List of geometrical descriptors calculated by DRAGON

 

Geometrical descriptors are defined in several different ways but always derived from the three-dimensional structure of the molecule. Generally, geometrical descriptors are calculated either on some optimised molecular geometry obtained by the methods of the computational chemistry or on crystallographic coordinates.

 

Since a geometrical representation of a molecule involves the knowledge of the relative positions of the atoms in 3D space, i. e., the (x,y,z) atomic coordinates of the molecule atoms, geometrical descriptors usually provide more information and discrimination power also for similar molecular structures and molecule conformations than topological descriptors. Despite their high information content, geometrical descriptors usually show some drawbacks. They require geometry optimisation and therefore the overhead to calculate them. Moreover, for flexible molecules, several molecule conformations can be available: on one hand, new information is available and can be exploited, but, on the other hand, the problem complexity can significantly increase.

 

Many molecular descriptors calculated by DRAGON and belonging to the block of geometrical descriptors are commonly known as topographic indices, being calculated on the graph representation of molecules but using the geometric distances between atoms instead of the topological distances.

These are mainly derived from the geometry matrix which is a square symmetric matrix collecting the geometric distances (Euclidean distances) between all pairs of atoms. A row sum of the geometry matrix, called geometric distance degree, is the sum of the geometric distances from an atom to any other atom in the molecule.

The distance/distance matrix is a square symmetric matrix whose entries are the ratios of the geometric over topological distance between all pairs of atoms [M. Randic, A.F. Kleiner, L.M. DeAlba, J.Chem.Inf.Comput.Sci. 1994, 34, 277-286].

 

• The 3D-Wiener index (W3D) is calculated as the sum of all the interatomic geometric distances in a molecule [O. Mekenyan, D. Peitchev, D. Bonchev, N. Trinajstic, I.P. Bangov, Arzneim.Forsch. 1986, 36, 176-183; B. Bogdanov, S. Nikolic, N. Trinajstic, J.Math.Chem. 1989, 3, 299-309]. Even hydrogens are considered. This index is obviously more discriminant than the Wiener index W and shows different values for different molecular conformations, the largest values corresponding to the most extended conformations, the smallest to the most compact conformations. Therefore, it is considered among shape descriptors since it decreases with increasing sphericity of a structure [S. Nikolic, N. Trinajstic, Z. Mihalic, S. Carter, Chem.Phys.Lett. 1991, 179, 21-28].
• The 3D-Balaban index (J3D) is calculated by the same formula as the Balaban distance connectivity index J using the geometric distance degrees in place of topological distance degrees [Z. Mihalic, S. Nikolic, N. Trinajstic, J.Chem.Inf.Comput.Sci. 1992, 32, 28-37].
• The 3D-Harary index (H3D) is calculated as the sum of all the reciprocal geometric distances in a molecule.
• The average geometric distance degree (AGDD) is calculated dividing the sum of all geometric distance degrees by the total number of molecule atoms (nAT).
• The 3D Petitjean shape index (PJI3) is calculated dividing the difference between geometric diameter and radius by the geometric radius [P.A. Bath, A.R. Poirrette, P. Willett, F.H. Allen, J.Chem.Inf.Comput.Sci. 1995, 35, 714-716]. The geometric radius of a molecule is defined as the minimum geometric eccentricity and the diameter is defined as the maximum geometric eccentricity in the molecule, the atom geometric eccentricity being the longest geometric distance from the considered atom to any other atom in the molecule.
• The absolute eigenvalue sum on geometry matrix (SEig) is the sum of the absolute eigenvalues of the geometry matrix.
• Sums of geometrical distances between X..Y (G(X..Y)), X and Y referring to any heteroatom, are simple molecular descriptors calculated by summing geometric distances between all pairs of specific atom-types.
• The D/D index (DDI) is calculated as the half sum of all distance/distance matrix entries [M. Randic, A.F. Kleiner, L.M. DeAlba, J.Chem.Inf.Comput.Sci. 1994, 34, 277-286; M. Randic, G. Krilov, Int.J.Quant.Chem. 1999, 75, 1017-1026].
• The average distance/distance degree (ADDD) is the average row sum of the distance/distance matrix. It encodes information on the molecular folding; in fact, in highly folded structures, it tends to be relatively small as the interatomic distances are small while the topological distances increase as the size of the structure increases [M. Randic, A.F. Kleiner, L.M. DeAlba, J.Chem.Inf.Comput.Sci. 1994, 34, 277-286; M. Randic, G. Krilov, Int.J.Quant.Chem. 1999, 75, 1017-1026].
• The folding degree index (FDI) is the largest eigenvalue of the distance/distance matrix, normalised dividing it by the number of atoms nAT [M. Randic, A.F. Kleiner, L.M. DeAlba, J.Chem.Inf.Comput.Sci. 1994, 34, 277-286; M. Randic, G. Krilov, Int.J.Quant.Chem. 1999, 75, 1017-1026]. This index tends to one for linear molecules (of infinite length) and decreases in correspondence with the folding of the molecule. Thus, it can be thought of as a measure of the folding degree of the molecule because it indicates the degree of departure of a molecule from strict linearity. 

