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

44 CHAPTER 5. REACTION NETWORKS
corresponds to an “elastic collision”. Otherwise the server picks a half-rule at
random for the first graph and then a corresponding half-rule for the second
graph. This corresponds to choosing a reaction channel if there is more than
one subgraph isomorphism. Instead of picking a subgraph isomorphism at
random from the list, it is possible to consider all reaction channels and to
compute a reactivity index for each of them. The Toy Model can then pick
the reaction channel (pair of subgraph isomorphisms for the two half-rules).
The procedure for a unimolecular reaction is straightforward: the Toy Model
sends one molecule to GRW for which in analogy to the previous description
all subgraph isomorphism are constructed, and unless there is none, one
isomorphism is picked according to its reactivity.
Using the aforementioned algorithm, the CRN is likely to grow very fast.
This corresponds for example to the very complex network of elementary
reactions constituting a polymerization. The repetitive Diels-Alder networks
in fig. 6.1 suffers from this problem. However, there are techniques to simplify
such a network called model reduction techniques [43]. Reactions may
be removed from the CRN during or after generation, or formally “lumped”
together before, on the basis of e.g. energetic criteria. It is particularly efficient
to perform CRN generation and reduction simultaneously [33, 43],
as uninteresting reactions generate products that may further undergo uninteresting
reactions and so on. Detailed model reduction eliminates such
reactions before they can develop further. In the Toy Model, detailed reduction
is performed by selecting reactions according to their enthalpy, their
activation energy, or both.
5.2 Representation
A chemical reaction network (CRN) is a set of molecules linked together by
reactions leading from one subset to another.
A chemical reaction
aA + bB + · · · → vV + wW + · · · (5.1)
can be described as a directed hypergraph [131]. A hypergraph H(V, E)
is a set of vertices and a set of hyperedges, where vertices are elements v i
and hyperedges are pairs of lists of vertices ({u i }, {v i }) with u i ∈ ∏ V and
v i ∈ ∏ V defining connected vertices. The term directed specifies that the
order of vertices in the definition of an edge or a hyperedge is important. The
chemical species are the vertices. A reaction forms a hyperedge ρ ∈ E that
connects educts with products. A CRN is then represented by a hypergraph
with hyperedges for each reaction.