Keywords: What is described here?

1. Introduction

Chem(o)informatics is for developing models linking chemical structure and various molecular properties. In this sense cheminformatics relates to two other modeling approaches – quantum chemistry and force-field simulations. These three complementary fields differ with respect to the form of their molecular models, their basic concepts, inference mechanisms and domains of application. Unlike the molecular models used in quantum mechanics (ensembles of nuclei and electrons) and force field molecular modeling (ensembles of “classical” atoms and bonds), cheminformatics treats molecules as molecular graphs or related descriptor vectors with associated features (physicochemical properties, biological activity, 3D geometry, etc.) [Varnek2011]. The ensemble of graphs or descriptor vectors forms a chemical space in which some relations between the objects must be defined. Unlike real physical space, a chemical space is not unique: each ensemble of graphs and descriptors defines its own chemical space. Thus, cheminformatics could be defined as a scientific field based on the representation of molecules as objects (graphs or vectors) in a chemical space [Varnek2011].

Cheminformatics considers a molecule as a graph or an ensemble of descriptors generated from this graph. A set of molecules forms a chemical space for which the relationships between the objects themselves, on one hand, and between their chemical structures and related properties, on the other hand, are established using two main mathematical approaches: graph theory and statistical learning. Due to the rapidity of such calculations, these structure-property relationships can be applied to fast screening of large databases [Varnek2011].

2. Basic Statistical Methods

Among a multitude of descriptors currently used in Structure–Activity Relationships/Quantitative Structure–Activity Relationships/Quantitative Structure–Property Relationships (SAR/QSAR/QSPR) studies, fragment descriptors (application as atoms and bonds increments in the framework of additive schemes) occupy a special place [Baskin2008].

The epoch of QSAR (Quantitative Structure–Activity Relationships) studies began in 1963–1964 with two seminal approaches: the σ-ρ-π analysis of Hansch and Fujita [Hansch1963, Hansch1964] and the Free–Wilson method [Free1964]. The former approach involves three types of descriptors related to electronic, steric and hydrophobic characteristics of substituents, whereas the latter considers the substituents themselves as descriptors. Both approaches are confined to strictly congeneric series of compounds. The Free–Wilson method additionally requires all types of substituents to be sufficiently present in the training set. A combination of these two approaches has led to QSAR models involving indicator variables, which indicate the presence of some structural fragments in molecules.

In organic chemistry, decomposition of molecules into substituents and molecular frameworks is a natural way to characterize molecular structures. In QSAR, both the Hansch–Fujita [Hansch1963, Hansch1964] and the Free–Wilson [Free1964] classical approaches are based on this decomposition, but only the second one explicitly accounts for the presence or the absence of substituent(s) attached to molecular framework at a certain position. While the multiple linear regression technique was associated with the Free–Wilson method, recent modifications of this approach involve more sophisticated statistical and machine-learning approaches, such as the principal component analysis [Fleischer2000] and neural networks [Hatrik1996]. Disconnected atoms represent the simplest type of fragments. Usually, the atom types account for not only the type of chemical element but also hybridization, the number of attached hydrogen atoms (for heavy elements), occurrence in some groups or aromatic systems, etc. Nowadays, atom-based methods are used to predict some physicochemical properties and biological activities. Chemical bonds are another type of simple fragment. Topological torsions are defined as a linear sequence of four consecutively bonded non-hydrogen atoms. The above-mentioned structural fragments – atoms, bonds and topological torsions – can be regarded as chains of different lengths.

They are used to assess a chemical or biological property P in the framework of an additive scheme based on chainlike contributions:

(10.1)

where n_i is the number of atoms, bonds or topological torsions of i-type, C_i is corresponding chainlike contributions.

Hansch pioneered the use of descriptors related to a molecule’s electronic characteristics and to its hydrophobicity [Leach2007]. This led him to propose that biological activity could be related to the molecular structure via equations of the following form:

(10.2)

(10.3)

To find values for the intercept b and slope a can be done by minimizing the sum of the squared differences between the values predicted by the equation and the actual observations:

For more than one independent variable, the method is referred to as multiple linear regression (the term simple linear regression applies where there is just one independent variable) [Leach2007].

If the data (or the measurement error) have (multiple) normal distribution, R² of zero means that the variation in the observations is not at all explained by the variation in the independent variables; while R² of one means the perfect explanation. However in other data (or error) distribution, R² statistic can be misleading, because correlation and linearity will be not the same entity:

Cross-validation methods provide a way to try and overcome some of the problems inherent in the use of the R² value alone [Leach2007]. Cross-validation involves the removal of some of the values from the data set, the derivation of a QSAR model using the remaining data, and then the application of this model to predict the values of the data that have been removed [Leach2007]. The simplest form of cross-validation is the leave-one-out approach (LOO), where just a single data value is removed [Leach2007]. Repeating this process for every value in the data set leads to a cross-validated R² (more commonly written Q² or q²):

(10.4)

Q² value is normally lower than the simple R². If there is a large discrepancy then it is likely that the data has been over-fit and the predictive ability of the equation will be suspect. A more rigorous procedure is to use an external set of molecules that is not used to build the model [Leach2007].

3. Introduction to the Advanced Statistical Methods

For many compounds and many descriptors the property matrix X can be defined:

(10.5)

(10.6)

The principal components (PCs) can be considered as a new orthogonal coordinate system, the projection of the original data matrix X to this new axes can be given by the following equation:

(10.7)

(10.8)

(10.9)