Minimal Cut Sets Analysis of Metabolic Pathways

Minimal Cut Sets and Its Application to Study
Metabolic Pathway Structures
Nguyen Vu-Ngoc Tung1,3 Beurton-Aimar Marie1
Colombié Sophie2
1Laboratoire Bordelais de Recherche en Informatique, UMR 5800
2
INRA Bordeaux Aquitaine, Fruit Biology and Pathology BP 81.
3
Faculty of Science and Technology, Hoa Sen University.
Nice’13 Thematic Research School, 2013
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 1 / 22

Motivation
One of the major current challenges in Systems Biology is how to
understand complex structure of metabolic networks.
Metabolic pathways are speciﬁc subsets into a metabolic network
identiﬁed as functional processes of cells.
Elementary Flux Modes (EFMs) are minimal sets of reactions
that represent feasible pathways under steady state condition
(Schuster,2000) .
Minimal Cut Sets (MCSs) are minimal sets of reactions that
inhibit the production of a certain objective reaction.

Outline
1 Context
2 Integrated Approach
Elementary Flux Modes Analysis
Minimal Cut Sets Analysis
3 Graph Cut Sets
Deﬁnitions and Notations
Algorithms
MCSs in Metabolic Networks
4 Application
Metabolic Network Description
Computing Tools
Results and Discussion
5 Conclusion and Perspective

Context
Context
Metabolic Pathways Analysis
Identifying pathways involved in speciﬁc production.
Discovering how to increase the yield of a product, to channel a
product into desired pathways or in functional reconstruction from
genomic data (Schuster,1999).
Predicting key aspects of network functionality, robustness and
gene regulation from network structure (Stelling,2002).

Integrated Approach Elementary Flux Modes Analysis
Analyze Reliability of Metabolite Production
Constraint-based approach (Schuster,1994).
Identifying all genetically independent pathways (Trinh,2009).
Being unique and non-decomposable set of reactions.
Selecting groups of reactions which interact together and
respecting the well-known steady-state mass balancing equation.

Integrated Approach Elementary Flux Modes Analysis
Analyze Reliability of Metabolite Production
Constraint-based approach (Schuster,1994).
Identifying all genetically independent pathways (Trinh,2009).
Being unique and non-decomposable set of reactions.
Selecting groups of reactions which interact together and
respecting the well-known steady-state mass balancing equation.
Steady-State Mass Balancing Assumption
dS
dt
= Nv (1)
S is a vector of concentration
values.
N is the stoichiometric matrix of
m metabolites × r reactions.
v is the r-dimensional (ﬂux)
vector of the reaction rates.
At the steady state: Nv = 0.

Integrated Approach Minimal Cut Sets Analysis
Analyze Fragility of Metabolic Networks
Mininal Cut Set Analysis
Finding all sets of reactions able to eliminate a given objective
functioning.
A Minimal Cut Set (MCS) is a unique and minimal set of
reactions (Klamt,2004).
EFMs and MCSs complement each other in a duality based
relationship (Klamt,2005;Ballerstein,2012).

Graph Cut Sets Deﬁnitions and Notations
Cut Sets deﬁnitions
Notations
Let G = (V, E) be an undirected graph with n = |V|, m = |E|.
Cut Sets
A cut C = {S, S} where S ∪ S = V(G) and S ∩ S = ∅.
∀u, v ∈ V, the set δ(S) = {(u, v) ∈ E ∧ S ⊂ V : u ∈ S, v ∈ S}
is a cut set since removal from G disconnects G into more than
one subgraphs.
The size of a cut set is |δ(S)| in unweighted graphs. Otherwise,
the cut set size equals to sum of the weights of the edges in δ(S).
A minimum cut set is a cut set of a certain minimum size.

Cut Set Deﬁnitions
Minimal Cut Set Example

Minimal Cut Set Example
s-t Cut Set Deﬁnition
A cut s − t of an undirected graph G is simply a cut C = {S, S}
with s ∈ S and t ∈ S.

