Talk at the Nice Spring School on ASSB
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Transcript of Talk at the Nice Spring School on ASSB
Minimal Cut Sets and Its Application to StudyMetabolic Pathway Structures
Nguyen Vu-Ngoc Tung1,3 Beurton-Aimar Marie1
Colombié Sophie2
1Laboratoire Bordelais de Recherche en Informatique, UMR 58002INRA Bordeaux Aquitaine, Fruit Biology and Pathology BP 81.
3Faculty 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 tounderstand complex structure of metabolic networks.Metabolic pathways are specific subsets into a metabolic networkidentified as functional processes of cells.Elementary Flux Modes (EFMs) are minimal sets of reactionsthat represent feasible pathways under steady state condition(Schuster,2000) .Minimal Cut Sets (MCSs) are minimal sets of reactions thatinhibit the production of a certain objective reaction.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 2 / 22
Motivation
One of the major current challenges in Systems Biology is how tounderstand complex structure of metabolic networks.Metabolic pathways are specific subsets into a metabolic networkidentified as functional processes of cells.Elementary Flux Modes (EFMs) are minimal sets of reactionsthat represent feasible pathways under steady state condition(Schuster,2000) .Minimal Cut Sets (MCSs) are minimal sets of reactions thatinhibit the production of a certain objective reaction.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 2 / 22
Motivation
One of the major current challenges in Systems Biology is how tounderstand complex structure of metabolic networks.Metabolic pathways are specific subsets into a metabolic networkidentified as functional processes of cells.Elementary Flux Modes (EFMs) are minimal sets of reactionsthat represent feasible pathways under steady state condition(Schuster,2000) .Minimal Cut Sets (MCSs) are minimal sets of reactions thatinhibit the production of a certain objective reaction.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 2 / 22
Motivation
One of the major current challenges in Systems Biology is how tounderstand complex structure of metabolic networks.Metabolic pathways are specific subsets into a metabolic networkidentified as functional processes of cells.Elementary Flux Modes (EFMs) are minimal sets of reactionsthat represent feasible pathways under steady state condition(Schuster,2000) .Minimal Cut Sets (MCSs) are minimal sets of reactions thatinhibit the production of a certain objective reaction.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 2 / 22
Outline
1 Context
2 Integrated ApproachElementary Flux Modes AnalysisMinimal Cut Sets Analysis
3 Graph Cut SetsDefinitions and NotationsAlgorithmsMCSs in Metabolic Networks
4 ApplicationMetabolic Network DescriptionComputing ToolsResults and Discussion
5 Conclusion and Perspective
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 3 / 22
Context
Context
Metabolic Pathways AnalysisIdentifying pathways involved in specific production.Discovering how to increase the yield of a product, to channel aproduct into desired pathways or in functional reconstruction fromgenomic data (Schuster,1999).Predicting key aspects of network functionality, robustness andgene regulation from network structure (Stelling,2002).
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 4 / 22
Context
Context
Metabolic Pathways AnalysisIdentifying pathways involved in specific production.Discovering how to increase the yield of a product, to channel aproduct into desired pathways or in functional reconstruction fromgenomic data (Schuster,1999).Predicting key aspects of network functionality, robustness andgene regulation from network structure (Stelling,2002).
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 4 / 22
Context
Context
Metabolic Pathways AnalysisIdentifying pathways involved in specific production.Discovering how to increase the yield of a product, to channel aproduct into desired pathways or in functional reconstruction fromgenomic data (Schuster,1999).Predicting key aspects of network functionality, robustness andgene regulation from network structure (Stelling,2002).
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 4 / 22
Integrated Approach Elementary Flux Modes Analysis
Analyze Reliability of Metabolite Production
Elementary Flux Modes AnalysisConstraint-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 andrespecting the well-known steady-state mass balancing equation.
Steady-State Mass Balancing Assumption
dSdt
= Nv (1)
S is a vector of concentrationvalues.
