Another example: 4WYV.

http://www.rcsb.org/pdb/explore/jmol.do?structureId=4WYV&bionumber=1

In this case, the point group would be D4, but the structure has an open side that breaks the symmetry. That is why the Helical detector is called instead of the Rotational detector.

Two requirements could be added to determine more specifically Helical symmetry:

1. In a multiple-start helical symmetry, the start of the nth helix is before the previous ends.
2. The helix rise has to allow for infinite helix construction, because if it is less than the structure height, then the first subunit of the next turn will clash with the first subunit of the current turn.

A note about the last case (4wyv): there is another structure of the same protein (1j1j) which does have a proper D4 without the open side. Apparently in 4wyv the difference is that they purify the protein in complex with RNA, which somehow produces an opening of the D4 (see paper). The RNA however is not seen in the density at all.

With loose thresholds, these are actually OK helical examples. First, I'll show why these are decent examples, then I'll describe bugs and shortcomings as I see it with the existing implementation and visualization.

3MOP is a tricky case because the subunits have three conformations. If you believe this is helical, this can be argued as coming about from the dramatically different crystal contacts at the "head" and "tail" of the short helix fragment in the asymmetric unit. Below, the "tail" (chains A-F) is green, "body" (G-J) is purple, and the "head" (K-N) is pink.

Superimposing chains A and B from the tail, we see that all subunits align fairly well to the next one up, so it does have a pseudo-helical structure with ~100 degrees between subunits.

3OQ9 contains 10 subunits, with ~140 degree rotations between sequential subunits. The chain order is CADBEJHKIL. Here are two asymmetric units stacked, which makes the axis clearer:

Problems

1. Visualization. Using straight lines between segments is difficult to follow, especially for rotation angles near 180°. This could be solved by fitting a true helix to the centroids. Potential links: Eberly, Enkhbayar 2008
2. Entity clustering. It's not entirely clear which entities are being considered equivalent here. Both examples have significant structural similarity and so could be considered equivalent, but I'm not sure if the symmetry code is doing so. PDB website gives uneven stoichiometries for both examples (A6B4C4 for 3MOP, corresponding to head/body/tail; A5B5 for 3OQ9, corresponding to head/tail). I would have throught that global helical symmetry would require that all subunits be included, but the breaks in the images suggest that maybe helices are being fit separately to each entity cluster. That would be a bug.
3. Multiple starts. This is a tough problem. For a set of points distributed evenly along a helix, you can fit an arbitrary number of other helices perfectly to the points, in the same way that you can index a grid of points using an infinite number of basis vectors. Thus, a double helix with equal major and minor grooves is equivalent to a single helix with a different pitch. However, it is not always possible to fit a single helix to points taken from a true double helix, so we do need to be able to detect this case. The problem is that we want to be quite tolerant to errors in the position of the centroid, so it will be a trade-off between overfitting many helices to the points and fitting a single helix with a lot of error. We may also want to take into account the interfaces between sequential subunits. For instance, it may be preferable to fit 3OQ9 using 5 helices (red) or 3 helices (yellow) since sequential subunits along each helix have a big interface, while sequential subunits along the the single helix (green) don't touch at all:

Great analysis Aleix and Spencer.

Regarding:

1. Drawing a helix rather than the straight line segments would really help
   with visualization. You just need to keep in mind the primitives you can
   draw in Jmol. You either could use a polyline with short segments or try to
   make the curve out of arcs.
2. This looks like a bug. The stoichiometry should be checked. To see which
   subunits are considered equivalent, color by "sequence" on the Jmol page.
   For global symmetry, it uses a 95% seq. identity cutoff to distinguish
   subunits. For lower seq. identity cutoffs (to determine pseudosymmetry), it
   does a structural alignment between non-identical sequences.
3. For multi-start helices, I've implemented a number of algorithms to find
   the "best" layerlines (see methods in HelixLayers). The one being used is
   based on the maximum number of contacts between sequential subunits.

On Mon, Aug 10, 2015 at 8:06 AM, Spencer Bliven notifications@github.com
wrote:

With loose thresholds, these are actually OK helical examples. First, I'll
show why these are decent examples, then I'll describe bugs and
shortcomings as I see it with the existing implementation and visualization.

3MOP is a tricky case because the subunits have three conformations. If
you believe this is helical, this can be argued as coming about from the
dramatically different crystal contacts at the "head" and "tail" of the
short helix fragment in the asymmetric unit. Below, the "tail" (chains A-F)
is green, "body" (G-J) is purple, and the "head" (K-N) is pink.

