Winter School in Mathematical & Computational Biology

Transcription

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2 Network reconstruction, topology and feasible solution space

3 From component to systems biology Component biology Systems biology Component view Systems view Needed homeostasis Function S+E X E+P Reaction network Time-dependent concentration Steady state flux map HT analytical chemistry Integrative analysis Compute flux Calculate C Calculate k Palsson (2000) Nat Biotech 20:649 3

4 E. coli on glycerol Objective Constraints Max growth rate on glycerol Network topology Steady state Maximum rates Ibarra et al, Nature 420, (2002) 4

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 5

6 Systems biology Components Gen- Transcript- Prote- Metabol- Systemic annotation Reconstruction of biochemical network (unique!) In silico modelling Topology Constraints Dynamics Sensitivity Noise Hypothesis generation & testing Phenotypic space is essentially infinite 6

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 7

8 Systemic (2D) annotation Palsson (2004) Nat Biotech 22:1218 8

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 9

10 Annotation workflow 10

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 11

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 H 2 O, H +, redox (difficult to pick up) Lumped reactions (e.g., PDH) Connectivity Membrane transporters Biomass drains 12

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 13

14 Model overview Unique gene-reaction associations 4617 Number of genes 1399 Mapped reactions 1757 Other reactions Reactions w/o gene association Membrane transporters Biomass reactions Autocatalytic Total number of reactions Rxns in mitochondria Metabolites

15 Topology S [m,n] Rows for m metabolites Columns for reactions

16 glc-d g6p f6p fdp dhap g3p 13dpg 3pg 2pg pep pyr lac-l atp adp pi h2o nadh nad h glc-d[e] lac-d[e] h[e] Enzyme Protein EC # Reaction HEX Hexokinase hk glc-d + atp -> g6p + adp + h v1 PGI1 Phosphoglucose isomerase pgi g6p <-> f6p v2 PFKA Phosphofructokinase pfka f6p + atp -> fdp + adp + h v3 FBA Fructose-1,6-bisphosphatate aldolase fba fdp <-> dhap + g3p v4 TPIA Triosphosphate Isomerase tpia dhap <-> g3p v5 GAPA G3P dehydrogenase-a complex gapa g3p + pi + nad <-> 13dpg + nadh + h v6 PGK Phosphoglycerate kinase pgk dpg + adp <-> 3pg + atp v7 GPMA Phosphoglycerate mutase 1 gpma pg <-> 2pg v8 ENO Enolase eno pg <-> pep + h2o v9 PYKF Pyruvate Kinase I pykf pep + adp + h -> pyr + atp v10 LDH_L L-Lactate dehydrogenase Ldh pyr + nadh + h <-> lac-l + nad v11 ATP hydrolysis atp + h2o -> adp + pi + h v12 GLCt glucose exchange glc-d[e] <-> glc-d b1 L-LAC-t lactate transport lac-l + h <-> h[e] + lac-l[e] b2 16

17 glc-d g6p f6p fdp dhap g3p Reaction S' glc-d + atp -> g6p + adp + h v g6p <-> f6p v2-1 1 f6p + atp -> fdp + adp + h v fdp <-> dhap + g3p v dhap <-> g3p v5-1 1 g3p + pi + nad <-> 13dpg + nadh + h v dpg + adp <-> 3pg + atp v pg <-> 2pg v pg <-> pep + h2o v pep + adp + h -> pyr + atp v pyr + nadh + h <-> lac-l + nad v atp + h2o -> adp + pi + h v glc-d[e] <-> glc-d b1 1-1 lac-l + h <-> h[e] + lac-l[e] b dpg 3pg 2pg pep pyr lac-l atp adp pi h2o nadh nad h glc-d[e] lac-d[e] h[e] 17

18 S v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 b1 b2 glc-d -1 1 g6p 1-1 f6p 1-1 fdp 1-1 dhap 1-1 g3p dpg 1-1 3pg 1-1 2pg 1-1 pep 1-1 pyr 1-1 lac-l 1-1 atp adp pi -1 1 h2o 1-1 nadh 1-1 nad -1 1 h glc-d[e] -1 lac-d[e] 1 h[e] 1 18