N is the stoichiometric matrix ofm metabolites × r reactions.
v is the r-dimensional (flux)vector of the reaction rates.
At the steady state: Nv = 0.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 5 / 22
Integrated Approach Elementary Flux Modes Analysis
Analyze Reliability of Metabolite Production
Elementary Flux Modes AnalysisConstraint-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 andrespecting the well-known steady-state mass balancing equation.
Steady-State Mass Balancing Assumption
dSdt
= Nv (1)
S is a vector of concentrationvalues.
N is the stoichiometric matrix ofm metabolites × r reactions.
v is the r-dimensional (flux)vector of the reaction rates.
At the steady state: Nv = 0.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 5 / 22
Integrated Approach Minimal Cut Sets Analysis
Analyze Fragility of Metabolic Networks
Mininal Cut Set AnalysisFinding all sets of reactions able to eliminate a given objectivefunctioning.A Minimal Cut Set (MCS) is a unique and minimal set ofreactions (Klamt,2004).EFMs and MCSs complement each other in a duality basedrelationship (Klamt,2005;Ballerstein,2012).
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 6 / 22
Graph Cut Sets Definitions and Notations
Cut Sets definitions
NotationsLet 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 thanone 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.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 7 / 22
Graph Cut Sets Definitions and Notations
Cut Set Definitions
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.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 8 / 22
Graph Cut Sets Definitions and Notations
Cut Set Definitions
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.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 8 / 22
Graph Cut Sets Definitions and Notations
Cut Set Definitions
Examples-t cut set betwen s = a and t = d .
MCSs in Directed GraphsIn 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.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 9 / 22
Graph Cut Sets Algorithms
Algorithms to Compute MCS
Finding one MCSOriginated from the well-known max flow theorem(Elias,1956;Ford,1956).Gomory and Hu (1961) introduced a tree structure to findminimum s − t cuts for all pairs of s and t . Improving by Hao Orlinin 1992.The first deterministic minimum cut algorithm: Nagamochi andIbaraki (NI)(1992): O(|V ||E |+ |V |2log|V |.Stoer and Wagner (1997) simplified NI and implemented it inJGraphT library.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 10 / 22
Graph Cut Sets Algorithms
Algorithms to Compute MCSs
Finding all MCSsApplied in the field of reliability engineering (Ariyoshi,1972;Arunkumar:1979).Constructing a binary relation associated with an optimalmaximum flow (Curet,2002).
Definition of MCS in Metabolic Network ContextA set of reactions is called a cut set (with respect to a definedobjective reaction) if after the removal of these reactions fromthe network no feasible balanced flux distribution involves theobjective reaction (Klamt,2004).
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 11 / 22
Graph Cut Sets MCSs in Metabolic Networks
Klamt’s algorithm
Main IdeasIn small networks it is relatively easy to calculate the MCSs but .....
For larger networks, we need a systematic computationscheme.The algorithm needs to guarantee:
MCSs are real cut sets.MCSs are minimal.All MCSs are found.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 12 / 22
Graph Cut Sets MCSs in Metabolic Networks
Klamt’s algorithm
Main IdeasIn small networks it is relatively easy to calculate the MCSs but .....For larger networks, we need a systematic computationscheme.The algorithm needs to guarantee:
MCSs are real cut sets.MCSs are minimal.All MCSs are found.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 12 / 22
Graph Cut Sets MCSs in Metabolic Networks
Klamt’s algorithm
Main IdeasIn small networks it is relatively easy to calculate the MCSs but .....For larger networks, we need a systematic computationscheme.The algorithm needs to guarantee:
MCSs are real cut sets.MCSs are minimal.All MCSs are found.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 12 / 22
Graph Cut Sets MCSs in Metabolic Networks
Algorithm for computing MCSs
Preparatory phase1 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 toprecutsets. {j} is essential if efm_obR[i][j] = 1 for each EFM_i
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 13 / 22
Graph Cut Sets MCSs in Metabolic Networks
Algorithm for computing MCSs
Main phase1 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 setswith reaction j and store the new preliminary cut sets intemp_precutsets.
