Robustness and Modularity in Metabolic Networks Alexander Ullrich Bioinformatics University of Leipzig Bled, February 17
Robustness and Modularity
in Metabolic Networks
Alexander Ullrich
BioinformaticsUniversity of Leipzig
Bled, February 17
Metabolic Networks
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Metabolic Networks
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Robustness
• Ability to function despite changes
• Genetic Changes: Mutations,...
• Epigenetic Changes: Fluctuations in Molecule concetrations
• Complex Systems are highly robust
• Scale-free Networks are particularly robust
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Modularity
• Module: (structural) subsystem with distinct function
• Key organizing principle of biological networks
• High Clustering Coefficient suggests Modularity
• However, Origin and Preservation of Modularity notunderstood
• Changing Goals or Environments
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Motivation
• Biological systems develop desired properties• Robustness, Flexibility, Modularity, Evolvability, ...• Properties are connected
• Well studied, but their emergence is less well understood• Investigate the evolution of metabolic networks• Analyse network structure and metabolic functions
• Answers beyond analyzing real-world data
→ a multi-scale computational model of early metabolism→ appropriate measures for network properties
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Simulation
• Protocellular entity
• Bag of ribozymes
• Algebraic chemistry model
• Exchange of molecules with the environment
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Simulation - Overview
O O
flux v
B
flux v
C
flux vA
A
C
B
D
E
U U
UA
G
C G
CG
CC
GG
A
GG
UCC
UA
G
AA
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Simulation - Growth
O
O
O
O44.3
O
O
O
O
O
44.3
O
O
O
O
O
66.4
O
O
O
O
O
O66.4
N
N N
O
O
384.
O
O
O
O
O
O
O
70.0
O
O
O
O
O
O
O
O
70.0
N
N
N
N
O
O
25.4
N
N
O
O
O141. N
N
O
O
O
O
141.
O
O
O
O
O
O
66.4
O
O
O
O
O
O
O
66.4
O
O
O
O
O
O
O
66.4
O
O
O
O
O
O
O
O
66.4
N
N N
N
N
O
O
34.4
N
N N
O
O
O
329.
N
N N
O
O
O
O
329.
N
N
N
N
O
O
N
21.0
N
N
N
O
O
O
50.4
N
N
N
O
O
O
O
50.4
N
100
18.6
27.0
6.80
N
C−
−106
−92.
−185
O
46.3
O
O70.0
4.12O
O
−8.4
153.
70.0
70.0384.
52.8
N
N
O
O
O
4.12
O
O
O
O4.12
N
N N
41.7
N N
N
N
N
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
N
N N N
N
N
N N
N
N
N
N
N
N N
O
O
N
N
N
O
O
N
N
N
66.4
66.4
4.12
66.4
66.4
4.12
−2.4−2.4
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
cyanide, formaldehyde glycol; aldolcondensation, tautomerization
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Simulation - Stochastic Network Generator
Faulon, J-L, (2001) J Chem Inf Comput Sci 41:894-908
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General network analysis
• Connectivity Distribution• small vs big• early vs evolved
• Clustering Coefficient, Centrality, Entropy, ...• simulated vs real world
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after 10 Generations
1 10 100connectivity
0.0001
0.001
0.01
0.1
1fr
eque
ncy
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after 50 Generations
0.0001
0.001
0.01
0.1
1
freq
uenc
y
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after 100 generations
1 10 100connectivity
0.0001
0.001
0.01
0.1
1fr
eque
ncy
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after 250 generations
1 10 100connectivity
0.0001
0.001
0.01
0.1
1fr
eque
ncy
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after 500 generations
1 10 100connectivity
0.0001
0.001
0.01
0.1
1fr
eque
ncy
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Changing Environment - after 100 generations
1 10 100connectivity
0.0001
0.001
0.01
0.1
1fr
eque
ncy
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after 250 generations
1 10 100connectivity
0.0001
0.001
0.01
0.1
1fr
eque
ncy
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after 500 generations
1 10 100connectivity
0.0001
0.001
0.01
0.1
1fr
eque
ncy
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after 750 generations
1 10 100connectivity
0.0001
0.001
0.01
0.1
1fr
eque
ncy
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after 1000 generations
1 10 100connectivity
0.0001
0.001
0.01
0.1
1fr
eque
ncy
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Metabolic network analysis
We have sets of edges forming meaningful complex entities↓
pathways
• number of pathways → flexibilty
• change in case of single/multiple knockouts → robustness
• number of acceptable knockouts → robustness
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Metabolic Pathway Analysis
A
D
B C
F
E
R1
R6
R2
R7
R3 R4
R5
R8
R9
→
A
B
C
D
E
F
R1 R2 R3 R4 R5 R6 R7 R8 R90
B
B
B
B
B
@
1 −1 0 0 −1 0 −1 0 00 1 −1 0 0 0 0 0 00 1 0 1 −1 0 0 0 00 0 0 0 0 1 −1 0 00 0 0 0 0 0 1 −1 00 0 0 0 1 0 0 1 −1
1
C
C
C
C
C
A
←R1R2R3R4R5R6R7R8R9
↓
S ∗ vi = 0v1 v2 . . . vn−1 vn
0
B
B
B
B
B
B
B
B
B
B
B
@
2 −1 . . . 0 01 −1 . . . −1 −11 −1 . . . −1 −10 1 . . . 1 01 0 . . . 0 −10 0 . . . 1 20 0 . . . 1 20 0 . . . 1 21 0 . . . 1 1
1
C
C
C
C
C
C
C
C
C
C
C
A