19 S tot, S exch, S int Internal fluxes Exchange Secondary Primary External S tot 19

20 S tot v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 b1 b2 glc-d -1 1 g6p 1-1 f6p 1-1 fdp 1-1 dhap 1-1 g3p dpg 1-1 3pg 1-1 2pg 1-1 pep 1-1 pyr 1-1 lac-l 1-1 atp adp pi -1 1 h2o 1-1 nadh 1-1 nad -1 1 h glc-d[e] -1 lac-l[e] 1 h[e] 1 dhap glc-d[e] glc-d g6p f6p fdp g3p 13dpg 3pg 2pg pep pyr lac-l lac-l[e]

21 S tot, S exch, S int Internal fluxes Exchange Primary Secondary S exch External S tot 21

22 S exch v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 b1 b2 glc-d -1 1 g6p 1-1 f6p 1-1 fdp 1-1 dhap 1-1 g3p dpg 1-1 3pg 1-1 2pg 1-1 pep 1-1 pyr 1-1 lac-l 1-1 atp adp pi -1 1 h2o 1-1 nadh 1-1 nad -1 1 h glc-d[e] -1 lac-l[e] 1 h[e] 1 dhap glc-d[e] glc-d g6p f6p fdp g3p 13dpg 3pg 2pg pep pyr lac-l lac-l[e]

23 S tot, S exch, S int Internal fluxes Exchange Primary Secondary S int S exch External S tot 23

24 S int v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 b1 b2 glc-d -1 1 g6p 1-1 f6p 1-1 fdp 1-1 dhap 1-1 g3p dpg 1-1 3pg 1-1 2pg 1-1 pep 1-1 pyr 1-1 lac-l 1-1 atp adp pi -1 1 h2o 1-1 nadh 1-1 nad -1 1 h glc-d[e] -1 lac-d[e] 1 h[e] 1 dhap glc-d[e] glc-d g6p f6p fdp g3p 13dpg 3pg 2pg pep pyr lac-l lac-l[e]

25 Reaction map: S Metabolites are nodes For reaction draw edge between substrates (- entry) to products (+entry) Highly non-linear map Participation: 4 is typical 25

26 Compound map: -S T 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 26

27 Open or closed systems A B v 1 C S int A B v 1 C b 1 A b v 1 b 1 A 3 C C S b v 1 2 exch b 3 B b 2 B A e B e b 1 A b v b 1 A 1 3 C b C C e v 2 S tot 1 b 3 B B e b 2 B A e C e 27

28 Binary S sˆ Sˆ : s ˆ ij ij = 0 = 1 if if s s ij ij = 0 0 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 b2 glc-d -1 1 glc-d g6p 1-1 g6p f6p 1-1 f6p fdp 1-1 fdp dhap 1-1 dhap g3p g3p dpg dpg pg 1-1 3pg pg 1-1 2pg pep 1-1 pep pyr 1-1 pyr lac-l 1-1 lac-l atp atp adp adp pi -1 1 pi h2o 1-1 h2o nadh 1-1 nadh nad -1 1 nad h h glc-d[e] -1 glc-d[e] lac-l[e] 1 lac-l[e] h[e] 1 h[e]

29 Reaction adjacency matrix, A v = ˆ T S Sˆ diag( A v Av=Sb'Sb v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 b1 b2 v v v v v v v v v v v v b b ) i ˆ ˆ T = i i = 2 s s ski = k k ˆ sˆ ki = π π i = 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 i

30 Compound adjacency matrix, A x = SS ˆ ˆ T ( a x ) ii glc-d g6p ˆ f6p fdp dhap Ax=SbSb' glc-d g6p f6p fdp dhap g3p dpg pg pg pep pyr lac-l atp adp pi h2o nadh nad h glc-d[e] lac-l[e] h[e] = 2 sik = k k g3p sˆ ik 13dpg = 3pg ρ 2pg i pep pyr lac-l ρ i = 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. atp adp pi h2o nadh nad h glc-d[e] lac-l[e] h[e]

31 Singletons 31

32 Network topology 32

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 33

34 S as a linear transformation v = dx dt i dx dt Range T ( v v K v ) x = ( x x K x ) = = v R 1 n k Sv s 2 ik v k S dx [ m, n] R dt Domain n (Dynamic mass balance) m 1 2 m T 34