3 Drop all temp_precutsets which are a superset of any of the alreadydetermined minimal cut sets stored in mcs.
4 Find all retained temp_precutsets which do now cover allEFMs and append them to mcs. Append all others to new_precutsets
3 IF isempty(new_precutsets) BREAK; ELSE precutsets =new_precutsets;
2 return mcs;
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 13 / 22
Graph Cut Sets MCSs in Metabolic Networks
Algorithm for computing MCSs
Simple Example
efm: [{R1,R2,objR}, {R3,objR}]mcs:[{objR}, {R1,R3}, {R2,R3}]
Source: reused a simple example from M. Bader.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 13 / 22
Application Metabolic Network Description
Application to 5 Networks
PurposeTo verify the hypothesis: MCS computing can provide a smallernumber of solutions than EFMs.
Data3 networks to model energetic metabolism of mitochondria into 3tissues: muscle, liver, yeast, approx. 40 reactions (Pérès Sabine,PhD thesis, 2005).2 networks to model the central metabolism of heterotrophic plantcells, approx. 80 reactionsa including several biologicalpathways: glycolysis, Pentose Phosphate pathway, Starch andSucrose synthesis and degradation.
adescribed more detail in Beurton-Aimar M. et al., BMC Sys. Bio., 2011.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 14 / 22
Application Metabolic Network Description
Application to 5 Networks
PurposeTo verify the hypothesis: MCS computing can provide a smallernumber of solutions than EFMs.
Data3 networks to model energetic metabolism of mitochondria into 3tissues: muscle, liver, yeast, approx. 40 reactions (Pérès Sabine,PhD thesis, 2005).2 networks to model the central metabolism of heterotrophic plantcells, approx. 80 reactionsa including several biologicalpathways: glycolysis, Pentose Phosphate pathway, Starch andSucrose synthesis and degradation.
adescribed more detail in Beurton-Aimar M. et al., BMC Sys. Bio., 2011.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 14 / 22
Application Metabolic Network Description
Mitochondrial Network
ACoA
Cit
HB
AcylCoA Carnitine
AcylCarnitine
ATP
Pi−
HB_ext
T10
Citulline
CarbmoylPOrni
AACoA
HMGCoA
AA
2
AA_ext
T11
Intermenbran space
Matrix
NAD
H
PyrNAD
NADH
R7
Mal
Fum
Suc−CoASuc
Akg
Isocit
R12
OAA
ATP
ADP
NAD
NADH
NADHADPATP
FAD
FADH2
NAD
NADHASP
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
R22R6
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_extT1
T2
H
T19
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 15 / 22
Application Metabolic Network Description
Metabolic Network of Heterotrophic Plant Cell
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 16 / 22
Application Computing Tools
Computing Tools for EFMs and MCSs
CellNetAnalyzer
CellNetAnalyzer (CNA)a derived from Metatoolb
Package for MATLAB containing several modules to visualizenetworks 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 withCNA (running on a linux server).
ahttp://www.mpi-magdeburg.mpg.de/projects/cna/cna.htmlbhttp://pinguin.biologie.uni-jena.de/bioinformatik/networks/
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 17 / 22
Application Computing Tools
Computing Tools for EFMs and MCSs
CellNetAnalyzer
CellNetAnalyzer (CNA)a derived from Metatoolb
Package for MATLAB containing several modules to visualizenetworks 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 withCNA (running on a linux server).
ahttp://www.mpi-magdeburg.mpg.de/projects/cna/cna.htmlbhttp://pinguin.biologie.uni-jena.de/bioinformatik/networks/
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 17 / 22
Application Computing Tools
Computing Tools for EFMs and MCSsEfmtool
A new implementation to compute EFMs in Java (Terzer,2008) a.Supporting multi-threading and seems to be robust to computelarge networks.
