GETAWAY[4,5,6]

The GETAWAY (GEometry, Topology, and Atom Weights AssemblY) try to match 3D molecular geometry provided by the molecular influence matrix and atom relatedness by topology with chemical information by using different atomic weighting schemes (unit weights, mass, polarizability, electronegativity).

The molecular influence matrix, $\varmathbb{H}$ is defined by

$$\varmathbb{H} = \varmathbb{M} \cdot ( \varmathbb{M}^T \cdot \varmathbb{M}) \cdot \varmathbb{M}^T$$

where $\varmathbb{M}$ is the molecular matrix. The resultant $A \times A$ matrix is invariant to rotation of the molecular coordinates. The diagonal elements $h_{ii}$ are termed leverages and represent the influence of each atom in determining the shape of the molecule. Each off diagonal element $h_{ij}$ represents the degree of of accessibility of the j'th atom to interactions with the i'th atom.

H-GETAWAY

H-GETAWAY descriptors are obtained by manipulating the $\varmathbb{H}$ matrix

Simple H-GETAWAY

The simple H-GETAWAY is defined by

$$H_{GM} = 100\cdot \left( \prod^{A}_{i=1} h_{ii} \right)^{1/A}$$

and is essentially the geometric mean of the leverage magnitude.

Informational Indices

There are 3 types of informational indices which consider the diagonal elements of $\varmathbb{H}$ as indicative of molecular complexity (since they are sensitive to the whole molecule structure).

• Total information content
  
  $$I_{TH} = A_0 \cdot \log_2 A_0 - \sum^{G}_{g=1} N_g \cdot \log_2 N_g$$
  
  

• Standardized information content on the leverage equality represented by,
  $$\begin{eqnarray*} I_{SH} & = & \frac{I_{TH}}{A_0 \cdot \log_2 A_0} \\ & = & 1... ...rac{\sum^{G}_{g=1} N_g \cdot \log_2 N_g}{A_0 \cdot \log_2 A_0} \end{eqnarray*}$$
  
  
  
  
  

where $N_g$ is the number of atoms with the same leverage value and $G$ is the number of equivilance classes into which the atoms are partitioned according to leverage equality and $A_0$ is the number of non hydrogen atoms. These indices encode information regarding the molecule entropy.

• Mean information content on the leverage magnitude is defined as
  $$\begin{eqnarray*} HIC & = & \overline{I}_H \\ & = & - \sum^{A}_{i=1} \frac{h_{ii}}{D} \cdot \log_2 \frac{h_{ii}}{D} \end{eqnarray*}$$
  
  
  
  
  
  where $D$ is the matrix rank defined by $D = \sum^{A}_{i=1} h_{ii}$ and $A$ is the total number of atoms, hydrogens included.

Autocorrelation indices

The H-GETAWAY autocorrelation descriptor utilizes the geometric information implicit in the leverage values and atomic weighting schemes to give rise to a new set of descriptors. By analogy with the Moreau - Broto descriptors the HATS indices are defined by weighting each atom in a molecule by

$$w^{'}_{i} = w_{i} \cdot h_{ii}$$

thus giving a weight vector $\varmathbb{W}^{'}$. The HATS are then defined as

$$HATS_0(w) = \sum^{A}_{i=1} (w_i \cdot h_{ii})^2$$

and

$$HATS_k(w) = \sum^{A-1}_{i=1} \sum_{j>i} (w_i \cdot h_{ii}) \cdot (w_j \cdot h_{jj}) \cdot \deltaup(k;d_{ij})$$

where $d_{ij}$ is the topological distance between the i'th and j'th atoms and $k = 1, 2, \cdots, d$ with

$$\deltaup(k;d_{ij}) = \left\{ \begin{array}{lcl} 1 & if & d_{ij} = k \\ 0 & if & d_{ij} \neq k \end{array} \right.$$

The HATS total index is defined by summing the individual $HATS_k$ indices for increasing $k$ and is given by

$$\begin{eqnarray*} HATS(w) & = & (\varmathbb{W}^{'})^T \cdot \varmathbb{U} \cdot... ... \cdot h_{jj} \\ & = & HATS_0(w) + 2 \sum_{k=1}^{d} HATS_k(w) \end{eqnarray*}$$

where $\varmathbb{U}$ is the $A \times A$ unit matrix.

R-GETAWAY

This subset replaces $\varmathbb{H}$ with other types of matrices such as a combination of the molecular influence matrix and the geometry matrix to give the influence/distance matrix, $\varmathbb{R}$ defined as

$$[ \varmathbb{R} ]_{ij} = \left[ \frac{\sqrt{h_{ii} \cdot h_{jj}}}{r_{ij}} \right]_{ij}$$

where $i \neq j$ Using the elements of $\varmathbb{R}$ we can get several descriptors

RARS

The average row sum of $\varmathbb{R}$ is defined as

$$\begin{eqnarray*} RARS & = & \frac{1}{A} \sum_{i=1}^{A} \sum_{j=1}^{A} \frac{\... ...dot h_{jj}}}{r_{ij}} \\ & = & \frac{1}{A} \sum_{i=1}^{A} RS_i \end{eqnarray*}$$

where $RS_i$ is the i'th row sum.

RCON

The $\varmathbb{R}$ connectivity index is defined by analogy with the Randic connectivity index and is defined as

$$RCON = \sum^{B}_{b=1} \sqrt{(RS_i \cdot RS_j)_b}$$

REIG

This descriptor is designed by analogy with the Lovasz-Pelikan index and describes molecular branching and is defined as the first eigenvalue of $\varmathbb{R}$

Autocorrelation indices

These are defined analogously to the H-GETAWAY autocorrelational indices and so we have the w weighted k'th order autocorrelation index

$$R_k(w) = \sum^{A-1}_{i=1} \sum_{j>i} \frac{\sqrt{h_{ii} \cdot h_{jj}}}{r_{ij}} w_i \cdot w_j \cdot \deltaup(k;d_{ij})$$

where $k = 1, 2, \cdots, d$. We can also calculate the a R total index defined as

$$\begin{eqnarray*} RT(w) & = & \varmathbb{W}^T \cdot \varmathbb{R} \cdot \varma... ...}} \cdot w_i \cdot w_j \\ & = & 2 \cdot \sum_{k=1}^{d} R_k(w) \end{eqnarray*}$$