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Dear Eaton Lattman,
many thanks for this extremely clear explanation!
Best regards,
Dirk.
Eaton Lattman wrote:
It was well established years ago that a rotation function including
all molecules related by *crystallographic *symmetry has the same signal to
noise ratio as the rotation function carried out with a single molecule.. To
see this consider a P2 crystal with one molecule in the asymmetric unit. Let
*F*1 * *and *F2* be the complex diffraction patterns of the 2 molecules. Then
I(*h*), the intensity at point *h *is given by
I(*h*) = |*F1(h)|*^2 + |*F2(h)*|^2 + *F1*(*h*)*F2**(*h*) +
*F**1**(*h*)*F2*(*h*)
Now let *Fm *be the diffraction pattern (Fourier transform) of the probe
molecule. The rotation function can then be expressed as
R(*M*) = Sum(I(*h*)•|*Fm(Mh)|*^2)
where the matrix *M* contains the rotation angles. Consider the value of
*M* that correctly aligns the probe and crystal molecule 1. This means
that |*F1(h)|*^2 ~ |*Fm(Mh)|*^2. Therefore in the rotation function the signal
terms are
|*F1(h)|*^2• |*Fm(Mh)|*^2 and the noise terms are
(|*F2(h)*|^2 + *F**1*(*h*)*F2**(*h*) + *F**1**(*h*)*F2*(*h*))•|*Fm(Mh)|*^2
If we had searched with two copies of *Fm, * called* Fm1 *and *Fm2, *related by
the crystallographic two fold, then the signal in the RF would be twice as large
|*F1(h)|*^2• |*Fm1(Mh)|*^2 + |*F2(h)|*^2• |*Fm2(Mh)|*^2
but the noise would be twice as large as well.
(|*F2(h)*|^2 + *F**1*(*h*)*F2**(*h*) + *F**1**(*h*)*F2*(*h*))•|*Fm1(Mh)|*^2
+ (|*F1(h)*|^2 + *F**1*(*h*)*F2**(*h*) + *F**1**(*h*)*F2*(*h*))•|*Fm2(Mh)|*^2
1. Adding in crystal rotation symmetry does not help
2 Above example shows why RF works less well as the symmetry is higher. Alwas
get one signal term, but noise terms depend on the number of
equivalent position in the unit cell.
3. If the crystal and probe molecules have non crystallograpic symmetry, then
using this symmetry in the rotation search can increase signal to noise. This
is the so-called locked rotation function.
Ed
Eaton E. Lattman
Dean of Research and Graduate Education
Professor of Biophysics
Zanvyl Krieger School of Arts and Sciences
410.516.8215 (voice); 410.516.4100 (fax)
Mail address:
Johns Hopkins University
3400 North Charles Street
237 Mergenthaler Hall
Baltimore, MD 21218
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On Jan 11, 2007, at 7:37 AM, Nicholas M Glykos wrote:
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It this point that I had in my mind. I would like to account simultaneously
for_all_ symmetry-equivalent intra-molecular Patterson vectors at once. Since
the intra-molecular Patterson vectors of all symmetry copies of the true
structure fall into the same sphere around the origin, any true orientation of
the search molecule will have its P1 Patterson self-vectors overlapping both
with the self-vectors of the corresponding true molecule, which will add to
the signal, _and_ with the self-vectors from its symmetry copies, which will
add to the noise. Fitting them all simultaneously by applying the rotational
symmetry of the space group to the P1 self-vectors of the search molecule
would result in a Patterson overlap with all self-vectors of the true
structure at once. This should enhance the signal-to-noise ratio of the
orientation search.
What do you and others think about this?
Apologies for the unashamed plug, but multidimensional molecular
replacement search methods (like Qs :-) do exactly what you described
[plus more, since (in the vector-space-based metaphor) they simultaneously
try to match all intra- and inter-molecular vectors, with the full
crystallographic symmetry explicitly accounted for]. On a related side
note, I believe that the major source of problems with the results from
the cross rotation function is not overlapping self-vectors, but
'overlapping' cross-vectors.
Nicholas
--
Dr Nicholas M. Glykos, Department of Molecular
Biology and Genetics, Democritus University of Thrace,
Dimitras 19, 68100 Alexandroupolis, Greece, Fax ++302551030613
Tel ++302551030620 (77620), http://www.mbg.duth.gr/~glykos/
--
****************************************
Dirk Kostrewa
Paul Scherrer Institut
Biomolecular Research, OFLC/110
CH-5232 Villigen PSI, Switzerland
Phone: +41-56-310-4722
Fax: +41-56-310-5288
E-mail: [EMAIL PROTECTED]
http://sb.web.psi.ch
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