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

52 CHAPTER 6. COMPUTATIONAL RESULTS
from a mixture of formaldehyde and glycol. The first condensation step
yielding glycol from formaldehyde is assumed implicitly. The rewrite rule
used is a variant of the one described in fig. 4.3. In order to account for
cyclization, and so to reduce the network, carbon chains with more than four
members, or two chains with four members were not permitted to further
undergo aldol condensation. Furthermore, although more than 40 different
sugars have been identified in the experimental reaction mixture [24], condensation
was restricted to enoles with a primary acid hydrogen. Due to
this model reduction, the algorithm converged after only two iterations. The
Toy Model could be made more accurate by a Monte Carlo CRN generation.
It would have the same limits implicitly because unimolecular cyclization is
by far faster than bimolecular aldol condensation. This is equivalent to the
systematic model reduction described by [43].
6.3 Graph-theoretic properties
For assessing 〈L〉 and 〈C〉, we need to compare them to the results for random
Erdös-Renyi graphs:
〈L rand 〉 ≈ ln n
ln 〈k〉
〈k〉
〈C rand 〉 =
(n − 1)
Tab. 6.1 compares the network characteristics of the Diels-Alder, the
Formose, and the E. coli metabolic network. They are all sparse graphs, i.e.
they have much fewer edges than complete graphs, reflected by m ≪ n(n−1)
2
or 〈k〉 ≪ n. Sparse networks are very common, ranging from the network
of acquaintances to a neural network. In both cases, there are only few
connections at each node.
The small-world phenomenon [118, 119] is also widespread, and applies
for instance to the network of acquaintances. In the latter example, it describes
the fact that every person is acquainted to another over “six degrees
of separation”, in average. More generally, it means that the average shortest
path between two nodes in a network is small. An important application is
the propagation of diseases, where the small-world property leads to a rapid
spread. From the networks of tab. 6.1, only Diels-Alder and E. coli fulfill the
conditions 〈C〉 ≫ 〈C rand 〉 and 〈L〉 ≤ 〈L rand 〉 and thus are strictly small-world
networks. The Formose reaction network includes many non-reactive species
with respect to keto-enol condensation, which leads to small cliquishness and
longer paths.