1. Chapter 10: The Left Null Space of S - or - Now we’ve got S. Let’s do some Math and see what happens. - 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 -S T : 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 =S T L T to make some progress: LS = 0 (LS) T = 0 T = 0 S T L T = 0

7. Matlab: why we’re not afraid of a big S S T L T = 0 means that L T is the basis for the null space of S T. Let b = S T. Then the Matlab command a = null(b) will return a basis for the null space of L T. 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, a i L x = a is true for many different values of x, such as Lx ref = a. So whatever x we pick, we can also pick a x ref such that L (x - x ref ) = 0. This transformation changes the basis of the concentration space. Whereas x is not orthogonal to L, x - x ref is.

13. The reference state of x The new basis of the concentration space from (x - x ref ) 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?