九大数理集中講義
Comparison, Analysis, and Control of
Biological Networks (6)
Boolean and Flux Balance Analyses of
Metabolic Networks
Tatsuya Akutsu
Bioinformatics Center
Institute for Chemical Research
Kyoto University
Contents






Boolean Model of Metabolic Networks
Impact Degree
Flux Balance Analysis
Elementary Mode Analysis
FBID: Flux-Balance Impact Degree
Summary
Boolean Model
of Metabolic Networks
Boolean Model of Metabolic Networks

Metabolic Network


Set of chemical reactions in a cell
Two kinds of nodes

Compound node： OR of inputs


Active if at least one of input nodes is active
Reaction node： AND of inputs

A
Active only if all input nodes are active
AND
C
D
B
A+B→C+D
OR
Minimum Knockout Problem

A measure of robustness: the minimum number of
knockout reactions to inactivate production of the
target compound
Robustness=2
Target compound
Robustness=１
[Tamura, Takemoto & Akutsu、IJKDB, 2010]
Utilization of Integer Linear Programming
Minimum Knockout: NP-hard
⇒ Use of Integer Linear Programming







Optimization of linear objective function
Under linear inequalities
Under integer constraint
⇒ Each variable takes either 0 or 1 (for BN)
ＡＮＤ
z  x AND y
z  x, z  y , z  x  y  1
OR
z  x OR y
z  x, z  y, z  x  y
NOT
z  NOT x
z 1  x
Comparison of Robustness via Minimum Knockout
• Use of a KEGG pathway combining Glycolysis, Citrate
cycle (TCA cycle), Pentose phosphate
• Robustness: minimum number of knockout reactions to
inactivate production of important target compound(s)
C00022
C00024
C00033
C00036
C00074
Pyruvic
acid
AcetylCoA
Acetic
acid
Oxaloacet
ic acid
Phosphoenol
pyruvic acid
All five
compound
s
Human（hsa）
2
2
2
4
3
4
Yeast （sce）
2
1
1
3
2
3
Archaea（hal）
1
2
2
2
0
2
E.coli（eco）
2
2
2
3
2
4
•
Robustness of E.coli is similar to human: it may because eukaryotes do not have
any metabolic pathways such as the ED pathway and the ascorbate metabolism
Impact Degree
Impact Degree

Proposed by Jiang et al.
[Jiang et al., Biotech. & Bioeng., 103:361-
369, 2009]

Similar measure was also proposed by Smart et
al. [Smart et al., PNAS, 2008]

Impact Degree = number of inactivated
reactions caused by knockout of single reaction

Useful to measure the importance of each
reaction/gene
We also applied Branching Process theory to
analysis of distribution of impact degree [Takemoto et al.,

Physica A, 2012]
Example of Impact Degree (1)
Impact degree = 4
Example of Impact Degree (2)
Impact degree = 10
Flux Balance Analysis
FBA: Flux Balance Analysis



Analysis of steady states of metabolic networks
Use of linear equalities to represent balance of
input and output fluxes
Widely used in metabolic engineering
v1
v2
v6
X1
v3
X2
v5
X4
X3
v4
FBA: Example
dX 1
dt
dX 2
dt
dX 3
dt
dX 4
dt
dX 1 dX 2 dX 3 dX 4



0
dt
dt
dt
dt
v1
v6
v2
X1
v3
X2
v5
X4
X3
v4
1

0
0

0

 v1  v2  v3
 v2  v3  v5  v6
 v3  v4
 v4  v5
 v1 
 
 1  1 0 0 0  v2   0 
   
1  1 0 1  1 v3
0
  
0 1  1 0 0  v4   0 
   
0 0 1  1 0  v5   0 
v 
 6
FBA: Application
Non-unique solutions in general
 1 1
 Unique solution if v1, v6 are

0 1
specified
0 0
⇒ Internal states can be inferred 
0 0
from external states （v1,v6)

 v1 
 
 1 0 0 0  v2   0 
   
 1 0 1  1 v3
0
  
1  1 0 0  v4   0 
   
0 1  1 0  v5   0 
v 
 6
v1
v6
v2
v2  v6
X1
v3
X2
v5
X4
X3
v4
v3  v1  v6
v4  v1  v6
v5  v1  v6
Elementary Mode Analysis
Elementary Mode


Any flux can be represented by a superposition
(linear combination) of elementary fluxes (modes)
Elementary mode


Steady state
Minimal number of reactions
Metabolic network
v4
A
v3
v1
C
v6
B
v2
v5
An elementary mode
A
B
C
Elementary Mode: Example (1)
M1
A
B
M2
A
C
M3
A
C
B
C
B
M4
A
B
C
Elementary Mode: Example (2)
r1
r2
r4
r1
X1
r2
r3
X2
r7
X3
X4
r5
r4
r7
r7
X1
X4
X3
X4
r5
r6
Vector Representation of EM
r1
r3
X2
r3
X2
r6
r1
r2
r4
X1
X3
r6
r5
r2
r3
r4
r5
r6
r7
EM1 1
1
0
1
0
0
0
EM2 2
1
1
0
1
0
0
EM3 1
0
1
0
0
1
1
Robustness Analysis via Elementary Mode
（Ａ） Two EMs are inactivated when r1 is inactivated
（B） One EM is inactivated when r4 is inactivated
⇒ r1 is more important.
(A)
Aext
Dext
A
r1
Ｂ
(B)
r2
Aext
r3
r4
Ｂ
r5
C
A
Eext
r6
r7
Dext
Eext
Definition of the network robustness via EM
: #EM m : #reaction
zi : #EM not affected by inactivation of ri
Ri : robustness of ri
R : robustness of the network
z
zi
Ri 
z

R
m
z
i 1 i
m z
Wilhelm et al.:
Syst. Biol. 1:114-120, 2004
FBID:
Flux Balance Impact Degree
[Zhao, Tamura, Akutsu. Vert: Bioinfromatics, 29:2178-2185, 2013]
FBID: Flux Balance Impact Degree

Boolean-based impact degree

Cannot properly handle reversible reactions
⇒ FBA-based modeling

FBID
Number of reactions inactivated by knockout of
reaction ri
 rj is inactivated by knockout of ri
⇔ flux of rj is always 0 if flux of ri is 0

FBID: EM-based Method

rj is inactivated by knockout of ri
⇔Flux of rj is always 0 if flux of ri is 0
⇔Every EM containing rj must contain ri
FBID can be calculated by examining all EMs
 However, #EMs grows exponentially
⇒ Applicable to small-size networks only

FBID： LP-based Method

Use of Linear Programming

For each reaction ri, solve two LP instances
（R: set of knockout reactions）

min xi=max xi ⇔ flux of ri is always 0
⇔ ri is impacted by R


Need to solve 2n LP instances (n: #reactions)
However, it still works in polynomial time
⇒ applicable to large-scale networks
Advantages of FBID
Proper treatment of reversible reactions
（vs. Boolean approach）
 Objective function is not needed
（vs. FBA-based approach）
 Work in polynomial time
（vs. EM-based approach）
 At least comparable accuracy

Summary
Summary

Boolean Model of Metabolic Networks


Reaction＝AND, Compound＝OR
Impact degree



Number of reactions inactivated by knockout of reaction(s)
Measure of importance of a reaction (or, a gene, a set of
reactions)
FBID (Flux Balance Impact Degree)

New definition of impact degree based on flux
balance analysis




Proper treatment of reversible reactions
Objective function is not required
Work in polynomial time
Comparative accuracy