Example
s-t cut set betwen s = a and t = d.
MCSs in Directed Graphs
In directed graphs, cut sets are deﬁned similarly.
The MCS value: summing all the weights of all the crossed edges
(between the two subsets) coming out S.

Graph Cut Sets Algorithms
Algorithms to Compute MCS
Finding one MCS
Originated from the well-known max flow theorem
(Elias,1956;Ford,1956).
Gomory and Hu (1961) introduced a tree structure to find
minimum s − t cuts for all pairs of s and t. Improving by Hao Orlin
in 1992.
The first deterministic minimum cut algorithm: Nagamochi and
Ibaraki (NI)(1992): O(|V||E| + |V|2log|V|.
Stoer and Wagner (1997) simplified NI and implemented it in
JGraphT library.

Graph Cut Sets Algorithms
Algorithms to Compute MCSs
Finding all MCSs
Applied in the field of reliability engineering (Ariyoshi,1972;
Arunkumar:1979).
Constructing a binary relation associated with an optimal
maximum flow (Curet,2002).
Definition of MCS in Metabolic Network Context
A set of reactions is called a cut set (with respect to a defined
objective reaction) if after the removal of these reactions from
the network no feasible balanced flux distribution involves the
objective reaction (Klamt,2004).

Graph Cut Sets MCSs in Metabolic Networks
Klamt’s algorithm
Main Ideas
In small networks it is relatively easy to calculate the MCSs but .....

Klamt’s algorithm
Main Ideas
In small networks it is relatively easy to calculate the MCSs but .....
For larger networks, we need a systematic computation
scheme.
The algorithm needs to guarantee:
MCSs are real cut sets.
MCSs are minimal.
All MCSs are found.

Algorithm for computing MCSs
Preparatory phase
1 Calculate the EFMs in the given network.
2 Deﬁne the objective reaction obR.
3 Choose all EFMs where reaction obR is non-zero and store it in
the binary array efm_obR.
4 Initialize the arrays mcs and precutsets as follows:
Append {j} to mcs if reaction {j} is essential, otherwise to
precutsets. {j} is essential if efm_obR[i][j] = 1 for each EFM_i

Main phase
1 FOR i=2 TO MAX_CUTSETSIZE
1 new_precutsets = [];
2 FOR j = 1 TO r
1 Remove all sets from precutsets where reaction j participates.
2 Find all sets of reactions in precutsets that do not cover any EFM in
efm_obR where reaction j participates. Combine each of these sets
with reaction j and store the new preliminary cut sets in
temp_precutsets.
3 Drop all temp_precutsets which are a superset of any of the already
determined minimal cut sets stored in mcs.
4 Find all retained temp_precutsets which do now cover all
EFMs and append them to mcs. Append all others to new_precutsets
3 IF isempty(new_precutsets) BREAK; ELSE precutsets =
new_precutsets;
2 return mcs;

Simple Example
efm: [{R1, R2, objR}, {R3, objR}]
mcs:
[{objR}, {R1, R3}, {R2, R3}]
Source: reused a simple example from M. Bader.

Application Metabolic Network Description
Application to 5 Networks
Purpose
To verify the hypothesis: MCS computing can provide a smaller
number of solutions than EFMs.

Application to 5 Networks
Purpose
To verify the hypothesis: MCS computing can provide a smaller
number of solutions than EFMs.
Data
3 networks to model energetic metabolism of mitochondria into 3
tissues: muscle, liver, yeast, approx. 40 reactions (Pérès Sabine,
PhD thesis, 2005).
2 networks to model the central metabolism of heterotrophic plant
cells, approx. 80 reactionsa including several biological
pathways: glycolysis, Pentose Phosphate pathway, Starch and
Sucrose synthesis and degradation.
a
described more detail in Beurton-Aimar M. et al., BMC Sys. Bio., 2011.

