Synthon graphs

The extension of the DU-model, based on the graph theoretical notions and concepts, was developed by our group [16]. This new model of organic chemistry represented by the so-called synthon model, also serves as a theoretical basis for the implementation of the program PEGAS [17]. 

Starting from a given educt synthon, we can form a set of all its possible synthon precursors/successors (SPS) that are generated by simple reaction graphs. It means that an over-all chemical reaction, passing through several SPS can be expressed as a sequence of the above mentioned reaction graphs. 

The topic of the present section is a formulation of an alternative graph-theoretical model of synthons [16,24]. Such an approach makes it possible to build-up all concepts and notions of synthon theory [18-22] in a very transparent and easily understandable level. 

Each chemical transformation of two preselected synthons St(yl) and S2(/l) will be represented by the so called reaction graph [16,29] denoted by GR. 

The reaction distance [16,18,21,25] between two isomeric synthons Si (A) and S2(A) will be used as a proper tool for the construction of reaction graphs [29]. The reaction graph obtained corresponds to the minimal number of the so-called elementary chemical transformations, the number of which determines the reaction distance between the synthons SjfA) and S2(A). 

For a fixed family 3 (A) of isomeric synthons we construct the so-called graph of reaction distances [18, 21, 16, 25] denoted by RD(A). The vertex set of this graph is formally identical with the family A) without forbidden synthons, its two distinct vertices v and v, assigned to the synthons S(d) and S (/l), are connected by an edge [u, v ] if such an elementary transformation i = a, p exists so that the synthon S(/l) is transformed into the synthon i.e. 

Whether the graph Rd A) is connected or not, depends on the choice of forbidden synthons. When no synthons are forbidden, one can simply demonstrate [30] that the graph RD(A) is connected, between an arbitrary pair of synthons S(A), S (A) e S (A) there exists a finite path connecting them. 

Definition 2.3 The reaction distance D(S(/1), S (T)) between two synthons S(4) and S 04) is equal to the graph (topological) distance between those vertices that are assigned to them in %d A). 

Fig. 7. An example of subfamily J (A), where A = landO < s 3, it contains 14 synthons indexed by 1 to 14. The corresponding graph of reaction distances 9R0(A) is composed of 14 vertices (synthons)...

Sn(A). Applying the similar procedure for all synthons from the subfamily of A) we arrive at 9 XD(A) given in Fig. 7. From this graph one can simply evaluate the reaction distance for an arbitrary pair of synthons, e.g. S4(A)) = 6... 

The above outlined method can serve as an almost exact approach for the evaluation of reaction distance between two isomeric synthons S(A) and S (A). Its almost exactness follows from the fact that there can be no previously prescribed mapping of vertices and than the reaction graph is not unique. There can be constructed corresponding reaction graphs for every mapping and the reaction distance should be obtained as a minimum of minimal coverings of those reaction graphs. 

We shall assume a priori that the family, (A) has at least two stable synthons and the graph %D(A) is connected. As a consequence each synthon S(A) has a nonempty stable neighborhood, i.e. S(S(A)) 4= 0. 

The notion of a stable neighborhood of a synthon will be illustrated by using the subfamily J fA) displayed in Fig. 7, the corresponding graph of reaction distances is given in Fig. 8. Now we will assume that the synthons S3(A), S9(A), and Si 3(A) are the only stable synthons (they are denoted by encircled vertices), see... 

Fig. 11. Modified graph of reaction distances RD(A) from Fig. 7, now the synthons indexed by 3, 9, and 13 are declared as the only stable ones (encircled vertices)...

Since the paths and P2 contain as internal vertex the stable synthon S13b4), the stable synthon S3(T) is not an element of the stable neighborhood of the synthon S12(A). The stable neighborhood of S12(/l) is S(S12(/1)) = S9(/l), S13(v4). In other words, a stable neighbourhood of a synthon S(T) is composed only of those synthons S (A), for which there exists no shortest path between S( 4) and S ( 4) in graph %D(A) containing as an intermediate some stable synthon. 

Definition 2.5 A stable synthon S (A) is called synthon precursor successor (SPS) of the stable synthon S(A) iff the reaction graph GR of the transformation S(A) => S (A) is of a cyclic or linear form, see Fig. 12. Both forms must have alternating evaluation of edges by +/— and the linear graph must have terminal virtual vertices. The set of all possible SPS of the synthon S(A) is denoted by S(S(A)). 

Let us consider a stable synthon S (A) e S(S(A)), the transformation S(A) => S (A) is represented by a reaction graph either of the linear or cyclic form. In order to verify that the synthon S (A) belongs to the stable neighborhood of the stable synthon S(A) it is sufficient to verify that each connected subgraph of GR produces the unstable synthon. As this subgraph would be of linear form, at least one nonvirtual atom would gain or loose one valence electron and therefore its valence state would be unstable and the synthon S (A) produced should not be stable. 

Fig. 13. Schematic illustration of definition 2.6. In the first row a chemical transformation is displayed in which a synthon with two nonincident vertices from the reaction set X is transformed into another synthon with already incident vertices which were initially nonincident. In the second row a possible reaction graph is displayed...

A global picture of electronic changes between synthons of FIS(A) is given by the graph of FIS(A), GF[S(i4). It is defined analogically to the graph of elementary conversions of valence states, GECVs (see Sect. 3.3) as an ordered couple... 

It should be mentioned that the notion of reaction distance, as defined for valence states of atoms, can be used also for synthons. It means that RD(S(A), S (A)) is defined as the length of the shortest path between S(A) and S (A) in the graph Gfis[Pg.160]

Example 3.7 Let us consider synthons shown in Scheme 7. It can be seen from the graph that ... 

Given two isomeric synthons S(A) and S (A) construct all synthon-reaction graphs (SR-graphs) for the transformation... 

Balaban, A.T. (1980). Chemical Graphs. XXXVIII. Synthon Graphs. MATCH (Comm.Math. Comp.Chem.), 8,159-192. 

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