Mike, I know this is a bit obtuse -- we're still working on the manuscript
for this... There's a PowerPoint presentation I gave on the subject at last
year's ACS meeting. See
http://chemapps.stolaf.edu/jmol/presentations/acs2009
On Mon, Sep 20, 2010 at 11:19 PM, Michael Evans <[email protected]>wrote:
> Bob + other Lords of Jmol—I'm looking for some clarification of the "plot
> ramachandran r" specification from the Interactive Script Docs; particularly
> the definition of theta. It makes sense to me that theta is approximately
> the sum of delta phi and delta psi. I get lost with two points:
>
> - How theta relates to dq[i]/dq[i-1] (and what exactly is represented
> by this quantity...does it have something to do with the rotation of
> residues [i] and [i-1] w.r.t. one another? I'm thinking about the
> definition
> of quaternion division described in your wonderful post about quaternions,
> which I happened to come across today, as describing the relationship
> between two rotational states)
>
>
Each residue is assigned a frame -- an xyz axis set -- based on one or
another definition. This is the "C" or "P" or "N" in "set quaternionFrame"
...
"C" -- alpha carbon
"P" -- peptide plane
"N" -- peptide nitrogen, specifically for solid state NMR
The orientation of these frames relative to the reference frame (the xyz
axes of the model) can be referred to by a quaternion -- four numbers.
dq[i] is the quaternion difference (often written as division,not
subtraction) between the quaternion at residue i and the quaternion at
residue i - 1.
>
> - The distinction between "C" and "P" straightness, and how the "P"
> straightness approximation for theta was derived
>
>
If you load a protein and then issue
wireframe only
set quaternionframe "C"
draw quaternion
I think you will see what we are talking about. The xyz axes for each
residue are shown, and a yellow arrow with a number shows the axis and angle
required to rotate the reference frame (axes molecular) to this orientation.
I'll have to dig up that proof. It originated here at St. Olaf. We
discovered the relationship during the summer of 2008 (that's what my
research over the past two years has been about).
> Related to point 1, I can't see how this is a second derivative. It looks
> like a derivative of the quaternion of residue [i] w.r.t. residue [i-1].
> What am I missing here? I'm trying to dumb down a description of the 3-D
> Rama plot as much as possible, to add it to a tooltip in an interface. With
> just a tooltip I don't have enough space to launch into a full-blown
> mathematical derivation, but if that's what it's going to take to really get
> at theta, I'll probably just leave it out :-)
>
>
If you call q[i-1] and q[i] the two quaternions for residues i-1 and i, then
dq[i] is q[i]/q[i-1], another quaternion. dq[i] indicates the axis and angle
in the standard reference plane that would rotate residue i-1 into the
orientation of residue i. That's the "first derivative." It also can be
thought of as the "local helical axis".
If you load a protein and then issue
draw quaternion difference
you will see that the vectors produced pretty well define the axes of
helices and sheet strands. In fact, I might suggest that they be used to
DEFINE those axes, because it's a relatively simple matter to determine the
"average" quaternion and to specify the root mean square deviation of those.
--Straightness--
Now you have to take two of those differences and take THEIR difference --
ddq[i] = dq[i+1] / dq[i] -- to get the "second" derivative. That's also a
quaternion. It is the first term of that quaternion (the quaternion dot
product of dq[i+1] and dq[i]) that we use in the straightness calculation:
S[i] = 1 - arcCos(abs(dq[i+1] dot dq[i])) / (pi/2)
This is in some ways analogous to what had been proposed earlier by others
[1] as a definition of "straightness", but they used the dot product of
regular three-dimensional vectors instead of quaternions, which makes
perfect sense, except that it doesn't allow the nice relationship to
Ramachandran angles. The definition of straightness we are using - defined
in terms of quaternions instead of just three-dimensional vectors - was the
(dare I say brilliant?) suggestion of Dan Kohler (St. Olaf '09, now at U.
Wisc.).
Bob
[1] Kneller GR, Calligari P. Efficient characterization of protein secondary
structure in terms of
screw motions. Acta Crystallogr, Sect. D: Biol Crystallogr 2006; 62:
302-311.
> Cheers and thanks as always, Mike
>
> --
> Mike Evans
> Organic Chemistry Graduate Student
> Moore Group
> University of Illinois, Urbana-Champaign
>
>
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--
Robert M. Hanson
Professor of Chemistry
St. Olaf College
1520 St. Olaf Ave.
Northfield, MN 55057
http://www.stolaf.edu/people/hansonr
phone: 507-786-3107
If nature does not answer first what we want,
it is better to take what answer we get.
-- Josiah Willard Gibbs, Lecture XXX, Monday, February 5, 1900
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