lkn/2009 winter school in mathematical & 1 computational...

LKN/2009 Winter School in Mathematical & Computational Biology

1

Network reconstruction, topology and feasible solution space


3

From component to systems biology

Palsson (2000) Nat Biotech 20:649

HT analytical chemistry Integrative analysis

Component biology Systems biology Component view

Time-dependentconcentration

Compute flux

FunctionS+E ↔ X → E+P

Systems view

Steady stateflux map

Neededhomeostasis

Calculate C Calculate k

Reactionnetwork


4

E. coli on glycerol

Ibarra et al, Nature 420, 186 - 189 (2002)

Objective Max growth rate on glycerolConstraints Network topology

Steady stateMaximum rates


5

Permanence of networks• The permanent feature of life is networks

– Interconnections or links between components define the essence of a living process

• Components have finite turn-over time– Metabolites: ~1 min– mRNA: ~2 hour– Cells: a few years (in human)– Yet, organisms remain essentially identical (if older)

• Kinetics is secondary– Networks are relatively insensitive to kinetics– Kinetics can evolve rapidly to realise network

potential


6

Systems biology

Components Gen- Transcript- Prote- Metabol-

Reconstruction of biochemical network (unique!)Systemicannotation

Hypothesisgeneration &testing

Phenotypic space is essentially infinite

In silicomodelling Topology Constraints Dynamics Sensitivity Noise


7

Reconstruction• All cellular networks

– Metabolic– Regulatory– Signaling

• are (bio)chemical• The chemical nature is important

– Defines a stoichiometric relationship between components (invariable, integer)

– Defines fundamental constraints for the systems• Thermodynamics: irreversibility and relative rates, maximum concentrations• Mass transfer: maximum rates• Spatial constraints: maximum concentrations, maximum rates

• Chemical networks are readily described by a stoichiometric matrix


8

Systemic (2D) annotation

Palsson (2004) Nat Biotech 22:1218


9

Reconstruction process• Genome sequence• ORF prediction• Gene annotation

– BLAST, Phylogeny, context

• Pathway reconstruction– Synthesis of all biomass components– Missing genes

• Functional validation– Historical data, phenotype arrays– Metabolomics

• Additional information– Regulation, e.g., array analysis


10

Annotation workflow


11

Automated model compilation• KEGG: Mus musculus release 46

– Primary: pathway maps• GENE & RN files plus COORD files• Covers 50% of genes in mmu-genome LST file

– Secondary: global files• mmu-enzyme LST file• Less specific EC entries

– Reaction attributes: LIGAND• Reaction & reaction-name LST• Compound names and ID• Reaction-mapformula: reversibility

• UniProtKB: Localisation– Default: cytoplasm


12

Manual curation

• Technical (KEGG issues)– Inconsistent compound or reaction labels (network

gaps)– Reactions violating atom conservation

• E.g., DNA + nucleotide = DNA• Generic molecules: R• H2O, H+, redox (difficult to pick up)

– Lumped reactions (e.g., PDH)• Connectivity

– Membrane transporters– Biomass drains


13

Network gaps• Visual inspection of KEGG maps

– Good for synthesis pathways– 6 essential reactions identified w/o known gene association– All found in human GSM

• Linear programming– Test generation of each biomass component– 3 reactions w/o gene association found (all had irreversible

counterparts in opposite direction)– 5 reactions mapped to mitochondria but needed in cytosol (3

cytosolic in human GSM)• Literature data

– 43 reactions w/o gene association added– 21 subcellular localisations corrected


14

Model overview

Unique gene-reaction associations 4617

Metabolites 2104

Number of genes 1399

Other reactionsReactions w/o gene associationMembrane transportersBiomass reactionsAutocatalytic

5268217

Mapped reactions 1757

Total number of reactionsRxns in mitochondria

2037387

Topology

• S[m,n]–Rows for m metabolites–Columns for reactions


16

glc-Dg6pf6pfdpdhapg3p13dpg3pg2pgpeppyrlac-Latpadppih2onadhnadhglc-D[e]lac-D[e]h[e]

Enzyme Protein EC # ReactionHEX Hexokinase hk 2.7.1.1 glc-D + atp -> g6p + adp + h v1PGI1 Phosphoglucose isomerase pgi 5.3.1.9 g6p <-> f6p v2PFKA Phosphofructokinase pfkA 2.7.1.11 f6p + atp -> fdp + adp + h v3FBA Fructose-1,6-bisphosphatate aldolase fba 4.1.2.13 fdp <-> dhap + g3p v4TPIA Triosphosphate Isomerase tpiA 5.3.1.1 dhap <-> g3p v5GAPA G3P dehydrogenase-A complex gapA 1.2.1.12 g3p + pi + nad <-> 13dpg + nadh + h v6PGK Phosphoglycerate kinase pgk 2.7.2.3 13dpg + adp <-> 3pg + atp v7GPMA Phosphoglycerate mutase 1 gpmA 5.4.2.1 3pg <-> 2pg v8ENO Enolase eno 4.2.1.11 2pg <-> pep + h2o v9PYKF Pyruvate Kinase I pykF 2.7.1.40 pep + adp + h -> pyr + atp v10LDH_L L-Lactate dehydrogenase Ldh 1.1.1.28 pyr + nadh + h <-> lac-L + nad v11

ATP hydrolysis atp + h2o -> adp + pi + h v12GLCt glucose exchange glc-D[e] <-> glc-D b1L-LAC-t lactate transport lac-L + h <-> h[e] + lac-L[e] b2


17

Reaction S' glc-

D

g6p

f6p

fdp

dhap

g3p

13dp

g

3pg

2pg

pep

pyr

lac-

L

atp

adp

pi h2o

nadh

nad

h glc-

D[e

]

lac-

D[e

]

h[e]

glc-D + atp -> g6p + adp + h v1 -1 1 -1 1 1g6p <-> f6p v2 -1 1f6p + atp -> fdp + adp + h v3 -1 1 -1 1 1fdp <-> dhap + g3p v4 -1 1 1dhap <-> g3p v5 -1 1g3p + pi + nad <-> 13dpg + nadh + h v6 -1 1 -1 1 -1 113dpg + adp <-> 3pg + atp v7 -1 1 1 -13pg <-> 2pg v8 -1 12pg <-> pep + h2o v9 -1 1 1pep + adp + h -> pyr + atp v10 -1 1 1 -1 -1pyr + nadh + h <-> lac-L + nad v11 -1 1 -1 1 -1atp + h2o -> adp + pi + h v12 -1 1 1 -1 1glc-D[e] <-> glc-D b1 1 -1lac-L + h <-> h[e] + lac-L[e] b2 -1 -1 1 1


18

S v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 b1 b2glc-D -1 1g6p 1 -1f6p 1 -1fdp 1 -1dhap 1 -1g3p 1 1 -113dpg 1 -13pg 1 -12pg 1 -1pep 1 -1pyr 1 -1lac-L 1 -1atp -1 -1 1 1 -1adp 1 1 -1 -1 1pi -1 1h2o 1 -1nadh 1 -1nad -1 1h 1 1 1 -1 -1 1 -1glc-D[e] -1lac-D[e] 1h[e] 1


