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2
Generic Chemical Reaction Network Properties
Gil Benko
Institute for Theoretical Chemistry and Structural Biology
Bled, 2003
3
Why ?
Combinatorial Atmospheric Combustion Metabolicchemistry processes networks
Chemical reaction networks
4
What ?
What are the generic properties of a CRN :
Are average minimum path lengths short ?(small-world property)
What is the scaling behavior of the graph?
How robust are these properties?
5
Organization of the Toy Model
ODE Stochasticsimulation simulation
↘ ↙CRN
Tankl
reactor↗ ↖
EHMO Graphcalculation rewriting
6
Electronic Energy Calculation
Schrodinger equationHΨ = EΨ
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
�
Born-Oppenheimer
LCAO and Extended Huckel
VSEPR and Tight Binding
Generalized Eigenvalue Problem
. . .HAOi−AOj
. . .
C =
. . .SAOi−AOj
. . .
CE
7
Atom Orbitals
sp3 hybridized sp2 hybridized + one p
sp hybridized + two p8
Implemented Overlaps
big σ-overlap lesser overlap
between sp2 orbitals between sp2 orbitals
9
Orbital graph
2sp2
2sp222sp
2sp2
2sp2
2sp2
2sp2
2sp2
2sp2
2sp2 2sp2
2sp2
2sp2
2sp222sp
p
p
p
p
p
C
C
O
N
C
1s
1s
1s
H
H
H H
H
1s 1s
10
Overlap matrix S of propenamide
C1(H2)(H3) = C4(H5)C6(= O7)N8(H9)H10
� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
�� � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � �� � � �� � � �� � �� � � � � � �
� � �� � � � � �� � � � � � � � � � � � � � � � � � � � �
� �� � � � � � � � � � � � � � � � � � � � � � � � �
� � � � �� � � �� � � � � � � � � � � � � � � � � � � � �
� � � �� � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � �� � � � �� � � � � � � � � � � � � � � � � �
� �� � � � �� � � � � � � � � � � �� � � � � � � � � � � � � � � � �
� � � � � �� � � � � �� � � �� � � � � � � � � � � � � � � � �
� � � � � � � �� � � � � � � � � � � � � � � � � � �
� � � � � �� � � � � � � � � � �� � � � �� � � � � � � � � � � � � �
� � � � � � �� � � �� � � � � � � � � � �� � � �� � � � � � � � � � � � �
� � � � � � � � � �� � � � � �� � � �� � � � � � � � �� � � �� � � � � �
� � � � � � � � � � � �� � �� � � � �� � � � � � � � � � � � �
� � � � � � � � � �� � � � � � �� � � � � � �� � � � �� � � � � � � � � �
� � � � � � � � � � �� � � �� � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � �� � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � �� � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � �� � � � � � � � � � � � � � �
� � � � � � � � � � � � �� � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � �� � � �
� � � � � � � � � � � � � � � � � � � � � �� � � �� � �
� � � � � � � � � � � � � � � � � � � � � � �� � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � �
� � � � � � � � � � � � � � � � � � � � � � � � � � �
11
Wave function analysis
By definition, any molecular property can be calculated from the
wave function:
• Energy
• Charge distribution
• Reactivities
12
1520 1530 1540 1550Experimental TAE (kcal/mol)
1550
1560
1570
1580
1590
1600
1610
Cal
cula
ted
TA
E (
kcal
/mol
)
Calculated vs Experimental TAE of C6H10 isomers(Total Atomization Energy)
13
Structure representation I : SMILES;)
CH4 CH3 CH3
C CC
or C(H)(H)(H)H or C(H)(H)(H)C(H)(H)(H)
CC(C)C orC(H)(H)(H)C(H)(C(H)(H)(H))C(H)(H)(H)
OH
OC(C = C1) = CC = C1 orHOC(C(H) = C1H) = C(H)C(H) = C1H
14
Structure representation II : GML
# Isobutane
graph [
node [ id 1 label "C" ]
node [ id 2 label "C" ]
node [ id 3 label "C" ]
node [ id 4 label "C" ]
edge [ source 1 target 2 label "-" ]
edge [ source 1 target 3 label "-" ]
edge [ source 1 target 4 label "-" ]
]
15
What is Graph Rewrite ?
A grammar which operates on graphs instead of strings.
A grammar is a finite set of rules describing a formal language.
