DOMINANT-EDGE PATHWAY : A Weighted Graph Algorithm for Identifying Dominant Metabolic Pathways

1. DOMINANT-EDGE PATHWAY: A Weighted Graph Algorithm for IdentifyingDominant Metabolic Pathways Ehsan Ullah & Soha Hassoun Department of Computer Science Kyongbum Lee Department of Chemical and Biological Engineering Tufts University

2. Ethanol Synthesis in E. Coli (Wlaschin, et al., 2006)

3. R1 : A → B R2 : B → C R3 : C → D R4 : B → E + F R5 : B → 2G R6 : G → H R7 : F → H R8 : H → I A R1 R2 R3 B C D R5 R4 G F E R7 R6 H R8 I Graph Representation of Metabolic Networks

4. Elementary Flux Mode • An elementary flux mode is a minimal set of reactions that can operate at steady state (Schuster & Hilgetag, 1994) • Metabolic flux distributions in living cells can be represented as non-negative linear combinations of elementary modes

5. A A A R1 R1 R1 R2 R3 B B B C D R5 R4 G F E R7 R6 H H R8 R8 I I Elementary Flux Mode Analysis: An Example A → D A → E + I A → 2I

6. A A A R1 R1 R1 R2 R3 B B B C D R5 R4 G F E R7 R6 H H R8 R8 I I Elementary Modes Can be Combined to Produce Several Possible Metabolic Flux Distributions

7. Our Approach for Pathway Analysis • Weighted graph search for a path containing a “dominant edge” that provides the most restrictive bottleneck • Dominance defined using edge weights that represent Gibbs free energy change

8. Electric Circuits Chemical Reactions Energy I Chemical Potential Low Chemical Potential High Chemical Potential A ∆G < 0 ∆G > 0 Spontaneous B Low Chemical Potential High Chemical Potential

9. Why Gibbs Free Energy? • ∆G gives estimate of reaction likelihood • For most metabolic reactions ∆G ≈ 0 • Flux through an entire pathway will be heavily influenced by a smaller subset of reactions that have very negative or positive ∆G

10. Gibbs Free Energy • Gibbs Free Energy (G) • Maximum amount of work that can be extracted from a closed system • Calculated using Group Contribution Theory (Mavrovouniotis, 1990) • Gibbs Free Energy Change (∆G) • Difference of Gibbs Free Energy of products and substrates

11. Dominant-Edge Pathway • Dominant reactions • for a series of reactions, it has the most positive ∆G • for parallel reactions, it has the most negative ∆G • Stoichiometrically balanced pathway

12. Weights WR4 = -0.5 WR7 = -2.5 R4 dominates R7 A R1 B R4 F R7 H R8 I Series Reactions

13. Parallel Reactions Weights WR4 = -0.5 WR5 = -2.5 R5 dominates R4 A A R1 R1 B B R5 R4 G F R7 R6 H H R8 R8 I I 13

14. Problem Statement Given a metabolic network graph Gm = (V, E), and starting and ending vertices s and t, find a dominant-edge pathway from s to t.

15. Algorithm • Identify dominant-edge path from s to t • Use modified Dijsktra’s algorithm to identify dominant-edge pathways • Augment the found path to realize a stoichiometrically balanced pathway • Use modified Dijsktra’s algorithm starting with the intermediate byproducts

16. S S A R1 R1 R1 R2 B B B R5 R4 G F E R7 R6 Intermediate Byproducts source S and target T R3 C D H H R9 R8 T T 16

17. A S S R1 R1 R1 R2 B B B R5 R4 G F E R7 R6 Intermediate Byproducts Dominant-Edge Path from S to T with E as abyproduct. The path is not stoichiometrically balanced R3 C D H H R9 R8 T T 17

18. Modified Dijkstra’s Algorithm • Dijkstra’s Shortest Path Algorithm • Finds shortest paths from a given source vertex vV to all other vertices in weighted graph G=(V, E), such that all edge weights are nonnegative • Modified Dijkstra’s Algorithm (Bottleneck Problem) (D.R.Fulkerson, 1966) • Selects reaction with most negative ∆G in parallel reactions • Select reaction with most positive ∆G in series reaction

