Thanks Bob! The PPT really cleared things up. I'm going to link to it in the
tooltip, if you don't mind, as a way for users to find out more.
Cheers, Mike
On Tue, Sep 21, 2010 at 6:49 AM, Robert Hanson <[email protected]> wrote:
> 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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>
--
Mike Evans
Organic Chemistry Graduate Student
Moore Group
University of Illinois, Urbana-Champaign
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