Mathematical Representation of Reconstructed Networks The Left Null space The Row and column spaces...
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Mathematical Representation of Reconstructed Networks
The Left Null space The Row and column spaces of S
Introduction
The system biology paradigm:
“components network in silico models phenotype”
System biology focuses on the nature of the links and their associated functional states (~phenotypes).
Cells must select useful ‘functional states’.
Functional states
E coli:
1. Outside the body (low O2, low temp)
2. Inside the body (high temp)
3. Stomach (Low PH)
4. Small intestine (low O2)
H pylori:
1. Human gastric (Low PH)
Reaction / links ‘key properties’
1. Stoichiometry. The stoichiometry is fixed, invariant between organisms for the same reactions and condition independent (pressure, pH, temp ..)
2. Relative rates. Fixed by basic thermodynamic properties which depend on conditions such as pressure, pH, temp ..
3. Absolute rates. In contrast to stoichiometry and thermodynamics, highly manipulated by the cell enzymes.
Reminder The dynamic mass balance equation:
Where,X is a vector of m metabolites.V represent vector of n ‘reaction rates’.S is m x n matrix of stoichiometric coefficients, rows represent metabolites and
columns represent reactions.
The right null space
1. For most practical purposes metabolism is in a steady state.2. The null space contains all the steady state flux distributions and is thus of
special importance to us.
0SV
dt
dXSV
nnsvsvsvXdt
d ..2211
Reminder
Constraint-based analysis
Linear programming
(Simplex)
• Each one of the generating vectors corresponds to an extreme pathway which the cell could theoretically control to reach every point in the flux cone.• A particular point within this flux cone corresponds to a given flux distribution which represents a particular metabolic phenotype.
• The analysis of the left null space of S allow us to define the achievable states of the cell and their physiological relevance.
• We look for ‘metabolic pools’ that have physiological meaningful interpretation.
Definition: The Left Null space of S
],1[
0,
rmi
sl ji
0..0
:
0..0
|||
..
|||
__
:
__
21
1
n
rm
sss
l
l
li
Span the left null space of
S
sj
All are in the column space.
(rank= r)
0LS
The Time invariants
A linear combination of individual metabolic concentrations that do not change over time is called a metabolic pool.
A dynamic motion along a reaction vector in the column space do not change the total mass in the pool.
00 Lxdt
d
dt
dxli
The concentrations space
a is a vector that gives the total concentrations of the pools. i.e.
is the conservation vector. The rows of L ( i.e. ) that span the left null space
define a concentration space. The time invariant metabolic pools resides in this
concentration space. defines an affine hyper plane. This plane does not go through the origin. The concentration vector x resides in this space.
aLx
aLx
ii axl
sli 'il
Classifying the pools
Co-Factors,
carriers
e.g. the carbon backbone in glycolysis
Reaction map Vs. Compound map
Groupings of chemical elements that move together.
TSS
Classifying the pools
Futile cycle
Internal cycle
Through flux pathways
Cofactor conservation
Primary& secondary moieties conservation
Primary moieties conservation
Simple reversible state
PAAP
11,1
11
lLS
PAAP
One Type A Pool:
Comment: The pool ‘AP+PA’ is constant both in SS and dynamic.
Reference states
We can choose that lie in the left null space. This reference state is orthogonal to
x is not orthogonal to the left null space, whereas
and are.
Now we can span the concentration space using the reaction vectors
aLx
0)( refxxL
dt
dx
)( refxx
is
refxx is
2/1
2/1
011
011
,2
,1
,22
,11
,2
,1
ref
ref
ref
ref
ref
ref
X
X
XX
XX
X
X)1(
)2(
Two conditions:
Bilinear association
APPA
The Pools Interpretation Type
1 .A+AP : Total cofactor : A
2 .P+AP : Total energy : B
110
101,
1
1
1
LS
Ordered by: A, P, AP
Carrier coupled reactionACPAPC
The pools:
1. C+CP : conservation of the substrate C.
2. A+AP : conservation of the cofactor A.
3. CP+AP : occupancy of P / total energy
4. C+A : vacancy of P / low energy state
1010
0101
1100
0011
,
1
1
1
1
Ls
The entries of x ordered by: (CP, C, AP, A)
Redox carrier coupled reaction
01010
10010
00110
01001
10001
00101
,
1
1
1
1
1
2
LS
HNADHRNADRH
The pools The pools interpretation Type
1. : Total R. : A
2. : Redox occupancy 1. : B
3. : Redox occupancy 2. : B
4. : Redox vacancy : B
5. : Total redox carrier : C
6. :Total redox carrier : C
HNAD
NADHNAD
RNAD
HRH
NADHRH
RRH
2
2
2
Multiple redox coupled reaction
NADHR
v
v
HNADHR
R
v
v
R
HNADHR
v
v
NADRH
2
6
5
4
3
2
1
2
''
'
The pools The pools interpretation Type
1. : Total R. : A
2. : Redox occupancy 1. : B
3. : Redox occupancy 2. : B
4. : Redox vacancy : B
5. : Total redox carrier 1 : C
6. :Total redox carrier 2 : C
HNAD
NADHNAD
RRNAD
HRHRH
HRNADHRH
HRRRRH
'
'
'
''
22
22
22
(1)
(2)
(3)
Glycolysis
PCPCPC
APCPCPCC
PAPPCPCPCPCPC
APPCPCPCPCPCC
3323
23366
332313266
3323132666
22
22
22432Type B pools:
High energy
Conservation of P
Low energy
Stand alone inorganic P
TCA cycle
NHN
NC
NHHCCHCH
HCCHC
CHCCHCH
56222
5624
56222
22
22 • Exchanging carbon group.
• Recycled C4 moiety which ‘carries’ the two carbon group that is oxidized.
• H group that contains the redox inventory in the system.
• Redox vacancy.
• Total cofactor pool.
Summary: Left Null space of S
Contains dynamic invariants. A convex basis for this space is biological meaningful and can be
found. Three basic types of convex basis vectors can be defined. The metabolic pools can be displayed on the compound map –
similar to pathways in a flux map. Integration of time derivatives leads to bounded affine space of
concentrations.
The affine space of concentrations1. All the concentrations states, dynamic and steady, lie in this space.2. A suitable reference state can be defined (parallel to the left null
space and orthogonal to the column space). The shifted concentration space is spanned by the Si’s.
The column space
Contain the time derivatives of the concentrations. Spanned by the reaction vectors. Change in the flux levels determine the location of
in the column space. Fast reactions that quickly come to SS reduce the
column space dimension on slower time scales. Reduction in the columns space dimension leads to
effective additional dimension in the left null space. Constraints on the fluxes induce constraints on the
‘s. Hence the column space is a closed space.
nnsvsvsvxdt
d ..2211
dt
dx
dt
dx
Example 1 OHOOH 2222 22
22OH 2O OH 2
O 2 2 1
H 2 0 2
1
222
2222
2222
2
2
22
2
1
0
0
9
22
22
22
202
122
2121
v
H
O
R
dt
d
OHOH
OHOOH
OHOOH
dt
d
OH
O
OH
dt
dx
dt
d
l
l
sT
The left null space will be spanned by the elemental matrix
LS = ES = 0
R is a group of concentrations changing over time
Example 2
3
2
1
21 X
v
v
XX
21
2
1
32
31
321
0
0
4
VV
l
l
R
dt
d
XX
XX
XXX
dt
d
110
101
111
2
1
1
l
l
sT
Example 3
AMPATPADP
ATPADP
2
(Ignore the P for a moment)
A
R
R
dt
d
AMPADPATP
AMPADPATP
ADPATP
AMP
ADP
ATP
dt
d2
1
2
111
121
011
Note: If one reaction is fast compare to the other, we get ‘L’ shape
The row space
The row space contain all the thermodynamic driving forces (i.e. fluxes).
The individual reaction fluxes form an orthogonal basis for the raw space.
Each reaction has a natural thermodynamic basis vector. Since the fluxes are constrained, All the fluxes are in a rectangle
in the positive orthant. The null space lies within the rectangle and its orthogonal
complement is the row space.
Constraints on the flux values The magnitude of the individual fluxes is constrained. These constraints are derived from:1. The limitation on the concentration.2. Upper limit on the kinetic constants. The turnover rate of an enzyme complex X:
Where the total amount of enzyme ( ) present is limited to X + e. Bilinear association of substrate to an enzyme:The rate is:
Where is the size of the most limiting conservation pool of which Xi is a member.
The total amount of enzyme (alone) limits the flux through enzymatic pathway.
The release step of a product from enzyme is often the rate limiting step in enzyme catalysis.
totalekeXkXkV 111 )(
totale
totalibib eakeXkV
ia
Thermodynamic driving forces If the fluxes are imbalanced, there will be a net generation or
elimination of compounds in the network.
Since the r’s are fixed and the V’s are bounded, the inner product is also bounded.
)cos(, iiii VrVr
dt
dX
The column space is naturally spanned by the reaction vectors.
The row space can be represented by an orthogonal basis formed by the individual fluxes with values only in the positive orthant.
The magnitude of the individual fluxes is limited by kinetics and caps on concentration values.
This limitation also limits the possible value of the time derivatives and thus the column space.
The column and row spaces are closed.
Summary: The row and column spaces of S
Thanks