1. Lecture #14.a Properties of the Null Space of S deciphered through the use of basis vectors

2. Basis vectors can describe every point in a space The null space describes flux states, that are candidate physiological states

3. Outline Introduction – The extreme pathway matrix P (SP=0) – 4 key properties – Example systems (simple, RBC, core E. coli, genome-scale studies) Pathway “length” (size) Reaction participation – Co-sets; a ‘module’ Input-output feasibility – Cross-talk The effects of regulation – Regulatory rules, expression profiling data – Elimination of ExPas

4. THE PATHWAY MATRIX (P) AND ITS FEATURES

5. Use of Basis Vector Basis vectors span a space They can be used to determine all of its properties; such as 1.Pathway length 2.Reaction participation/co-sets 3.Input-output analyses 4.Incorporation of regulation

6. A simple example system EP3 EP1 EP2

7. The Red Blood Cell A Model System for in silico Biology Relatively small metabolic network – 39 metabolites – 32 internal reactions Well studied, well understood system A full kinetic model has been developed in Mathematica ® (BE 213) Copyright Dennis Kunkel

8. Red Blood Cell Metabolic Network 32 internal reactions 19 exchange fluxes 39 metabolites Extreme Pathway Structure 36 ‘Through’ Pathways (Type I) 3 Futile Cycle Pathways (Type II) 17 Reversible Reaction Pathways (Type III) Currency exchanges Biophys.J, 83(2): pp. 808- 818 (2002).

9. P for the RBC Biophys.J, 83(2): pp. 808-818 (2002).

10. The core E. coli model: Number of rxns = 95 Number of cmpds= 72

11. P for the core E. coli with glucose as the input Anaerobic – Glucose input 2006 extreme pathways Number that produce acetate: 174 Number that produce co2: 506 Number that produce lactate: 249 Number that produce succinate: 1625 Aerobic – Glucose input 16688 extreme pathways Number that produce acetate: 1745 Number that produce co2: 11981 Number that produce lactate: 1420 Number that produce succinate: 7162

12. STAT1 rIFNγ JAK2 IFNγ ADP rIFNγ JAK2 IFNγ STAT1 rIFNγ JAK2 IFNγ STAT1 ADP STAT1 P rIFNγ JAK2 IFNγ STAT2 JAK1 IFNα/β ADP rIFNα/β JAK1 IFNα/β STAT2 rIFNα/β JAK1 IFNα/β STAT2 ADP STAT2 P rIFNα/β JAK1 IFNα/β rIFNα/β STAT1 P STAT2 P P P P P Input Output Metabolic Network Energy Generation Transcriptional Regulatory Network P P rIFNγ P P P P P P P P rIFNα/β P P P P P P The JAK-STAT system in B cells

13. PATHWAY LENGTH Property #1

14. Adjacency matrix Pathway Length and Reaction Participation Pathway Length Matrix EP1 EP2 EP3

15. Distribution of Pathway Lengths for RBC The figure shows the pathway lengths for the 39 Type I & II pathways

16. Example Extreme Pathways ExPa with max ATP yield Classical Glycolysis ExPa that requires ATP as input

17. Example Extreme Pathways 3 optimal paths for NADPH yield of 6 Equivalent overall states

18. Pathway lengths from glucose input for the core E. coli Mean pathway length = 35.6 Median pathway length = 37 Anaerobic Mean pathway length = 39.8 Median pathway length = 40 Aerobic

19. Pathway Length from glucose Anaerobic Aerobic A set of ExPas with the same yield These particular pathways are optimality properties

20. REACTION PARTICIPATION Property #2

21. Adjacency matrix Pathway Length and Reaction Participation Papin et al., Genome Research, 2002 Pathway Length Matrix Reaction participation matrix EP1 EP2 EP3

22. Reaction Participation for the RBC H,ATP, ADP and Pi – the primary currency 51 net reactions  68 elementary reactions

23. Participations for the core E. coli: consumption of glucose (growth and no growth) Anaerobic – Glucose input Aerobic – Glucose input ENO, FBP, GAPD, GLCpts, PFK, PGI, PGK, PGM, TPI, Glc exchange GLCpts, Glc exchange Never used under these growth conditions 95 net reactions  133 elementary reactions

24. Reaction Participation Amino Acid Synthesis Groups reactions into sets that are: Always necessary (I) represent essential core set of reactions Sometimes necessary (II) Represent variability, redundancy in the metabolic network Never utilized (III) These groups each have important implications for metabolic engineering and understanding of biological systems I II III Papin et al., Genome Research, 2002

