforce seems to preselect a hand in nature, but it is a feeble force.
• Polarization. Optically active organic molecules being synthesized in the presence of polarized light will be chiral. Sunlight is slightly polarized just before sunrise and after sunset. This averages out to zero, but chemical activity that dominates in the morning/evening or occurs in the presence of shadows could feel a net polarization effect. Also plausible are astronomical sources of polarized light outside the solar system. Supernovae have been known to emit circularly polarized light as have star forming or nebulae regions. These sources while weak would create conditions necessary for the synthesis of homochiral biomolecules.
• Vorticity. A chemical solution being stirred or agitated results in the synthesis of homochiral biomolecules, however, the handedness of the chirality is random. Certaintly there were chaotic turbulent conditions on the early Earth.

There is no conclusion to be drawn as to how life became homochiral. I found more questions than answers. So I asked Robert Compton for some answers.

1) Enantiomers are allotropes then?

Interesting question but allotropes are "two or more existing forms of an element" such as graphite, diamond and fullerenes. There are a number of fullerenes which are chiral, e.g. C84. The C84 molecule has an R and and S enantiomer as well as the meso- form (the meso is a combination of the R- and S- form). This is illustrated below. One goes from the left to the right by rotating two of the carbons twice and reconnecting the dots going through the achiral meso-form.

2) Life on earth is overwhelmingly 'left-handed' homochiral. Is this merely a coincidence of history? Do you think a fundamental effect like PVED or an environmental effect like polarized light from a local supernova or nebula was responsible for primordial chirality on Earth?

This is the 64 million dollar question. Did life end up as “left-handed” (or better L-amino acids) due to some fundamental bias or was it pure chance and there may be a planet out there that is made up of D-amino acids. I have discussed the possibilites in my review article on “Chirality of Biomolecules” but to summarize it may be due to 1) the influence of circularly-polarized light on biomolecules giving rise to life over long periods of time, or 2)the chiral weak force making one enantiomer be lower in energy than the other ( this is certainty true but the effect is really, really small) or 3) chiral beta – rays interacting with matter to produce handed biomolecules. At the present time I favor the interaction of circularly polarized light on molecules in interstellar space (maybe chiral microwaves). Meteors then bring these molecules to Earth to begin life forms.

I really feel that PVED is too small. My bet is on circularly polarized light from a local supernova or nebula.

3) Subsequent proteins formed in the presence of a particular enantiomer protein will be preferentially homochial to that primordial protein. Assuming 'the spark of life' only occurred once with a particular enantiomer protein present, would you agree that it isn't surprising that life is homochiral? Is this assumption realistic?

Homochirality is understandable. Everyone agrees that life molecules are preordained to be homochiral. But there is no reason that they will be of only one handedness, of a specific chirality. That’s the rub. It has always been a big surprise to me that ALL life is made of L-amino acids ( and chiral sugars in the backbone of the DNA).

4) Imagine we went to planet X and found the same flora and fauna as Earth, but planet X's organisms were dominated by the opposite chiral form as that found on Earth. Could we mate with or consume nutrients from creatures on this planet?

No. I don’t think I would like to try this.

5) It is obvious in biology that chirality is a function of emergent complexity in the system. In physics chirality is often considered to be a function of the deepest laws of nature. Could chirality in physics also be a function of emergent complexity from unseen laws of nature?

I believe the current view is that the Universe began as both matter and anti-matter but bifurcated into matter due to some fundamental reason. Thus the Weak force gives rise to left spinning beta particles. There are a lot of scientists trying to understand how the Universe ended up as matter. This may be the “emergent complexity from unseen laws of nature” that you asked about.

6) In biology and chemistry symmetry breaking on macro scales is understood through chiral selection processes which act on micro scales. That symmetry breaking on micro scales may be traced back to an asymmetry in fundamental physics. Is the universe fundamentally asymmetric?

 I think this is ture. You may have seen the book “The Left Hand of Creation” by Barrow and Silk or “The New Ambidextrous Univesre” by Martin Gardner. They touch on this subject. Gardner is fun reading and can be (should be) read by high school students.

Robert N. Compton, Richard M. Pagni, & Volume 48, 2002, Pages 219-261 (2002). The chirality of biomolecules Advances In Atomic, Molecular, and Optical Physics, 48, 219-261