Lecture #16

The Left Null Space of S

Outline

1. Definition2. Convex basis – formation of non-

negative pools3. Alignment of the affine

concentration space with LN(S)4. Three types of pools5. Examples of extreme pools6. (Tilting to form a new basis)

DEFINITION OF LN(S)

The Left Null Space of S

S•R=0

P , pij ≥0

( )( )=0

calculating convex basis:

LS=0(LS)T=0STLT=0

ST( )=0use ExPaprogram

L•S=0( )( )=0

lij ≥0 convex basis

reaction column sj

row vectors li

<lisj>=0

C(S)

LN(S)

x’

Dynamic mass balance

Multiply with L from the left

( )()( ) ( )

linear combination of the concentrations that always add up to ai

time invariants (pools)

want a set of basis vectorsli, where lik≥0 convex set

v2

v1N(S)R(S)

S

C(S)

LN(S)

dx1/dt

dx2/dt

S(•)dt

x2

x1

a1

a1

Keq line

The affine concentration space

ALIGNMENT OF THE LN(S) AND THE AFFINE CONCENTRATION SPACE

Finding a reference point

A reference state that aligns the affine concentration space with the null space

DEFINITION OF EXTREME POOLS

Refresher on compound maps

Definition of extreme pools

• Type A pools that are composed only of the primary compounds;

• Type B pools that contain both primary and secondary compounds internal to the system; and

• Type C pools are comprised only of secondary compounds.

• Type B pools generally represent the conserved moieties (or currencies) that are exchanged from one compound to another, such as a hydroxyl or phosphate group.

Analogy to extreme pathways

Classification of pools based on the structure of the matrix L

EXAMPLES OF EXTREME POOLS

The bi-linear reaction

• A+B ->AB• The pools are

– A+AB (x1+x3)

– B+AB (x1+x3)

• Very clear conservations

The exchange reaction:AP+C -> CP+A ; x=(CP,C,AP,A)

• The first pool is a conservation of the primary substrate pool C (=C+CP) and is a Type A pool.

• The second pool is a conservation of the cofactor A (=A+AP) and is a Type C pool.

• The third pool is a conservation of the phosphorylated compounds (=CP+AP) and represents the total energy inventory, or occupancy in the system.

• The last pool is a, vacancy pool (C+A) that represents the low energy state of the participating compounds. This pool is linearly redundant but convexly independent.

The exchange reaction:AP+C -> CP+A ; x=(CP,C,AP,A)

Simple redox exchange

Definition of Extreme Redox Pools

RH2 R

NAD+ NADH

H+

R’ R’H2

NADH NAD+

v1

NAD+

NADHH+

RRH2

v3

v2

R’R’H2

v1

v2

v3

Reaction map Compound map

RH2 R

NAD+ NADH

H+

R’ R’H2

NADH NAD+

v1

v2

v3

RH2 R

NAD+ NADH

H+

R’ R’H2

NADH NAD+

v1

v2

v3

RH2 R

NAD+ NADH

H+

R’ R’H2

NADH NAD+

v1

v2

v3

RH2 R

NAD+ NADH

H+

R’ R’H2

NADH NAD+

v1

v2

v3

RH2 R

NAD+ NADH

H+

R’ R’H2

NADH NAD+

v1

v2

v3

RH2 R

NAD+ NADH

H+

R’ R’H2

NADH NAD+

v1

v2

v3

v1

NAD+

NADHH+

RRH2

v2

R’R’H2

v1

NAD+

NADHH+

RRH2

v3

v2

R’R’H2

v1

NAD+

NADHH+

RRH2

v3

v2

R’R’H2

v1

NAD+

NADHH+

RRH2

v3

v2

R’R’H2

v1

NAD+

NADHH+

RRH2

v3

v2

R’R’H2

v1

NAD+

NADHH+

RRH2

v3

v2

R’R’H2

v3

Pool mapPool #1 (A)

Pool #3 (B)

Pool #5 (B)

Pool #2 (B)

Pool #4 (B)

Pool #6 (B)

#1 #2

#3 #4

#5 #6

Linked Redox Pools

Skeleton glycolysis

Interpretation of glycolytic pools

• l1, total carbon pool

• l2, high-energy conservation pool: – 2C6 + 3C6P + 4C6P2 + 2C3P1 + 2C3P2 + C3P + AP3

• l3, conservation of elemental P: – C6P + 2C6P2 + C3P1 + 2C3P2 + C3P + AP3 + P$

• l4, low-energy conservation pool:– 2C6 + C6P + C3P + 2C3 + AP2

• l5, potential to incorporate the stand-alone moiety P; – C3P2 + C3P + C3 + P

• l6, total carrier pool of A

Interpretation of TCA pools• l1 exchanging carbon group

– 2H2C2 + 2H2C6 + HC5 + C

• l2, recycled four-carbon moiety which `carries' the two carbon group that is oxidized

– C4 + H2C6 + HC5

• l3 , hydrogen group that contains the redox inventory in the system

– 2H2C2 + 2H2C6 + HC5 + NH

• l4 , redox vacancy – C + N

• l5 , total cofactor pool – N + NH

Other Examples

Glycolysis as an open system:many

conserved pools

disappear

The stoichiometric matrix and the left null space vectors

GENOME-SCALE STUDIES

iJR904

• Developed Minimal Conserved Pool Identification (MCPI) approach– Elucidating the conserved

pools for target metabolites without computing the entire basis conservation relationships.

– MILP formulationBiophys J, 88: 37-49 (2005)

Conserved pools spanning central metabolism of iJR904

Rotating the bases vectors of LN(S) for iAF1260

• The LN(S) basis vectors correspond to time invariant pools

• The pools found are:– Amino acyl tRNAs – tRNAs– Charge Carriers (NADH.

NAD)– Co-factor Pools– Apolipoprotein-lipoprotein

Fac

tor

Lo

adin

g

Summary

• The left null of S contains time invariant pools

• A convex basis can be found for LN(S)• Good basis can be found by tilting methods• Examples show the formation of meaningful

pools• The LN(S) has not been extensively studied