1. 21. Lecture WS 2003/04Bioinformatics III1 Metabolic Pathway Analysis: Elementary Modes The technique of Elementary Flux Modes (EFM) was developed prior to extreme pathways (EP) by Stephan Schuster, Thomas Dandekar and co-workers: Pfeiffer et al. Bioinformatics, 15, 251 (1999) Schuster et al. Nature Biotech. 18, 326 (2000) The method is very similar to the „extreme pathway“ method to construct a basis for metabolic flux states based on methods from convex algebra. Extreme pathways are a subset of elementary modes, and for many systems, both methods coincide. Are the subtle differences important?

2. 21. Lecture WS 2003/04Bioinformatics III2 Review: Metabolite Balancing For analyzing a biochemical network, its structure is expressed by the stochiometric matrix S consisting of m rows corresponding to the substances (metabolites) and n rows corresponding to the stochiometric coefficients of the metabolites in each reaction. A vector v denotes the reaction rates (mmol/g dry weight * hour) and a vector c describes the metabolite concentrations. Due to the high turnover of metabolite pools one often assumes pseudo-steady state (c(t) = constant) leading to the fundamental Metabolic Balancing Equation: (1) Flux distributions v satisfying this relationship lie in the null space of S and are able to balance all metabolites. Klamt et al. Bioinformatics 19, 261 (2003)

3. 21. Lecture WS 2003/04Bioinformatics III3 Review: Metabolic flux analysis Metabolic flux analysis (MFA): determine preferably all components of the flux distribution v in a metabolic network during a certain stationary growth experiment. Typically some measured or known rates must be provided to calculate unknown rates. Accordingly, v and S are partioned into the known (v b, S b ) and unknown part (v a, S a ). (1) leads to the central equation for MFA describing a flux scenario: 0 = S  v = S a  v a + S b  v b. The rank of S a determines whether this scenario is redundant and/or underdetermined. Redundant systems can be checked on inconsistencies. In underdetermined scenarios, only some element of v a are uniquely calculable. Klamt et al. Bioinformatics 19, 261 (2003)

4. 21. Lecture WS 2003/04Bioinformatics III4 Software: FluxAnalyzer A network project constructed by FluxAnalyzer. Here, v b consists of R1, R2,  and v a of R3 - R7, whereof R3, R4, R7 can be computed. 10 0000.8 0100000 0001 101 00101 11.8 R1 R2 R3 R4 R5 R6 R7 biomass synthesis S = Klamt et al. Bioinformatics 19, 261 (2003) Biomass component 1: BC1[g] = 2[mmol]A + 1 [mmol]C Biomass component 2: BC2[g] = 1[mmol]C + 3[mmol]D

5. 21. Lecture WS 2003/04Bioinformatics III5 Review: structural network analysis (SNA) Whereas MFA focuses on a single flux distribution, techniques of Structural (Stochiometric, Topological) Network Analysis (SNA) address general topological properties, overall capabilities, and the inherent pathway structure of a metabolic network. Basic topological properties are, e.g., conserved moieties. Flux Balance Analysis (FBA9 searches for single optimal flux distributions (mostly with respect to the synthesis of biomass) fulfilling S  v = 0 and additionally reversibility and capacity restrictions for each reaction (  i  v i   i ). Klamt et al. Bioinformatics 19, 261 (2003)

6. 21. Lecture WS 2003/04Bioinformatics III6 Review: Metabolic Pathway Analysis (MPA) Metabolic Pathway Analysis searches for meaningful structural and functional units in metabolic networks. The most promising, very similar approaches are based on convex analysis and use the sets of elementary flux modes (Schuster et al. 1999, 2000) and extreme pathways (Schilling et al. 2000). Both sets span the space of feasible steady-state flux distributions by non- decomposable routes, i.e. no subset of reactions involved in an EFM or EP can hold the network balanced using non-trivial fluxes. MPA can be used to study e.g. - routing + flexibility/redundancy of networks - functionality of networks - idenfication of futile cycles - gives all (sub)optimal pathways with respect to product/biomass yield - can be useful for calculability studies in MFA Klamt et al. Bioinformatics 19, 261 (2003)

7. 21. Lecture WS 2003/04Bioinformatics III7 Elementary Flux Modes Start from list of reaction equations and a declaration of reversible and irreversible reactions and of internal and external metabolites. E.g. reaction scheme of monosaccharideFig.1 metabolism. It includes 15 internal metabolites, and 19 reactions.  S has dimension 15  19. It is convenient to reduce this matrix by lumping those reactions that necessarily operate together.  {Gap,Pgk,Gpm,Eno,Pyk},  {Zwf,Pgl,Gnd} Such groups of enzymes can be detected automatically. This reveals another two sequences {Fba,TpiA} and {2 Rpe,TktI,Tal,TktII}. Schuster et al. Nature Biotech 18, 326 (2000)