25. Reaction participation: JAK-STAT network ATP/ADP  primary currency small number of reactions  diversity in network function STAT1 and STAT3 reactions with specific functions  drug targeting Papin, Palsson, Biophys. J., 2004. If EPO is removed from culture, erythrocyte progenitors (CFC-Es) rapidly undergo apoptosis. (Alberts, et al., Mol. Biol. Cell, 2004)

26. © 2004 Continuing Bioengineering Education, Inc. Correlated Reaction Sets Trends Biochem. Sci., 2003 ABCD E R1R1 R2R2 R7R7 R3R3 R4R4 R6R6 R5R5 System Boundary ABCDEABCDE R 1 R 2 R 3 R 4 R 5 R 6 R 7 S = P = R1R2R3R4R5R6R7R1R2R3R4R5R6R7 EP 1 EP 2 ABCD E R1R1 R2R2 R7R7 R3R3 R4R4 R6R6 R5R5 System Boundary EP 1 EP 2 Identical rows

27. Co-Sets for RBC 9 Co-Sets in RBC

28. Anaerobic co-sets 1 9 9 2 3 4 4 5 67 8

29. Rhamnose Rhamnulose Rhamnose 1-phosphate Dihydroxyacetone phosphate rhaA rhaB rhaD rhaT atp adp rhaD rhaA rhaB rhaS rhaR rhaT intracellular extracellular transcription & translation operon Lactaldehyde h h Arginine N2-succinyl-L-arginine N2-succinyl-L-ornithine astA astB astC arcD h 2 o, h co 2, nh 4 astE astB astD astA astC intracellular extracellular transcription & translation operon Ornithine N2-succinyl-L-glutamate 5-semialdehyde a-ketoglutarate Glutamate N2-succinyl-L-glutamate h 2 o, nad nadh, h Glutamate astE Succinate Succinyl-CoA CoA, h h2oh2o astD operon speB speA operon arcD ydgB AgmatinePutrescine Ureah2oh2o co 2 h Urea speA or adiAspeB glpF operon glpK glpF Known regulatory structure Unknown regulatory structure Correlated reaction sets E. coli metabolic network Example 1 – 3 operons – 1 correlated reaction set – 1 regulon Example 2 – 4 operons – 1 correlated reaction set – no known regulatory rules – Genes are co- expressed Papin et al., Trends Biochem. Sci., 2004 ref

30. Jamshidi, et al Molec. System Biol. (2006) Co-Sets: A way to correlate SNPs?

31. Systems Biology Research Group http://systemsbiology.ucsd.edu

32. components small-scale modules large-scale modules phenotype (physiology) mRNA protein products translation ABC genotype Correlated reaction sets Hierarchy in biological networks Papin et al., Trends Biochem. Sci., 2004

33. INPUT-OUTPUT ANALYSIS: THE IOFA Property #3

34. © 2004 Continuing Bioengineering Education, Inc. Network crosstalk: need to understand interactions Dumont, et al., Cell. Signal., 2001 e.g., cAMP inhibits proliferation in fibroblasts, and stimulates proliferation in epithelial cells Black lines – “textbook” pathways Green & red lines – interactions described over previous 2 years Localization, differentiated state, etc. need to be considered Overlap & specificity “Pathways are concepts, Networks are the reality” Uwe Sauer, 2005

35. Extreme Pathway 1 Pathway Redundancy These extreme pathways have the same “external state.” input: 2 A output: 1 E and 1 byp However, the internal flux distribution is very different in all three pathways. Pathway Redundancy = 3 for this network. Extreme Pathway 2Extreme Pathway 3

36. Pathway Redundancy Reconstructed Metabolic Map of H. influenzae Reconstructed Metabolic Map of H. pylori INPUTSOUTPUTS Alanine Arginine Oxygen Fructose Glutamate Ammonia Oxygen Acetate Succinate Carbon dioxide Ammonia Amino acid Acetate Succinate Carbon dioxide Reconstructed Metabolic Map of H. influenzae Reconstructed Metabolic Map of H. pylori INPUTSOUTPUTS Alanine Arginine Oxygen Fructose Glutamate Ammonia Oxygen Lysine Acetate Succinate Carbon dioxide Ammonia Amino acid Acetate Succinate Carbon dioxide Price et al., Genome Research, 2002 461 390 internal states external state 2 46 Similar components – very different network properties!

