Bands in a D2 Symmetry puzzle do not work as intended
|Opened on:||Saturday, June 12, 2021 - 21:17|
|Last modified:||Monday, June 14, 2021 - 17:32|
The D2 Tetramer symmetry is very interesting to work with, and it gives really different possibilities compared to the cyclical (C3 or C4) symmetry that we have seen before. But up to now, every attempt I made to really show the beauty of this D2 symmetry has failed miserably, because some mysterious force completely wrecked my monomer.
It has taken me up to the 6th day of the 3rd D2 Symmetry puzzle (Puzzle 2000) to find out what was wrong. At first, I thought this phenomenon was related to the Hydrogen Bond Network issue, for which Boots made a Feedback item. But since that issue has been resolved (see my comment to the DevPrev D2 puzzle), it is clear that this mysterious force is a totally different problem.
As I see it now, what happens is the following: any band that crosses over from Chain 0 (where all bands start) to either Chain 1 (purple) or Chain 3 (brown) will exert a force onto Chain 0 in the wrong direction.
For a band that starts and ends on Chain 0, there is no problem. Depending on the Actual Length of the band, compared to the Goal Length, a force is applied to both end points, and that has the desired effect on Wiggle, and on Shake if the band is connected to a sidechain.
For a band that starts on Chain 0 and ends on a different chain, it is a bit more complicated. The force that is exerted by the end point has to be converted into a force acting on the corresponding point of Chain 0, because that is the only chain that Wiggle and Shake are concerned with.
As an example, let's say we have a band from Segment 1 in Chain 0 to Segment 5 in Chain 1.
In a "regular" Symmetry puzzle, with cyclical symmetry (C3 or C4), that band will be "mirrored" as a band from the last chain (i.e. Chain 3 in a C4 Tetramer) to Chain 0. In the picture above, this is the grey band in the lower left, between the brown Segment 1 and the blue Segment 5.
In a D2 Symmetry puzzle, the situaltion is different. Our band from Segment 1 on Chain 0 to Segment 5 on Chain 1 should now be mirrored as a band from Segment 1 on Chain 1 to Segment 5 on Chain 0. That is shown below as the skinny band at the top left, just behind the Hydrogen Bond. But apparently, it is actually modeled, just like in the example above, from Chain 3 to Chain 0: the long grey band at the left side of the picture below.
As a result, the force on Segment 5 of Chain 0 (top left) is applied in the wrong direction: in this picture, down instead of to the right.
Coincidentally, bands from Chain 0 to Chain 2 (green) do not have this problem. The explanation for that is simple: such a band will be mirrored from Chain 2 to Chain 0, and that is correct in both sorts of Tetramers (C4 as well as D2).
As a workaround for D2 Symmetry puzzles, I have made a script (BandD2) that replaces bands to Chain 1 or Chain 3 by one or two Bands in Space (orange in the picture above). You will then have to manually adjust the end points of these bands. In the picture above, the one on the left still needs to be moved, to make it point along the skinny band, parallel to the Hydrogen Bond.
After running this script and manually adjusting the bands, Wiggle will work approximately as intended. But from time to time, you will have to adjust the Bands in Space by hand, because they will not move along with the residues that they are supposed to point to. When you do not remove the original bands (BandD2 will only disable them), you can use them as a guide to easily re-align the Bands in Space.