chapter 10: the left null space of s - or - now we’ve got s. let’s do some math and see what...
TRANSCRIPT
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.
- 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, dx/dt = Sv
Networks from S S: a network showing
how reactions link metabolites
-ST: a network showing how compounds link reactions
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)
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
What we really do to find L:a little bit of mathRemember 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
Matlab: why we’re not afraid of a big SSTLT = 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.
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!
Doing Math to find the meaning of L
dx/dt = Sv
L dx/dt = LSvsince LS = 0,
L dx/dt = 0
Palsson writes this as d/dt Lx = 0 (eq 10.5)
We can integrate to find Lx = a
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.
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
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.
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.
Intermission…
Until next week?