A formal language is a set of strings over some fixed alphabet.
Graph Rewriting Step
Step 1: find isomorphic subgraph (match left graph).
Step 2: remove subgraph (don’t touch context; keep dangling ends!!).
Step 3: insert new subgraph (right graph).
Step 4: rewire new subgraph (with respect to the dangling ends).
16
Graph rewriting steps
left graph context right graph
host graph product graph
find left graph and context reconnect
remove left graph insert right graph
17
Network Generation
Start perform all unimolecular and bimolecular reactions on the
molecules in L0 and put the products in a new set L′1, eli-
minating all duplicates. This is summarized by the notation
L′1 = L0 ⊗ L0. Calculate L1 = L′
1 \ L0.
Recursion (1) L′k+1 =
(
⋃k−1j=0 Lj
)
⊗ Lk ∪ (Lk ⊗ Lk)
(2) and Lk+1 = L′k+1 \
⋃
Lk.
18
Network Representation
Hypergraph and bipartite graph
for O3 + NO2 −→ O2 + NO3
3O
NO2
O2
NO3
3O
NO2
O2
NO3
one-mode projection −→ Substrate graph
3O
NO2
O2
NO3
19
Application I : Repetitive Diels-Alder
1.44
O
O
O
O
OO
O
O
O
O
OO
O
O
O
O
OO
O
O
O
O
O
O
O
OO
O
O
O
O
O
O
O
O
O
O
OO
O
O
O
O
OO
O
OO
O
OO
O
OO
O
O
O
O
OO
O
O
O
O
OO
O
O
O
O
OO
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
3.721.33
3.37
5.71
2.13
1.17
1.45
1.17
4.46
4.47
2.66
2.66
1.44
1.17
1.17
1.17
4.46
4.47
4.46
4.47
4.464.47
2.66
2.66
2.66
2.66
2.66
2.66
1.44
1.44
1.44
1.44
1.44
1.44
20
Application II : Formose reaction
O
O
O
O
O
O
O
O
O
O
O
O
O O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
OO
OO
O
OO
OO
O
O
O
OO
O
O
O
O
O
O
O
O
O
O
O
O
O
OO
O
O
O
O
OO
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
OO
OO
O
O
O
OO
OO
O
O
O
O
O
OO
O
O
O
OO
OO
O
O
O
OO
OO
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
0.04
0.36
0.03
0.03
0.26
0.26
0.36
4.17
0.36
4.18
0.26
0.26
2.51
0.26
2.51
2.51
21
Graph properties
• < k >= 2 edgesnodes is the average node degree.
• < L > is the average length of the shortest path between to
nodes.
• Ci = 2 edges between i-neighboursi-neighbours(i-neighbours−1)
is the clustering coefficient.
22
Computational results
For comparing: results for random Erdos-Renyi graphs
< Lrand > ≈lnn
ln< k >
< Crand > =< k >
(n − 1)
nodes < k > < L > < Lrand > < C > < Crand >
Diels-Alder 40 4.65 2.15 2.40 0.72 0.11Formose 48 3.25 3.55 3.28 0.15 0.068E. coli 282 7.35 2.9 3.04 0.32 0.026
23
Degree distribution
Repetitive Diels-Alder Formose reaction
1 10 100Degree
1
10
100
Ran
k
1 10Degree
1
10
100
Ran
k� : datapoints
solid : power-law regressiondashed : Poisson distribution
24
Threshold of 112
25
Threshold of 108
26
Threshold of 107
27
Threshold of 91.2
28
Threshold of 55.4
29
A larger CRN . . .
0 50 100 150 200 250Number of nodes
0
1
2
3
4
5
6
Ave
rage
deg
ree
0 50 100 150 200 2500
20
40
60
80
100
120
log(
rate
thre
shol
d)
< k > and reaction rate threshold vs. CRN size n
30
. . . with constant minimum path length < L > . . .
0 50 100 150 200 250Number of nodes
1
2
3
4
Ave
rage
min
imum
pat
h le
ngth
0 50 100 150 200 250
CRN size dependency of < L > and < Lrand >
31
. . . and clustering coefficient < C >
0 50 100 150 200 250Number of nodes
0
0.2
0.4
0.6
0.8
1
Ave
rage
clu
ster
ing
coef
fici
ent
CRN size dependency of < C > and < Crand >
−→ robust small-world property
32