8. 21. Lecture WS 2003/04Bioinformatics III8 Elementary Flux Modes Lumping the reactions in any one sequence gives the following reduced system: Construct initial tableau by combining S with identity matrix: 10...000100 01 00020 00...00001 00...0-2021 00...00000 00...010000 00 00100 00...00100 00...10000 Pgi {Fba,TpiA} Rpireversible {2Rpe,TktI,Tal,TktII} {Gap,Pgk,Gpm,Eno,Pyk} {Zwf,Pgl,Gnd} Pfkirreversible Fbp Prs_DeoB Schuster et al. Nature Biotech 18, 326 (2000) Ru5P FP 2 F6P GAP R5P T (0) =

9. 21. Lecture WS 2003/04Bioinformatics III9 Elementary Flux Modes Aim again: bring all entries of right part of matrix to 0. E.g. 2*row3 - row4 gives „reversible“ row with 0 in column 10 New „irreversible“ rows with 0 entry in column 10 by row3 + row6 and by row4 + row7. In general, linear combinations of 2 rows corresponding to the same type of directio- nality go into the part of the respective type in the tableau. Combinations by different types go into the „irreversible“ tableau because at least 1 reaction is irreversible. Irreversible reactions can only combined using positive coefficients. Schuster et al. Nature Biotech 18, 326 (2000) 100100 10020 1 0001 1-2021 1000 0 110000 101 00 10 100 10000 100100 10 020 2 00-23 1000 0 101 00 10 100 10000 1100001 120021 T (1) = T (0) =

10. 21. Lecture WS 2003/04Bioinformatics III10 Elementary Flux Modes Aim: zero column 11. Include all possible (direction-wise allowed) linear combinations of rows. continue with columns 12- 14. Schuster et al. Nature Biotech 18, 326 (2000) 100100 10020 2 00-23 1000 0 101 00 10 100 10000 1100001 120021 100100 2 00-23 1000 0 10000 1100001 120021 1100 20 1001-20 1100000 T (2) = T (1) =

11. 21. Lecture WS 2003/04Bioinformatics III11 Elementary Flux Modes In the course of the algorithm, one must avoid - calculation of nonelementary modes (rows that contain fewer zeros than the row already present) - duplicate modes (a pair of rows is only combined if it fulfills the condition S(m i (j) )  S(m k (j) )  S(m l (j+1) ) where S(m l (j+1) ) is the set of positions of 0 in this row. - flux modes violating the sign restriction for the irreversible reactions. Final tableau T (5) = This shows that the number of rows may decrease or increase in the course of the algorithm. All constructed elementary modes are irreversible. Schuster et al. Nature Biotech 18, 326 (2000) 1100201000... 0 -201113000... 021153200 001001001 514-200106 -52206010... 0000001100 0

12. 21. Lecture WS 2003/04Bioinformatics III12 Elementary Flux Modes Graphical representation of the elementary flux modes of the monosaccharide metabolism. The numbers indicate the relative flux carried by the enzymes. Fig. 2 Schuster et al. Nature Biotech 18, 326 (2000)

13. 21. Lecture WS 2003/04Bioinformatics III13 Two approaches for Metabolic Pathway Analysis? The pathway P(v) is an elementary flux mode if it fulfills conditions C1 – C3. (C1) Pseudo steady-state. S  e = 0. This ensures that none of the metabolites is consumed or produced in the overall stoichiometry. (C2) Feasibility: rate e i  0 if reaction is irreversible. This demands that only thermodynamically realizable fluxes are contained in e. (C3) Non-decomposability: there is no vector v (unequal to the zero vector and to e) fulfilling C1 and C2 and that P(v) is a proper subset of P(e). This is the core characteristics for EFMs and EPs and supplies the decomposition of the network into smallest units (able to hold the network in steady state). C3 is often called „genetic independence“ because it implies that the enzymes in one EFM or EP are not a subset of the enzymes from another EFM or EP. Klamt & Stelling Trends Biotech 21, 64 (2003)