19

Stot, Sexch, Sint

Stot

Prim

ary

Sec

onda

ryE

xter

nal

Internal fluxes Exchange

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 b1 b2glc-D -1 1g6p 1 -1f6p 1 -1fdp 1 -1dhap 1 -1g3p 1 1 -113dpg 1 -13pg 1 -12pg 1 -1pep 1 -1pyr 1 -1lac-L 1 -1atp -1 -1 1 1 -1adp 1 1 -1 -1 1pi -1 1h2o 1 -1nadh 1 -1nad -1 1h 1 1 1 -1 -1 1 -1glc-D[e] -1lac-L[e] 1h[e] 1

glc-D[e]

glc-D

g6p

f6p

fdp

dhap g3p

13dpg

3pg

2pg

pep

pyr

lac-L

lac-L[e]

Stot


21

Stot, Sexch, Sint

Stot

Prim

ary

Sec

onda

ryE

xter

nal


Sexch

Sexchv1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 b1 b2

glc-D -1 1g6p 1 -1f6p 1 -1fdp 1 -1dhap 1 -1g3p 1 1 -113dpg 1 -13pg 1 -12pg 1 -1pep 1 -1pyr 1 -1lac-L 1 -1atp -1 -1 1 1 -1adp 1 1 -1 -1 1pi -1 1h2o 1 -1nadh 1 -1nad -1 1h 1 1 1 -1 -1 1 -1glc-D[e] -1lac-L[e] 1h[e] 1

glc-D[e]

glc-D

g6p

f6p

fdp

dhap g3p

13dpg

3pg

2pg

pep

pyr

lac-L

lac-L[e]


23

Stot, Sexch, Sint

Stot

Prim

ary

Sec

onda

ryE

xter

nal


SexchSint

Sintv1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 b1 b2

glc-D -1 1g6p 1 -1f6p 1 -1fdp 1 -1dhap 1 -1g3p 1 1 -113dpg 1 -13pg 1 -12pg 1 -1pep 1 -1pyr 1 -1lac-L 1 -1atp -1 -1 1 1 -1adp 1 1 -1 -1 1pi -1 1h2o 1 -1nadh 1 -1nad -1 1h 1 1 1 -1 -1 1 -1glc-D[e] -1lac-D[e] 1h[e] 1

glc-D[e]

glc-D

g6p

f6p

fdp

dhap g3p

13dpg

3pg

2pg

pep

pyr

lac-L

lac-L[e]


25

Reaction map: SMetabolites are nodesFor reaction draw edge between substrates (- entry) to products (+entry)

Highly non-linear mapParticipation: 4 is typical


26

Compound map: -ST

• Reaction as nodes• Compounds as links• Connectivity number

– 2 is common– High for ATP etc

• Soft link– metabolites flowing

through reactions not fixed by stoichiometry


27

Open or closed systemsA

BC

v1

A

BC

v1

v1

A

B

C

b1

b2

b3 v1

A

B

Cb1

b2

b3

A

BC

v1

b1

b2

b3v1

A

B

Cb1

b2

b3

Ae

Be

Ce

Ae

Ce

Be

Sint

Sexch

Stot

Binary S

S v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 b1 b2 Sb v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 b1 b2glc-D -1 1 glc-D 1 0 0 0 0 0 0 0 0 0 0 0 1 0g6p 1 -1 g6p 1 1 0 0 0 0 0 0 0 0 0 0 0 0f6p 1 -1 f6p 0 1 1 0 0 0 0 0 0 0 0 0 0 0fdp 1 -1 fdp 0 0 1 1 0 0 0 0 0 0 0 0 0 0dhap 1 -1 dhap 0 0 0 1 1 0 0 0 0 0 0 0 0 0g3p 1 1 -1 g3p 0 0 0 1 1 1 0 0 0 0 0 0 0 013dpg 1 -1 13dpg 0 0 0 0 0 1 1 0 0 0 0 0 0 03pg 1 -1 3pg 0 0 0 0 0 0 1 1 0 0 0 0 0 02pg 1 -1 2pg 0 0 0 0 0 0 0 1 1 0 0 0 0 0pep 1 -1 pep 0 0 0 0 0 0 0 0 1 1 0 0 0 0pyr 1 -1 pyr 0 0 0 0 0 0 0 0 0 1 1 0 0 0lac-L 1 -1 lac-L 0 0 0 0 0 0 0 0 0 0 1 0 0 1atp -1 -1 1 1 -1 atp 1 0 1 0 0 0 1 0 0 1 0 1 0 0adp 1 1 -1 -1 1 adp 1 0 1 0 0 0 1 0 0 1 0 1 0 0pi -1 1 pi 0 0 0 0 0 1 0 0 0 0 0 1 0 0h2o 1 -1 h2o 0 0 0 0 0 0 0 0 1 0 0 1 0 0nadh 1 -1 nadh 0 0 0 0 0 1 0 0 0 0 1 0 0 0nad -1 1 nad 0 0 0 0 0 1 0 0 0 0 1 0 0 0h 1 1 1 -1 -1 1 -1 h 1 0 1 0 0 1 0 0 0 1 1 1 0 1glc-D[e] -1 glc-D[e] 0 0 0 0 0 0 0 0 0 0 0 0 1 0lac-L[e] 1 lac-L[e] 0 0 0 0 0 0 0 0 0 0 0 0 0 1h[e] 1 h[e] 0 0 0 0 0 0 0 0 0 0 0 0 0 1

⎭⎬⎫

⎩⎨⎧

≠===

0 if1ˆ0 if0ˆ

:ˆijij

ijij

ssss

S Binary S

Reaction adjacency matrix,

ik

kik

kiiTii ssdiag π===⋅= ∑∑ ˆˆˆˆ)( 2ssAv

πi = participation number for reaction i, i.e., number of compounds participating in reaction i. Off-diagonal elements indicates how many compounds two reactions i and j have in common

Av=Sb'Sb v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 b1 b2v1 5 1 3 0 0 1 2 0 0 3 1 3 1 1v2 1 2 1 0 0 0 0 0 0 0 0 0 0 0v3 3 1 5 1 0 1 2 0 0 3 1 3 0 1v4 0 0 1 3 2 1 0 0 0 0 0 0 0 0v5 0 0 0 2 2 1 0 0 0 0 0 0 0 0v6 1 0 1 1 1 6 1 0 0 1 3 2 0 1v7 2 0 2 0 0 1 4 1 0 2 0 2 0 0v8 0 0 0 0 0 0 1 2 1 0 0 0 0 0v9 0 0 0 0 0 0 0 1 3 1 0 1 0 0v10 3 0 3 0 0 1 2 0 1 5 2 3 0 1v11 1 0 1 0 0 3 0 0 0 2 5 1 0 2v12 3 0 3 0 0 2 2 0 1 3 1 5 0 1b1 1 0 0 0 0 0 0 0 0 0 0 0 2 0b2 1 0 1 0 0 1 0 0 0 1 2 1 0 4

SSAvˆˆ T=

Compound adjacency matrix,

ik

ikk

ikii ss ρ=== ∑∑ ˆˆ)( 2xa

ρi = connectivity number for compound i, i.e., number of reactions in which compound i participates. Off-diagonal elements indicates how many reactions both compounds i and j participate in.