37. Classifying the I/O signature

38. Overlaps between I/O signatures of ExPas Crosstalk

39. Input/output relationships Papin, Palsson, J. Theor. Biol., 2004. Expression arrays from combinations of IFN , - , or –  stimulation indicated novel regulation (Der, et al., PNAS, 1998). Mathematical framework is needed for studying “combinations.”

40. Network Crosstalk Non-overlapping  determined network Redundant output signals Significant network economization Crosstalk: the non-negative linear combination of extreme signaling pathways. Evaluate phenotypic effects of combinations of functional states, like conflicting cAMP signals. Papin, Palsson, Biophys. J., 2004.

41. IOFA for RBC

42. IOFA for the core E. coli CO2 GLC H2O NH4 LAC H 2700000 ETOHFOR PYR H 2700000 ETOHFOR PYRSUCC H 800000 ETOHFOR SUCC H 8101154609 ETOHFOR LAC SUCC H 800000 ETOH CO2H 8100000 ACALD ETOHFOR CO2H 1800000 ETOHFORGLU CO2H 0330000 ETOHFOR CO2H 0027000 ETOHFOR SUCCCO2H 69089000 ACALD FOR H 2700000 AC FOR SUCC H 0001100 ACALD ETOHFOR SUCC H 800000 ETOHFORGLU H 0510000 ETOHFORGLU SUCC H 0150000 ACALD SUCC HH2O1100300 ETOH SUCC HH2O5700800 PYRSUCC HH2O0001400 ETOHFOR SUCC HH2O8001000 LAC SUCC HH2O0002700 AC LAC SUCC HH2O000800 AC SUCC HH2O0001400 ETOHFORGLU SUCC HH2O02100170 ETOH GLU SUCC HH2O06600100 ETOH LAC SUCC HH2O3100000 AC ETOHFOR SUCC H 3508000 AC ETOHFOR CO2H 0018000 AC ETOHFOR H 0027000 ACALD FOR SUCC HH2O0002600 FORGLULAC SUCC HH2O0000170 AKGETOHFOR SUCCCO2H 2300000 AKGETOHFOR CO2H 0018000 ACALD ETOHFOR SUCCCO2H 500000 ETOHFORGLU SUCCCO2H 090000 ACALD FORGLU SUCC HH2O0000170 ACALDAKG FOR SUCC HH2O000900 AKGETOHFOR H 0027000 AKGETOHFOR SUCC H 3508 00 FOR LAC SUCC HH2O0001700 FORGLU SUCC HH2O000020 FOR SUCC HH2O0001400 AKG FOR SUCC HH2O000100 ACALDAKG SUCC HH2O000100 ACALD GLU SUCC HH2O000020 AC ETOH SUCC HH2O3500000 FORGLU PYRSUCC HH2O000020 AKG PYRSUCC HH2O000100 FOR PYRSUCC HH2O000100 GLU PYRSUCC HH2O000020 AKG FOR PYRSUCC HH2O000100 ETOH PYRSUCC HH2O12700000 ETOH GLU PYRSUCC HH2O0420000 AKGETOH PYRSUCC HH2O2300000 AC ETOHFOR SUCC HH2O500500 AC FORGLU SUCC HH2O000020 AC FOR SUCC HH2O000200 AC AKG FOR SUCC HH2O000100 AKGETOH SUCC HH2O35001000 ETOH SUCCCO2HH2O4600000 AC AKG SUCC HH2O000100 AC GLU SUCC HH2O000020 ETOH GLU SUCCCO2HH2O0420000 GLULAC SUCC HH2O0000170 ETOH GLULAC SUCC HH2O0420090 AKG FOR LAC SUCC HH2O000900 AKGETOHFOR SUCC HH2O51001400 AKGETOH SUCCCO2HH2O2300000 AKG LAC SUCC HH2O000900 AKGETOH LAC SUCC HH2O2300500 ETOHFOR PYRSUCC HH2O500000 ETOHFOR SUCCCO2HH2O500000 Under anaerobic conditions (there are many more I/O combinations for aerobic) Glucose is the primary input 27 ways to make lactate and a proton from glucose Only a fraction of ExPas give a growth like I/O signature

