Anslyn pp. 355 – 420 Part I (pp. 355 – 366)


Chapter 7:  Truly a monster chapter, with lots of stuff in it that didn’t exist  the early 60’s.  Even though Part 1 concerns only the first 11 pages it contains three questions for the cognoscenti, some calculations to be checked  and plenty of material.   Still waiting to see how potential energy surfaces are actually calculated as opposed to posited — said to be coming up in chapter 14.  

p. 356 — First line of 7.1 “measurments”

p. 356 — What would be an example of a reaction NOT involving an energy barrier needing to be surmounted?  I can’t really think of one.  Supercooled water perhaps?  But that’s a phase change not a reactions.  

p. 357 — It isn’t a far jump mathematically from saddle point (well explained in the book) to homoclinic points and orbits, where chaos was first glimpsed by Poincare. 

p. 357 — Where does 3N – 6 come from? I suppose it arises from the fact that you can fix the position and momentum of one of the atoms (removing 6 degrees of freedom), with the rest dancing around this fixed point.   Also,  degree of freedom should be defined (it is next page), so a note saying so should be placed in the sentence on this page where the term first appears.

p. 358 — “Internal energy can be both potential and kinetic” — true enough for an oscillating spring, but somehow putting something high off the ground, while increasing its potential energy, doesn’t seem very internal to the object itself.  

p. 358 — Mnemonic — exerGonic — Gibbs free energy of product lower than starting material.  exotHermic — Heat given off — enthalpy of product lower than starting material.

p. 359 — “The minimum energy pathway or the  pathway we depict as the weighted average of all the pathways, is called the reaction coordinate.”  OK–but how do you weight the pathways? 

p. 360 — Nice to see Patch clamping mentioned.  It’s taught neuroscientists and clinicians a tremendous amount — particularly how anticonvulsants (drugs against epilepsy) actually work.  Unfortunately Neher is a chemical ignoramus, talking about nanoDomains near the membrane where concentrations of a particular ion are in the microMolar region, but the domain is so small that there are only 8 calcium ions in this domain.

      Calculation for the cognoscenti to criticize.      [ Proc. Natl. Acad. Sci. vol. 100 p. 7341 ’03 ] Entry of calcium through a calcium channel produces a steep concentration gradient (since the overall intracellular calcium concentration is so low).  Levels of over 100 microMolar are obtained (1000 times the normal intracellular concentration) are reached within these nanoDomains — within 500 Angstroms of the channel pore. 

       My calculation: How many calcium ions is this?  1 milliLiter is 1 cm. on a side.  At a 1 molar concentration 1 milliLiter contains 6 x 10^20 ions.  At a 1 milliMolar concentration it contains 6 x 10^17) ions, at a 100 microMolar concentration a milliLiter contains 6 x 10^16 ions.  How many cubes 50 nanoMeters on a side are there in a cube 10 milliMeters on a side? There are twenty 50 nanoMeter lengths in 1 microMeter, 20,000 lengths in a millimeter, and 200,000 of them in 10 milliMeters so there are 8 x 10^15 50 nanoMeter sided cubes in 1 milliLiter which contains 6 x 10^16 ions at a concentration of 100 microMolar.  So the 500 Angstrom (50 nanoMeter) sided  cube with the calcium channel in the middle of one face contains less than 8 calcium ions.

        What is the volume of a hemisphere of radius 50 nanometers?   It is .5 * (4/3) * pi * (50) * 50 * 50 == 261666 cubic nanometers.  There are 10(7) nanometers in a centimeter, so there are 10(21) cubic nanoMeters in a cubic centimer (or milliliter).   Thus there are 10(21)/2.6 x 10(5) such hemispheres in a cc.  There are 4 x 10(15) of them.   Dividing this into the 6 x 10(16) ions in a cc. at 100 microMolar gives 15 ions in this hemisphere.   Is concentration a meaningful concept in such a small area? (bold !)  500 Angstroms is a long distance where proteins are concerned.  

       Question for the cognoscenti #1:  I look forward to understanding ion velocities and mean free path and diffusion constants by the end of this year (2011).  Perhaps 8 – 16 calcium ions in an area this small isn’t meaningless.  Think of a 3 lane freeway carrying heavy traffic.  On any given 100 foot stretch, there probably aren’t more than 8 – 16 cars (but they are all going one way). So if the ions are zinging around fast enough, the average number of the ions in question contained in this volume at any instant might be quite stable, even if the number is only 8 – 16 ions. If any PChem maven has anything to say on this subject, please enlighten me.

p. 362 — “Rigorous methods for creating reaction coordinate diagrams also exist.  High level computational methods such as we present in Chapter 14 can be used”  Can’t wait to see if they are actually able to calculate the details of a potential energy surface.  But Ch. 14 is a long 445 pages away

       p. 363 — Interesting to know that the activated complex has a lifetime no longer than a vibration (period of a vibration).  This is either new, or not emphasized when I studied transition state theory 50 years ago.   On p. 366 this is elaborated.