TSSAxˆˆ=

Ax=SbSb' glc-

D

g6p

f6p

fdp

dhap

g3p

13dp

g

3pg

2pg

pep

pyr

lac-

L

atp

adp

pi h2o

nadh

nad

h glc-

D[e

]

lac-

L[e]

h[e]

glc-D 2 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0g6p 1 2 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0f6p 0 1 2 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0fdp 0 0 1 2 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0dhap 0 0 0 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0g3p 0 0 0 1 2 3 1 0 0 0 0 0 0 0 1 0 1 1 1 0 0 013dpg 0 0 0 0 0 1 2 1 0 0 0 0 1 1 1 0 1 1 1 0 0 03pg 0 0 0 0 0 0 1 2 1 0 0 0 1 1 0 0 0 0 0 0 0 02pg 0 0 0 0 0 0 0 1 2 1 0 0 0 0 0 1 0 0 0 0 0 0pep 0 0 0 0 0 0 0 0 1 2 1 0 1 1 0 1 0 0 1 0 0 0pyr 0 0 0 0 0 0 0 0 0 1 2 1 1 1 0 0 1 1 2 0 0 0lac-L 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 1 1 2 0 1 1atp 1 1 1 1 0 0 1 1 0 1 1 0 5 5 1 1 0 0 4 0 0 0adp 1 1 1 1 0 0 1 1 0 1 1 0 5 5 1 1 0 0 4 0 0 0pi 0 0 0 0 0 1 1 0 0 0 0 0 1 1 2 1 1 1 2 0 0 0h2o 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 2 0 0 1 0 0 0nadh 0 0 0 0 0 1 1 0 0 0 1 1 0 0 1 0 2 2 2 0 0 0nad 0 0 0 0 0 1 1 0 0 0 1 1 0 0 1 0 2 2 2 0 0 0h 1 1 1 1 0 1 1 0 0 1 2 2 4 4 2 1 2 2 7 0 1 1glc-D[e] 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0lac-L[e] 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1h[e] 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1


31

Singletons


32

Network topology


33

Active, essential and zero-flux reactions

• 950 singletons of 2104 metabolites– Minor biomass components (e.g., spermidine)– “C-unconnected”, e.g., xenobiotics– Annotation errors

• 987 reactions linked to singletons (dead-end metabolites)

• 1050 active (i.e., non-zero flux) reactions– Approximately 270 essential– 409 degrees of freedom


34

S as a linear transformation

( ) ( )

DomainRange

balance) mass (Dynamic

],[

2121

mn

kkik

i

Tm

Tn

dtd

dtd

vsdtdx

xxxvvv

nm ℜ∈⎯⎯ →⎯ℜ∈

=

=

==

∑

xv

Svx

xv

S

KK


35

Four fundamental spaces

Row(S)

Null(S)

Col(S)

Left null(S)

vdyn

vss

v dx/dt

ℜn ℜm

S[m,n]


36

Dimensions of subspaces

• r = Rank(S) = # linearly, independent relationships between compounds and reactions

• dim(Col(S)) = dim(Row(S)) = r• dim(Right null(S)) = n – r• dim(Left null(S)) = m – r


37

(Right) null space

• v = vss + vdyn, where Svss=0• vss is in (right) null space of S• Null(S) contains all allowable steady-state

flux distributions.


38

Row space

• Together vss + vdyn span ℜn

• vdyn orthogonal to null space, i.e., in row(S)• row(S) contains all dynamic flux

distributions, i.e., the thermodynamic driving forces that changes state


39

Column space

• dx/dt = s1v1 + s2v2 …+ snvn, where– si is the ith column in S

• Hence, dx/dt is in the column space of S• Contains all allowable time derivatives of

the concentrations vector and hence how the thermodynamic driving forces move concentration state of network


40

Left column space

• Together Left null(S) and Row(S) span ℜm

– Dim(Left null(S)) + Dim(Col(S)) = m• Vectors in the left null space of S are

orthogonal to Col(S)• Left null(S) contains all the conservation

relationships, i.e., time invariants, defined by the network. This defines conserved metabolic pools as combinations of metabolites.


41

Four fundamental spaces

Row(S)

Null(S)

Col(S)

Left null(S)

vdyn

vss

v dx/dt

ℜn ℜm

S[m,n]


42

Basis for vector spaces• A basis for a space is a set of

vectors that can be used to span the space, e.g.,– b1=(1,0,0), b2=(0,1,0) and

b3=(0,0,1)– Any vector v ∈ ℜ(3) can be

decomposed as– v = w1b1 + w2b2 + w3b3– So [b1,b2,b3] is a basis for ℜ(3)

• Many types of bases– Linear basis– Orthonormal (linear) basis for

linear spaces– Convex basis for finite linear

spaces

x

y

z

(w1,w2,w3)

b1

b2

b3


43

Singular value decomposition

[ ] [ ] [ ] [ ]TVΣUS nnnmmmnm ,,,, =

• U and V are orthonormal matrices– UTU = I(mxm) and VTV = I(nxn)

– UT = U-1 and VT = V-1

• Σ = diag(σ1, σ2,…, σr), where σ1≥ σ2 ≥,…, ≥ σr>0


44

Orthonormal bases for subspaces

Col(S)Left

null(S)

Row(S)

Null(S)

U VT

rxr

mxn

=

S Σ

Columns in U defines orthonormal basis for Col(S) and Left null(S)Columns in V defines orthonormal basis for Row(S) and Null(S)


45

U-1.20E-01 3.85E-02 5.39E-03 1.44E-01 -4.77E-03 3.96E-01 -2.98E-01 5.77E-01 2.03E-01 -9.68E-02 1.99E-01 -3.07E-01 -4.48E-01 1.87E-16 5.02E-17 -2.01E-17 3.01E-17 -1.00E-17 -8.03E-171.28E-01 -4.34E-02 2.66E-03 -1.66E-01 -1.49E-01 -5.71E-01 2.75E-01 -1.66E-16 -1.44E-02 1.49E-02 2.76E-01 -5.06E-01 -3.18E-01 2.63E-01 -7.39E-02 -7.28E-02 -6.70E-02 6.70E-02 -9.15E-02

