Molecular Quantum Similarity in QSAR and Drug Design

Molecular similarity attempts to give a quantitative answer to the question: how similar are two molecules? It is clear that this is an interesting problem, and that it has no unique answer. The possible solutions will be associated to the type of molecular aspect that one wants to analyze. Due to the fact that molecules are objects ruled by the laws of quantum mechanics, it seems that one of the satisfactory answers to the question ought to be found within this specific discipline. Following this line of thought, the first quantitative measure of the similarity between two molecules, based on quantum-mechanical basic elements, was formulated by Carbó in 1980 [1]. Carbó proposed that a numerical comparative measure between two molecules could be derived from the superposed volume between their respective electronic distributions. This original definition still holds, and constitutes the fundamental tool of the present work. The seminal idea was developed by this author and collaborators [2‐6], and the present state-of-the-art can be obtained from various review articles [7‐10]. These papers deepen in the quantum-mechanical nature of the definition, connect it with several subjects of chemical and mathematical interest and show a broad amount of possible applications.

In this chapter, the elementary basis of Quantum Similarity framework is presented in an elementary way. Here, by defining in a rigorous way the concept of quantum object, the quantum mechanical concept of Quantum Similarity is described. This leads to a discussion about the role of density functions in the chemical description of molecular structures. Some definitions -tagged, Boolean and functional tagged sets, as well as vector semispaces- are previously introduced, in order to produce the adequate formalism from where Quantum Similarity can be easily deduced and after this computational algorithms can be developed.

One of the most progressing subjects in present-day chemistry is the establishment of quantitative relationships between biological or pharmacological properties and molecular structure. This topic has become a solid subject matter, usually known as quantitative structure-activity relationships (QSAR). Since Hansch and Fujita [142] performed the pioneering studies on QSAR, the advances in this matter have not ceased. The predictive capabilities of the earliest models were substantially improved when 3D structural descriptors were introduced, providing a powerful alternative to the use of extra-thermodynamical parameters in QSAR studies [143]. In addition, the definition of different quantitative similarity measures between two molecules proved a great aid in order to a source of 3D QSAR parameters acting as molecular descriptors.

In this chapter, a scheme of the application of molecular quantum similarity matrices to describe a molecular property of interest is exposed. Quantum similarity matrices need to be conveniently transformed when employed as descriptor source in QSAR procedures. In order to describe the usual transformations, dimensionality reduction and variable selection techniques will be discussed. Combination of different quantum similarity matrices, constituting the Tuned QSAR model, is also discussed. Since the only relevant test for the procedure protocol is its application on real cases, quantum similarity matrices will be used to study three different molecular sets in order to provide the reader with reliable quantitative equations for activity prediction.

In this chapter, the expectation value of the interelectronic repulsion energy operator, 〈V_ee〉 is presented as a kind of QS-SM, which consequently can be used as a molecular descriptor in QSAR applications. The efficiency of this parameter in QSAR for different molecular sets will be here examined.

So far, Quantum Similarity has only been applied to a chemical environment. However, its theoretical basis is flexible enough to allow a satisfactory extension to other quantum objects sets. In this chapter, application of Quantum Similarity to atomic nuclei and its use in quantitative structure-property relationships are discussed.