A Toy Model of Chemical Reaction Networks - TBI - Universität Wien

4.1. GRAPH REWRITING 37
Graphical transformations can be described in terms of graphical pre- and
post-conditions. It is possible to use a rule-based notation for this transformation.
Such a rewrite rule is a tuple r = (g l , g r , M), where the graph g l is
the left-hand side, g r is the right-hand side, and M is the set of embedding
descriptions (important for the dangling ends). The implementation used in
the Toy Model defines g l and g r by the elements of the graph no longer and
only present after the transformation, respectively, and by a context which
contains the constant element, see figs. 4.1 and 4.2.
The execution of the rewrite rule is decomposed into four steps:
• find an isomorphic subgraph (to g l )
• remove that subgraph (keeping the dangling ends in M)
• insert a new subgraph (g r )
• connect it to the rest of the graph (respecting the dangling ends from
M)
This graph rewriting formalism is very flexible and can be used to represent
chemical reactions as well as chemically impossible yet strategically
interesting reactions. It may be interesting to use scaffold replacement rules
for building a library in combinatorial chemistry, or to simulate deprotonation
implicitly, but in reality, chemical reactions do not create or destroy
atoms. A chemical reaction is the breaking, forming and changing of bonds.
Thus the number and type of atoms must remain constant, which can be
implemented by conservation of vertex labels. In analogy, the conservation of
the number of valence electrons can be imposed on rewrite rules by ensuring
conservation of total bond order. Both principles stem from the fact that
chemical reactions are stoichiometric [31, 113]:
• conservation of vertex labels : V l = V r
• conservation of total bond order : ∑ e∈E l
BO(e) = ∑ e∈E r
BO(e),
where V i and E i are the vertices and the edges of g i , and BO(e) is the bond
order of the edge e. In the example of the Diels-Alder reaction in fig. 4.2, the
first condition is met because all atoms are in the constant set, the context.
Counting the bond orders of g l (2+2+1+2) and g r (1+1+1+2+1+1) verifies
that the second condition is also satisfied, thus this rewrite rule is chemically
meaningful.
This reaction representation has the advantage of representing the reaction
itself also by graphs, and thus does not have the inherent limitations