Continuous symmetry analyses: C_nv and D_n measures of molecules, complexes, and proteins

Abstract

The continuous symmetry methodology has been developed to provide a quantitative estimation for the degree of a selected symmetry point-group in a molecular structure. Previous developed measures included the C(S), C(n), S(n) (including C(i)) symmetries, several polyhedral symmetries, and the related chirality measure. Motivated by the abundance and importance of C(nv) symmetry in many key molecules and by the abundance of D(n) symmetry in many organometallic complexes and in many protein clusters, measures for these symmetries, based on the general definition of the continuous symmetry measure, were developed. Detailed description of the derivation of the measures, along with plenty of examples for their application, is provided.

The near‐symmetry of proteins

“…The majority of our analyses concentrate on the most abundant oligomeric near‐symmetry, namely of C n . Few examples deal with the less abundant point‐group symmetry D n , using a recently developed computational tool developed for that symmetry …”

Continuous symmetry measures for complex symmetry group

“…The implicit approximation is that the directions of the symmetry elements of separate cyclic subgroups and the related permutations are identical to the corresponding directions and permutations in the complex group. This implicit approximation is usually fine and the results are usually exact, as was demonstrated by the authors, but in cases where the deviations from perfect symmetries are not small, the approximation can break and the results will not be accurate.…”

“…For complex symmetry groups, which contain more than one symmetry element, analytical solution cannot be formulated as the dependencies between the different symmetry elements make the expressions too complicated. Recently, an article was published where the explicit expressions for the C nv and D n groups were rearranged and solved numerically . Although this method was proven to be successful, it has some limitations: First of all, this scheme was designed especially for the C nv and D n groups and it cannot be applied or generalized to other complex symmetry groups.…”

“…We ran several tests to estimate the error of the methodology. First, we compared the results of our program with the results of the C nv / D n program, which is based on the numerical solution . We calculated the C nv and D n symmetry of 500,000 different structures using both methodologies.…”

“…For more complex groups, the exact expression is too complex to be solved analytically. A specific numeric solution for the C nv and D n groups was described recently, but this solution cannot be applied to other complex symmetry groups.…”