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Introduction to Steady State Metabolic Modeling
ConceptsFlux Balance Analysis
ApplicationsPredicting knockout phenotypesQuantitative Flux Prediction
Lab PracticalFlux Balance Analysis of E. coli central metabolismPredicting knockout phenotypesModeling M. tuberculosis Mycolic Acid Biosynthesis
Why Model Metabolism?
• Predict the effects of drugs on metabolism– e.g. what genes should be disrupted to prevent
mycolic acid synthesis
• Many infectious disease processes involve microbial metabolic changes– e.g. switch from sugar to fatty acid metabolism in
TB in macrophages
Genome Wide View of Metabolism
Streptococcuspneumoniae
• Explore capabilities of global network• How do we go from a pretty picture to
a model we can manipulate?
Metabolic Pathways
Metabolitesglucose
Enzymesphosphofructokinase
Reactions & Stoichiometry1 F6P => 1 FBP
Kinetics
Regulationgene regulation
metabolite regulation
hexokinase
phosphoglucoisomerase
phosphofructokinase
aldolase
triosephosphate isomerase
G3P dehydrogenase
phosphoglycerate kinase
phosphoglycerate mutase
enolase
pyruvate kinase
Metabolic Modeling: The Dream
up
take
Steady State Assumptions
• Dynamics are transient• At appropriate time-
scales and conditions, metabolism is in steady state
A Buptake conversion secretion
Two key implications1. Fluxes are roughly
constant2. Internal metabolite
concentrations are constant
constant
0
uptake
d A
dt
t Co
nv
ers
ion
t
Steady state
Metabolic Flux
Output fluxes
Input fluxes
Volume of pool of water =
metaboliteconcentration
Slide Credit: Jeremy Zucker
Reaction Stoichiometries Are Universal
The conversion of glucose to glucose 6-phosphate always follows this stoichiometry :
1ATP + 1glucose = 1ADP + 1glucose 6-phosphate
This is chemistry not biology.
Biology => the enzymes catalyzing the reaction
Enzymes influence rates and kinetics• Activation energy• Substrate affinity• Rate constants
Not required for steady state modeling!
Metabolic Flux Analysis
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
From Genomes to the S Matrix
ABCDEFGHI
R1 R2 R3 R4 R5 R6 R7 R8 R9 R10
-100-200101
0-10-100010
Columns encode reactions
Relationships btw genes and rxns-1 gene 1 rxn-1 gene 1+ rxns-1+ genes 1 rxn
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
configuration that maximizes the F
( ) i ii
F v v
0 0,i j
dxS V v v
dt
(all i) (some j)
Optimizing E. coli Growth
Z = 41.257vATP - 3.547vNADH + 18.225vNADPH + 0.205vG6P + 0.0709vF6P +0.8977vR5P + 0.361vE4P + 0.129vT3P + 1.496v3PG + 0.5191vPEP +2.8328vPYR + 3.7478vAcCoA + 1.7867vOAA + 1.0789vAKG
Metabolite (mmol)
ATP 41.257
NADH -3.547
NADPH 18.225
G6P 0.205
F6P 0.0709
R5P 0.8977
E4P 0.361
T3P 0.129
3PG 1.496
PEP 0.5191
PYR 2.8328
AcCoA 3.7478
OAA 1.7867
AKG 1.0789
For one gram of E. coli biomass, you need this ratio of metabolites
Assuming a matched balanced set of metabolite fluxes, you can formulate this objective function
FBA Overview
p1
p2
p3
p4
flux v1
flux v2
flux v3
p1
p2
p3
p4
flux v1
flux v2
flux v3
p1
p2
p3
p4
flux v1
flux v2
flux v3
p1
p2
p3
p4
flux v1
flux v2
flux v3
Stoichiometric MatrixGene annotation
Enzyme and reaction catalog
Feasible SpaceS*v=0
Add constraints:vi>0
i>vi>i
Optimal FluxGrowth objective
Z=c*v
Solve with linear programming
Flux solution
Applications
• Knockout Phenotype Prediction
• Quantitative Flux Predictions
in silico Deletion Analysis
Can we predict gene knockout phenotype based on their simulated
effects on metabolism?
Q: Why, given other computational methods exist?(e.g. protein/protein interaction map connectivity)
A: Other methods do not directly consider metabolic flux or specific metabolic conditions
in silico Deletion Analysis
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
“wild-type”
“mutant”
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
Gene knockouts modeled by removing a reaction
Mutations Restrict Feasible Space
• Removing genes removes certain extreme pathways
• Feasible space is constrained
• If original optimal flux is outside new space, new optimal flux is predicted
Calculate new optimal flux
No change in optimal flux
Difference of F(V) (i.e growth) between optima is a measure of the KO
phenotype
Mutant Phenotypes in E. coli
PNAS| May 9, 2000 | vol. 97 | no. 10
Model of E. coli central metabolism436 metabolites720 reactions
Simulated mutants in glycolysis, pentose phosphate, TCA, electron transport
Edward & Palsson (2000) PNAS
E. coli KO simulation resultsIf Zmutant/Z =0, mutant is no growth (-)
growth (+) otherwise
Compare to experiment(in vivo / prediction)86% agree
Used FBA to predict optimal growth of mutants (Zmutant) versus non-mutant (Z)
Simulated growth on glucose, galactose, succinate, and acetate
“lethal”
“reduced growth”
Edward & Palsson (2000) PNAS
in vivo
in silico
+ -
+ 36 2
- 9 32
Condition specific prediction
What do the errors tell us?
• Errors indicate gaps in model or knowledge
• Authors discuss 7 errors in prediction– fba mutants inhibit stable RNA synthesis (not
modeled by FBA)– tpi mutants produce toxic intermediate (not
modeled by FBA)– 5 cases due to possible regulatory
mechanisms (aceEF, eno, pfk, ppc)
Edward & Palsson (2000) PNAS
Quantitative Flux Prediction
• Measure some fluxes, say A & B– Controlled uptake rates
• Predict optimal flux of A given B as input to model
Can models quantitatively predict fluxes and/or growth rate?
NATURE BIOTECHNOLOGY VOL 19 FEBRUARY 2001
Growth vs Uptake Fluxes
Predict relationship between
- growth rate - oxygen uptake - acetate or succinate
uptake
Compare to experiments from batch reactors
Edwards et al. (2001) Nat. Biotech.
Acetate Uptake
Oxy
gen
Up
take
Oxy
gen
Up
take
Succinate Uptake
Uptake vs Growth
Specify glucose uptake
Predict
- growth
- oxygen uptake
- acetate secretion
Compare to chemostat experiment
Varma et al. (1994) Appl. Environ. Microbiol.
Just An Intro!
Lecture drawn heavily from
Palsson Lab work
http://gcrg.ucsd.edufor more info
Price, Reed, & Palsson (2004)Nature Rev. Microbiology
Interpreting Array Data in Metabolic Context
Clustering,GSEA
Kegg,PathwayExplorer
Expression to Flux?
Resources
• Tools and Databases– Kegg
– BioCyc
– PathwayExplorer (pathwayexplorer.genome.tugraz.at)
• Metabolic Modeling– Palsson’s group at UCSD (http://gcrg.ucsd.edu/)
– www.systems-biology.org
– Biomodels database (www.ebi.ac.uk/biomodels/)
– JWS Model Database (jjj.biochem.sun.ac.za/database/index.html)
CellNetAnalyzer
Tomorrow’s Lab
• Tools– CellNetAnalyzer– Flux Balance Analysis
• Applications– Predict effects of gene knockouts– Central metabolism– TB mycolic acid biosynthesis
Recommended