-1.28E-01 3.53E-02 -2.82E-02 6.28E-02 3.88E-01 3.41E-01 4.06E-02 -5.77E-01 -1.69E-01 3.06E-02 -8.19E-02 -7.39E-02 -4.87E-01 2.63E-01 -7.39E-02 -7.28E-02 -6.70E-02 6.70E-02 -9.15E-021.25E-01 1.92E-02 8.38E-02 1.57E-01 -6.01E-01 5.17E-02 -2.65E-01 -1.94E-15 -3.77E-02 7.98E-02 -2.33E-01 2.54E-01 -6.39E-02 5.25E-01 -1.48E-01 -1.46E-01 -1.34E-01 1.34E-01 -1.83E-01

-8.64E-03 -3.13E-03 -3.36E-02 -8.17E-02 3.93E-01 -5.47E-02 3.39E-01 5.77E-01 -3.72E-01 1.27E-01 -2.81E-01 2.33E-01 -3.91E-02 2.63E-01 -7.39E-02 -7.28E-02 -6.70E-02 6.70E-02 -9.15E-02-7.26E-02 -3.42E-01 -1.64E-01 -3.92E-01 2.93E-01 -1.38E-01 -2.15E-01 1.08E-14 5.72E-01 -1.83E-01 5.97E-02 2.89E-01 7.33E-03 2.63E-01 -7.39E-02 -7.28E-02 -6.70E-02 6.70E-02 -9.15E-021.32E-01 2.21E-01 3.18E-02 4.22E-01 2.32E-01 -1.29E-01 -4.77E-02 -9.36E-17 -7.90E-03 -4.89E-01 1.49E-01 -6.78E-03 2.09E-01 1.77E-01 1.79E-01 1.87E-01 -3.72E-01 3.72E-01 3.71E-02

-8.06E-02 4.11E-02 1.89E-01 -4.61E-01 -2.50E-01 2.50E-01 2.24E-01 -3.12E-15 -1.85E-01 -1.67E-01 3.60E-01 2.93E-01 -1.36E-01 -7.07E-02 4.40E-01 -4.95E-02 -1.76E-01 1.76E-01 -8.98E-025.55E-03 -9.29E-03 -4.79E-01 2.88E-01 -3.27E-02 -2.94E-02 1.66E-01 5.20E-15 2.77E-01 5.40E-01 4.55E-02 9.44E-02 -8.66E-02 -7.07E-02 4.40E-01 -4.95E-02 -1.76E-01 1.76E-01 -8.98E-021.07E-01 -6.08E-02 5.09E-01 -8.15E-02 2.18E-01 9.31E-02 -1.82E-01 1.54E-15 6.79E-02 3.84E-01 3.68E-02 -2.59E-01 2.88E-01 -1.34E-01 -2.71E-02 -2.75E-01 -2.46E-01 2.46E-01 -3.33E-01

-6.41E-02 2.66E-01 -3.10E-01 -2.54E-01 -6.21E-02 -1.77E-01 -2.39E-01 -2.46E-15 -9.00E-02 -2.54E-01 -4.85E-01 -1.37E-01 -1.50E-01 -3.96E-01 4.67E-02 -2.02E-01 -1.79E-01 1.79E-01 -2.42E-01-9.36E-03 -1.93E-01 1.72E-01 2.22E-01 -9.67E-02 1.69E-01 5.05E-01 6.79E-15 3.35E-01 -3.61E-01 -3.30E-01 -1.54E-01 6.16E-02 1.44E-02 1.88E-01 -3.09E-01 1.29E-01 -1.29E-01 -2.18E-01-5.15E-01 2.32E-01 1.24E-02 -8.70E-02 -9.17E-02 3.92E-02 1.47E-01 3.21E-15 1.44E-01 7.73E-02 -5.19E-02 -1.16E-01 1.72E-01 1.24E-01 -1.10E-01 5.77E-01 -3.36E-03 3.36E-03 -4.57E-015.15E-01 -2.32E-01 -1.24E-02 8.70E-02 9.17E-02 -3.92E-02 -1.47E-01 -3.26E-15 -1.44E-01 -7.73E-02 5.19E-02 1.16E-01 -1.72E-01 -1.24E-01 1.52E-01 3.41E-01 1.92E-01 -1.92E-01 -5.84E-015.06E-02 -3.29E-01 -2.40E-01 -2.39E-01 -3.13E-02 2.23E-01 -1.70E-01 -3.91E-15 -2.29E-01 -4.35E-03 -1.95E-01 -4.49E-01 3.07E-01 3.26E-01 3.94E-01 1.52E-01 3.01E-03 -3.01E-03 1.52E-01

-1.07E-01 7.67E-02 4.98E-01 -1.06E-02 8.24E-02 -3.15E-01 -1.72E-01 1.89E-15 1.45E-01 1.21E-01 -3.59E-01 6.86E-02 -2.94E-01 6.31E-02 4.68E-01 2.25E-01 7.00E-02 -7.00E-02 2.43E-019.98E-02 4.58E-01 -5.96E-02 -8.61E-02 1.05E-01 8.47E-03 -7.79E-02 1.50E-15 6.20E-02 1.38E-02 1.06E-01 -1.53E-02 1.25E-01 2.13E-01 1.65E-01 -2.08E-01 7.19E-01 2.81E-01 -9.76E-02

-9.98E-02 -4.58E-01 5.96E-02 8.61E-02 -1.05E-01 -8.47E-03 7.79E-02 -1.50E-15 -6.20E-02 -1.38E-02 -1.06E-01 1.53E-02 -1.25E-01 -2.13E-01 -1.65E-01 2.08E-01 2.81E-01 7.19E-01 9.76E-025.73E-01 2.71E-01 8.66E-03 -2.85E-01 -2.73E-02 2.76E-01 2.51E-01 6.73E-15 3.23E-01 7.91E-02 -2.15E-01 -6.14E-02 -6.93E-02 -1.44E-02 -1.88E-01 3.09E-01 -1.29E-01 1.29E-01 2.18E-01

Col(S) Left Null(S)


46

Σ4.059613 0 0 0 0 0 0 0 0 0 0 0 0 0

0 2.851793 0 0 0 0 0 0 0 0 0 0 0 00 0 2.120736 0 0 0 0 0 0 0 0 0 0 00 0 0 1.979295 0 0 0 0 0 0 0 0 0 00 0 0 0 1.878411 0 0 0 0 0 0 0 0 00 0 0 0 0 1.715997 0 0 0 0 0 0 0 00 0 0 0 0 0 1.667575 0 0 0 0 0 0 00 0 0 0 0 0 0 1.414214 0 0 0 0 0 00 0 0 0 0 0 0 0 1.377889 0 0 0 0 00 0 0 0 0 0 0 0 0 1.171902 0 0 0 00 0 0 0 0 0 0 0 0 0 1.082776 0 0 00 0 0 0 0 0 0 0 0 0 0 0.953417 0 00 0 0 0 0 0 0 0 0 0 0 0 0.603711 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0

r = 13, i.e., 13 linearly, independent relationships between compounds and reactions


