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

Chapter 5
Reaction networks
5.1 Network generation
The starting point for generation a CRN is an initial set of molecules L 0 .
At the present stage of development, the CRN is built exhaustively from the
network “seed” L 0 . The algorithm used can be termed “orderly generation”
for its resemblance to an algorithm of this name used for enumeration of
isomers [93, 33]. It generates the CRN by the following recursion:
Start (1) perform all unimolecular reactions on each molecule M ∈ L 0 and
put the products in a new set L ′ 1, eliminating all duplicates.
(2) perform all bimolecular reactions with each pair of molecules
(M 1 , M 2 ) ∈ L 0 × L 0 and add the products to set L ′ 1 , eliminating again
all duplicates. The last two steps are summarized by the notation
L ′ 1 = L 0 ⊗ L 0 . L i ⊗ L j means that step 1 is applied only to the first
operand L i , and step 2 on pairs (M 1 , M 2 ) ∈ L i × L j .
(3) this first iteration is completed by calculating L 1 = L ′ 1 \ L 0 .
(
Recursion (1) L ′ k+1 = ⋃k−1
)
j=0 L j ⊗ L k ∪ (L k ⊗ L k )
(2) and L k+1 = L ′ k+1 \ ⋃ L k .
Let us now consider the rewriting step for a bimolecular reaction in detail.
First the Toy Model sends the two educt graphs to the GRW. The server
then constructs all subgraph isomorphisms for the left hand side of both
half-rules for both graphs. If the list of subgraph isomorphisms for one of
the two half-rules is empty for both graphs, the rule is not applicable and
the server sends the two graphs unaltered back to the Toy Model. This case
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