Section 13.1.5.5. Noncrystallographic symmetry in atomic coordinate refinement

In atomic coordinate refinement, noncrystallographic symmetry again provides a useful increase in the ratio of the number of observed quantities to the number of atomic parameters to be refined. As is discussed by Cruickshank in Chapter 18.5 , the application of restraints in refinement (on quantities like bond lengths, bond angles and the elimination of short contacts) is formally equivalent to an increase in the number of observational equations. However, if these restraints are tightly applied, they act more like constraints, and their effect is more like a reduction in the number of parameters to be determined. Meaningful refinement is not possible unless the number of observations exceeds the number of parameters, and in practice it usually needs to do so by a factor of 2 or so. If noncrystallographic symmetry is imposed, the number of observations required to define the structure is reduced, because the volume of unknown structure is reduced.

Noncrystallographic symmetry can thus provide a crucial advantage in leading to unambiguous interpretation of structure at relatively poor resolution (say, 3.0 to 3.8 Å), where the ratio of refined parameters to the number of observations is marginal. Consider two crystals of the same material, one of which has one subunit per asymmetric unit and the asymmetric volume is . The other has N subunits in an asymmetric unit of volume . To the same resolution, the available number of observed reflections is increased in the ratio , in order to obtain the same number of parameters if noncrystallographic symmetry is imposed.

What effect do these N subunits have on the precision of the final coordinates? The crystal allows the determination of N sets of atomic coordinates. If the errors were independent of each other, the precision of the mean value of each coordinate could be improved in the ratio (compared to a well refined structure).

This improvement will be lost when constraints are applied to the mean coordinates (to make them conform to given bond lengths and angles, for example). If this is done, the errors are no longer independent, and the increase of precision will be less.

Cruickshank (1999a,b and Chapter 18.5 ) shows that at high resolution (examples at 0.94 and 1.0 Å) and for atoms of low B factor (less than say 10 Ų), restraints make little difference to the precision of refinement. Under these conditions, N independent subunits in the asymmetric unit might improve the precision of the mean coordinates by a factor approaching . But at such good resolution, it is very possible that the differences between the calculated subunit conformations are not due to error, but reflect real structural differences. If so, the precision of the mean coordinates is less significant.

At less high resolution (example given at 1.7 Å), Cruickshank has shown that the precision of unrestrained refinement is significantly worse than the precision of the restraints. In this case, imposing noncrystallographic symmetry on the structure should provide some improvement. But because the coordinate errors then cease to be independent, the improvement in the mean coordinates would be less.