       Question for the cognoscenti #2The notion of eigenvector as a force is certainly new to me. Is it in fact correct? Quantum states are eigenvectors, but are they forces?  Seems quite strange !  Any cognoscenti out there?  It’s clear that force is the negative first derivative of energy.  Where do matrices come in?  Eigenvalues as a second derivative of energy was never mentioned in the QM course I took.  Some elaboration could be used here.  I can see why the second derivative must be negative at the top of a hill, that’s just calculus 101 after all, but why do the authors call this an eigenvalue?    Hopefully all will be explained in the far distant Chapter 14 (440+ pages away. 

       p. 366 — The terms are enshrined by usage, but cognitively it would be better to have better differentiated symbols for k, K, k†, K†, K†’ and kappa.

       Henry Eyring !  He came to speak to us as undergraduates in the late 50’s.  It was my first exposure to a Western type guy.  Everything was Aw shucks, I jes’ happened to back into this (amazing) result.  In the West, the one unforgivable sin is bragging, followed by telling people what you are ‘really’ like.  Out there, people are so thin on the ground, that they’ll find out what you’re like without you having to tell them.  Of course I went out and bought Eyring Walter and Kimball, even though I was far from having the background to understand it.

      “We invoke statistical mechanics because a transition state does not have a Boltzmann distribution of states (see the next section), because its lifetime is so fleeting.”  If the Boltzmann distribution isn’t statistical mechanics what is?  Question for the cognoscenti #3:  Does this sentence really make sense? 

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  • MJ  On July 26, 2011 at 7:25 pm

    Regarding the comment about eigenvectors – it sounds as if the authors are describing some sort of classical mechanics-inspired model for understanding oscillations of coupled bodies (in relation to a discussion of vibrational spectroscopy, I presume), where the eigenvalue problem does legitimately crop up. In this sort of scenario, the eigenvalues correspond to the oscillation frequency while the eigenvectors describe the motion of the normal modes.

    Insofar as where the 3N-6 comes from….again, presuming the authors are talking about vibrational degrees of freedom for a polyatomic molecule (N> 2). The general notion via classical mechanics is that a particle has 3 degrees of freedom in three-dimensional space (translation in x, y, & z). For N particles, there are 3N degrees of freedom. For a molecule, three degrees of freedom are used for translation in three dimensions and another three for rotation. This leaves the remainder for vibrational degrees of freedom.

    While I know who Erwin Neher is (patch clamping with Sakmann), I am not familiar with his (?) nanodomain idea. Is there any evidence for it (fluorescent dyes lighting up in some sort of assay, perhaps? Other measurements?) or just postulated to exist since it would explain observed phenomena?

    I’m going to figure that one can fit a single calmodulin protein into a sphere of 50 nm diameter. (I have this vague recollection that its radius of gyration is somewhere between 20 to 25 nm.) Calmodulin binds four calcium ions. The dissociation constant is somewhere on the order of nanoMolar, as I recall. It would seem possible that a calmodulin protein passing through a volume as defined with 8 to 15 calcium ions would be able to fully bind at all four sites and thereby effect the conformational change needed for it to then bind other proteins in its role as a calcium sensor and signal transduction component.

    Statistical mechanics does deal with the issue of rare events, including transition states, and there are computational ways to explore them via simulations by introducing a weighting bias so the simulation explores parameter space that would be otherwise undersampled. But without having read A&D, can’t say for sure.

    (FYI – If I get more time I will recheck the calculations, but I presumed they weren’t too far off. Mostly as I have my own to do tonight for a project – speaking of which, back to it.)

  • luysii  On July 27, 2011 at 8:00 am

    Thanks a lot. There’s plenty of room in a 500 Angstrom (50 nanoMeter) sphere for most proteins. The bacterial ribosome with its 54 proteins is only 200 Angstroms wide.

    The nanoDomain idea is all over the neuroscience literature, and widely assumed to exist, but I’m far from sure that talking about concentration in such a small volume is meaningful. Clearly. when volume gets small enough (say a 10 Angstrom sphere) concentration becomes meaningless.

  • MJ  On July 27, 2011 at 9:05 pm

    I figured I’d be conservative with my size estimate of calmodulin (and, for what it’s worth, its radius of gyration is on the order of ~ 20 nm from a quick scan of PubMed abstracts).

    It has been years since I last thought about neuronal cell biology of any sort (and even then, it was part of a course), but ion channels are known to cluster to a degree as I understand it. Certainly, transmembrane protein/receptor clustering has been observed in other systems to varying extents, so that in and of itself does not seem outrageous. It would suggest to me, though, that a 50 nm diameter nanoDomain per single ion channel is not quite the same as a 50 nm nanoDomain in the vicinity of a cluster of ion channels. This is a factor that should be accounted for, at least I’d hope so.

    I think your observation regarding concentration at small length/volume scales certainly is an interesting one, although perhaps there one might reframe concentration as an occupancy factor (e.g., “will host an average of two ions per unit time”). Something to think about!

  • MJ  On August 14, 2011 at 5:08 pm

    I feel kind of bad it took this long to remember, but I suppose better later rather than never –

    With regard to forces in quantum mechanics, there is a fairly classic finding known as the Hellmann-Feynman theorem. Essentially, what is states is that the force on a nucleus in a collection of nuclei and electrons is of a classical electrostatic nature, once you’ve solved for the electron charge distribution via quantum mechanics. Of course, as originally formulated, it’s done within the Born-Oppenheimer approximation, so the nuclei themselves are fixed as originally demonstrated, but researchers have made efforts to extend it (the status of which I am not familiar with off the top of my head).

    Anyway, something else to dig into whenever you get a chance.

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