Back to Chapter 10:Sections 10.3-10.7

Ben Heavner May 10, 2007

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.

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

How to Pick xref

xref is orthogonal to si

si . xref = 0

x – xref is orthogonal to li

li . (x – xref) = 0

Finding the Bounded Concentration SpaceExample 1: “Simple reversible reaction”

PCCP

11

11S =

Finding L

Matlab:

EDU>> S=[-1 1; 1 -1]S = -1 1 1 -1

EDU>> b = S'b = -1 1 1 -1

EDU>> a=null(b,'r')a = 1 1

EDU>> L=a'L = 1 1

PCCP

L = (1 1)

11

11S =

Toward Finding xref – start with x

Suppose a1 = 1 Remember Lx = a

PCCP

L = (1 1)

11

11S =

1

1

0

1x

Then one parameterization of x is:

1Lx

10*11*10

111

11*10*11

011

That is, from

or

Finding xref:Systems of Linear Equations

First criteria for xref:

si . xref = 0

or

PCCP

L = (1 1)

11

11S =

1

1

0

1x

0112

1

ref

ref

x

x

(-1*x1ref) + (1*x2ref) = 0

x1ref = x2ref

Finding xref:Systems of Linear Equations

Second criteria for xref:

li . (x – xref) = 0

or

PCCP

L = (1 1)

11

11S =

1

1

0

1x

01122

11

ref

ref

xx

xx

[1*(x1-x1ref)] + [1*(x2-x2ref)] = 0x1-x1ref=-x2+x2ref

x1+x2=2xref

Since (x1+x2) = a = 1x1ref = x2ref = 1/2

Reparamatarizing the Concentration Space: x-xref

Since

1

1

0

1x

x1ref

x 2ref

1212

And

x xref 12

12

11

[0,1] Then

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

Further Transformation Examples and Pool Interpretation

“Bilinear association” (“Bimolecular association” in reaction space):

APA + P

2

1a

1

1

1

S

110

101L

1

1

1

1

1

1

refxx

1

1

1

0

2

1

x

Further Transformation Examples and Pool Interpretation

“Carrier-coupled reaction” (“Cofactor-coupled reaction” in reaction space):

CP + AC + AP

1

1

1

1

43434343

refxx

1

1

1

1

S

1010

0101

1100

0011

L

1

2

2

1

a

More Pool Interpretation “Rodox carrier coupled reactions”:

R + NADH + H+RH2 + NAD+

01010

10010

00110

01001

10001

001012 NADHHRNADRH

L

11

11

11

11

11

S

Redox carrier coupled reactions

R + NADH + H+RH2 + NAD+

L =

RH2 NAD R H NADH

1 0 1 0 0

1 0 0 0 1

1 0 0 1 0

0 1 1 0 0

0 1 0 0 1

0 1 0 1 0

Combining pools

R + NADH + H+RH2 + NAD+

R’R

R’H2 + NAD+R’ + NADH + H+

Combining pools

0001010

0010010

0100110

1001001

1010001

1100101

'' 22 HRRNADHHRNADRH

L

R + NADH + H+RH2 + NAD+

R’R

R’H2 + NAD+R’ + NADH + H+

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