 

Gravitational indices are molecular descriptors reflecting the mass distribution in a molecule, defined as [A.R. Katritzky, L. Mu, V.S. Lobanov, M. Karelson, J.Phys.Chem. 1996, 100, 10400-10407]:

 

             

 

where mi and mj are the atomic masses of the considered atoms, rij the corresponding interatomic distances, nAT and nBT the number of atoms and bonds of the molecule, respectively. The G1 index takes into account all atom pairs in the molecule while the G2 index is restricted to pairs of bonded atoms. These indices are related to the bulk cohesiveness of the molecules accounting, simultaneously, for both atomic masses (volumes) and their distribution within the molecular space.

 

The radius of gyration (RGyr) is a size descriptor for the distribution of atomic masses in a molecule [G.A. Arteca, Molecular Shape Descriptors in Reviews in Computational Chemistry - Vol. 9, K.B. Lipkowitz, D. Boyd (Eds.), VCH Publishers, New York (NY), pp. 191-253, 1991], calculated as:

 

 

where ri is the distance of the ith atom from the centre of mass of the molecule, mi is the corresponding atomic mass, nAT the atom number and MW the molecular weight.

 

The span R (SPAN) is a size descriptor defined as the radius of the smallest sphere, centred on the centre of mass, completely enclosing all atoms of a molecule [G.A. Arteca, Molecular Shape Descriptors in Reviews in Computational Chemistry - Vol. 9, K.B. Lipkowitz, D. Boyd (Eds.), VCH Publishers, New York (NY), pp. 191-253, 1991]:

 

 

where ri is the distance of the ith atom from the centre of mass.

 

The average span R (SPAM) is the root square of the ratio of SPAN over the number of atoms.

 

The molecular eccentricity (MEcc) is a shape descriptor calculated from the eigenvalues l of the molecular inertia matrix [G.A. Arteca, Molecular Shape Descriptors in Reviews in Computational Chemistry - Vol. 9, K.B. Lipkowitz, D. Boyd (Eds.), VCH Publishers, New York (NY), pp. 191-253, 1991]:

 

 

It ranges from 0 to 1, value 0 corresponding to spherical top molecules and value 1 to linear molecules.

 

The spherosity (SPH) is an anisometry descriptor calculated as a function of the eigenvalues of the covariance matrix calculated from the molecular matrix:

 

 

Spherosity index varies from zero for flat molecules, such as benzene, to one for totally spherical molecules [D.D. Robinson, T.W. Barlow, W.G. Richards, J.Chem.Inf.Comput.Sci. 1997, 37, 939-942].

 

The asphericity (ASP) is an anisometry descriptor which measures the deviation from the spherical shape [G.A. Arteca, Molecular Shape Descriptors in Reviews in Computational Chemistry - Vol. 9, K.B. Lipkowitz, D. Boyd (Eds.), VCH Publishers, New York (NY), pp. 191-253, 1991], calculated from the eigenvalues l of the molecular inertia matrix as follows:

 

 

Asphericity varies from 0 for spherical top molecules to 1 for linear molecules. For prolate molecules (cigar-shaped), l1 » l2 > l3 and ASP » 0.25, whereas for oblate molecules (disk-shaped), l1 > l2 » l3 and ASP » 1.

