Weighted Holistic Invariant Molecular Descriptors[16,17,18,19]

WHIM descriptors are based on statistical indices calculated on the projections of atoms along principal axes ($\mathbf{t}_m$). The aim is to capture 3D information regarding size, shape, symmetry and atom distributions with respect to invariant reference frames.

The algorithm essentially carries out a PCA on the centered cartesian coordinates of a molecule by using a weighted covariance matrix:

$$s_{jk} = \frac{\sum^{A}_{i=1} w_i \cdot (q_{ij} - \overline{q}_j) \cdot (q_{ik} - \overline{q}_k)}{\sum^{A}_{i=1} w_i}$$

where $s_{jk}$ is the weighted covariance between the j'th and k'th atomic coordinates, $A$ is the number of atoms, $w_i$ is the weight of the i'th atom and $q_{ij}$ & $q_{ik}$ represent the j'th and k'th coordinate ($j,k = x,y,z$) of the i'th atom and $\overline{q}$ is the corresponding average value.

The weighted covariance matrix is obtained from different weighting schemes for the atoms. Size schemes are proposed

1. The unweighted case where $w_i = 1$
2. atomic mass
3. van der Waals volume
4. Sanderson atomic electronegativity
5. atomic polarizability
6. electrotopological state indices

Depending on the weighting scheme different covariances matrices and hence different principal axes are obtained. Essentially the WHIM descriptors provide a variety of principal axes with respect to a defined atomic property. For each weighting scheme, a set of statistical indices are calculated on the atoms projected onto the principal axes (ie principal components).

Directional WHIM descriptors

directional WHIM descriptors - these are univariate statistical indices calculated on the scores of the individual prinicpal components

• directional WHIM size - these are the eigenvalues $\lambda_1, \lambda_2, \lambda_3$ of the weighted covariance matrix of the atomic coordinates and account for the molecular size along the principal axes
• directional WHIM shape - these are denoted as $v_1, v_2, v_3$ and are defined as
  
  $$v_m = \frac{\lambda_m}{\sum_m \lambda_m}$$
  
  
  where $m = 1, 2, 3$
• directional WHIM symmetry - these are denoted by $\gamma_1, \gamma_2, \gamma_3$ and are calculated as mean information content on the symmetry along each component wrt centre of the scores
  $$\begin{eqnarray*} \gamma^{'}_{m} & = & - \left[ \frac{n_s}{n} \cdot \log_2 \fr... ...} \right) \right] \\ \gamma_m & = &\frac{1}{1+\gamma^{'}_{m}} \end{eqnarray*}$$
  
  
  
  
  
  where $n_s, n_a$ and $n$ are the number of central symmetric (along the m'th component), unsymmetric and total atoms of the molecule.
• directional WHIM density - these are denoted by $\eta_1, \eta_2, \eta_3$ and are the inverse kurtosis calculated from fourth order moments of the scores ($\mathbf{t}_m$) and describes the atom distribution and density around the origin and along the principal axes. Thus
  $$\begin{eqnarray*} \kappa_m & = & \frac{\sum_i t^{4}_{i}}{\lambda^2 \cdot n} \\ \eta_m & = & \frac{1}{\kappa_m} \\ m & = &1, 2, 3 \end{eqnarray*}$$
  
  
  
  
  
  The the $\eta$'s relate to the quantity of unfilled space per projected atom - higher values of $\eta_m$ indicate larger values of unfilled space

This for each weighting scheme we get a set of 11 directional whim descriptors ($v_3$ is excluded since it is a linear combination of $v_1$ and $v_2$) giving a total of 66 directional WHIM descriptors

Global WHIM descriptors

These are calculated by combining the directional WHIM descriptors

• WHIM size - these consist of 3 descriptors representing the total dimensions of the molecule
  $$\begin{eqnarray*} T & = & \lambda_1 + \lambda_2 + \lambda_3 \\ A & = & \lambd... ..._1 \lambda_3 \\ V & = & T + A + \lambda_1 \lambda_2 \lambda_3 \end{eqnarray*}$$
  
  
  
  
  

• WHIM shape - this is defined as
  
  $$K = \sum_m \left\vert \frac{\lambda_m}{\sum_m \lambda_m} - \frac{1}{3} \right\vert / (4/3)$$
  
  
  where $m = 1, 2, 3$ and $0 \leq K \leq 1$. We only retain 3 $G$ descriptors ($Gu, Gm, Gs$) - that is those descriptors for the unitary, mass and electrotopological weighting schemes (since these are the only ones where the symmetry values are different)
• WHIM symmetry - this accounts for the total molecular symmetry and is defined as
  
  $$G = (\gamma_1 \gamma_2 \gamma_3)^{1/3}$$
  
  
  Thus $G = 1$ when the molecule shows a central symmetry along each axis and tends to 0 when there is a loss of symmetry along at least one axis
• WHIM density - describes the total density of atoms in a molecule and is defined as
  
  $$D = \eta_1 + \eta_2 + \eta_3$$
  
  

Thus for each weighting scheme there are 5 global WHIM descriptors plus the 3 symmetry descriptors giving a total of 33 global WHIM descriptors.