Mitochondrial Network
ACoA
Cit
HB
AcylCoA Carnitine
AcylCarnitine
ATP
Pi−
HB_ext
T10
Citulline
CarbmoylP
Orni
AACoA
HMGCoA
AA
2
AA_ext
T11
Intermenbran space
Matrix
NAD
H
Pyr
NAD
NADH
R7
Mal
Fum
Suc−CoA
Suc
Akg
Isocit
R12
OAA
ATP
ADP
NAD
NADH
NADH
ADPATP
FAD
FADH2
NAD
NADH ASP
Glu
NAD
NAD
NADH
Glutamine
NH3
2 ATP
2 ADP
R21
Suc−CoA
Suc
8
7 NAD
7 FAD
7 NADH
7 FADH2
FADH2
FAD
ADP
NADH2
R2
R1
R3
R22
R6
R15
R14
R13
R11
R10
R9
R8
R24
R25
R17
R16
R26
R27
R30
R31
R28
H
H
Pi2−
R5
T8
Citru_ext
Ornit_ext
H_ext
T6
Mal
Pyr_ext + H_ext
Akg
Mal
Mal
Cit + H
Glu + HAsp MalFum
NH3
R23
H
H_ext
Akg
T21Orni_ext
ATP_ext
ADP_ext
H_ext
Pi_ext
T5
T4
Akg_extPi2−_ext Mal_ext
T7
Pi2−_ext Glu_ext + H Asp_ext
T12 T13
Fum_extMal_ext Glutamine_ext H_ext Glu_ext
T20
Carnitine_ext
T3
AcylCarnitine_ext
Mal_ext
Akg_ext
Cit_ext + H_ext
Mal_ext
T1
T2
H
T19

Metabolic Network of Heterotrophic Plant Cell

Application Computing Tools
Computing Tools for EFMs and MCSs
CellNetAnalyzer
CellNetAnalyzer (CNA)a derived from Metatoolb
Package for MATLAB containing several modules to visualize
networks and to analyze their structures.
CNA enables users to compute both EFMs and MCSs.

CellNetAnalyzer
CellNetAnalyzer (CNA)a derived from Metatoolb
Package for MATLAB containing several modules to visualize
networks and to analyze their structures.
CNA enables users to compute both EFMs and MCSs.
Problem: Taking more than 10 days to obtain MCSs of PCA with
CNA (running on a linux server).
a
http://www.mpi-magdeburg.mpg.de/projects/cna/cna.html
b
http://pinguin.biologie.uni-jena.de/bioinformatik/networks/

Efmtool
A new implementation to compute EFMs in Java (Terzer,2008) a.
Supporting multi-threading and seems to be robust to compute
large networks.

Efmtool
large networks.
Problem: open source but not easy to use (many parameters)
and lacks of manuals.
a
http://www.csb.ethz.ch/tools/efmtool/

Efmtool
large networks.
Problem: open source but not easy to use (many parameters)
and lacks of manuals.
a
http://www.csb.ethz.ch/tools/efmtool/
regEfmtool
Written by C. Jungreuthmayer (Jung,2012)a.
Containing several scripts clearly documented.
New available operations: possibility to deﬁne genetics
constraints as logical rules to compute EFMs.
a
http:
//www.biotec.boku.ac.at/regulatoryelementaryfluxmode.html

Application Results and Discussion
Results of Computation of 5 Networks
Tissues Nb.React Nb. Int.Meta Nb.EFMs Nb.MCSs
Muscle 37 31 3, 253 (17.7) 42, 534 (10.2)
Liver 44 36 2, 307 (16.7) 47, 203 (11.4)
Yeast 40 34 4, 627 (15.3) 90, 318 (11.6)
PCA 78 55 114, 614 (37.7) 93, 009 (11.1)
PCC 89 50 9, 319, 997 (33.1) 2, 815, 375(11.8)

Muscle 37 31 3, 253 (17.7) 42, 534 (10.2)
Liver 44 36 2, 307 (16.7) 47, 203 (11.4)
Yeast 40 34 4, 627 (15.3) 90, 318 (11.6)
PCA 78 55 114, 614 (37.7) 93, 009 (11.1)
PCC 89 50 9, 319, 997 (33.1) 2, 815, 375(11.8)
Correlation between Number of EFMs and MCSs
The size of the EFMs set is a measure of the network robustness
(Stelling 2004).