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Metabolic Pathway Analysis
A
D
B C
F
E
R1
R6
R2
R7
R3 R4
R5
R8
R9
A
D
B C
F
E
R1
R6
R2
R7
R3 R4
R5
R8
R9A
D
B C
F
E
R1
R6
R2
R7
R3 R4
R5
R8
R9
A
D
B C
F
E
R1
R6
R2
R7
R3 R4
R5
R8
R9A
D
B C
F
E
R1
R6
R2
R7
R3 R4
R5
R8
R9
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Knockout effects
single R1 =
∑r
i=1 z i
r ∗ zdepletion R2 =
∑n
i=1 R i1
n
multiple R3(k) =
∑s(k)i=1 z i
s(k) ∗ zoverall R3(≤ K ) =
K∑
k=1
R3(k)pk
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Minimal Knockout sets
A
D
B C
F
E
R1
R6
R2
R7
R3 R4
R5
R8
Target
A
D
B C
F
E
R1
R6
R2
R7
R3 R4
R5
R8
Target A
D
B C
F
E
R1
R6
R2
R7
R3 R4
R5
R8
Target A
D
B C
F
E
R1
R6
R2
R7
R3 R4
R5
R8
Target
{R1} {R5, R6} {R3, R4, R8}
NO knockout sets
A
D
B C
F
E
R1
R6
R2
R7
R3 R4
R5
R8
Target A
D
B C
F
E
R1
R6
R2
R7
R3 R4
R5
R8
Target
{R2, R3, R4, R5} {R6, R7, R8}
Knockout set size distribution → Robustness (bigger is better)
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after 10 generations
0 5 10 15 20 25 30knockout set size
0
0.05
0.1
0.15
0.2
0.25
freq
uenc
y
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after 20 generations
0 5 10 15 20 25 30knockout set size
0
0.05
0.1
0.15
0.2
0.25
freq
uenc
y
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after 50 generations
0 5 10 15 20 25 30knockout set size
0
0.05
0.1
0.15
0.2
0.25
freq
uenc
y
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after 100 generations
0 5 10 15 20 25 30knockout set size
0
0.05
0.1
0.15
0.2
0.25
freq
uenc
y
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after 250 generations
0 5 10 15 20 25 30knockout set size
0
0.05
0.1
0.15
0.2
0.25
freq
uenc
y
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after 500 generations
0 5 10 15 20 25 30knockout set size
0
0.05
0.1
0.15
0.2
0.25
freq
uenc
y
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Changing Environment - after 10 generations
0 5 10 15 20 25 300
0.05
0.1
0.15
0.2
0.25
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after 50 generations
0 5 10 15 20 25 300
0.05
0.1
0.15
0.2
0.25
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after 250 generations
0 5 10 15 20 25 300
0.05
0.1
0.15
0.2
0.25
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after 500 generations
0 5 10 15 20 25 300
0.05
0.1
0.15
0.2
0.25
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after 1000 generations
0 5 10 15 20 25 300
0.05
0.1
0.15
0.2
0.25
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Chemical Organizations
Self-maintaining and closed sets of molecules and reactions↓
chemical organizations
• Hierarchies of organizations
• Shape of Hierarchies → robustness
• Size distribution of organizations → robustness, modularity
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Chemical Organizations
0
1 2 4
3 6 7 5
8 9 10
11
0
1 2 4
3 5 6 11
7 13 8 10
9 12
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Level Size Distribution - after 500 generations
0 2 4 6 80
0.2
0.4
0.6
0.8
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Chemical Organizations
0
1 2 3 19
4 5 18 6 17 16
7 14 15 12 13
9 11 10
8
0
1 2 19
4 18 17 16 12
15 13 9 14 10 7
11 8 6 5
3
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Changing Environment - after 500 generations
0 1 2 3 4 5 6 7organizations per level
0
0.1
0.2
0.3
0.4fr
eque
ncy
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Organization Size Distribution - after 500 generations
0 2 4 6 80
0.2
0.4
0.6
0.8
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Changing Environment - after 1000 generations
0 1 2 3 4 5 6 7level
0
0.1
0.2
0.3
0.4re
lativ
e or
gani
zatio
n si
ze
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Work in Progress
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FBA
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Flux barrier analysis
• linear optimization: EMs modeled as system of linearequations
• constraints: limits on reactions, exclusion of combinations ofEMs
• barrier tree
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Reaction barrier analysis
• linear optimization: stoichiometrix matrix
• constraints: limits on reactions, exclusion of combinations ofreactions
• barrier tree
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Flux similarity
• Compute pairwise similarity of elementary modes
• similarity between metabolites (in+out / all) throughtopological indices
• similarity between enzymes/reactions by comparing transitionstate structure
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Conclusion
• Summary• Structural and Functional measures for Robustness and
Modularity• Follow the Law (Connectivity Distribution)• Size Matters (Knockout set Size)• Shape too (Chemical organization Hierarchy)
• Outlook• Investigate single networks (flux barriers, flux similarity)• Different scenarios (Horizontal Gene Transfer)• Structural modularity (Clustering Coefficient)
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Acknowledgements
Christoph Flamm
Peter Stadler
Konstantin Klemm
Martin Mann
Markus Rohrschneider
Peter Dittrich
Dennis Goerlich
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