Search for non-crystallographic symmetry (NCS) with Patterson
self-rotation maps does not reveal any new peaks in cases where the NCS axis is parallel
(or nearly so) to a crystallographic axis of same or higher symmetry in the respective
spacegroup. Occurrence of NCS parallel to crystallographic axes is not uncommon. Such a
NCS axis can be found by inspecting native Patterson maps. All you need to do is create a Patterson map from the data and a large peak will
occur. The position of the axis can be calculated from the Patterson peak position. I
found no description in a basic textbook, but see *Meth. Enz. 277, p26*,
and the reference therein

Let us look at a monoclinic cell, *P*2 (No.3) for example, with two
molecules (red-green square protein, RGSP-1) in the asymmetric unit related by a twofold
NCS axis (blue symbol) parallel to the twofold crystallographic axis (green). Standard
setting, unique axis *b*. The origin could be at any of the 2-fold axes.

Applying any of the equivalent 2-fold operations creates a second copy of the 2 molecules (RGSP-1 dimer) in the unit cell :

As you may have seen already, there is now a set of translational vectors relating each atom in the 2 copies of the molecule (I have shown only a few vectors) :

The large number of same (or very similar, in non-perfect NCS) distance vectors of course gives rise to a whopping peak in the native Patterson map :

The blue peak is symmetry related to the purple one, and directly gives
the position of the NCS axis from the general Patterson peak position at *u,v,w* =
2x, 0, 2z. From the way I drew the vectors between the molecules in the unit cell the
purple peak actually yields the origin shifted solution: translate origin to
1/2,0,1/2 and then add the x and z derived from the purple vector (the origin can be
chosen at any of the 2-fold axes in *P*2).

Click here for a real life example (native
Patterson NCS peak in *P*2_{1} , 2E8 Fab
antibody fragment) 1

**QUIZ**

Where would the NCS peak be in space group

*P*2_{1 }?

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