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Lecture #16. The Left Null Space of S. Outline. Definition Convex basis – formation of non-negative pools Alignment of the affine concentration space with LN(S) Three types of pools Examples of extreme pools (Tilting to form a new basis). DEFINITION OF LN(S). The Left Null Space of S. - PowerPoint PPT Presentation
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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