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
x-xref could just be S, for example.

4. How to Pick xref xref is orthogonal to si x – xref is orthogonal to li
si . xref = 0
x – xref is orthogonal to li
li . (x – xref) = 0

5. Finding the Bounded Concentration Space Example 1: “Simple reversible reaction”PC CP
Consider the Rxn CP <-> PC (in reaction space, reversible conversion; in concentration space, a “simple reversible reaction)
S: rows are compounds, columns are reactions
S and L are as on 10.9 (p. 158). Suppose a1 = 1, then x is as in (see linear algebra book p. 20). the two criteria above give x1 and x2, so we can get x-xref as in This gives a bounded concentration space (fig again).
S =

6. Finding L Matlab: CP PC S = L = (1 1) EDU>> S=[-1 1; 1 -1] S =
EDU>> b = S'
b =
EDU>> a=null(b,'r')
a = 1
EDU>> L=a'
L = PC CP
S =
see help null for the difference between null(b) and null(b,’r’)
Interpreting L: 2 pools – CP and PC (?)
Matlab:
L = (1 1)

7. Toward Finding xref – start with x
Suppose a1 = 1
Remember Lx = a PC CP
S =
Then one parameterization of x is:
Recall that a is the total pool size:
L = (1 1)
That is, from or

8. Finding xref: Systems of Linear EquationsPC CP
First criteria for xref:
si . xref = 0 or
S =
L = (1 1)
(-1*x1ref) + (1*x2ref) = 0
x1ref = x2ref

9. Finding xref: Systems of Linear EquationsPC CP
Second criteria for xref:
li . (x – xref) = 0 or
S =
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. Reparamatarizing the Concentration Space: x-xref
Since
And
Then

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. Further Transformation Examples and Pool Interpretation
“Bilinear association” (“Bimolecular association” in reaction space): AP
A + P
As a slightly more complex example, consider Rxn C + P <-> CP (in reaction space, bimolecular association; in concentration space, bilinear assocation)
Book typo on P. 160? – Should S only have one column – 1 direction?
S and L are as on (p. 160). Suppose a1 = (1, 2)T, then x is as in and fig 104 (p. 160). we can get x-xref from L(x-xref) = 0 and <s1 . xref> = 0. Again, this process gives a bounded concentration space (which is just a translation of x to a more convenient basis - fig. 10.5).
PROBLEM:
Matlab - trying to find L, get
L =
Which is different than (p. 160) - not all positive entries. But a if add these two rows, get Palsson’s L. How fix?
Note: L pools: first row is “A” pool, second is “P” pool

13. Further Transformation Examples and Pool Interpretation
“Carrier-coupled reaction” (“Cofactor-coupled reaction” in reaction space):
CP + A
C + AP
These translations of x can be accomplished in higher dimensional spaces as well. Consider the Rxn C + AP <-> CP + A (in reaction space, cofactor-coupled reaction; in concentration space, carrier-coupled reaction).
S ordered (CP, C, AP, A)
a = (1,2,2,1)T
Pools of L:
1 - primary substrate, C
2 - cofactor, A
3 - phosphorylated compounds (CP + AP)
4 - vacancy pool (C + A)
? Why no P pool? Perhaps b/c no free P?

14. More Pool Interpretation
“Rodox carrier coupled reactions”:
R + NADH + H+
RH2 + NAD+
Doesn’t do translation of x this time.
Pools of L:
1 - Total R
2 - Redox occupancy 1
3 - Redox occupancy 2
4 - Redox vacancy
5 - total redox carrier 1
6 - total redox carrier 2

15. Redox carrier coupled reactions
R + NADH + H+
RH2 + NAD+
L =
Doesn’t do translation of x this time.
Pools of L:
1 - Total R
2 - Redox occupancy 1
3 - Redox occupancy 2
4 - Redox vacancy
5 - total redox carrier 1
6 - total redox carrier 2
See Palsson p.164 for figure

16. Combining pools R + NADH + H+ RH2 + NAD+ R’ R R’H2 + NAD+
Pools of L:
1 - Total R
2 - Redox occupancy 1
3 - Redox occupancy 2
4 - Redox vacancy
5 - total redox carrier 1
6 - total redox carrier 2

17. Combining pools R + NADH + H+ RH2 + NAD+ R’ R R’H2 + NAD+
Pools of L:
1 - Total R
2 - Redox occupancy 1
3 - Redox occupancy 2
4 - Redox vacancy
5 - total redox carrier 1
6 - total redox carrier 2
These combinations can lead to glycolysis/TCA cycle (section 10.5)
See Palsson p.167 for figure

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