1. Seminar in bioinformatics Computation of elementary modes: a unifying framework and the new binary approach Elad Gerson, Spring 2006, Technion. Julien Gagneur and Steffen Klamt BMC Bioinformatics 2004, 5:175

2. Agenda Quick overview of last week’s lecture. Extension of the EP concept. –Enter EM. General framework for EM computation. –Reversible reactions split. –Network compression. –Post processing. Some implementation tweaks.

3. Last week on bioinformatics seminar! Given a metabolic network we wish to find all the possible flux distributions which results in a steady state. Meaning, the overall flux in a pathway is 0.

4. Last week on bioinformatics seminar! This is done by describing the pathway as a stoichiometric matrix S, solving the equation –

5. Last week on bioinformatics seminar! Notice that we are interested only in solutions where (sign suggests reaction’s direction). Solution space is spanned by linearly independent vectors. We look for a spanning set s.t. every solution can be written as a linear combination of the spanning vector where all coefficients are non-negative (Genetically independent). Those solutions are called Extreme pathways (EP). Can be found using the Null Space Approach (NSA) Algorithm.

6. Problem Biology suggests some reaction are reversible. Consider the following network for instance – R5 can work in both directions (Not simultaneously!)

7. Solution ? Remove the restriction, signs suggests direction.. Bad idea.. Not all reactions are reversible. Solutions no longer take the form of a polyhedral cone.

8. Solution ! Split the reversible reactions.. Find Extreme Pathways using the NSA algorithm. Post process found EPs, merge split reactions (“opposite direction” should be set with a negative sign). Post processed EPs are now called - Elementary Modes (EM). R5aR5b

9. Compressing the network Removing redundancies Can be united..

10. Compressing the network Removing redundancies R1 is null in any feasible steady state

11. Compressing the network Removing redundancies Contradict each other.. Can be eliminated.

12. Compressing the network Removing redundancies Active in any stead state.

13. Compressing the network Removing redundancies Some redundancies can be detected as dependent linear rows in the kernel matrix. Iterative approach, remove redundancies until non detected. –Produce better results.

14. Preprocessing - Metabolic networks yield deeper insight of organisms metabolism. Failure modes analysis will provide Crucial parts identification. Suitable targets for repressing undesired metabolic functions. Apply NSA algorithm. Post process. General framework

15. The authors offers an efficient implementation to the NSA and CBA (Combined basis – Schuster et. al.) algorithms. –Using binary representation for vectors. Fast bit operators. Efficient memory usage (up to 1.6% of original!) One more tweak

16. Seminar in bioinformatics Minimal cut sets in biochemical reaction networks Elad Gerson, Spring 2006, Technion. Steffen Klamt and Ernst Dieter Gilles Bioinformatics Vol. 20 no. 2 2004, pages 226–234

17. Abstract Motivation Metabolic networks yield deeper insight of organisms metabolism. Failure modes analysis will provide Crucial parts identification. Suitable targets for repressing undesired metabolic functions. Results The biochemical networks minimal cut sets concept. Algorithm which computes MCS with respect to an objective reaction. Potential applications includes phenotype predictions. Network verifications. Structural robustness and fragility assessment. Metabolic flux analysis. Target identification in drug discovery.

18. Introduction Assume we wish to prevent the production of metabolite X. i.e. there is no balanced flux distribution possible which involves obR. Can be done by gene deletion or enzyme inhibition.

19. Introduction Definition - We call a set of reactions a cut set (with respect to a defined objective reaction) if after the removal of these reactions from the network no feasible balanced flux distribution involves the objective reaction.

20. Introduction That’s easy.. Consider C0 = {obR} One might wish to cut the reaction at the beginning. What if there are numerous obR’s ? Simultaneous failure might be achieved more efficiently.

21. Introduction Take two – Remove all reactions except for oBR. Not efficient. Not intelligent.

22. Introduction Consider C1 = {R5, R8} Sufficient. Neither the removal of R5 nor R8 is sufficient. No subset of C1 is a valid cut set → C1 is minimal.

23. Introduction Definition - A cut set C (related to a defined objective reaction) is a minimal cut set (MCS) if no proper subset of C is a cut set. Can you spot all the MCS in the network ?

24. Introduction Is C2 = {R2, R4, R6} minimal ?

25. Introduction Is C3 = {R2, R5, R7} ?

26. Introduction How about C1 = {R1} ?

27. Introduction OK, what about Graph disconnectivity algorithms ? No good, They don’t take the hypergraph nature of metabolic pathways into account.