35 Four fundamental spaces S [m,n] Row(S) R n R m Col(S) v dyn v dx/dt v ss Null(S) Left null(s) 35

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 36

37 (Right) null space v = v ss + v dyn, where Sv ss =0 v ss is in (right) null space of S Null(S) contains all allowable steady-state flux distributions. 37

38 Row space Together v ss + v dyn span R n v dyn 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 38

39 Column space dx/dt = s 1 v 1 + s 2 v 2 + s n v n, where s i is the i th 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 39

40 Left column space Together Left null(s) and Row(S) span R m 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. 40

41 Four fundamental spaces S [m,n] Row(S) R n R m Col(S) v dyn v dx/dt v ss Null(S) Left null(s) 41

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., b 1 =(1,0,0), b 2 =(0,1,0) and b 3 =(0,0,1) Any vector v R (3) can be decomposed as v = w 1 b 1 + w 2 b 2 + w 3 b 3 So [b 1,b 2,b 3 ] is a basis for R (3) Many types of bases Linear basis Orthonormal (linear) basis for linear spaces Convex basis for finite linear spaces b 3 z b 1 b 2 (w 1,w 2,w 3 ) y x 42

43 Singular value decomposition S = U Σ V T [ m, n] [ m, m] [ m, n] [ n, n] U and V are orthonormal matrices U T U = I (mxm) and V T V = I (nxn) U T = U -1 and V T = V -1 Σ = diag(σ 1, σ 2,, σ r ), where σ 1 σ 2,, σ r >0 43

44 Orthonormal bases for subspaces S U Σ V T = rxr Row(S) mxn Null(S) Col(S) Left null(s) 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) 44

45 U -1.20E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-01 Col(S) Left Null(S) 45

46 Σ r = 13, i.e., 13 linearly, independent relationships between compounds and reactions 46

47 V 4.56E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-10-2E E E E E E E E-10 Row(S) Null(S) 47

48 Eigen-reaction dx T T = Sv = UΣV v uk x = uk1x1 + uk 2x2 + L+ ukmxm ( pool) dt T v T dx k v = vk1v1 + vk 2v2 + L+ vknvn pathway T T U = U UΣV v dt Eigen reaction : ( T d U x) Σ( T = V v) vkjv j for vkj > 0 dt for k r : u kixi ukixi ( T d u x) for uki < 0 for uki > 0 ( T v v) k = σ vkjv j k k for vkj < 0 dt ( ) 48

49 1 st Mode (σ = 4.060) v8 2.12E-02 v6 2.28E-01 v E-01 dhap v1 4.56E-01 2pg lac-l v3 4.57E-01 pi pyr nadh g3p pep pg fdp nad v9-1.32e-03 g6p h2o v5-1.58e-02 13dpg glc-d b1-2.94e-02 adp f6p v4-5.08e-02 h atp v2-6.30e-02 b2-1.39e-01 v e-01 v7-3.06e-01 v e h adp atp u 2 v v 3 v 1 v 10 v

50 Mode 1 50

51 Mode 2 51

52 Null spaces glc-d + atp -> g6p + adp + h v g6p <-> f6p v f6p + atp -> fdp + adp + h v fdp <-> dhap + g3p v dhap <-> g3p v g3p + pi + nad <-> 13dpg + nadh + h v dpg + adp <-> 3pg + atp v pg <-> 2pg v pg <-> pep + h2o v pep + adp + h -> pyr + atp v pyr + nadh + h <-> lac-l + nad v atp + h2o -> adp + pi + h v glc-d[e] <-> glc-d b lac-l + h <-> h[e] + lac-l[e] b Conserved metabolite pools Steady state flux balance 52

53 0 v 1 10 A 0 v v 3 8 dx dt = Sv da dt = v ( 1 1 1) v 2 v3 1 S = = ()( ) 0 Eigen reaction : u v T 1 T 1 x = 1 A ( pool) v = v v v 3 ( pathway) v v v A 53

54 Sv v SS 0 SS = w 1 + w 2 v 10 0 v 6 0 v 8 1 = Orthonormal basis for Null(S) 0 v 1 10 A 0 v v 3 8 vss = α 1 + α 2 v 10 0 v v 3 8 Convex basis for Null(S) 54

55 vss = α 1 + α 2 v 10 0 v v 3 8 Convex basis for Null(S) Since all v i 0 and all basis elements positive: α i 0 0 v 10 : 0 α1 + α v 6 : 0 α1 2 0 v 8: 0 α