14. 21. Lecture WS 2003/04Bioinformatics III14 Two approaches for Metabolic Pathway Analysis? The pathway P(e) is an extreme pathway if it fulfills conditions C1 – C3 AND conditions C4 – C5. (C4) Network reconfiguration: Each reaction must be classified either as exchange flux or as internal reaction. All reversible internal reactions must be split up into two separate, irreversible reactions (forward and backward reaction). (C5) Systemic independence: the set of EPs in a network is the minimal set of EFMs that can describe all feasible steady-state flux distributions. Klamt & Stelling Trends Biotech 21, 64 (2003)

15. 21. Lecture WS 2003/04Bioinformatics III15 Two approaches for Metabolic Pathway Analysis? Klamt & Stelling Trends Biotech 21, 64 (2003) ACP B D A(ext)B(ext)C(ext) R1R2R3 R5 R4R8 R9 R6 R7

16. 21. Lecture WS 2003/04Bioinformatics III16 Reconfigured Network Klamt & Stelling Trends Biotech 21, 64 (2003) ACP B D A(ext)B(ext)C(ext) R1R2R3 R5 R4R8 R9 R6 R7bR7f 3 EFMs are not systemically independent: EFM1 = EP4 + EP5 EFM2 = EP3 + EP5 EFM4 = EP2 + EP3

17. 21. Lecture WS 2003/04Bioinformatics III17 Property 1 of EFMs Klamt & Stelling Trends Biotech 21, 64 (2003) The only difference in the set of EFMs emerging upon reconfiguration consists in the two-cycles that result from splitting up reversible reactions. However, two-cycles are not considered as meaningful pathways. Valid for any network: Property 1 Reconfiguring a network by splitting up reversible reactions leads to the same set of meaningful EFMs.

18. 21. Lecture WS 2003/04Bioinformatics III18 Software: FluxAnalyzer What is the consequence of when all exchange fluxes (and hence all reactions in the network) are irreversible? Klamt & Stelling Trends Biotech 21, 64 (2003) EFMs and EPs always co-incide!

19. 21. Lecture WS 2003/04Bioinformatics III19 Property 2 of EFMs Klamt & Stelling Trends Biotech 21, 64 (2003) Property 2 If all exchange reactions in a network are irreversible then the sets of meaningful EFMs (both in the original and in the reconfigured network) and EPs coincide.

20. 21. Lecture WS 2003/04Bioinformatics III20 Reconfigured Network Klamt & Stelling Trends Biotech 21, 64 (2003) ACP B D A(ext)B(ext)C(ext) R1R2R3 R5 R4R8 R9 R6 R7bR7f 3 EFMs are not systemically independent: EFM1 = EP4 + EP5 EFM2 = EP3 + EP5 EFM4 = EP2 + EP3

21. 21. Lecture WS 2003/04Bioinformatics III21 Comparison of EFMs and EPs Klamt & Stelling Trends Biotech 21, 64 (2003) ProblemEFM (network N1)EP (network N2) Recognition of 4 genetically indepen-Set of EPs does not contain operational modes:dent routes all genetically independent routes for converting (EFM1-EFM4)routes. Searching for EPs exclusively A to P.leading from A to P via B, no pathway would be found.

22. 21. Lecture WS 2003/04Bioinformatics III22 Comparison of EFMs and EPs Klamt & Stelling Trends Biotech 21, 64 (2003) ProblemEFM (network N1)EP (network N2) Finding all the EFM1 and EFM2 are One would only find the optimal routes:optimal because theysuboptimal EP1, not the optimal pathways foryield one mole P peroptimal routes EFM1 and synthesizing P duringmole substrate AEFM2. growth on A alone.(i.e. R3/R1 = 1), whereas EFM3 and EFM4 are only sub- optimal (R3/R1 = 0.5).

23. 21. Lecture WS 2003/04Bioinformatics III23 Comparison of EFMs and EPs Klamt & Stelling Trends Biotech 21, 64 (2003) EFM (network N1) 4 pathways convert A to P (EFM1-EFM4), whereas for B only one route (EFM8) exists. When one of the internal reactions (R4- R9) fails, for production of P from A 2 pathways will always „survive“. By contrast, removing reaction R8 already stops the production of P from B alone. EFM (network N1) Only 1 EP exists for producing P by substrate A alone, and 1 EP for synthesizing P by (only) substrate B. One might suggest that both substrates possess the same redundancy of pathways, but as shown by EFM analysis, growth on substrate A is much more flexible than on B. Problem Analysis of network flexibility (structural robustness, redundancy): relative robustness of exclusive growth on A or B.