19. Example Problem Find Dominant-Edge Pathway from S to T A -24 B S -9 -18 -6 -2 E C -19 -30 -11 D -15 -5 -6 -25 -20 T F

20. Example Problem S = { } Q = { S, A, B, C, D, E, F, T }   A -24 B  S -9 -18  -6 -2 E  C  -19 -30 -11 D -15 -5 -6 -25  -20 T F 

21. Example Problem S = { } Q = { S, A, B, C, D, E, F, T } Source   A -24 B  S -9 -18  -6 -2 E  C  -19 -30 -11 D -15 -5 -6 -25  -20 T F 

22. Example Problem S = { S } Q = { A, B, C, D, E, F, T } update   -9 X A -24 B  update S -9 -18  -9 X -6 -2 E  C  -19 -30 -11 D -15 -5 -6 -25  update -20 T F  -15 X

23. Example Problem S = { S } Q = { A, B, C, D, E, F, T }   -9 X A -24 B  S -9 -18  -9 X -6 -2 E  C  -19 -30 -11 D -15 -5 -6 -25  -20 T F Min energy  -15 X

24. Example Problem S = { S, F } Q = { A, B, C, D, E, T }   -9 X A - 24 B  S -9 -18  -9 X -6 -2 E  C  -19 -30 -11 D -15 -5 -6 -25  -20 T F  -15 X

25. Example Problem S = { S, T, A, B, C, D, E, F } Q = { } -9  X  -9 X A -24 B  S -9 -18  -9 X -6 -2 E  -11 X  -15 C -30 X -19 -11 D -15 -5 -6 -25  -15 X -20 T F  -15 X

26. Dominant-Edge Path Node D is an intermediate byproduct -9  X  -9 X A -24 B  S -9 -18  -9 X -6 -2 E  -11 X  -15 C -30 X -19 -11 D -15 -5 -6 -25  -15 X -20 T F  -15 X

27. Augmenting Path Path from byproducts to final product -9  X  -9 X A -24 B  S -9 -18  -9 X -6 -2 E  -11 X  -15 C -30 X -19 -11 D -15 -5 -6 -25  -15 X -20 T F  -15 X

28. Runtime • Dominant Path: O(|V|2 + |E|) • Augmenting Path: O(|V|2 + |E|)

29. Test Cases • Pentose Fermentation in Zymomonas mobilis (bacterium) • Escherichia coli • Liver Cell

30. Test Case 1 (Altintas, et al., 2006)

31. Test Case 2 (Wlaschin, et al., 2006)

32. Test Case 2 Total elementary flux modes : 33,000 Each DOMINANT-EDGE pathway between an input sugar and ethanol exactly corresponds to the maximal conversion pathway engineered through gene knockouts by Wlaschin et al., 2006

33. Test Case 3 • Total elementary flux modes : 188

34. Runtime • Windows Machine • C++ Implementation • 4GB Memory

35. ∆G Used in Recent Studies • ∆G analysis of a E. Coli resulted in identifying four reactions representing thermodynamic bottlenecks in the production of growth (Henry et al., 2006) • Using thermodynamics-based constraints to enforce the exclusion of thermodynamically infeasibilities when calculating flux distribution (e.g. a flux cannot be positive unless ∆G is negative) (Henry et al., 2007)

36. Conclusion • An Efficient way of finding a maximum conversion pathway with runtime advantages over enumeration-based approaches • The Dominant-Edge Pathway found is identical, proper subset, or partially overlaps with EFM results

37. Applications & Future Work • Pathway Analysis has promising applications • Understanding living systems • Efficient in silicoexperimentation • Engineering microorganisms to perform specific synthesis tasks • Future Work • Find an energetically favored pathway (most negative overall Gibbs energy change) • Find alternative pathways with (nearly) equivalent bottleneck or pathway weights

38. Acknowledgements • Gautham Sridharan (BCE Tufts University)

39. Questions

40. Thank You

41. Questions for Ehsan? • Why use EFM instead of Extreme Pathways? How would your results be different? • How did you calculate the delta_Gs? • How do you ensure that your paths are stoichiometrically balanced in terms of conversion?