47

V4.56E-01 -9.62E-02 -8.89E-03 -2.13E-01 6.42E-03 -4.48E-01 3.18E-01 -4.08E-01 -1.32E-01 3.08E-02 -3.17E-02 -2.93E-02 -4.71E-01 1.62E-01

-6.30E-02 2.76E-02 -1.45E-02 1.16E-01 2.86E-01 5.31E-01 -1.41E-01 -4.08E-01 -1.12E-01 1.34E-02 -3.30E-01 4.53E-01 -2.79E-01 1.62E-014.57E-01 -7.31E-02 4.52E-02 -8.68E-03 -4.43E-01 -5.36E-02 -2.09E-01 4.08E-01 1.21E-01 -2.24E-02 -2.42E-01 5.23E-01 1.47E-02 1.62E-01

-5.08E-02 -1.28E-01 -1.33E-01 -3.19E-01 6.85E-01 -1.42E-01 2.33E-01 4.08E-01 1.73E-01 -1.16E-01 1.03E-02 2.81E-01 5.33E-02 1.62E-01-1.58E-02 -1.19E-01 -6.14E-02 -1.57E-01 -5.33E-02 -4.84E-02 -3.32E-01 -4.08E-01 6.85E-01 -2.65E-01 3.15E-01 5.92E-02 7.68E-02 1.62E-012.28E-01 7.29E-01 1.53E-01 3.01E-01 8.15E-02 4.57E-02 2.60E-01 1.82E-15 6.95E-02 -1.66E-01 2.61E-01 6.45E-02 1.26E-01 3.24E-01

-3.06E-01 9.95E-02 8.58E-02 -5.34E-01 -3.54E-01 2.66E-01 3.39E-01 1.79E-15 8.04E-02 4.07E-01 9.94E-02 7.10E-02 -6.62E-04 3.24E-012.12E-02 -1.77E-02 -3.15E-01 3.79E-01 1.16E-01 -1.63E-01 -3.52E-02 6.07E-15 3.35E-01 6.03E-01 -2.91E-01 -2.09E-01 8.13E-02 3.24E-01

-1.32E-03 8.83E-03 7.01E-01 -1.92E-01 1.77E-01 -1.12E-01 -3.12E-01 -1.18E-15 -4.64E-02 -2.96E-02 -3.40E-01 -2.99E-01 1.34E-01 3.24E-01-4.37E-01 1.82E-01 -3.79E-01 -3.10E-02 -2.32E-01 -2.72E-01 -8.84E-03 -3.37E-15 -1.40E-01 -4.80E-01 -3.80E-01 -5.18E-02 -4.05E-02 3.24E-01-1.77E-01 -5.78E-01 2.80E-01 4.71E-01 -1.16E-01 3.12E-02 3.89E-01 -1.19E-16 -1.59E-02 -1.82E-01 1.46E-01 7.91E-02 5.00E-02 3.24E-014.34E-01 -2.10E-01 -3.56E-01 -1.71E-01 2.26E-02 4.29E-01 -2.42E-02 -4.14E-15 -2.46E-01 -1.71E-01 4.92E-02 -3.64E-01 3.11E-01 3.24E-01

-2.94E-02 1.35E-02 2.54E-03 7.30E-02 -2.54E-03 2.31E-01 -1.79E-01 4.08E-01 1.47E-01 -8.26E-02 1.84E-01 -3.22E-01 -7.41E-01 1.62E-01-1.39E-01 -2.74E-02 -8.52E-02 3.17E-02 6.60E-02 -2.59E-01 -4.53E-01 -9.55E-15 -4.77E-01 2.40E-01 5.03E-01 2.26E-01 1.28E-02 3.24E-01

1 -9.76E-10 -3.19E-10 6.62E-10 -1.89E-11 3.06E-10 -6.16E-12 -2.86E-10 2.38E-10 1.28E-10 6.45E-12 -3.61E-10 7.65E-11 -2.34E-101 -5.12E-10 -6.94E-11 -3.04E-10 -3.12E-10 -2E-10 -1.22E-10 6.17E-11 4.35E-10 2.74E-10 -3.97E-11 -2.35E-11 -1.52E-10

Row(S) Null(S)


48

Eigen-reaction

( ) ( )

( ) ( )vvxu

vVΣxU

vVUΣUxU

vVUΣSvx

TT

TT

TTT

T

:rkfor

kkk

dtd

dtd

dtd

dtd

σ=

≤

=

=

== ( )( )

∑∑>

<

>

<

∑ −⎯⎯←⎯→⎯

∑

−

−+++=

+++=

0for

0for

0for

0for

2211T

2211T

:reactionEigenpathway

pool

ki

kj

kj

ki uiki

vjkj

vjkj

uiki

nknkkk

mkmkkk

xu

vv

vv

xu

vvvvvv

xuxuxu

L

L

vv

xu

1st Mode (σ = 4.060)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 5 10 15 20

v8 2.12E-02v6 2.28E-01v12 4.34E-01

dhap -0.008642 v1 4.56E-01 2pg 0.005551lac-L -0.00936 v3 4.57E-01 pi 0.050597pyr -0.064063 nadh 0.099802g3p -0.072627 pep 0.1073193pg -0.08055 fdp 0.125139nad -0.099802 v9 -1.32E-03 g6p 0.127779h2o -0.107144 v5 -1.58E-02 13dpg 0.131546glc-D -0.119502 b1 -2.94E-02 adp 0.514654f6p -0.128144 v4 -5.08E-02 h 0.573352atp -0.514654 v2 -6.30E-02

b2 -1.39E-01v11 -1.77E-01v7 -3.06E-01v10 -4.37E-01

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15

h

atpadp

u2 v2

v3 v1

v10 v12


50

Mode 1


51

Mode 2


52

Null spaces

glc-D + atp -> g6p + adp + h v1 0.1622 1g6p <-> f6p v2 0.1622 1f6p + atp -> fdp + adp + h v3 0.1622 1fdp <-> dhap + g3p v4 0.1622 1dhap <-> g3p v5 0.1622 1g3p + pi + nad <-> 13dpg + nadh + h v6 0.3244 213dpg + adp <-> 3pg + atp v7 0.3244 23pg <-> 2pg v8 0.3244 22pg <-> pep + h2o v9 0.3244 2pep + adp + h -> pyr + atp v10 0.3244 2pyr + nadh + h <-> lac-L + nad v11 0.3244 2atp + h2o -> adp + pi + h v12 0.3244 2glc-D[e] <-> glc-D b1 0.1622 1lac-L + h <-> h[e] + lac-L[e] b2 0.3244 2

Conserved metabolite pools Steady state flux balance


53

A0≤v1 ≤10

0≤v2 ≤6

0≤v3 ≤8( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛−−==

3

2

1111

vvv

dtdA

dtd Svx

( )( )

A

vvv

A

vv

v

⎯⎯⎯⎯⎯ ⎯←⎯⎯⎯⎯⎯ →⎯

−−=

⋅=

−

+ 32

1

5774.05774.0

5774.0

321T1

T1

0

pathway5774.05774.05774.0

pool1:reactionEigen

vv

xu

( )( ) 0788702113057740211307887057740577405774057740

001.73211 =⎟⎟⎠

⎞⎜⎜⎝

⎛

−−−−

=.........