Muscle 37 31 3, 253 (17.7) 42, 534 (10.2)
Liver 44 36 2, 307 (16.7) 47, 203 (11.4)
Yeast 40 34 4, 627 (15.3) 90, 318 (11.6)
PCA 78 55 114, 614 (37.7) 93, 009 (11.1)
PCC 89 50 9, 319, 997 (33.1) 2, 815, 375(11.8)
(Stelling 2004).
But no obvious relationship between the number of reactions (or
internal metabolites) and of EFMs.

Muscle 37 31 3, 253 (17.7) 42, 534 (10.2)
Liver 44 36 2, 307 (16.7) 47, 203 (11.4)
Yeast 40 34 4, 627 (15.3) 90, 318 (11.6)
PCA 78 55 114, 614 (37.7) 93, 009 (11.1)
PCC 89 50 9, 319, 997 (33.1) 2, 815, 375(11.8)
(Stelling 2004).
The number of MCSs is unfortunately not at all lower than the
number of EFMs in the 3 mitochondrial networks.

Muscle 37 31 3, 253 (17.7) 42, 534 (10.2)
Liver 44 36 2, 307 (16.7) 47, 203 (11.4)
Yeast 40 34 4, 627 (15.3) 90, 318 (11.6)
PCA 78 55 114, 614 (37.7) 93, 009 (11.1)
PCC 89 50 9, 319, 997 (33.1) 2, 815, 375(11.8)
(Stelling 2004).
The number of MCSs is unfortunately not at all lower than the
number of EFMs in the 3 mitochondrial networks.
When the number of EFMs is huge (PCA, PCC), the number of
MCSs begins to be lower than EFM number.

Comparison of EFMs and MCSs Length
The average length of EFMs increases with the number of
reactions while the average length of MCSs remains stable.

EFM length: when the number of reactions doubles, the length
too. For example, the average length of EFMs muscle is 17.7,
comparing to the values obtained for the PCA network, 37.7.
MCS length: while the number of reactions doubles from
mitochondria to plant cell networks, the average length is only
10% point more.

EFM length: when the number of reactions doubles, the length
too. For example, the average length of EFMs muscle is 17.7,
comparing to the values obtained for the PCA network, 37.7.
MCS length: while the number of reactions doubles from
mitochondria to plant cell networks, the average length is only
10% point more.
To inhibit a speciﬁc functionning, the number of reactions to stop
is approximatively the same whatever the network size.

Conclusion and Perspective
Conclusion
Metabolic networks are complex networks - Evidence!
To study network architecture of a whole metabolism, automatic
tools are necessary. Connexion among several pathways are
impossible to manage only by hands.
As EFMs computing, computing MCSs could generate huge
results. Post treatments like classiﬁcation are mandatory.
Most of available algorithms require large capacity of computing:
memory size and processor speed.
New machines and types of programming: GPU and algorithm
improvements help to solve the problem and allow to analyze
networks larger and larger.

Conclusion
Metabolic networks are complex networks - Evidence!
To study network architecture of a whole metabolism, automatic
tools are necessary. Connexion among several pathways are
impossible to manage only by hands.
As EFMs computing, computing MCSs could generate huge
results. Post treatments like classiﬁcation are mandatory.
Most of available algorithms require large capacity of computing:
memory size and processor speed.
New machines and types of programming: GPU and algorithm
improvements help to solve the problem and allow to analyze
networks larger and larger.
Perspective
New technics to analyze results coming from graph theory and
data mining have to be implemented to provide tools to do it.
Connection with techniques like ﬂux balance analysis is our nextNguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 21 / 22

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Minimal Cut Sets Analysis of Metabolic Pathways

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