Molecular descriptors rotational invariance

Mak L, Grandison S, Morris RJ (2008) An extension of spherical harmonics to region-based rotationally invariant descriptors for molecular shape description and comparison. J Mol Graph Model 26 1035-1045... 

The orthogonality of a set of molecular descriptors is a very desirable property. Classification methodologies such as CART (11) (or other decision-tree methods) are not invariant to rotations of the chemistry space. Such methods may encounter difficulties with correlated descriptors (e.g., production of larger decision trees). Often, correlated descriptors necessitate the use of principal components transforms that require a set of reference data for their estimation (at worst, the transforms depend only on the data at hand and, at best, they are trained once from some larger collection of compounds). In probabilistic methodologies, such as Binary QSAR (12), approximation of statistical independence is simplified when uncorrelated descriptors are used. In addition,... 

Geometric and mass centres of a molecule are not molecular descriptors, but they are used to translate the molecule at the centre, thus providing a unique reference origin which allows invariance to translation and separation between translational and rotational motions. 

It must be noted that the invariance to rotation of G-WHIM descriptors, i.e. the independence of any molecular alignment rule, is obtained if the grid points are dense enough. In fact, a too sparse distribution of grid points represents an inadequate sampling of the ideal scalar field and is not able to guarantee that the calculated scalar field is representative of the ideal scalar field in such a way as to preserve rotational invariance. 

Two other important invariance properties, translational invariance and rotational invariance, are the invariance of a descriptor value to any translation or rotation of the molecules in the chosen reference frame. These invariance properties have to be considered when dealing with descriptors derived from - molecular geometry G. For... 

Rotational Invariance describes the independency of a molecular descriptor from partial or complete rotation of the molecule — independent of absolute Cartesian coordinates of the atoms. 

Molecular descriptors based on this kind oflocal vertex invariant are called MIS indices, being defined in the framework of the Method of Ideal Symmetry (MIS), based on a partial optimization procedure of the molecular geometry, where bond lengths and bond angles are kept fixed and only free rotations around C-C bonds are varied jToropov, Toropova et al, 1994]. 

In QSAR modeling, the question of molecular representation is central. For the modeling, a molecule is represented as a multidimensional vector, i.e., a molecule is a point in multidimensional representational space. An ideal representation should be unique, uniform, reversible, and invariant on rotation and translation of molecules. Unique means that different structures give different representations, uniform means that the dimension of representation is the same for all structures, reversible means that the structure can be unambiguously reconstructed from the representation vector. Furthermore, invariant means that the representation is not sensitive if a molecule is rotated or translated. It is not expected that we would find a general representation that fulfills all requirements simultaneously [26]. Nowadays, thousands of descriptors and structural representations are in use [26-29]. We will give a short overview, and references about descriptors that often appear in QSAR studies related to mutagenicity of aromatic amines are presented in more detail. 

Another descriptor based on eigenvalues is the EVA descriptor [66,67] (where EVA stands for EigenVAlue). The EVA descriptor is an example of a 3D descriptor that does not require alignment of the 3D structures (i.e., it is invariant to translation or rotation of the structure). It is based on the infrared spectrum of a molecule, which is related to the 3D structure. The molecular vibrations are calculated using a normal coordinate analysis (NCA) of the energy-minimized structure (e.g., using MOPAC AMI), and the eigenvalues from the NCA are used to derive the EVA descriptor (see Ref. 67). 

Relative and absolute descriptors also differ from each other at a more fundamental level. If we compare two molecules (either their nuclear geometries or their electron densities), the result will normally depend on how they are oriented relative to each other. In contrast to absolute descriptors, relative descriptors are not invariant if one of the molecules compared is rigidly translated or rotated. Therefore, the proper use of relative descriptors must be accompanied by some sort of optimization in the superposition between two structures. Yet, a maximum superposition (e.g., by minimizing the rms deviation of paired atoms) may not produce a relative orientation that is most relevant in a given comparison of molecular shapes. This is a problem with no unique solution and is still under much research. [See discussions in Ref. 27.]... 

The development and application of molecular shape descriptors is an active area in computational chemistry and biology. The main goal of our work is to develop mathematical descriptors that can determine whether two molecules have comparable shapes. In this chapter we present a series of molecular shape descriptors developed oti the basis of molecular vdW space. The molecules are treated in the hard sphere approximation, as a body composed from a collection of atomic fused spheres. Each sphere is centered in the corresponding nucleus and it is characterized by its Cartesian coordinates and by its vdW radius, r. These molecular vdW shape descriptors depend only on the internal structure of the molecule, being invariants to any translation and rotation movement. Consequently, they may inform us that two molecules have comparable shapes, but since they carry no information about the absolute orientation or position of the molecule, they are not useful for computing molecular superposition. 

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