S


54

80601007887.02113.0

5774.0

2113.07887.05774.0

321

21

≤≤∧≤≤∧≤≤

⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟

⎟⎠

⎞⎜⎜⎝

⎛

−=

=

vvv

wwSS

SS

v

0Sv

Orthonormal basis for Null(S)

A0≤v1 ≤10

0≤v2 ≤6

0≤v3 ≤8

8060100101

011

321

21

≤≤∧≤≤∧≤≤

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

vvv

ααSSv Convex basis for Null(S)


55

100:100 211 ≤+≤≤≤ ααv

60:60 12 ≤≤≤≤ αv

80:80 23 ≤≤≤≤ αv

8060100101

011

321

21

≤≤∧≤≤∧≤≤

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛=

vvv

ααSSv Convex basis for Null(S)

Since all vi≥0 and all basis elements positive: αi ≥0


56

α1

0 2 4 6 8 10

α2

0

2

4

6

8

10

12

0

5

10

15

20

25

0

2

4

6

8

10

02

46

8

v 1

v 2

v3


57

Linear vs convex spacesLinear space• Described by linear equations

• Vector space defined by set of linearly independent basis vectors (bi)

v = Σwibi -∞< wi <+∞

• Every point uniquely described by linear combination of bi

• Number of basis vectors equals dim(Null(S))

• Infinite number of bases can span space

Convex space• Described by linear equations

and inequalities• Convex polyhedral cone

defined by conically independent generating vectors (pi)

v = Σ αipi 0≤ αi <+∞

• Every point described by non-negative, linear combination of pi (non-unique)

• Number of generation vectors may exceed dim(Null(S))

• Unique set of generating vectors


58

Extreme pathways• vss = Σ αipi 0≤ αi < αi,max• Pi’s are unique & correspond to edges in (n-r)-

dimensional cone; αi’s not unique• Correspond to pathways on a flux map • Termed extreme pathways, since they define edges of

bounded null space in its conical representation


59

ExPa classificationp1 pn

v1

b1

c1

Cur

renc

yP

rimar

y

Internal fluxes

Exchange fluxes

Type IPrimary pathways

Type IIFutile cycles

Type IIIInternal cycles

≠0

≠0 ≠0

=0 =0

=0

Type I Type II Type III

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 b1 b2 b3 b4 c1 c2 c3 c4 c5 c6 c7glc-D -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0g6p 1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b1f6p 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 glc-Dfdp 0 0 0 1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v1

dhap 0 0 0 0 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g6pg3p 0 0 0 0 1 -1 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v2 v33pg 0 0 0 0 0 0 0 0 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f6ppep 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v4pyr 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 fdp

accoa 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 v5 v6acp 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 dhap g3plac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 v7/v8 v9/v10

etoh 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 3pgac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 -1 0 0 0 0 0 0 0 v11/v12for 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 pepatp -1 0 0 -1 0 0 0 0 1 -1 0 0 1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 -1 0 0 0 0 0 v13adp 1 0 0 1 0 0 0 0 -1 1 0 0 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 pyr lacpi 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 v14 v17/v18 b2

nadh 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 -1 1 -2 2 0 0 0 0 0 0 0 0 0 0 -1 0 0 accoa etohnad 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 1 -1 2 -2 0 0 0 0 0 0 0 0 0 0 0 -1 0 v15/16 v19/v20 b3coa 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 -1 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 acp ac

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 b1 b2 b3 b4 c1 c2 c3 c4 c5 c6 c7 v21/v22 b4

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 b1 b2 b3 b4 c1 c2 c3 c4 c5 c6 c7Type 3 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Type 3 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Type 3 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Type 3 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Type 3 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Type 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0glc-lac 1 1 0 1 1 0 1 0 2 0 2 0 2 0 0 0 2 0 0 0 0 0 1 2 0 0 0 2 -2 -2 0 0 0Type 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Type 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0lac-etoh 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 -1 1 0 1 0 0 0 -1 1 0glc-etoh 1 1 0 1 1 0 1 0 2 0 2 0 2 2 0 0 0 0 2 0 0 0 1 0 2 0 2 2 -2 -2 -2 2 0Type 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0ac-etoh 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 -1 0 -1 1 1 -2 2 0etoh-ac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 -1 1 0 1 -1 -1 2 -2 0lac-ac 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 -1 0 1 1 1 -1 -1 1 -1 0glc-ac 1 1 0 1 1 0 1 0 2 0 2 0 2 2 2 0 0 0 0 0 2 0 1 0 0 2 2 4 -4 -4 2 -2 0

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 b1 b2 b3 b4 c1 c2 c3 c4 c5 c6 c7glc-D -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 b1g6p 1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 glc-Df6p 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v1fdp 0 0 0 1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g6p

dhap 0 0 0 0 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v2 v3g3p 0 0 0 0 1 -1 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f6p3pg 0 0 0 0 0 0 0 0 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v4pep 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 fdppyr 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v5 v6

accoa 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 dhap g3pacp 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 v7/v8 v9/v10lac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 3pg

etoh 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 v11/v12ac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0 0 -1 0 0 0 0 0 0 0 pepfor 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 v13atp -1 0 0 -1 0 0 0 0 1 -1 0 0 1 0 0 -2 0 0 0 0 1 -1 0 0 0 0 0 -1 0 0 0 0 0 pyr lacadp 1 0 0 1 0 0 0 0 -1 1 0 0 -1 0 0 2 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 v14 v17/v18 b2pi 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 -1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 accoa etoh

nadh 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 -1 1 -2 2 0 0 0 0 0 0 0 0 0 0 -1 0 0 v15 v19/v20 b3nad 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 1 -1 2 -2 0 0 0 0 0 0 0 0 0 0 0 -1 0 acp accoa 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 -1 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 v21/v22 b4

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 b1 b2 b3 b4 c1 c2 c3 c4 c5 c6 c7

v16

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 b1 b2 b3 b4 c1 c2 c3 c4 c5 c6 c7Type 3 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Type 3 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Type 3 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Type 3 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Type 3 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0Type 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0glc-lac 1 1 0 1 1 0 1 0 2 0 2 0 2 0 0 0 2 0 0 0 0 0 1 2 0 0 0 2 -2 -2 0 0 0ac-etoh 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 -1 0 -2 2 2 -2 2 0Type 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0lac-etoh 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 -1 1 0 1 0 0 0 -1 1 0glc-etoh 1 1 0 1 1 0 1 0 2 0 2 0 2 2 0 0 0 0 2 0 0 0 1 0 2 0 2 2 -2 -2 -2 2 0Type 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0

