1. Lecture #16 The Left Null Space of S

2. Outline 1.Definition 2.Convex basis – formation of non- negative pools 3.Alignment of the affine concentration space with LN(S) 4.Three types of pools 5.Examples of extreme pools 6.(Tilting to form a new basis)

3. DEFINITION OF LN(S)

4. The Left Null Space of S SR=0 P, p ij ≥0 ( ) =0 calculating convex basis: LS=0 (LS) T =0 S T L T =0 STST ( )=0 use ExPa program LS=0 ( ) =0 l ij ≥0 convex basis reaction column s j row vectors l i =0 C(S) LN(S) x’x’

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

6. v2v2 v1v1 N(S) R(S) S C(S) LN(S) dx 1 /dt dx 2 /dt S()dt x2x2 x1x1 a1a1 a1a1 K eq line The affine concentration space

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

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 (x 1 +x 3 ) –B+AB (x 1 +x 3 ) 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. RH 2 R NAD + NADH H+H+ R’R’ R’H2R’H2 NAD + v1v1 NADHH+H+ R RH 2 v3v3 v2v2 R’R’ R’H2R’H2 v1v1 v2v2 v3v3 Reaction mapCompound map RH 2 R NAD+NADH H+ R’R’ R’H2R’H2 NADHNAD+ v1v1 v2v2 v3v3 RH 2 R NAD+NADH H+ R’R’R’H2R’H2 NADHNAD+ v1v1 v2v2 v3v3 RH 2 R NAD+NADH H+H+ R’R’R’H2R’H2 NAD+ v1v1 v2v2 v3v3 RH2R NAD + NADH H+ R’R’ R’H2 NADHNAD + v1v1 v2v2 v3v3 RH2R NAD + NADH H+ R’R’R’H2 NADHNAD + v1v1 v2v2 v3v3 RH2R NAD + NADH H+H+ R’R’R’H2 NADHNAD + v1v1 v2v2 v3v3 v1v1 NADHH+H+ R RH 2 v2v2 R’R’ R’H2R’H2 v1v1 NAD + NADHH+H+ R RH 2 v3v3 v2v2 R’R’R’H2R’H2 v1v1 NAD + NADHH+H+ R RH 2 v3v3 v2v2 R’R’R’H2R’H2 v1v1 NAD + NADHH+H+ R RH 2 v3v3 v2v2 R’R’ R’H2R’H2 v1v1 NAD + NADHH+H+ R RH 2 v3v3 v2v2 R’R’R’H2R’H2 v1v1 NAD + NADHH+H+ R RH 2 v3v3 v2v2 R’R’R’H2R’H2 v3v3 Pool map Pool #1 (A) Pool #3 (B) Pool #5 (B) Pool #2 (B) Pool #4 (B) Pool #6 (B) #1#2 #3#4 #5#6

21. Linked Redox Pools

22. Skeleton glycolysis

23. Interpretation of glycolytic pools l 1, total carbon pool l 2, high-energy conservation pool: –2C 6 + 3C 6 P + 4C 6 P 2 + 2C 3 P 1 + 2C 3 P 2 + C 3 P + AP 3 l 3, conservation of elemental P: –C 6 P + 2C 6 P 2 + C 3 P 1 + 2C 3 P 2 + C 3 P + AP 3 + P$ l 4, low-energy conservation pool: –2C 6 + C 6 P + C 3 P + 2C 3 + AP 2 l 5, potential to incorporate the stand-alone moiety P; –C 3 P 2 + C 3 P + C 3 + P l 6, total carrier pool of A

24. Interpretation of TCA pools l 1 exchanging carbon group –2H 2 C 2 + 2H 2 C 6 + HC 5 + C l 2, recycled four-carbon moiety which `carries' the two carbon group that is oxidized –C 4 + H 2 C 6 + HC 5 l 3, hydrogen group that contains the redox inventory in the system –2H 2 C 2 + 2H 2 C 6 + HC 5 + NH l 4, redox vacancy –C + N l 5, 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