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Back to Chapter 10: Sections 10.3-10.7

Back to Chapter 10: Sections 10.3-10.7. Ben Heavner May 10, 2007. Review: Last Week – Mostly doing Math. From S , we found L such that LS = 0 By definition, d x /dt = Sv , so d/dt Lx = 0 L represents conserved quantities, called pools . Pools are like extreme pathways.

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Back to Chapter 10: Sections 10.3-10.7

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  1. Back to Chapter 10:Sections 10.3-10.7 Ben Heavner May 10, 2007

  2. Review: Last Week – Mostly doing Math • From S, we found L such that LS = 0 • By definition, dx/dt = Sv, so d/dt Lx = 0 • L represents conserved quantities, called pools. Pools are like extreme pathways. • Integrating, we found Lx = a. a is a matrix which gives the size of the pools.

  3. More Review • Different values of x satisfy Lx = a. • We can pick xref such that L(x – xref) = 0 • We know such an xref exists because LS = 0. • This transformation changes basis of x (concentration space) to one that is orthogonal to L. • transformed concentration space is bounded • boundaries are extreme concentration states

  4. How to Pick xref • xref is orthogonal to si • si.xref = 0 • x – xref is orthogonal to li • li . (x – xref) = 0

  5. PC CP S = Finding the Bounded Concentration SpaceExample 1: “Simple reversible reaction”

  6. Matlab: EDU>> S=[-1 1; 1 -1] S = -1 1 1 -1 EDU>> b = S' b = -1 1 1 -1 EDU>> a=null(b,'r') a = 1 1 EDU>> L=a' L = 1 1 PC CP S = Finding L • L = (1 1)

  7. PC CP • Then one parameterization of x is: S = • That is, from or Toward Finding xref – start with x • Suppose a1 = 1 • Remember Lx = a L = (1 1)

  8. First criteria for xref: si.xref = 0 or PC CP S = Finding xref:Systems of Linear Equations L = (1 1) (-1*x1ref) + (1*x2ref) = 0 x1ref = x2ref

  9. Second criteria for xref: li . (x – xref) = 0 or PC CP S = Finding xref:Systems of Linear Equations L = (1 1) [1*(x1-x1ref)] + [1*(x2-x2ref)] = 0 x1-x1ref=-x2+x2ref x1+x2=2xref Since (x1+x2) = a = 1 x1ref = x2ref = 1/2

  10. And • Then Reparamatarizing the Concentration Space: x-xref • Since

  11. What we gain by transforming x • Move from unbounded dx/dt = Sv space to bounded L(x-xref)=0 space • Note: • x-xref spanned by s1 • concentration space through origin

  12. A + P AP Further Transformation Examples and Pool Interpretation • “Bilinear association” (“Bimolecular association” in reaction space):

  13. C + AP CP + A Further Transformation Examples and Pool Interpretation • “Carrier-coupled reaction” (“Cofactor-coupled reaction” in reaction space):

  14. RH2 + NAD+ R + NADH + H+ More Pool Interpretation • “Rodox carrier coupled reactions”:

  15. RH2 + NAD+ R + NADH + H+ Redox carrier coupled reactions L =

  16. R R’ RH2 + NAD+ R + NADH + H+ R’ + NADH + H+ R’H2 + NAD+ Combining pools

  17. RH2 + NAD+ R + NADH + H+ R R’ R’ + NADH + H+ R’H2 + NAD+ Combining pools

  18. Summary • L contains “dynamic invariants” • Integrating d/dt (Lx) = 0 gives the pool sizes (a “bounded affine space”) • Three types of convex basis vectors span this space (like extreme pathways) • A reference state can be found to make this space parallel to L and be orthogonal to the column space • Metabolic pools can be displayed on a compound map

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