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One of the most interesting fields of mathematically oriented chemical research is the so-called computer-assisted organic synthesis design. These lecture notes elaborate the mathematical model of organic chemistry, which offers formal concepts for unambiguous description of computer algorithms for organic synthesis design including retrosynthesis and reaction mechanisms. All definitions and theorems are supplemented by many illustrative examples. The model is closely related to the course of thinking of organic chemists. These notes will be useful for all theoretically oriented organic chemists who are interested in mathematical modelling of organic chemistry and computer-assisted organic synthesis design.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
During recent decades hundreds of thousands of new organic compounds have been synthesized every year, each of which may react with many others. This “jungle” of organic compounds continues to grow unremittingly. Therefore, new methods leading to better orientation in this “jungle” should be of great importance for introducing a logic and order into organic chemistry. Mathematics and mathematical models are now playing a principal role in these attempts. In general, the models may be classified as physical and nonphysical, where the difference between them is, however, of a very relative nature and the trend is towards a physicalization of nonphysical models. A typical representative of physical models is quantum chemistry. The nonphysical models are usually based on discrete mathematics, employing, in particular, graph theory, many different algebras, and group theory.
J. Koča, M. Kratochvíl, L. Matyska, V. Kvasnička, J. Pospíchal

2. The Molecular Graphs

Abstract
The purpose of this chapter is to present basic concepts of graph theory [1] as applied to organic chemistry [2–6]. The importance of graph theory in this large branch of chemistry consists in the existence of the phenomena of the structural formula and its isomerism. Organic chemists have unconsciously used graph theory concepts for more than one hundred years, just as Molière’s “bourgeois gentilhomme” was speaking prose without realizing it (cited after Ba-laban [7]).
J. Koča, M. Kratochvíl, L. Matyska, V. Kvasnička, J. Pospíchal

3. S-Graphs and Synthons

Abstract
The concepts of a graph and molecular graph were introduced in the previous chapter. These terms will be generalized in such a way that some vertices are distinguished from others and are called the virtual vertices. We remember that in graph theory [1] such a process is called rooting and the distinguished vertices are called the roots. In the original approach employing the molecular graphs the concept of chemical transformation is determined for pairs of graphs that are isomeric, i.e. in the course of the chemical transformation the numbers of atoms and valence electrons are conserved; the chemical transformations are strictly stoichiometric. This prominent feature of the theory may cause some formal difficulties should we wish to implement in the model the fruitful idea of general organic chemistry, in particular, the notion of the synthon. The description of chemical transformations may be limited to so-called synthons that represent the necessary minimal fragment of the molecular system for the given chemical transformation. The concept of synthon was initially formulated by Corey [2] in 1967 for the purposes of computer-assisted organic synthesis design.
J. Koča, M. Kratochvíl, L. Matyska, V. Kvasnička, J. Pospíchal

4. The Applied Synthon Model

Abstract
In the preceding chapter we have defined the notion of a synthon. Its main difference from the molecular graph (Chap. 2) is that some of its vertices are distinguished from others by virtue of being so-called virtual vertices. In our forthcoming considerations we shall often use the term atom instead of the mathematical term vertex. The virtual vertex, or atom, represents a “virtual” functional group or a molecular fragment irrelevant for the chemical transformations of interest. Let us consider a synthon constructed over an atomic set A, this synthon is chemically described by a structural formula in which some bonds are not terminated by atoms. Loosely speaking, they are sticking out of the structural formula and formally terminate in virtual atoms. A synthon constructed over the atomic set A will be denoted by the symbol S (A).
J. Koča, M. Kratochvíl, L. Matyska, V. Kvasnička, J. Pospíchal

5. Mathematical Model of Synthon Reactions

Abstract
The synthon matrices and synthon graphs are well-developed formal tools for the description of synthons as a static objects. For the study of conversions of synthons, that is for the dynamic part of synthon model, the basic notion is that of the isomerism of synthons.
J. Koča, M. Kratochvíl, L. Matyska, V. Kvasnička, J. Pospíchal

6. The Synthon Model and Organic Synthesis

Abstract
Chemical synthesis is one of the most important tasks both of pure as well as of applied organic chemistry. Since the middle of the 1960s, attempts have been made to explore the initial stages of organic synthesis design (e.g. searching for the most effective synthetic steps to prepare the target compound from the available substrate and reagents) using computers. This so-called computer-assisted organic synthesis formed a new branch of theoretical (or rather computational) chemistry. There are three basic approaches of interest in this field: (i) Synthesis in the forward direction, (ii) retrosynthesis, and (iii) reaction mechanisms of particular reactions. The third of these items has already been discussed in Chap. 4.
J. Koča, M. Kratochvíl, L. Matyska, V. Kvasnička, J. Pospíchal

7. Conclusions

Abstract
The purpose of this paper was to elaborate a mathematical model of organic chemistry potentially applicable as a basis for computer-assisted organic synthesis design. Particular attention was devoted to the close correspondence of the proposed model with the thinking of organic synthesis chemists, since the model has been built up mainly as a basic methodological starting point for the implementation of computer programs processing the design of organic synthesis.
J. Koča, M. Kratochvíl, L. Matyska, V. Kvasnička, J. Pospíchal

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