28. The algorithm Initialization (1)Calculate the EMs in the given network (2)Define the objective reaction obR (3) Choose all EMs where reaction obR is non-zero and store it in the binary array em_obR (em_obR[i][j]==1 means that reaction j is involved in EM i) (4) Initialize arrays mcs and precutsets as follows (each array contains sets of reaction indices): append {j } to mcs if reaction j is essential (em_obR[i][j]=1 for each EM i), otherwise to precutsets

29. The algorithm (5) FOR i=2 TO MAX_CUTSETSIZE (5.1) new_precutsets=[ ]; (5.2) FOR j = 1 TO q (q: number of reactions) (5.2.1) Remove all sets from precutsets where reaction j participates (5.2.2) Find all sets of reactions in precutsets that do not cover at least one EM in em_obR where reaction j participates; combine each of these sets with reaction j and store the new preliminary cut sets in temp_precutsets (5.2.3) Drop all temp_precutsets which are a superset of any of the already determined minimal cut sets stored in mcs (5.2.4) Find all retained temp_precutsets which do nowcover all EMs and appendthem to mcs; append all others to new_precutsets ENDFOR (5.3) If isempty(new_precutsets) (5.3.1) Break ELSE (5.3.2) precutsets=new_precutsets ENDIF ENDFOR (6) result: mcs contains the MCSs

30. Running example Initialization – Calculate EM We are only interested in EM containing obR

31. Running example Initialization mcs = {{1}}, precutsets = {{2},{3},{4},{5},{6},{7},{8}} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

32. Running example I = 2, j = 1 mcs = {{1}}, precutsets = {{2},{3},{4},{5},{6},{7},{8}} new_precutsets = {} temp_precutsets = {} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

33. Running example I = 2, j = 1 mcs = {{1}}, precutsets = {{2},{3},{4},{5},{6},{7},{8}} new_precutsets = {} temp_precutsets = {{1 2}} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

34. Running example I = 2, j = 1 mcs = {{1}}, precutsets = {{2},{3},{4},{5},{6},{7},{8}} new_precutsets = {} temp_precutsets = {{1 2}, {1 3}, {1 4}, {1 5} {1 6}} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

35. Running example I = 2, j = 1 mcs = {{1}}, precutsets = {{2},{3},{4},{5},{6},{7},{8}} new_precutsets = {} temp_precutsets = {{1 2}, {1 3}, {1 4}, {1 5} {1 6} {1 7} {1 8}} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

36. Running example I = 2, j = 1 mcs = {{1}}, precutsets = {{2},{3},{4},{5},{6},{7},{8}} new_precutsets = {} temp_precutsets = {{1 2}, {1 3}, {1 4}, {1 5}, {1 6}, {1 7}, {1 8}} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

37. Running example I = 2, j = 2 mcs = {{1}}, precutsets = {{2},{3},{4},{5},{6},{7},{8}} new_precutsets = {} temp_precutsets = {} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

38. Running example I = 2, j = 2 mcs = {{1}}, precutsets = {{3},{4},{5},{6},{7},{8}} new_precutsets = {} temp_precutsets = {} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

39. Running example I = 2, j = 2 mcs = {{1}}, precutsets = {{3},{4},{5},{6},{7},{8}} new_precutsets = {} temp_precutsets = {{2 4}} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

40. Running example I = 2, j = 2 mcs = {{1}}, precutsets = {{3},{4},{5},{6},{7},{8}} new_precutsets = {} temp_precutsets = {{2 4},{2 6},{2 7},{2 8}} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

41. Running example I = 2, j = 2 mcs = {{1}}, precutsets = {{3},{4},{5},{6},{7},{8}} new_precutsets = {{2 4}} temp_precutsets = {{2 6},{2 7},{2 8}} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

42. Running example I = 2, j = 2 mcs = {{1}}, precutsets = {{3},{4},{5},{6},{7},{8}} new_precutsets = {{2 4},{2 6},{2 7},{2 8}} temp_precutsets = {} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

43. Running example I = 2, j = 5 mcs = {{1}}, precutsets = {{5},{6},{7},{8}} new_precutsets = {{2 4},{2 6},{2 7},{2 8},..} temp_precutsets = {} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

44. Running example I = 2, j = 5 mcs = {{1}}, precutsets = {{6},{7},{8}} new_precutsets = {{2 4},{2 6},{2 7},{2 8},..} temp_precutsets = {{5 6},{5 7},{5 8}} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

45. Running example I = 2, j = 5 mcs = {{1}}, precutsets = {{6},{7},{8}} new_precutsets = {{2 4},{2 6},{2 7},{2 8},..} temp_precutsets = {{5 6},{5 7},{5 8}} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