[image: 3MOP chains - free superposition]
https://cloud.githubusercontent.com/assets/595872/9173160/9fe44c40-3f78-11e5-8a87-2205f0ef772e.png

Superimposing chains A and B from the tail, we see that all subunits align
fairly well to the next one up, so it does have a pseudo-helical structure
with ~100 degrees between subunits.

[image: 3MOP helices - superposition of A and B]
https://cloud.githubusercontent.com/assets/595872/9173121/5c819e1c-3f78-11e5-8378-3487f4ec0c66.png

3OQ9 contains 10 subunits, with ~140 degree rotations between
sequential subunits. The chain order is CADBEJHKIL. Here are two asymmetric
units stacked, which makes the axis clearer:

[image: screen shot 2015-08-10 at 15 16 02]
https://cloud.githubusercontent.com/assets/595872/9172306/171dbf86-3f73-11e5-9a8b-84a63556d308.png

Problems

Visualization. Using straight lines between segments is difficult to
follow, especially for rotation angles near 180°. This could be solved by
fitting a true helix to the centroids. Potential links: Eberly
http://www.geometrictools.com/Documentation/HelixFitting.pdf, Enkhbayar
2008 http://www.ncbi.nlm.nih.gov/pubmed/18467178
2.

Entity clustering. It's not entirely clear which entities are being
considered equivalent here. Both examples have significant structural
similarity and so could be considered equivalent, but I'm not sure if the
symmetry code is doing so. PDB website gives uneven stoichiometries for
both examples (A6B4C4 for 3MOP, corresponding to head/body/tail; A5B5 for
3OQ9, corresponding to head/tail). I would have throught that global
helical symmetry would require that all subunits be included, but the
breaks in the images suggest that maybe helices are being fit separately to
each entity cluster. That would be a bug.
3.

Multiple starts. This is a tough problem. For a set of points
distributed evenly along a helix, you can fit an arbitrary number of other
helices perfectly to the points, in the same way that you can index a grid
of points using an infinite number of basis vectors. Thus, a double helix
with equal major and minor grooves is equivalent to a single helix with a
different pitch. However, it is not always possible to fit a single helix
to points taken from a true double helix, so we do need to be able to
detect this case. The problem is that we want to be quite tolerant to
errors in the position of the centroid, so it will be a trade-off between
overfitting many helices to the points and fitting a single helix with a
lot of error. We may also want to take into account the interfaces between
sequential subunits. For instance, it may be preferable to fit 3OQ9 using 5
helices (red) or 3 helices (yellow) since sequential subunits along each
helix have a big interface, while sequential subunits along the the single
helix (green) don't touch at all:

[image: 3OQ9 potential helices]
https://cloud.githubusercontent.com/assets/595872/9173795/bc10dd44-3f7c-11e5-9508-2da2b25edd1a.png

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#306 (comment).

Peter Rose, Ph.D.
Site Head, RCSB Protein Data Bank West (http://www.rcsb.org)
San Diego Supercomputer Center (http://bioinformatics.sdsc.edu)
University of California, San Diego
+1-858-822-5497

I agree with @sbliven, after looking into detail at the examples they have helical symmetry. The stoichiometry and the axes' line segments were confusing in the first place.

I still do not understand why the line segments are not all connected (there are discontinuities), for example in the 3MOP example subunits 6 and 7 (in the order of the helix from the bottom) are not connected, nor 10 and 11. This makes the impression that the helix is broken, or that the symmetry is not conserved between these subunits, which is not true. The same happens in the 3OQ9 example, where there are 3 discontinuities. Maybe fixing that is easier than drawing a helix rather than straight line segments and it will improve the visualization.

About the last example (4WYV), since it is a broken D4 symmetry it has some translation component in addition to the rotation that makes a helix rise, but it is not sufficient to continue the helix more than 4 subunits because of clashes. I think we could check that the helix rise is sufficient enough (larger than the subunit diameter in the direction of helix axis) to discard these cases.

If I run CeSymm on the structure 3MOP I get the correct axis line segment connections, corresponding to the Spencer's red lines (4-start helix). The only thing that changes between running CeSymm or the QuatSymmetryDetector is the alignment part, because the drawing part is the same. It might give a clue to what is going on:

With this visualization the helical symmetry is clearer.