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Metabolic Pool Transformations: Understanding Conservation in Dynamic Systems

Explore how metabolic pool transformations impact dynamic systems, identifying bounded concentration spaces and extreme pathways. Learn how to select reference states for orthogonal transformation. Discover practical examples and interpretations of different pool interactions in a biochemical context.

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Metabolic Pool Transformations: Understanding Conservation in Dynamic Systems

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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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