Futile cycle 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 -1 1 1 0 0 0etoh-ac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 -1 1 0 1 -1 -1 2 -2 0lac-ac 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 -1 0 1 1 1 -1 -1 1 -1 0glc-ac 1 1 0 1 1 0 1 0 2 0 2 0 2 2 2 0 0 0 0 0 2 0 1 0 0 2 2 4 -4 -4 2 -2 0

glc-etoh/ac 2 2 0 2 2 0 2 0 4 0 4 0 4 4 2 0 0 0 2 0 2 0 2 0 2 2 4 6 -6 -6 0 0 0

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 b1 b2 b3 b4 b5glc-D -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 b1g6p 1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 glc-Df6p 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v1fdp 0 0 0 1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g6p

dhap 0 0 0 0 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v2 v3g3p 0 0 0 0 1 -1 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f6p3pg 0 0 0 0 0 0 0 0 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v4pep 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 fdppyr 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 v5 v6

accoa 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 0 0 -1 1 0 0 0 0 0 0 0 0 dhap g3pacp 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 1 0 0 0 0 0 0 v7/v8 v9/v10lac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 -1 0 0 0 3pg

etoh 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 -1 0 0 v11/v12ac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0 0 0 -1 0 pepfor 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 v13atp -1 0 0 -1 0 0 0 0 1 -1 0 0 1 0 0 -2 0 0 0 0 1 -1 -1 0 0 0 0 0 pyr lacadp 1 0 0 1 0 0 0 0 -1 1 0 0 -1 0 0 2 0 0 0 0 -1 1 1 0 0 0 0 0 v14 v17/v18 b2pi 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 -1 2 0 0 0 0 0 0 1 0 0 0 0 0 accoa etoh

nadh 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 -1 1 -2 2 0 0 0 0 0 0 0 0 v15 v19/v20 b3nad 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 1 -1 2 -2 0 0 0 0 0 0 0 0 acp accoa 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 -1 0 0 1 -1 0 0 0 0 0 0 0 0 v21/v22 b4

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 b1 b2 b3 b4 b5

v16

t3 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0t3 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0t3 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0t3 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0t3 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0t3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0t3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0t3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0

glc-lac 1 1 0 1 1 0 1 0 2 0 2 0 2 0 0 0 2 0 0 0 0 0 2 1 2 0 0 0glc-lac 1 1 0 1 1 0 1 0 2 0 2 0 2 0 2 2 2 0 0 0 2 0 0 1 2 0 0 0

lac-etoh/ac 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 2 1 0 1 0 1 0 -2 1 1 2lac-etoh/ac 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 1 0 2 1 0 2 0 0 0 -2 1 1 2glc-etoh/ac 1 1 0 1 1 0 1 0 2 0 2 0 2 2 1 0 0 0 1 0 1 0 3 1 0 1 1 2glc-etoh/ac 1 1 0 1 1 0 1 0 2 0 2 0 2 2 4 3 0 0 1 0 4 0 0 1 0 1 1 2

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 b1 b2 b3 b4 b5

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 b1 b2 b3 b4 b5glc-D -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0g6p 1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b1f6p 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 glc-Dfdp 0 0 0 1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v1

dhap 0 0 0 0 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g6pg3p 0 0 0 0 1 -1 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v2 v33pg 0 0 0 0 0 0 0 0 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f6ppep 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v4pyr 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 fdp

accoa 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 0 0 -1 1 0 0 0 0 0 0 0 0 v5 v6acp 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 1 0 0 0 0 0 0 dhap g3plac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 -1 0 0 0 v7/v8 v9/v10

etoh 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 -1 0 0 3pgac 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0 0 0 -1 0 v11/v12for 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 pepatp -1 0 0 -1 0 0 0 0 1 -1 0 0 1 0 0 -2 0 0 0 0 1 -1 -1 0 -1 0 0 0 v13adp 1 0 0 1 0 0 0 0 -1 1 0 0 -1 0 0 2 0 0 0 0 -1 1 1 0 1 0 0 0 pyr lacpi 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 -1 2 0 0 0 0 0 0 1 0 1 0 0 0 v14 v17/v18 b2

nadh 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 -1 1 -2 2 0 0 0 0 0 0 0 0 accoa etohnad 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 1 -1 2 -2 0 0 0 0 0 0 0 0 v15 v19/v20 b3coa 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 -1 0 0 1 -1 0 0 0 0 0 0 0 0 acp ac

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 b1 b2 b3 b4 b5 v21/v22 b4

v16

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 v17 v18 v19 v20 v21 v22 v23 b1 b2 b3 b4 b51 1 0 1 1 0 1 0 2 0 2 0 2 0 0 0 2 0 0 0 0 0 0 1 2 0 0 01 1 0 1 1 0 1 0 2 0 2 0 2 2 1 0 0 0 1 0 1 0 3 1 0 1 1 21 1 0 1 1 0 1 0 2 0 2 0 2 2 4 3 0 0 1 0 4 0 0 1 0 1 1 2


64

v ∈ ℜ(n) v ∈ subspace of ℜ(n)

Sv = 0

StoichiometryLinear algebra

Eigenreactions

v = Σαipi , αi ≥ 0convex cone

ExPa

Reaction directionConvex analysis

v ≥ 0


65

Pathway matrix, P, and Binary P

Binary P

( ) ( )

⎭⎬⎫

⎩⎨⎧

≠===

==

0 if1ˆ0 if0ˆ

:ˆ

|||

ijij

ijij

pppp

P

pppP 321 K

PPP)) T

LM = Pathway length matrixDiagonal elements = # reactions in each ExPaOff-diagonal elements = # shared reactions between two ExPa

TPM PPR ˆˆ= Reaction participation matrix

Diagonal elements = # ExPa in which a given reaction is found Off-diagonal elements = # ExPa that contains given pair of reactions


66

JAK-STAT signaling network• Input

– 15 receptors with ligands• Output

– 7 STAT homo- and heterodimers• 297 reactions

– 216 internal– 81 irreversible exchange

• 147 extreme pathways

http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B94RW-4V4MF7P-4&_image=B94RW-4V4MF7P-4-C&_ba=&_user=331728&_rdoc=1&_fmt=full&_orig=search&_cdi=56421&view=c&_isHiQual=Y&_acct=C000016898&_version=1&_urlVersion=0&_userid=331728&md5=38aadd5b7808cfaa6028ab35ac734426

http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B94RW-4V4MF7P-4&_image=B94RW-4V4MF7P-4-8&_ba=&_user=331728&_rdoc=1&_fmt=full&_orig=search&_cdi=56421&view=c&_isHiQual=Y&_acct=C000016898&_version=1&_urlVersion=0&_userid=331728&md5=a8ab5a7fa81adfead84ecd05c3de9c3c


