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Use universal reaction stoichiometries to predict network
metabolic capabilities at steady state
Stoichiometry As Vectors
• We can denote the stoichiometry of a reaction by a vector of coefficients
• One coefficient per metabolite– Positive if metabolite is produced– Negative if metabolite is consumed
Example:
Metabolites:
[ A B C D ]T
Reactions:
2A + B -> C
C -> D
Stoichiometry Vectors:
[ -2 -1 1 0 ]T
[ 0 0 -1 1 ]T
The Stoichiometric Matrix (S)
R1 R2 R3 R4 R5 R6 R7 R8 R9 R10
-100-200101
0-10-100010
ABCDEFGHI
ReactionsM
etab
oli
tes
A (Very) Simple System
We have introduced two new things
• Reversible reactions – are represented by two reactions that proceed in each direction (e.g. v4, v5)
• Exchange reactions – allow for fluxes from/into an infinite pool outside the system (e.g. vin and vout). These are frequently the only fluxes experimentally measured.
A
B
D
Cvin
v1
v3
v2
v5
vout
v4
ABCD
v1 v2 v3 v4 v5 vin vout-1001
00-10
1000
-1100
0-110
001-1
00-11
ExchangeReactions
Calculating changes in concentration
ABCD
v1 v2 v3 v4 v5 vin vout-1001
00-10
1000
-1100
0-110
001-1
00-11
A
B
D
Cvin
v1
v3
v2
v5
vout
v4
What happens ifvin is 1 “unit” per
second
1
0
0
0
0
00
0 v10 v20 v30 v40 v51 vin0 vout
=
1000
dA/dtdB/dtdC/dtdD/dt
A grows by 1 “unit”
per “second”
We can calculate this with S
Given these fluxes
These are the changes in metabolite concentration
The Stoichiometric Matrix
dxS V
dt
v1v2v3v4v5v6v7v8v9
v10
v1v2v3v4v5v6v7v8v9
v10
V is a vector of fluxes through each reaction
Then S*V is a vector describing the change in concentration of each metabolite per unit time
R1 R2 R3 R4 R5 R6 R7 R8 R9 R10
-100-200101
0-10-100010
ABCDEFGHI
dA/dtdB/dtdC/dtdD/dtdE/dtdF/dtdG/dtdH/dtdI/dt
dA/dtdB/dtdC/dtdD/dtdE/dtdF/dtdG/dtdH/dtdI/dt
=
Some advantages of S
• Chemistry not Biology: the stoichiometry of a given reaction is preserved across organisms, while the reaction rates may not be preserved
• Does NOT depend on kinetics or reaction rates
• Depends on gene annotations and mapping from gene to reactionsDepends on information we frequently already have
Genes to Reactions
• Expasy enzyme database
• Indexed by EC number
• EC numbers can be assigned to genes by– Blast to known
genes– PFAM domains
Online Metabolic Databases
Pathlogic/BioCyc
Kegg
There are several online databases with curated and/or automated EC number assignments for sequenced genomes
The same reaction can be included as multiple roles (paralogs)
Gene A Gene B Gene C
Examples
Gene D Gene E Gene E’
Enzyme A Enzyme B/C Enyzme D Enzyme E Enzyme E’
Same rxn
What Can We Use S For?
From S we can investigate the metabolic capabilities of the system.
We can determine what combination of fluxes (flux configurations) are possible at
steady state
Flux Configuration, VImagine we have another simple system:
v1v2v3
VFlux
Configuration
A Bv1
C Dv3v2
flux v1
flux v2
flux v3
1
2
3
4
1
1 2 3
Rate of change of C to D per
unit time113
VFlux
Configuration
1 1 3
We want to know what regionof this space contains feasible
fluxes given our constraints
The Steady State Constraint
• We have
• But also recall that at steady state, metabolite concentrations are constant: dx/dt=0
dxS V
dt
0dx
S Vdt
Positive Flux Constraint
0 0, 1..i
dxS V v i n
dt
*recall that reversible reactions are represented by two unidirectional fluxes
*
All steady state flux vectors, V, must satisfy these constraints
What region do these V live in?
The solution through convex analysis
flux v1flux v2
flux v3
?
The Flux Cone
• Every steady state flux vector is inside this cone
• Edges of the cone are circumscribed by Extreme Pathways
v
p1
p2 p3
p4
flux v1
flux v2
flux v3
Solution is a convex flux cone
At steady state,the organism “lives in” here
Extreme Pathways
Extreme pathways are “fundamental modes” of the metabolic system at steady
state
0i i ii
V p
They are steady state flux vectors
All other steady state flux vectors are non-negative linear combinations
v
p1
p2 p3
p4
flux v1
flux v2
flux v3
Example Extreme Patways
A
B
D
Cvin
v1
v3
v2
v4
vout
ABCD
v1 v2 v3 v4 vin vout-1001
00-10
1000
-1100
0-110
001-1
b1110011
001111
b2v1v2v3v4vinvout
All steady state fluxes configurations are combinations of these extreme pathways
A
B
D
Cvin
v1
v3
v2
v4
voutb1
b2
1
1 1
1
1
1 1
1
Capping the Solution Space• Cone is open ended, but no reaction can have infinite flux
• Often one can estimate constraints on transfer fluxes – Max glucose uptake measured at maximum growth rate– Max oxygen uptake based on diffusivity equation
• Flux constraints result in constraints on extreme pathways
p1
p2
p3
p4
flux v1
flux v2
flux v3
p1
p2 p3
p4
max flux v3
flux v1
flux v2
flux v3
The Constrained Flux Cone
p1
p2
p3
p4
flux v1
flux v2
flux v3
• Contains all achievable flux distributions given the constraints:– Stoichiometry– Reversibility– Max and Min Fluxes
• Only requires:– Annotation – Stoichiometry– Small number of flux
constraints (small relative to number of reactions)
Selecting One Flux Distribution
p1
p2
p3
p4
flux v1
flux v2
flux v3 • At any one point in time,
organisms have a single flux configuration
• How do we select one flux configuration?
We will assume organisms are trying to maximize a “fitness” function that is a function of fluxes, F(v)?
Optimizing A Fitness Function
p1
p2
p3
p4
NADH->NADPH(v1)AMP->ADP
(v2)
ADP->ATP(v3)Imagine we are trying to optimize ATP
production
Then a reasonable choice for the fitness function is F(V)=v3
Goal: find a flux in the cone that maximizes v3
7
5
If we choose F(v) to be a linear function of V:
The optimizing flux will always lie on vertex or edge of the coneLinear Programming
( ) i ii
F v v
Flux Balance Analysis
p1
p2
p3
p4
flux v1
flux v2
flux v3 Start with stoichiometric matrix and constraints:
Choose a linear function of fluxes to optimize:
Use linear programming we can find a feasible steady state flux