24. 21. Lecture WS 2003/04Bioinformatics III24 Comparison of EFMs and EPs Klamt & Stelling Trends Biotech 21, 64 (2003) EFM (network N1) R8 is essential for producing P by substrate B, whereas for A there is no structurally „favored“ reaction (R4-R9 all occur twice in EFM1-EFM4). However, considering the optimal modes EFM1, EFM2, one recognizes the importance of R8 also for growth on A. EFM (network N1) Consider again biosynthesis of P from substrate A (EP1 only). Because R8 is not involved in EP1 one might think that this reaction is not important for synthesizing P from A. However, without this reaction, it is impossible to obtain optimal yields (1 P per A; EFM1 and EFM2). Problem Relative importance of single reactions: relative importance of reaction R8.

25. 21. Lecture WS 2003/04Bioinformatics III25 Comparison of EFMs and EPs Klamt & Stelling Trends Biotech 21, 64 (2003) EFM (network N1) R6 and R9 are an enzyme subset. By contrast, R6 and R9 never occur together with R8 in an EFM. Thus (R6,R8) and (R8,R9) are excluding reaction pairs. (In an arbitrary composable steady-state flux distribution they might occur together.) EFM (network N1) The EPs pretend R4 and R8 to be an excluding reaction pair – but they are not (EFM2). The enzyme subsets would be correctly identified. However, one can construct simple examples where the EPs would also pretend wrong enzyme subsets (not shown). Problem Enzyme subsets and excluding reaction pairs: suggest regulatory structures or rules.

26. 21. Lecture WS 2003/04Bioinformatics III26 Comparison of EFMs and EPs Klamt & Stelling Trends Biotech 21, 64 (2003) EFM (network N1) The shortest pathway from A to P needs 2 internal reactions (EFM2), the longest 4 (EFM4). EFM (network N1) Both the shortest (EFM2) and the longest (EFM4) pathway from A to P are not contained in the set of EPs. Problem Pathway length: shortest/longest pathway for production of P from A.

27. 21. Lecture WS 2003/04Bioinformatics III27 Comparison of EFMs and EPs Klamt & Stelling Trends Biotech 21, 64 (2003) EFM (network N1) All EFMs not involving the specific reactions build up the complete set of EFMs in the new (smaller) sub- network. If R7 is deleted, EFMs 2,3,6,8 „survive“. Hence the mutant is viable. EFM (network N1) Analyzing a subnetwork implies that the EPs must be newly computed. E.g. when deleting R2, EFM2 would become an EP. For this reason, mutation studies cannot be performed easily. Problem Removing a reaction and mutation studies: effect of deleting R7.

28. 21. Lecture WS 2003/04Bioinformatics III28 Comparison of EFMs and EPs Klamt & Stelling Trends Biotech 21, 64 (2003) EFM (network N1) For the case of R7, all EFMs but EFM1 and EFM7 „survive“ because the latter ones utilize R7 with negative rate. EFM (network N1) In general, the set of EPs must be recalculated: compare the EPs in network N2 (R2 reversible) and N4 (R2 irreversible). Problem Constraining reaction reversibility: effect of R7 limited to B  C.

29. 21. Lecture WS 2003/04Bioinformatics III29 Software: FluxAnalyzer FluxAnalyzer has both EPs and EFMs implemented. Allows convenient studies of metabolic systems. Klamt et al. Bioinformatics 19, 261 (2003)

30. 21. Lecture WS 2003/04Bioinformatics III30 Software: FluxAnalyzer Representation of stochiometric matrix. Klamt et al. Bioinformatics 19, 261 (2003)

31. 21. Lecture WS 2003/04Bioinformatics III31 Application of elementary modes Metabolic network structure of E.coli determines key aspects of functionality and regulation Compute EFMs for central metabolism of E.coli. Catabolic part: substrate uptake reactions, glycolysis, pentose phosphate pathway, TCA cycle, excretion of by-products (acetate, formate, lactate, ethanol) Anabolic part: conversions of precursors into building blocks like amino acids, to macromolecules, and to biomass. Stelling et al. Nature 420, 190 (2002)

32. 21. Lecture WS 2003/04Bioinformatics III32 Metabolic network topology and phenotype The total number of EFMs for given conditions is used as quantitative measure of metabolic flexibility. a, Relative number of EFMs N enabling deletion mutants in gene i (  i) of E. coli to grow (abbreviated by µ) for 90 different combinations of mutation and carbon source. The solid line separates experimentally determined mutant phenotypes, namely inviability (1–40) from viability (41–90). Stelling et al. Nature 420, 190 (2002) The # of EFMs for mutant strain allows correct prediction of growth phenotype in more than 90% of the cases.