Problem: open source but not easy to use (many parameters)and lacks of manuals.
ahttp://www.csb.ethz.ch/tools/efmtool/
regEfmtoolWritten by C. Jungreuthmayer (Jung,2012)a.Containing several scripts clearly documented.New available operations: possibility to define geneticsconstraints as logical rules to compute EFMs.
ahttp://www.biotec.boku.ac.at/regulatoryelementaryfluxmode.html
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 18 / 22
Application Computing Tools
Computing Tools for EFMs and MCSsEfmtool
A new implementation to compute EFMs in Java (Terzer,2008) a.Supporting multi-threading and seems to be robust to computelarge networks.Problem: open source but not easy to use (many parameters)and lacks of manuals.
ahttp://www.csb.ethz.ch/tools/efmtool/
regEfmtoolWritten by C. Jungreuthmayer (Jung,2012)a.Containing several scripts clearly documented.New available operations: possibility to define geneticsconstraints as logical rules to compute EFMs.
ahttp://www.biotec.boku.ac.at/regulatoryelementaryfluxmode.html
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 18 / 22
Application Computing Tools
Computing Tools for EFMs and MCSsEfmtool
A new implementation to compute EFMs in Java (Terzer,2008) a.Supporting multi-threading and seems to be robust to computelarge networks.Problem: open source but not easy to use (many parameters)and lacks of manuals.
ahttp://www.csb.ethz.ch/tools/efmtool/
regEfmtoolWritten by C. Jungreuthmayer (Jung,2012)a.Containing several scripts clearly documented.New available operations: possibility to define geneticsconstraints as logical rules to compute EFMs.
ahttp://www.biotec.boku.ac.at/regulatoryelementaryfluxmode.html
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 18 / 22
Application Results and Discussion
Results of Computation of 5 NetworksTissues Nb.React Nb. Int.Meta Nb.EFMs Nb.MCSsMuscle 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 MCSsThe size of the EFMs set is a measure of the network robustness(Stelling 2004).
But no obvious relationship between the number of reactions (orinternal metabolites) and of EFMs.The number of MCSs is unfortunately not at all lower than thenumber of EFMs in the 3 mitochondrial networks.When the number of EFMs is huge (PCA, PCC), the number ofMCSs begins to be lower than EFM number.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 19 / 22
Application Results and Discussion
Results of Computation of 5 NetworksTissues Nb.React Nb. Int.Meta Nb.EFMs Nb.MCSsMuscle 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 MCSsThe size of the EFMs set is a measure of the network robustness(Stelling 2004).
But no obvious relationship between the number of reactions (orinternal metabolites) and of EFMs.The number of MCSs is unfortunately not at all lower than thenumber of EFMs in the 3 mitochondrial networks.When the number of EFMs is huge (PCA, PCC), the number ofMCSs begins to be lower than EFM number.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 19 / 22
Application Results and Discussion
Results of Computation of 5 NetworksTissues Nb.React Nb. Int.Meta Nb.EFMs Nb.MCSsMuscle 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 MCSsThe size of the EFMs set is a measure of the network robustness(Stelling 2004).
But no obvious relationship between the number of reactions (orinternal metabolites) and of EFMs.The number of MCSs is unfortunately not at all lower than thenumber of EFMs in the 3 mitochondrial networks.When the number of EFMs is huge (PCA, PCC), the number ofMCSs begins to be lower than EFM number.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 19 / 22
Application Results and Discussion
Results of Computation of 5 NetworksTissues Nb.React Nb. Int.Meta Nb.EFMs Nb.MCSsMuscle 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 MCSsThe size of the EFMs set is a measure of the network robustness(Stelling 2004).But no obvious relationship between the number of reactions (orinternal metabolites) and of EFMs.
