Overview

1
Amino acids
Nucleotides
Lipids fatty acids
Cofactors
Precursors
Carriers
Amino acid
Catabolism
Lipids fatty acid
Cofactor
Nucleotide
2
32
Other metabolites
86
Nucleotides
Vitamins
106
Other metabolites
Catabolism
Sugars
Lipids
138
Amino Acids
158
Precursors
190
205
Enzymes
Carriers
267
Precursors
276
289
Carriers
Vertical decomposition
58
133
190
240
Reactions
3
32
86
106
138
Amino acid
158
Nucleotide
Lipid fatty acid
190
205
Cofactor
267
276
289
58
133
190
240
Catabolism
Biosynthesis
Nucleotides
Vitamins
Catabolism
Sugars
Lipids
Horizontal decomposition
Amino Acids
Precursors
4
Metabolites
Carriers
33
3
Catabolism
10
Precursors
65
all metabolites
78
Rank
2
10
Amino acids
132
152
Carriers
184
Nucleotides
1
10
204
Lipids fatty acids
236
251
0
10
1
10
100
Cofactors
313
58
133
190
240
Number of reactions
H. Pylori
Reactions
5
All Metabolites
3
10
2
10
Rank
10
1
1
2
10
10
1
Number of reactions
6
Carriers
Catabolism
Precursors
Amino acids
Nucleotides
Lipids fatty acids
Cofactors
7
Carriers
Catabolism
Precursors
Reactions
Amino acids
Nucleotides
Lipids fatty acids
Cofactors
8
Metabolites
Carriers
33
3
Catabolism
10
Precursors
65
all metabolites
78
Rank
2
10
Amino acids
132
152
Carriers
184
Nucleotides
1
10
204
Lipids fatty acids
236
251
0
10
1
10
100
Cofactors
313
58
133
190
240
Number of reactions
H. Pylori
Reactions
9
Reactions
3
10
All Metabolites
2
10
Rank
10
1
1
2
10
10
1
Number of reactions
10
Power laws are ubiquitous

• This is no surprise, and requires no special
  explanation.
• Gaussians (normal) distributions are attractors
  for averaging (e.g Central Limit Theorem) so are
  also ubiquitous.
• Power laws are attractors for averaging too, but
  are also the only distributions invariant under
  maximizing, marginalization, and mixtures.
• More normal than Normal.

11
Power laws are normal

• Power laws are indeed ubiquitous, but not exotic
• Just like Gaussians, existence of power laws is
  not evidence for any particular mechanism (other
  than the strong statistical invariance
  properties)
• Easy to get statistical analysis wrong

12
All Metabolites
3
10
Others
2
10
Carriers
Rank
10
Precursors
1
1
2
10
10
1
Number of reactions
13
3
10
All Metabolites
2
Exponential
10
Others
Rank
Carriers
Exponential
10
Precursors
Exponential
1
1
2
10
10
0
20
40
60
1
Number of reactions
Number of reactions
14
All Metabolites
3
10
2
10
Others
Rank
Carriers
10
Mixture
Precursors
1
1
2
10
10
1
Number of reactions
15
All Metabolites
3
10
2
10
Others
Rank
Carriers
10
Scale-rich Self-dissimilar
Precursors
1
1
2
10
10
1
Number of reactions
16
Sugars
2
10
Amino Acids
Carriers
Nucleotides
Precursors
Rank
Fatty acids
Co-factors
1
10
Carriers
0
10
0
1
2
10
10
10
Number of reactions
In biosynthetic pathways
17
2
2
10
10
Carriers
Rank
Rank
? exponential
1
1
10
10
0
0
10
10
0
1
0
20
40
60
10
10
Number of reactions
Number of reactions
18
Reactions
3
All Metabolites
10
2
10
Others
Rank
Carriers
10
Precursors
1
1
2
10
10
1
Number of reactions
19
?
Polymerization
Taxis and transport
Proteins
Core metabolism
Sugars
Catabolism
Amino Acids
Nucleotides
Precursors
Nutrients
Trans
Fatty acids
Genes
Co-factors
Carriers
DNA replication
20
Amino Acids
Precursors
3
10
Carriers
2
10
All Metabolites
Others
Rank
Carriers
10
Carriers in groups
Precursors
1
0
5
10
1
2
10
10
1
Number of reactions
Number of reactions
21
1
catabolism precursors
12
23
amino acids biosyn
50
61
carriers
21
22
Emergence

• Edge-of-chaos
• Order for free
• Power laws
• Self-organized criticality
• Phase transitions
• Scale-free networks

Discard the extremely unlikely

• Models
• Cellular automata
• Boolean networks
• Lattices
• Spin glasses
• Graphs

General idea all of these emergent features are
invariant under random rewiring, provided their
macroscopic features are preserved (degree,
densities, etc)
23
The SF/SOC/EOC approach
Assume modules as given to start with, in this
case metabolites.
Metabolites
3
10
all metabolites
Rank
2
10
Try to reproduce statistics as emergent
phenomena of random ensembles, with minimal
tuning.
Carriers
1
10
0
10
1
10
100
Number of reactions
24
Random rewiring, even preserving all macroscopic
features, destroys functionality
21
25
NAD
ADP
NADP
COA
AMP
THF
PI
CO2
PPI
NH3
AC
H2S
NADH
ATP
NADPH
ACCOA
ATP
MTH

• Preserve
• degree
• carrier
• enzyme

Carriers
amino acids
precursors
Wild type
scale-free
26
Simplest possible model

• What do we need to extract from metabolism to
  create high variability in total metabolite
  degree?
• Then invariance of power laws will make them
  likely in practice.
• Features low degree enzymes and common carriers

27
Trivial model assumptions

• General minimal assumptions
• Metabolites have two features relative to
  reaction modules
• Local (other) versus global (prec-, carrier)
• High (carrier), medium (prec), low (other)
  degrees in modules
• Enzymes have few substrates
• 1-3 others or precursors
• 1-3 or more carriers

28
Simplest possible HOT model

• Model minimal assumptions
• Two kinds of metabolites
• Many local, each occurs in exactly 1 reaction
• One global carrier, occurs in every reaction
• Enzymes have 2 substrates
• 1 local other
• 1 global carriers
• Dont worry about reversibility, this is very
  abstract
• Can easily generalize to any distribution, once
  you see the basic idea

29
Simplest possible HOT s-matrix
r reactions
CVs
Others
Carrier
This holds no matter what the size of the modules
or the number of reactions.
Module 1
Module 2
30
Compute CV for r reactions
31
Discussion

• High variability created by mixing a few high
  degree global carriers with many low degree
  others
• Precursors and spread of degrees creates power
  laws
• High variability is more fundamental than power
  laws