33. 21. Lecture WS 2003/04Bioinformatics III33 Robustness analysis The # of EFMs qualitatively indicates whether a mutant is viable or not, but does not describe quantitatively how well a mutant grows. Define maximal biomass yield Y mass as the optimum of: e i is the single reaction rate (growth and substrate uptake) in EFM i selected for utilization of substrate S k. Stelling et al. Nature 420, 190 (2002)

34. 21. Lecture WS 2003/04Bioinformatics III34 Software: FluxAnalyzer Dependency of the mutants' maximal growth yield Y max ( i) (open circles) and the network diameter D( i) (open squares) on the share of elementary modes operational in the mutants. Data were binned to reduce noise. Stelling et al. Nature 420, 190 (2002) Central metabolism of E.coli behaves in a highly robust manner because mutants with significantly reduced metabolic flexibility show a growth yield similar to wild type.

35. 21. Lecture WS 2003/04Bioinformatics III35 Distribution of growth-supporting elementary modes in wild type (rather than in the mutants), that is, share of modes having a specific biomass yield (the dotted line indicates equal distribution). Stelling et al. Nature 420, 190 (2002) Multiple, alternative pathways exist with identical biomass yield. Growth-supporting elementar modes

36. 21. Lecture WS 2003/04Bioinformatics III36 Assume that optimization during biological evolution can be characterized by the two objectives of flexibility (associated with robustness) and of efficiency. Flexibility means the ability to adapt to a wide range of environmental conditions, that is, to realize a maximal bandwidth of thermodynamically feasible flux distributions (maximizing # of EFMs). Efficiency could be defined as fulfilment of cellular demands with an optimal outcome such as maximal cell growth using a minimum of constitutive elements (genes and proteins, thus minimizing # EFMs). These 2 criteria pose contradictory challenges. Optimal cellular regulation needs to find a trade-off. Can regulation be predicted by EFM analysis? Stelling et al. Nature 420, 190 (2002)

37. 21. Lecture WS 2003/04Bioinformatics III37 Compute control-effective fluxes for each reaction l by determining the efficiency of any EFM e i by relating the system‘s output  to the substrate uptake and to the sum of all absolute fluxes. With flux modes normalized to the total substrate uptake, efficiencies  i (S k,  ) for the targets for optimization  -growth and ATP generation, are defined as: Can regulation be predicted by EFM analysis? Stelling et al. Nature 420, 190 (2002) Control-effective fluxes v l (S k ) are obtained by averaged weighting of the product of reaction- specific fluxes and mode-specific efficiencies over all EFMs using the substrate under consideration: Y max X/Si and Y max A/Si are optimal yields of biomass production and of ATP synthesis. Control-effective fluxes represent the importance of each reaction for efficient and flexible operation of the entire network.

38. 21. Lecture WS 2003/04Bioinformatics III38 Prediction of gene expression patterns As cellular control on longer timescales is predominantly achieved by genetic regulation, the control-effective fluxes should correlate with messenger RNA levels. Compute theoretical transcript ratios  (S 1,S 2 ) for growth on two alternative substrates S 1 and S 2 as ratios of control-effective fluxes. Compare to exp. DNA-microarray data for E.coli growin on glucose, glycerol, and acetate. Excellent correlation! Stelling et al. Nature 420, 190 (2002) Calculated ratios between gene expression levels during exponential growth on acetate and exponential growth on glucose (filled circles indicate outliers) based on all elementary modes versus experimentally determined transcript ratios19. Lines indicate 95% confidence intervals for experimental data (horizontal lines), linear regression (solid line), perfect match (dashed line) and two-fold deviation (dotted line).

39. 21. Lecture WS 2003/04Bioinformatics III39 Predicted transcript ratios for acetate versus glucose for which, in contrast to a, only the two elementary modes with highest biomass and ATP yield (optimal modes) were considered. This plot shows only weak correlation. This corresponds to the approach followed by Flux Balance Analysis. Stelling et al. Nature 420, 190 (2002) Prediction of transcript ratios

40. 21. Lecture WS 2003/04Bioinformatics III40 Summary EFM are a robust method that offers great opportunities for studying functional and structural properties in metabolic networks. Klamt & Stelling suggest that the term „elementary flux modes“ should be used whenever the sets of EFMs and EPs are identical. In cases where they don‘t, EPs are a subset of EFMs. It remains to be understood more thoroughly how much valuable information about the pathway structure is lost by using EPs. Ongoing Challenges: - study really large metabolic systems by subdividing them - combine metabolic model with model of cellular regulation. Klamt & Stelling Trends Biotech 21, 64 (2003)