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.


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


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).


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.


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).


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.



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.



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.


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.


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).


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.



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.



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




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;




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

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.



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



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 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/



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/



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


http://www.csb.ethz.ch/tools/efmtool/

http://www.biotec.boku.ac.at/regulatoryelementaryfluxmode.html



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


http://www.csb.ethz.ch/tools/efmtool/



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.




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.




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.




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.



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.




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.




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.


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.


Thanks for your attention!Questions?

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