The Symmetricity of Normal Modes in Symmetric Complexes

TL;DR

This paper investigates the symmetry properties of normal modes in symmetric structures, revealing that modes group by symmetricity, which can be computed more efficiently, and exploring how structural symmetry relates to mode symmetricity.

Contribution

It introduces a theoretical framework for classifying normal modes by symmetricity in cyclic structures, enabling more efficient computations and insights into symmetry preservation.

Findings

Normal modes in ring structures form symmetricity groups of equal size.

Modes with perfect symmetry or anti-symmetry are non-degenerate.

Some symmetric modes retain symmetry even when structural symmetry is broken.

Abstract

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 $m^3$ less), thus making it applicable to larger complexes. We show that normal modes with perfect symmetry or anti-symmetry have no degeneracy while the rest of the modes have a degeneracy of two. We show also how symmetry in normal modes correlates with symmetry in structure. While a broken symmetry in structure generally leads to a loss of symmetricity in symmetric normal modes, the symmetricity of some symmetric normal modes is preserved even when symmetry in structure is broken. 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 symmetry is needed by a given complex to obtain certain symmetric motions that are functionally critical.