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

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

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

THE PATHWAY MATRIX (P) AND ITS FEATURES

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

A simple example system EP3 EP1 EP2

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

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 (2002).

P for the RBC Biophys.J, 83(2): pp (2002).

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

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 extreme pathways Number that produce acetate: 1745 Number that produce co2: Number that produce lactate: 1420 Number that produce succinate: 7162

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

PATHWAY LENGTH Property #1

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

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

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

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

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

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

REACTION PARTICIPATION Property #2

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

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

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

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

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., If EPO is removed from culture, erythrocyte progenitors (CFC-Es) rapidly undergo apoptosis. (Alberts, et al., Mol. Biol. Cell, 2004)

© 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

Co-Sets for RBC 9 Co-Sets in RBC

Anaerobic co-sets

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

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

Systems Biology Research Group

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

INPUT-OUTPUT ANALYSIS: THE IOFA Property #3

© 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

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

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, internal states external state 2 46 Similar components – very different network properties!

Classifying the I/O signature

Overlaps between I/O signatures of ExPas Crosstalk

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

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.

IOFA for RBC

IOFA for the core E. coli CO2 GLC H2O NH4 LAC H ETOHFOR PYR H ETOHFOR PYRSUCC H ETOHFOR SUCC H ETOHFOR LAC SUCC H ETOH CO2H ACALD ETOHFOR CO2H ETOHFORGLU CO2H ETOHFOR CO2H ETOHFOR SUCCCO2H ACALD FOR H AC FOR SUCC H ACALD ETOHFOR SUCC H ETOHFORGLU H ETOHFORGLU SUCC H ACALD SUCC HH2O ETOH SUCC HH2O PYRSUCC HH2O ETOHFOR SUCC HH2O LAC SUCC HH2O AC LAC SUCC HH2O AC SUCC HH2O ETOHFORGLU SUCC HH2O ETOH GLU SUCC HH2O ETOH LAC SUCC HH2O AC ETOHFOR SUCC H AC ETOHFOR CO2H AC ETOHFOR H ACALD FOR SUCC HH2O FORGLULAC SUCC HH2O AKGETOHFOR SUCCCO2H AKGETOHFOR CO2H ACALD ETOHFOR SUCCCO2H ETOHFORGLU SUCCCO2H ACALD FORGLU SUCC HH2O ACALDAKG FOR SUCC HH2O AKGETOHFOR H AKGETOHFOR SUCC H FOR LAC SUCC HH2O FORGLU SUCC HH2O FOR SUCC HH2O AKG FOR SUCC HH2O ACALDAKG SUCC HH2O ACALD GLU SUCC HH2O AC ETOH SUCC HH2O FORGLU PYRSUCC HH2O AKG PYRSUCC HH2O FOR PYRSUCC HH2O GLU PYRSUCC HH2O AKG FOR PYRSUCC HH2O ETOH PYRSUCC HH2O ETOH GLU PYRSUCC HH2O AKGETOH PYRSUCC HH2O AC ETOHFOR SUCC HH2O AC FORGLU SUCC HH2O AC FOR SUCC HH2O AC AKG FOR SUCC HH2O AKGETOH SUCC HH2O ETOH SUCCCO2HH2O AC AKG SUCC HH2O AC GLU SUCC HH2O ETOH GLU SUCCCO2HH2O GLULAC SUCC HH2O ETOH GLULAC SUCC HH2O AKG FOR LAC SUCC HH2O AKGETOHFOR SUCC HH2O AKGETOH SUCCCO2HH2O AKG LAC SUCC HH2O AKGETOH LAC SUCC HH2O ETOHFOR PYRSUCC HH2O ETOHFOR SUCCCO2HH2O 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

INCORPORATING REGULATION Property #4

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

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

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

Extreme Pathway Reduction Total number of extreme pathways is reduced from 80 to between 26 and %-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

Example: C1, C2, O2 (ExPA) All possible extreme pathways Pathway reduction -Remove all inconsistent pathways Environment-specific regulation: R5b, Tc2 o Environmental-dependent constraints Environment-specificity: C1, C2 and O o Environmental inconsistencies Environment-independent regulation 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

Complex medium: Regulation of pathways Environment-specific regulation: R2a, R5b, R7, R8a, Tc Environment-independent regulation Number of extreme pathways is only reduced to 26 More flexibility in the system Covert et al., Journal of Theoretical Biology, 2002

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

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

EXTRAS

A simple example system

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

Reaction participations in the simple example