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Chapter 10: The Left Null Space of S

Chapter 10: The Left Null Space of S. - or - Now we’ve got S. Let’s do some Math and see what happens. A review of S. Every column is a reaction Every row is a compound S transforms a flux vector v into a concentration time derivative vector, d x /dt = Sv. Networks from S.

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Chapter 10: The Left Null Space of S

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  1. Chapter 10: The Left Null Space of S - or - Now we’ve got S. Let’s do some Math and see what happens.

  2. A review of S • Every column is a reaction • Every row is a compound • S transforms a flux vector v into a concentration time derivative vector, dx/dt = Sv

  3. Networks from S • S: a network showing how reactions link metabolites • -ST: a network showing how compounds link reactions

  4. Introducing L • LS = 0 • Dimension of L is m-r • Rows are: • linearly independent • span L • Are orthogonal to the reaction vectors of S (columns)

  5. Finding L • “The convex basis for the left null space can be computed in the same way as the right null space by transposing S” - Palsson p. 155

  6. What we really do to find L:a little bit of math Remember we’re trying to find L from LS = 0. We might try to say that since SR = 0 and LS = 0, S = R. But matrix multiplication is generally not commutative. That is, LS SL, so that’s wrong. BUT, we can use the identity that (LS)T=STLT to make some progress: LS = 0 (LS)T = 0T = 0 STLT = 0

  7. Matlab: why we’re not afraid of a big S STLT = 0 means that LT is the basis for the null space of ST. Let b = ST. Then the Matlab command a = null(b) will return a basis for the null space of LT. Once we have a, the Matlab command L = a’ will return L. Note that this L is not a unique basis - there are infinitely many.

  8. So? What does L mean? • We’ve found a matrix, L, that when multiplied by S, gives the 0 matrix: LS = 0 • Recall the definition of S as a transformation: dx/dt = Sv • Let’s do more math!

  9. Doing Math to find the meaning of L dx/dt = Sv L dx/dt = LSv since LS = 0, L dx/dt = 0 Palsson writes this as d/dt Lx = 0 (eq 10.5) We can integrate to find Lx = a

  10. Pools are like Pathways. • Chapter 9: Using R (the right null space), found with the rows of S, to find extreme pathways on flux maps. • Chapter 10: Using L (the left null space), found with the columns of S, to find pools.

  11. Pathways and pools • 3 types of extreme pathways • through fluxes • futile cycles + cofactors • internal cycles • 3 types of pools • primary compounds • primary and secondary compounds internal to system • only secondary compounds

  12. Back to the Math: the reference state of x • In L x = a, there’s a few ways we can get x and a. For example, we can pick either initial or steady-state conditions to set the pool sizes, ai • L x = a is true for many different values of x, such as Lxref = a.So whatever x we pick, we can also pick a xref such that L (x - xref) = 0. • This transformation changes the basis of the concentration space. Whereas x is not orthogonal to L, x - xref is.

  13. The reference state of x • The new basis of the concentration space from (x - xref) allows us to transform our choice of x to a closed, or bounded, concentration space that has end points representing the extreme concentration states.

  14. Intermission… Until next week?

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