Microtubules
I got all excited when I came across a reference to Fibonacci numbers in microtubules in Roger Penrose's Shadows of the Mind. But now I'm not excited, or rather only excited about microtubules.
Microtubules are hollow tubes, around 25nm in diameter, which help form the cytoskeleton. Microtubules are a polymer of tubulin dimers, which are connected head to tail to form protofilaments which, when assembled side by side, form the cylinder, although that's probably not how they are actually made in vivo. Now the exciting thing was that one common arrangement is to make the cylinder from 13 parallel lines of protofilaments rolled together, and 13 is a Fibonacci number, right? Exciting because we understand enough about Fibonacci structure in plants to see that under the right circumstances you should be able to see it elsewhere (like Douady and Couder's oil drops), but in truth there are no other biological examples. There are some chemical ones I think.
So I read some papers: Li et al, Structure 10: 1317 (2002) and Inclan et al J Cell Science 114:413 (2000). These are why modern molcular biology is so exciting - sequence analysis, structure data, molecular modelling, functional studies, astonishing imaging coming together to offer the beginnings of a real explanation of cellular processes. It turns out that the 13 protofilament tubule has a rise of one and a half tubulin dimers per turn, so that there is a seam running along the tube where the two different monomers become neighbours. First problem for a phyllotaxis like explanation - with aperiodic boundary conditions you're lost. It also turns out that, while most microtubules have 13 protofilaments, they can have between 9 and 16. Second problem. Final problem: it looks as if the assembly mechanism is template driven in some way, which makes it hard to envisage a continuous change along a parastichy tree as an explanation.
So, lose enthusiasm for that idea. Let's go back to Penrose and have another look. He does to his credit say
However, one should not get carried away with such considerations; for example,the "9" that occurs in the bundles of microtubules in cilia and centrioles is not a Fibonacci number. Penrose, Shadows of the Mind
You might wonder what all the fuss was about 13, then. Also, Penrose gives a picture, Fig 7.8, imagining a microtubule split open and displaying a 5+8 parastichy (but not the 1 1/2 unit rise). Where did this picture come from? The only reference in the text is that 'it is apparently found (at least normally) that this pattern is made up of 5 right-handed and 8 left-handed helical arrangements' but there's no reference given for this claim.
Why I am going on about this? Well I already had a caricature view of Penroses's thesis, something like "we perform Turing uncomputable operations when we think, and we couldn't do that with deterministic, digital computer-like brains, and the way it can happen is because quantum mechanics, and quantum coherence in particular, are not averaged out up at the biophysical level of the brain". Always thought it seemed like nonsense, and now I have the book on my desk it's a chance to find out what this manifestly bright man knows that I don't that makes him say such apparently silly things.
Unfortunately I flunked my chance. Philosophy makes my eyes glaze, no matter how I try, and a benefit of my long mathematical education was to learn skip all formulae, so that's most of the book unread. It's towards the end of the book where Penrose tries to figure out where the QM is getting to act at this scale and points the finger at those microtubules. But that's an unconvincing discussion when it talks about things I know about, which is always pretty discouraging. So maybe Penrose will remain unread, and the mystery of the apparently silly will remain unsolved, at least until I'm in my philopause.
Dear Jonathan,
I have read your remarks on microtubules. I work on this topic for more than a year and a half, and I hope I will find in the end detailed description of what's going on in the brain. You could find a lot of neurobiology, biophysics and hypothesizing if you look at my papers [links at my web], but now I want to remark something on the microtubule lattice:
1. I don't know from where the Fibonacci connection comes from [I haven't seen the Penrose's picture you speak about of 8+5 helices] but I suppose that I could be result from considering 13A lattice [!? I don't know whether I am right, just suppose]. 13A lattice is fully symetrical, so I cannot say whether it is right or left handed, because of this symmetry].
2. Indeed brain microtubules have 13B lattice [Song & Mandelkow, 1993; Kikkawa et al., 1994] and have a seam! I still don't understand the seam role, but suppose it will be important [the biology is very optimized, so there is no place for useless things in evolution!!!]. Hameroff in the Orch OR model predicts 13A lattice, and he still does so [I had presentation at Q-mind 2003 conference in Tucson, Arizona, march 15-19] and there is one curious list of 20 testable predictions [generated in 1996]. The references I have provided [considered crucial in the current molecular biology of microtubules] are from 1993, 1994 so ..... no comment
Indeed my paper dealing with the "electric and magnetic fields in neurons and their impact upon the cytoskeletal microtubules"
http://cogprints.ecs.soton.ac.uk/archive/00003190/
is good review of microtubule biology and biophysics.
Regards,
D.G.
Posted by: Danko Georgiev, M.D. at October 16, 2003 06:19 PMErratum:
"I suppose that I could be result from considering.." -> "I suppose that it could be result from considering.."
I suppose the Fibonacci connection has something to do with 13A lattice.
D.G.
Posted by: D.G. at October 16, 2003 06:27 PMOn microtubule dynamic instability and GTP coherence pumping
Hameroff, Hagan, Tuszynski, Mershin et al. suppose that stable microtubule could use energy from GTP to "pump" coherence. This is biological non-sense because beta-tubulins in the microtubule have bound GDP that cannot be exchanged, alpha tubulins do not hydrolize its bound GTP, and the beta-tubulin GTP-cap if hydrolyzed leads to rapid microtubule depolymerization !!!!!
Each nucleotide in the tubulin protofilament is at an alpha-beta interface. In order 2 dimers to bind the upper beta-tubulin bound GTP hydrolises to GDP. The beta-tubulin in the plus end is with GTP cap that prevent microtubule from depolymerization. The inability of GTP to dissociate from the alpha-subunit is consistent with occlusion by a loop from the beta subunit. A similar occlusion would account for the inability of beta-tubulin within a protofilament to exchange bound GDP for GTP. So, there is no way stable microtubule to pump energy from GTP!
See the following link for nice illustrations and detailed explanation of microtubule biology.
http://www.dentistry.leeds.ac.uk/biochem/MBWeb/mb2/part1/microtub.htm
Regards,
D.G.
ON MICROTUBULE LATTICE
I want to add some "fresh" info to my comments above. ALL MICROTUBULES HAVE 3-START HELIX as can be found in virtually all recent papers on that topic [see for example Erickson, H.P. and D. Stoffler. (1996). Protofilaments and rings, two conformations of the tubulin family conserved from bacterial FtsZ to alpha/beta and gamma tubulin. J. Cell Biol. 135:5-8.]. I have seen a reference in paper published in 1980's to and older source claiming that MTs with A lattice could have 5-start and 8-start helices, but all this is ruled out. Current experimental data favors 3-start helix and B-type lattice - confirmed both in vivo and in vitro by Kikkawa et al. (1994).
What about the number of protofilaments it is shown that it depends on the beta-tubulin isotype. betaII and betaIII are most abundant in brain and indeed may favor the 13 PF microtubule formation. Also centrioles in vivo favor 13 MT nucleation! So at least 2 independent factors act to increase the number of 13 PF MTs in vivo.
On the issue of GTP-pumped MT coherence you can read the following e-paper:
Georgiev, D. (2003). Revisiting the microtubule based quantum models of mind: tubulin bound GTP cannot pump microtubule coherence or provide energy for alpha beta computation in stable microtubules. CogPrints ID 00003251.
http://cogprints.ecs.soton.ac.uk/archive/00003251/
I hope this info is interesting for those studying microtubule role in consciousness.
D.G.
Posted by: Danko Georgiev at November 26, 2003 02:17 PM