56 α 2 6 v α v v 2 56

57 Linear vs convex spaces Linear space Described by linear equations Vector space defined by set of linearly independent basis vectors (b i ) v = Σw i b i - < w i <+ Every point uniquely described by linear combination of b i 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 (p i ) v = Σα i p i 0 α i <+ Every point described by nonnegative, linear combination of p i (non-unique) Number of generation vectors may exceed dim(null(s)) Unique set of generating vectors 57

58 Extreme pathways v ss = Σ α i p i 0 α i < α i,max P i s are unique & correspond to edges in (n-r)- dimensional cone; α i 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 58

59 ExPa classification p 1 p n Internal fluxes v 1 Type I Primary pathways Type II Futile cycles Type III Internal cycles Exchange fluxes Currency Primary b 1 c =0 =0 0 =0 Type I Type II Type III 59

60 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 glc-d g6p b1 f6p glc-d fdp v1 dhap g6p g3p v2 v3 3pg f6p pep v4 pyr fdp accoa v5 v6 acp dhap g3p lac v7/v8 v9/v10 etoh pg ac v11/v12 for pep atp v13 adp pyr lac pi v14 v17/v18 b2 nadh accoa etoh nad v15/16 v19/v20 b3 coa 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 c7 Type Type Type Type Type Type glc-lac Type Type lac-etoh glc-etoh Type ac-etoh etoh-ac lac-ac glc-ac

61 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 glc-d b1 g6p glc-d f6p v1 fdp g6p dhap v2 v3 g3p f6p 3pg v4 pep fdp pyr v5 v6 accoa dhap g3p acp v7/v8 v9/v10 lac pg etoh v11/v12 ac pep for v13 atp pyr lac adp v14 v17/v18 b2 pi accoa etoh nadh v15 v19/v20 b3 nad acp ac coa 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 c7 Type Type Type Type Type Type glc-lac ac-etoh Type lac-etoh glc-etoh Type Futile cycle etoh-ac lac-ac glc-ac glc-etoh/ac

62 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 glc-d b1 g6p glc-d f6p v1 fdp g6p dhap v2 v3 g3p f6p 3pg v4 pep fdp pyr v5 v6 accoa dhap g3p acp v7/v8 v9/v10 lac pg etoh v11/v12 ac pep for v13 atp pyr lac adp v14 v17/v18 b2 pi accoa etoh nadh v15 v19/v20 b3 nad acp ac coa 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 t t t t t t t t glc-lac glc-lac lac-etoh/ac lac-etoh/ac glc-etoh/ac glc-etoh/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

63 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 glc-d g6p b1 f6p glc-d fdp v1 dhap g6p g3p v2 v3 3pg f6p pep v4 pyr fdp accoa v5 v6 acp dhap g3p lac v7/v8 v9/v10 etoh pg ac v11/v12 for pep atp v13 adp pyr lac pi v14 v17/v18 b2 nadh accoa etoh nad v15 v19/v20 b3 coa 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 b

64 Sv = 0 Stoichiometry Linear algebra v R (n) Eigenreactions v subspace of R (n) v 0 Reaction direction Convex analysis ExPa v = Σα i p i, α i 0 convex cone 64

65 Pathway matrix, P, and Binary P ( p p p ) = ( ) P = 2 3 ˆ pˆ ij = 0 if p P : pˆ ij = 1 if p P LM = 1 K ) T P ) P ij ij = 0 0 Binary P Pathway length matrix Diagonal elements = # reactions in each ExPa Off-diagonal elements = # shared reactions between two ExPa R PM ˆ ˆ T = PP Reaction participation matrix Diagonal elements = # ExPa in which a given reaction is found Off-diagonal elements = # ExPa that contains given pair of reactions 65

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 66

67 Reaction participation 168 reactions participate in only one extreme pathway Very specific function, i.e., ideal drug targets 67

68 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

69 Correlated reaction sets 69

70 Sv = 0 Stoichiometry Linear algebra v R (n) Eigenreactions v subspace of R (n) v 0 Reaction direction Convex analysis ExPa v = Σα i p i, α i 0 convex cone 70