43. INCORPORATING REGULATION Property #4

44. COBRA View: Regulation is a Constraint (a restraint) External Glucose External Signal (-) (+)(+) CRP Mlc Regulatory Proteins ptsHI, crr (+)(+) X glpK (-)(-) Transcriptional Regulation (-) (+)(+) Altered Network Capabilities (+)(+) (-) Restricted Solution Space Solution Shifts New Growth Behavior Time (hrs) Concentration (g/L) Growth Prediction Shifts Metabolic Network Reconstruction: Databases Literature General Solution Space Constraints: Mass Balance S. v = 0 Capacity i ≤vi≤ ii ≤vi≤ i Particular Solution Time course of growth (phenotype) Dynamics: Quasi Steady- State Assumption Integration Time (hrs) Concentration (g/L) Genome Extreme Pathways

45. Extreme Pathways and Regulatory Constraints Flux C Flux B Flux A P2 P1P3 P4 Consider the entire solution space of a metabolic network, bounded by extreme pathways P1-P4… P1 P2 P3 P4 P1 is not permitted due to regulatory constraints One or more of these pathways may not be feasible, depending on the environment and corresponding regulatory effects… Flux C Flux B Flux A P1 P2 P3 P4 P2 P3 P4 This leads to a reduced solution space bounded by fewer extreme pathways Covert et al., Journal of Theoretical Biology, 2002

46. Sample Network Characteristics 21 metabolic reactions 4 regulatory proteins 7 regulated reactions Boolean representation Regulation modeled Catabolite repression Amino acid biosynthesis Oxygen-dependent Carbon storage Analysis 80 Extreme pathways Forced growth output 5 environmental inputs 2 5 = 32 environments

47. Extreme Pathway Reduction Total number of extreme pathways is reduced from 80 to between 26 and 2 67.5%-97.5% reduction 21 of the extreme pathways are never available as solutions due to inconsistent regulation P1, P13-28 and P53-56 Infeasible! Covert et al., Journal of Theoretical Biology, 2002

48. Example: C1, C2, O2 (ExPA) 61 37 41 17 123421222324 567825262728 910111229303132 1314151633343536 181920383940 424344626364 4546474865666768 4950515269707172 5354555673747576 5758596077787980 All possible extreme pathways Pathway reduction -Remove all inconsistent pathways 46 7874 6270 5850 42 3430 Environment-specific regulation: R5b, Tc2 o Environmental-dependent constraints Environment-specificity: C1, C2 and O2 910111229303132 1314151633343536 1718192037383940 4142434461626364 4546474865666768 78 123421222324 567825262728 4950515269707172 5354555673747576 57585960777980 o Environmental inconsistencies Environment-independent regulation 910111229303132 1314151633343536 1718192037383940 4142434461626364 4546474865666768 123421222324 567825262728 4950515269707172 5354555673747576 5758596077787980 o Environment-independent constraints -Constrained solution space o 4 extreme pathways o Corresponds to Phenotypic Phase Plane P30 P34 P46 P50 LO Covert et al., Journal of Theoretical Biology, 2002

49. Complex medium: Regulation of pathways Environment-specific regulation: R2a, R5b, R7, R8a, Tc2 910111229303132 1314151633343536 1718192037383940 4142434461626364 4546474865666768 78 123421222324 567825262728 4950515269707172 5354555673747576 57585960777980 Environment-independent regulation 910111229303132 1314151633343536 1718192037383940 4142434461626364 4546474865666768 123421222324 567825262728 4950515269707172 5354555673747576 5758596077787980 Number of extreme pathways is only reduced to 26 More flexibility in the system Covert et al., Journal of Theoretical Biology, 2002

50. Regulation for the core E. coli With regulation, the reactions D_LACt2, FUMt2_2, ICL, MALS, MALt2_2, MDH, NADH16, and SUCCt2_2 are inactivated under anaerobic conditions w/regulation:118 ExPas are feasible w/o regulation: 2006 ExPas are feasible Mean pathway length = 28.9 Median pathway length = 31 Mean pathway length = 35.6 Median pathway length = 37 Anaerobic

51. Summary Basis vectors span a space and can describe all of its contents Some of the properties of P are: – Pathway lengths – Reaction participation Co-sets – I/O redundancy characteristics Cross talk – Incorporation of regulation Shrinking the space and excluding possible states

52. EXTRAS

53. A simple example system

54. A B ifng3 ifng4 ifng6 ifng12 ifn15 ifn14 ifn13 ifng13 ifng14 sd6

55. Reaction participations in the simple example