46. Running example I = 2, j = 5 mcs = {{1}, {5 6}}, precutsets = {{6},{7},{8}} new_precutsets = {{2 4},{2 6},{2 7},{2 8},..} temp_precutsets = {{5 7},{5 8}} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

47. Running example I = 2, j = 5 mcs = {{1}, {5 6}, {5 7}}, precutsets = {{6},{7},{8}} new_precutsets = {{2 4},{2 6},{2 7},{2 8},..} temp_precutsets = {{5 8}} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

48. Running example I = 2, j = 8 mcs = {{1}, {5 6}, {5 7}, {5 8}}, precutsets = {} new_precutsets = {{2 4},{2 6},{2 7},{2 8},..} temp_precutsets = {} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

49. Running example I = 3, j = 2 mcs = {{1}, {5 6}, {5 7}, {5 8}}, precutsets = {{2 4},{2 6},{2 7},{2 8},…{4 6},…} new_precutsets = {} temp_precutsets = {} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

50. Running example I = 3, j = 2 mcs = {{1}, {5 6}, {5 7}, {5 8}}, precutsets = {…{4 6},…} new_precutsets = {} temp_precutsets = {…} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

51. Running example I = 3, j = 2 mcs = {{1}, {5 6}, {5 7}, {5 8}}, precutsets = {…{4 6},…} new_precutsets = {…} temp_precutsets = {{2 4 6},…} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

52. Running example I = 3, j = 2 mcs = {{1}, {5 6}, {5 7}, {5 8}, {2 4 6}}, precutsets = {…{4 6},…} new_precutsets = {…} temp_precutsets = {…} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

53. Running example I = 3, j = 8 mcs = {{1}, {5 6}, {5 7}, {5 8}, {2 4 6},…}, precutsets = {} new_precutsets = {…} temp_precutsets = {…} R8R7R6R5R4R3R2R1em_obR 111000011 (EM2) 000101112 (EM3) 000110013 (EM4)

54. Complexity Let q be the number of reactions. Assuming |EM| << q. In initialization q singletons are generated and tested. In the i-th iteration –Overall number of temp_precutsets generated –O(p) comparisons are made. Hence, (All subsets of q items) –Yes.. exponential.. Maximal MCS size << q bounds polynomial approximation.

55. MCS in central metabolism of E. coli MCS calculated with ‘biomass synthesis’ as objective reaction (growth). –Network comprises 110 reactions and 89 metabolites. –Catabolic (material breakdown) part modeled in details. Enables excretion of 5 metabolites. Uptake of glucose, acetate, glycerol and succinate. Growth on each substrate was tested separately.

56. MCS in central metabolism of E. coli

57. Possible applications Structural fragility and robustness MCS can be used for “risk assessment” in metabolic pathways. –More EMs suggested a more robust and less fragile pathway. EMs number and MCSs size are strongly correlated. (More elements must fail). We seek a better criteria. Glucose is known to be the least fragile growth substrate having most EMs and apparently longest MCSs ‘Dangerous’ MCSs

58. Possible applications Structural fragility and robustness Definition – Reaction fragility factor Fi is the reciprocal of the average size of all the MCSs the reaction i participates.

59. Possible applications Structural fragility and robustness Definition – Reaction fragility factor Fi is the reciprocal of the average size of all the MCSs the reaction i participates. May suggest reaction’s importance.

60. Possible applications Structural fragility and robustness Definition – Reaction fragility factor Fi is the reciprocal of the average size of all the MCSs the reaction i participates. Is there a correlation between Fi and the number of EMs the reaction participates?

61. Possible applications Structural fragility and robustness

62. Possible applications Structural fragility and robustness Definition – Network fragility F is defined as where q is then number of reactions.

63. Possible applications Network verification and mutant phenotype predictions. Cutting an MCS is predicted to leave a metabolic pathway dysfunctional. Apply the algorithm with ‘growth’ as obR. – If a set of gene deletions (or mutants) contains an MCS a non-viable phenotype is expected. Viable phenotype would be a false negative. –Proof for incorrect or incomplete network. –Otherwise growth is possible. Non-viable phenotype would be a false positive. –May suggest a false assumption in the network structure. »One of the reactions in the MCS might be of regulatory nature.

64. Possible applications Target identification and repressing cellular functions. MCS offers a theoretical tool for target identification in drug discovery. –An irreducible set of interventions needed for pathway dysfunction. –Usually we will look for minimal size of MCS. –Other pathways should be weakly affected. Can be checked easily – set of untouched EM’s. MCS 0, 2, 3, 4 will not affect EM1