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?