67

Reaction participation

168 reactions participate in only one extreme pathwayVery specific function, i.e., ideal drug targets

http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B94RW-4V4MF7P-4&_image=B94RW-4V4MF7P-4-K&_ba=&_user=331728&_rdoc=1&_fmt=full&_orig=search&_cdi=56421&view=c&_isHiQual=Y&_acct=C000016898&_version=1&_urlVersion=0&_userid=331728&md5=cbe20039defd36d65e2b098c52deffe5

Cross-talk• Cross-talk

– 147 ExPAs– 10,731 pairwise

comparisons

• Observations– All pathways have single

output– 99.8% disjoint output– 0.2% identical output– 63.9% deterministic– 14.8% classical cross talk

http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B94RW-4V4MF7P-4&_image=B94RW-4V4MF7P-4-G&_ba=&_user=331728&_rdoc=1&_fmt=full&_orig=search&_cdi=56421&view=c&_isHiQual=Y&_acct=C000016898&_version=1&_urlVersion=0&_userid=331728&md5=5b1976b68a4acc56e467df7bf66f62ba


69

Correlated reaction sets


70


Sv = 0


Eigenreactions


ExPa


v ≥ 0

COBRA = Constraint-based reconstruction and analysis of

metabolic and regulatory networks


72


Sv = 0


Eigenreactions


ExPa


v ≥ 0


73

Link to real world• Defines feasible solution space in terms of balanced

pathways• Thermodynamic, regulatory, kinetic constraints

– Delete unfeasible ExPas to see true solution space• Gene KO and upregulation

– Removes all ExPas that use particular gene– Identify KO candidates among genes not required in desired

ExPas– Identify upregulation candidates among genes with high

coefficients or that are unique in desired ExPas• Max yields

– Yields calculated for individual ExPa(s)– Trade-off between biomass vs product


74


Sv = 0


Eigenreactions


ExPa


v ≥ 0

Union of convex subsets

RegulationThermoKinetics


75

Link to real world• Defines feasible solution space in terms of balanced

pathways• Thermodynamic, regulatory, kinetic constraints

– Delete unfeasible ExPas to see true solution space• Gene KO and upregulation

– Removes all ExPas that use particular gene– Identify KO candidates among genes not required in desired

ExPas– Identify upregulation candidates among genes with high

coefficients or that are unique in desired ExPas• Max yields

– Yields calculated for individual ExPa(s)– Trade-off between biomass vs product


76

In silico E. coli model

Network:71 reactions- 21 reversible55 internal metabolites13 external metabolites

Captures:- Substrate uptake- Glycolysis- PPP, EDP, TCA-Cycle- Anaplerosis- Respiration- Fermentation- Biomass formation- 4 different pathways

for product formation


773HP

4 natural pathways for 3HP production (KEGG)

http://www.genome.jp/Fig/compound/C00894.gif














78

Left (1) Right (2) Center(3) Center-right (4)

3-HP Yield (aerobic)

95.8 % 96.6 % 84.2 % 100 % **

3-HP Yield (anaerobic)

100 % 100 % *(50%)

85.7 % 100 % **

Yields are given in C‐mol / C‐mol [%]

In general the yields are higher under anaerobic conditions

The second pathway is strongly dependent on a reversible acetate kinase.

The third pathway underperforms the other in both scenarios.

Two unknown enzymes in the fourth pathway

Effect on product formation (anaerobic, aerobic)


79

3HP

#product synthesis ‐ left pathwayR54 : OAA + GLU = ASP + 2‐OXOR55 : ASP = bALA + CO2R56 : bALA + 2‐OXO = 3‐OXOPRO + GLUR57 : 3‐OXOPRO + NADH = 3‐HPA + NAD

In silico analysis of β-alanine pathway for the production of 3-HP in E. coli







80

Carbon yield for 3-HP

Car

bon

yiel

d fo

r bio

mas

s 16881 elementary modes2292 make the desired product

Mode #11048Max P with X>0

AnaerobicDoes not form acetate

Does not require PEP-carboxylase and GPIHighly dependent on malic enzyme (NADPH)

Elementary mode analysis


81Carbon yield for 3-HP

Car

bon

yiel

d fo

r bio

mas

s

464 elementary modes126 make the desired product

Knock-out of GPI, PEP-C and Acetate kinaseAnaerobic conditions


82


Sv = 0


Eigenreactions


ExPa


v ≥ 0

v = Σαipi , αi ≥ 0Bounded convex cone0 ≤ αi ≤ αi,max

Capacity constraintsv ≤ vmax

Union of convex subsets

RegulationThermoKinetics


83

Linear programming

nivvv

vwZ

iii

ii

,,1,and subject tomaximize

max,min, K=≤≤=

=⋅= ∑0Sv

vw Objectives• Linear

– Max growth (μ)– Max product (π)– Min substrate (σ)– Max w1μ+w2π

• Nonlinear– Min ||v||2 (QP)


84

LP solutions• Shadow price (dual)

– Increase in objective function for a unit increase in a constraint

• Reduced cost– Increase in objective

function for unit increase in flux

Unique

Degenerate

Unbounded


85

ExPa vs LP• ExPa

– Define all feasible points – Define all extreme points in unbounded problem

• Degenerate solutions are linear combinations of ExPas with identical objective function

– Does not define all points in bounded problem• Length constrained by capacities

– Computationally challenging• LP

– Computationally inexpensive even for large problems– Give one extreme solution point, reconstructing all more difficult– Many approaches developed


86

Mixed integer programming

(binary)integer

,,1,0,,1,

and

subject to

maximize

max,min,

i

ji

iii

y

mjnynivvv

Z

K

K

=≤≤

=≤≤≥=≤

⎥⎦⎤

⎢⎣⎡

⎥⎦⎤

⎢⎣⎡⋅=

byvA

yvw


87


88

PhPP


89


90

Phenotypic phase plane

Ibarra et al, Nature 420, 186 - 189 (2002)


91

AraGEMGene-reaction-association entries 5253ORFs (unique) 1419Metabolites 1748Unique reactions 1567

Cytosolic reactions 1265Mitochondrial reactions 60Plastidic reactions 159Peroxisomal reactions 98

Modified reactions 36Biomass drains and transporters 148

Biomass drains 47Transporters (Intercellular) 18Transporters (Inter-organelle) 83

Gaps (unique reactions ID) filled by manual curation 75Singleton metabolites 446


92


93

AssumptionsInputs, outputs and constraints Case 1 Case 2Photons uptake (free flux) + +Glutamine transporter (mitochondria) - -Glutamate transporter(mitochondria) - -Glutamine transporter (plastid) - -Glutamate transporter (plastid) + +RuBisCO; EC 4.1.1.39 (carboxylation:oxygenation; 3:1)

- +

Fd-GOGAT ; EC 1.4.7.1 (plastid) + +NADH-GOGAT; EC 1.4.1.14 (plastid) - -Optimization: minimize uptake of Photons PhotonsBiomass rate (estimated and fixed) Leaf Leaf


94

RuBisCO 3:1 C:O: ~40% increase in photon requirements (litt 30-50%)

lkn/2009 winter school in mathematical & 1 computational...

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