Understanding Time Invariant Pools in Convex Basis Formation for Systems Biology

1. Lecture #16 The Left Null Space of S

2. Outline • Definition • Convex basis – formation of non-negative pools • Alignment of the affine concentration space with LN(S) • Three types of pools • Examples of extreme pools • (Tilting to form a new basis)

3. DEFINITION OF LN(S)

4. The Left Null Space of S x’ P , pij ≥0 convex basis lij ≥0 C(S) S•R=0 LN(S) L•S=0 =0 ( ) ( ) =0 ( ) ( ) calculating convex basis: reaction column sj row vectors li ( )=0 ST LS=0 (LS)T=0 STLT=0 <lisj>=0 use ExPa program

5. Dynamic mass balance Multiply with L from the left ( ) ( ) ) ( ) ( linear combination of the concentrations that always add up to ai time invariants (pools) want a set of basis vectors li, where lik≥0 convex set

6. dx2/dt C(S) v2 LN(S) S dx1/dt R(S) v1 N(S) S(•)dt x2 eq line a1 K The affine concentration space x1 a1

7. Finding a reference point ALIGNMENT OF THE LN(S) AND THE AFFINE CONCENTRATION SPACE

8. A reference state that aligns the affine concentration space with the null space

9. DEFINITION OF EXTREME POOLS

10. Refresher on compound maps

11. Definition of extreme pools • Type A pools that are composed only of the primary compounds; • Type B pools that contain both primary and secondary compounds internal to the system; and • Type C pools are comprised only of secondary compounds. • Type B pools generally represent the conserved moieties (or currencies) that are exchanged from one compound to another, such as a hydroxyl or phosphate group.

12. Analogy to extreme pathways

13. Classification of pools based on the structure of the matrix L

14. EXAMPLES OF EXTREME POOLS

15. The bi-linear reaction • A+B ->AB • The pools are • A+AB (x1+x3) • B+AB (x1+x3) • Very clear conservations

16. The exchange reaction:AP+C -> CP+A ; x=(CP,C,AP,A) • The first pool is a conservation of the primary substrate pool C (=C+CP) and is a Type A pool. • The second pool is a conservation of the cofactor A (=A+AP) and is a Type C pool. • The third pool is a conservation of the phosphorylated compounds (=CP+AP) and represents the total energy inventory, or occupancy in the system. • The last pool is a, vacancy pool (C+A) that represents the low energy state of the participating compounds. This pool is linearly redundant but convexly independent.

17. The exchange reaction:AP+C -> CP+A ; x=(CP,C,AP,A)

18. Simple redox exchange

19. Definition of Extreme Redox Pools

20. Reaction map Compound map NAD+ v2 RH2 R R’ R’H2 RH2 R v1 v1 v3 H+ v2 H+ NADH NAD+ NADH NADH NAD+ v3 R’H2 R’ Pool map Pool #1 (A) Pool #2 (B) #1 #2 NAD+ NAD+ v2 v2 RH2 RH2 R R RH2 R R’ R’H2 RH2 R R’ R’H2 v1 v1 v1 v3 v1 v3 H+ H+ v2 v2 H+ NADH H+ NADH NAD+ NADH NADH NAD+ NAD+ NADH NADH NAD+ v3 v3 R’H2 R’ R’H2 R’ Pool #3 (B) Pool #4 (B) #3 #4 NAD+ NAD+ v2 v2 RH2 RH2 R R RH2 R R’ R’H2 RH2 R R’ R’H2 v1 v1 v1 v3 v1 v3 H+ H+ v2 v2 H+ NADH H+ NADH NAD+ NADH NADH NAD+ NAD+ NADH NADH NAD+ v3 v3 R’H2 R’ R’H2 R’ #5 #6 Pool #5 (B) Pool #6 (B) NAD+ NAD+ v2 v2 RH2 RH2 R R RH2 R R’ R’H2 RH2 R R’ R’H2 v1 v1 v1 v3 v1 v3 H+ H+ v2 v2 H+ NADH H+ NADH NAD+ NADH NADH NAD+ NAD+ NADH NADH NAD+ v3 v3 R’H2 R’ R’H2 R’

21. Linked Redox Pools

22. Skeleton glycolysis

23. Interpretation of glycolytic pools • l1, total carbon pool • l2, high-energy conservation pool: • 2C6 + 3C6P + 4C6P2 + 2C3P1 + 2C3P2 + C3P + AP3 • l3, conservation of elemental P: • C6P + 2C6P2 + C3P1 + 2C3P2 + C3P + AP3 + P$ • l4, low-energy conservation pool: • 2C6 + C6P + C3P + 2C3 + AP2 • l5, potential to incorporate the stand-alone moiety P; • C3P2 + C3P + C3 + P • l6, total carrier pool of A

24. Interpretation of TCA pools • l1 exchanging carbon group • 2H2C2 + 2H2C6 + HC5 + C • l2, recycled four-carbon moiety which `carries' the two carbon group that is oxidized • C4 + H2C6 + HC5 • l3 , hydrogen group that contains the redox inventory in the system • 2H2C2 + 2H2C6 + HC5 + NH • l4 , redox vacancy • C + N • l5 , total cofactor pool • N + NH

25. Other Examples

26. Glycolysis as an open system:many conserved pools disappear

27. The stoichiometric matrix and the left null space vectors

28. GENOME-SCALE STUDIES

29. iJR904 • Developed Minimal Conserved Pool Identification (MCPI) approach • Elucidating the conserved pools for target metabolites without computing the entire basis conservation relationships. • MILP formulation Biophys J, 88: 37-49 (2005)

30. Conserved pools spanning central metabolism of iJR904

31. Rotating the bases vectors of LN(S) for iAF1260 • The LN(S) basis vectors correspond to time invariant pools • The pools found are: • Amino acyl tRNAs – tRNAs • Charge Carriers (NADH. NAD) • Co-factor Pools • Apolipoprotein-lipoprotein Factor Loading

32. Summary • The left null of S contains time invariant pools • A convex basis can be found for LN(S) • Good basis can be found by tilting methods • Examples show the formation of meaningful pools • The LN(S) has not been extensively studied