 

The length-to-breadth ratio is commonly the ratio of the longest L to the shortest B side of a rectangle containing some molecular projection, once univocally defined a specific molecular orientation. The length-to-breadth ratio by WHIM (L/Bw) is calculated as the ratio between the first and second eigenvalue of the molecular inertia matrix. It can be noted that this shape parameter not only accounts for the distance between extreme atoms along the principal axes but also for the distribution of all atoms around the molecule centre.

 

Aromaticity indices encode information on the electron delocalisation degree of a molecule [T.M. Krygowski, A. Ciesielski, C.W. Bird, A. Kotschy, J.Chem.Inf.Comput.Sci. 1995, 35, 203-210]. In spite of the concept of aromaticity has never been defined unequivocally, the commonly accepted description of aromaticity is as a characterictic delocalisation of the p-electrons giving a stabilisation of cyclic and polycyclic conjugated molecules, i.e. the increased stability of conjugated rings compared to its classical localised structure.

Aromaticity indices are often calculated from bond lengths and bond orders of the compounds; therefore, their values closely depend on the molecule optimisation procedure and the codification of conjugated bonds.

 

• The Harmonic Oscillator Model of Aromaticity index (HOMA) is based on the degree of alternation of single/double bonds, measuring the bond length deviations from the optimal lengths attributed to the typical aromatic state [T.M. Krygowski, M. Cyranski, A. Ciesielski, B. Swirska, P. Leszczynski, J.Chem.Inf.Comput.Sci. 1996, 36, 1135-1141]. The HOMA index encodes information on conjugated systems, being calculated as follows: 

 

 

where the first sum runs over each conjugetd bond type, Bpk is the number of considered p-bond contributions of the kth conjugated bond type, rb is the actual bond length, ak and rkopt are a numerical constant and the typical conjugated bond length referring to the kth conjugated bond type (see values in the table below). Bp is the total number of bonds belonging to conjugated systems.

 

Bond	a	ropt	Bond	a	ropt
C » C (butadiene)	257.7	1.388	C » P	118.91	1.698
C » C	98.89	1.397	C » S	94.09	1.677
C » N	93.52	1.334	N » N	130.33	1.309
C » O	157.38	1.265	N » O	57.21	1.248

 

 

• The HOMA total (HOMT) is derived from the HOMA index and is calculated as follows: 

 

 

where Bp is the total number of conjugated bonds, the sum runs over each conjugated bond type, Bpk is the number of considered p-bond contributions of the kth conjugated bond type, rb is the actual bond length, ak and rkopt are a numerical constant and the typical aromatic bond length referring to the kth aromatic bond type (see values in the table above). This descriptor depends on the conjugation degree of a molecule as well as on the total number of p bonds.

 

• The Jug RC index (RCI) is an aromaticity index based on the idea of ring current whose magnitude is determined by its weakest link in the ring [K. Jug, Croat.Chem.Acta 1984, 57, 941-953]. The weakest link is calculated as the maximum bond distance of aromatic bonds (inversely proportional to the minimum total bond order).
• The Aromaticity (AROM) is derived from the general aromaticity indices defined by Bird [C.W. Bird, Tetrahedron 1986, 42, 89-92] and is calculated as follows: 

 

 

where the sum runs over the bonds belonging to aromatic rings, is the p bond average length and rp are the actual p bond lengths, Bp is the number of aromatic bonds.

 

Based on the same principles of the WHIMs, COMMA2 descriptors have recently been proposed [B.D.Silverman, J. Chem. Inf. Comput. Sci. 2000, 40, 1470-1476]. They consist of 11 descriptors given by moment expansions for which the zero-order moment of a property field is nonvanishing. DRAGON calculates 4 COMMA2 descriptors for 4 different molecular properties: mass (m), van der Waals volume (v), Sanderson electronegativity (e), polarizability (p). COMMA2 descriptors provided by DRAGON for each molecular property w are:

 

• the displacement (DISPw) between the geometric centre and the centre of the considered property field, calculated with respect to the molecular principal axes;
• the quadrupole moments (QXXw, QYYw, QZZw) with respect to the geometric centre, weighted by the considered molecular property.