Symmetry in normal modes and its strong dependence on symmetry in structure

In this work, we look at the symmetry of normal modes in symmetric structures, particularly structures with cyclic symmetry. We show that normal modes of symmetric structures have different levels of symmetry, or symmetricity. One novel theoretical result of this work is that, for a ring structure with m subunits, the symmetricity of the normal modes falls into m groups of equal size, with normal modes in each group having the same symmetricity. The normal modes in each group can be computed separately, using a much smaller amount of memory and time (up to m³ less). Lastly, we show that symmetry in normal modes depends strongly on symmetry in structure. This work suggests a deeper reason for the existence of symmetric complexes: that they may be formed not only for structural purpose, but likely also for a dynamical reason, that certain structural symmetry is needed to obtain certain symmetric motions that are functionally critical.