71 COBRA = Constraint-based reconstruction and analysis of metabolic and regulatory networks

72 Sv = 0 Stoichiometry Linear algebra v R (n) Eigenreactions v subspace of R (n) v 0 Reaction direction Convex analysis ExPa v = Σα i p i, α i 0 convex cone 72

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 73

74 Sv = 0 Stoichiometry Linear algebra v R (n) Eigenreactions v subspace of R (n) v 0 Reaction direction Convex analysis ExPa Union of convex subsets Regulation Thermo Kinetics v = Σα i p i, α i 0 convex cone 74

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 75

76 In silico E. coli model Network: 71 reactions - 21 reversible 55 internal metabolites 13 external metabolites Captures: - Substrate uptake - Glycolysis - PPP, EDP, TCA-Cycle - Anaplerosis - Respiration -Fermentation - Biomass formation - 4 different pathways for product formation 76

77 4 natural pathways for 3HP production (KEGG) 3HP 77

78 Effect on product formation 3-HP Yield (aerobic) 3-HP Yield (anaerobic) (anaerobic, aerobic) Left (1) Right (2) Center(3) Center-right (4) 95.8 % 96.6 % 84.2 % 100 % ** 100 % 100 % * (50%) Yields are given in C mol / C mol [%] 85.7 % 100 % ** 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 78

79 In silico analysis of β-alanine pathway for the production of 3-HP in E. coli #product synthesis left pathway R54 : OAA + GLU = ASP + 2 OXO R55 : ASP = bala + CO2 R56 : bala + 2 OXO = 3 OXOPRO + GLU R57 : 3 OXOPRO + NADH = 3 HPA + NAD 3HP 79

80 Elementary mode analysis Carbon yield for biomass elementary modes 2292 make the desired product Mode #11048 Max P with X>0 Carbon yield for 3-HP Anaerobic Does not form acetate Does not require PEP-carboxylase and GPI Highly dependent on malic enzyme (NADPH) 80

81 Knock-out of GPI, PEP-C and Acetate kinase Anaerobic conditions 464 elementary modes 126 make the desired product Carbon yield for biomass Carbon yield for 3-HP 81

82 Sv = 0 Stoichiometry Linear algebra v R (n) Eigenreactions v subspace of R (n) Capacity constraints v v max v = Σα i p i, α i 0 Bounded convex cone 0 α i α i,max v 0 ExPa Reaction direction Convex analysis Union of convex subsets Regulation Thermo Kinetics v = Σα i p i, α i 0 convex cone 82

83 Linear programming maximize subject to and Z = w v = wivi Sv = 0 v v v, i = 1, i, min i i,max K, n Objectives Linear Max growth (μ) Max product (π) Min substrate (σ) Max w 1 μ+w 2 π Nonlinear Min v 2 (QP) 83

84 LP solutions Unique Shadow price (dual) Increase in objective function for a unit increase in a constraint Degenerate Reduced cost Increase in objective function for unit increase in flux Unbounded 84

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 85

86 Mixed integer programming maximize subject to and Z = w y v A y v = b v v v i,min 0 y i y i i n j i,max, j = 1, K, m integer (binary), i = 1, K, n 86

87 87

88 PhPP 88

89 89

90 Phenotypic phase plane Ibarra et al, Nature 420, (2002) 90

91 AraGEM Gene-reaction-association entries 5253 ORFs (unique) 1419 Metabolites 1748 Unique reactions 1567 Cytosolic reactions 1265 Mitochondrial reactions 60 Plastidic reactions 159 Peroxisomal reactions 98 Modified reactions 36 Biomass drains and transporters 148 Biomass drains 47 Transporters (Intercellular) 18 Transporters (Inter-organelle) 83 Gaps (unique reactions ID) filled by manual curation 75 Singleton metabolites

92 92

93 Assumptions Inputs, outputs and constraints Case 1 Case 2 Photons uptake (free flux) + + Glutamine transporter (mitochondria) - - Glutamate transporter(mitochondria) - - Glutamine transporter (plastid) - - Glutamate transporter (plastid) + + RuBisCO; EC (carboxylation:oxygenation; 3:1) Fd-GOGAT ; EC (plastid) + + NADH-GOGAT; EC (plastid) - - Optimization: minimize uptake of Photons Photons Biomass rate (estimated and fixed) Leaf Leaf 93

94 RuBisCO 3:1 C:O: ~40% increase in photon requirements (litt 30-50%) 94