The number of MCSs is unfortunately not at all lower than thenumber of EFMs in the 3 mitochondrial networks.When the number of EFMs is huge (PCA, PCC), the number ofMCSs begins to be lower than EFM number.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 19 / 22
Application Results and Discussion
Results of Computation of 5 NetworksTissues Nb.React Nb. Int.Meta Nb.EFMs Nb.MCSsMuscle 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 MCSsThe size of the EFMs set is a measure of the network robustness(Stelling 2004).But no obvious relationship between the number of reactions (orinternal metabolites) and of EFMs.The number of MCSs is unfortunately not at all lower than thenumber of EFMs in the 3 mitochondrial networks.
When the number of EFMs is huge (PCA, PCC), the number ofMCSs begins to be lower than EFM number.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 19 / 22
Application Results and Discussion
Results of Computation of 5 NetworksTissues Nb.React Nb. Int.Meta Nb.EFMs Nb.MCSsMuscle 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 MCSsThe size of the EFMs set is a measure of the network robustness(Stelling 2004).But no obvious relationship between the number of reactions (orinternal metabolites) and of EFMs.The number of MCSs is unfortunately not at all lower than thenumber of EFMs in the 3 mitochondrial networks.When the number of EFMs is huge (PCA, PCC), the number ofMCSs begins to be lower than EFM number.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 19 / 22
Application Results and Discussion
Results of Computation of 5 Networks
Comparison of EFMs and MCSs LengthThe average length of EFMs increases with the number ofreactions while the average length of MCSs remains stable.
EFM length: when the number of reactions doubles, the lengthtoo. 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 frommitochondria to plant cell networks, the average length is only10% point more.To inhibit a specific functionning, the number of reactions to stopis approximatively the same whatever the network size.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 20 / 22
Application Results and Discussion
Results of Computation of 5 Networks
Comparison of EFMs and MCSs LengthThe average length of EFMs increases with the number ofreactions while the average length of MCSs remains stable.EFM length: when the number of reactions doubles, the lengthtoo. 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 frommitochondria to plant cell networks, the average length is only10% point more.
To inhibit a specific functionning, the number of reactions to stopis approximatively the same whatever the network size.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 20 / 22
Application Results and Discussion
Results of Computation of 5 Networks
Comparison of EFMs and MCSs LengthThe average length of EFMs increases with the number ofreactions while the average length of MCSs remains stable.EFM length: when the number of reactions doubles, the lengthtoo. 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 frommitochondria to plant cell networks, the average length is only10% point more.To inhibit a specific functionning, the number of reactions to stopis approximatively the same whatever the network size.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 20 / 22
Conclusion and Perspective
Conclusion and PerspectiveConclusion
Metabolic networks are complex networks - Evidence!To study network architecture of a whole metabolism, automatictools are necessary. Connexion among several pathways areimpossible to manage only by hands.As EFMs computing, computing MCSs could generate hugeresults. 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 algorithmimprovements help to solve the problem and allow to analyzenetworks larger and larger.
PerspectiveNew technics to analyze results coming from graph theory anddata mining have to be implemented to provide tools to do it.Connection with techniques like flux balance analysis is our nextstep to analyze plant metabolism and to charaterize behaviours.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 21 / 22
Conclusion and Perspective
Conclusion and PerspectiveConclusion
Metabolic networks are complex networks - Evidence!To study network architecture of a whole metabolism, automatictools are necessary. Connexion among several pathways areimpossible to manage only by hands.As EFMs computing, computing MCSs could generate hugeresults. 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 algorithmimprovements help to solve the problem and allow to analyzenetworks larger and larger.
PerspectiveNew technics to analyze results coming from graph theory anddata mining have to be implemented to provide tools to do it.Connection with techniques like flux balance analysis is our nextstep to analyze plant metabolism and to charaterize behaviours.
Nguyen Vu, Beurton-Aimar, Colombié (LaBRI, INRA and HSU)Minimal Cut Sets in Metabolic Networks aSSB 2013 21 / 22
Thanks for your attention!Questions?