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Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Flux Balance Analysis: Using MathProgramming to Model Metabolism
Paul Brooks Jill Hardin
Department of Statistical Sciences and Operations Research
Virginia Commonwealth University
BNFO 691 November 28, 2006
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Outline
1 Introduction to FBAAssumptions of FBAUses of FBA
2 Math Programming and FBAMaximization of Biomass ProductionGene Addition/Deletion Studies
3 Strengths/Weaknesses of FBA
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Assumptions of FBAUses of FBA
Metabolism
A
B
C
A
D
B
D
C
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Assumptions of FBAUses of FBA
Metabolism
A Varma, BW Boesch, and BO Palsson. Biochemical Production Capabilities of Escherichia coli. Biotechnology and
Bioengineering 42:59-73, 1993.
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Assumptions of FBAUses of FBA
Outline
1 Introduction to FBAAssumptions of FBAUses of FBA
2 Math Programming and FBAMaximization of Biomass ProductionGene Addition/Deletion Studies
3 Strengths/Weaknesses of FBA
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Assumptions of FBAUses of FBA
Assumptions of FBA
Cells make optimal use of scarce resources (metabolites,proteins, enzymes) in functioning to serve the needs of theorganism. Functions include production of biomass,production of energy, and promotion of cell growth.
The quantity of metabolites in a cell is constant over time;cells are in a steady-state.
Flux through a reaction is constrained only bystoichiometric considerations.
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Assumptions of FBAUses of FBA
Assumptions of FBA
Cells make optimal use of scarce resources (metabolites,proteins, enzymes) in functioning to serve the needs of theorganism. Functions include production of biomass,production of energy, and promotion of cell growth.
The quantity of metabolites in a cell is constant over time;cells are in a steady-state.
Flux through a reaction is constrained only bystoichiometric considerations.
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Assumptions of FBAUses of FBA
Assumptions of FBA
Cells make optimal use of scarce resources (metabolites,proteins, enzymes) in functioning to serve the needs of theorganism. Functions include production of biomass,production of energy, and promotion of cell growth.
The quantity of metabolites in a cell is constant over time;cells are in a steady-state.
Flux through a reaction is constrained only bystoichiometric considerations.
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Assumptions of FBAUses of FBA
Outline
1 Introduction to FBAAssumptions of FBAUses of FBA
2 Math Programming and FBAMaximization of Biomass ProductionGene Addition/Deletion Studies
3 Strengths/Weaknesses of FBA
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Assumptions of FBAUses of FBA
Uses of FBA
Predict a cell’s response to a changing environment.
Determine an upper bound on the production capacity of acell.
Determine which reactions should be targeted formetabolic engineering.
Explore the tradeoff (i.e., differing fluxes through reactions)for different objectives (i.e., biomass production versuspromotion of cell growth).
Determine which genes are necessary for maintaining agiven level of production.
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Outline
1 Introduction to FBAAssumptions of FBAUses of FBA
2 Math Programming and FBAMaximization of Biomass ProductionGene Addition/Deletion Studies
3 Strengths/Weaknesses of FBA
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Objective in Words
Decide the flux through each reaction in the network ofmetabolism reactions so that the production of biomass ismaximized subject to the following constraints:
Conservation of mass (stoichiometric coefficients).
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Objective in Words
Decide the flux through each reaction in the network ofmetabolism reactions so that the production of biomass ismaximized subject to the following constraints:
Conservation of mass (stoichiometric coefficients).
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Objective in Words
Decide the flux through each reaction in the network ofmetabolism reactions so that the production of biomass ismaximized subject to the following constraints:
Conservation of mass (stoichiometric coefficients).
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Objective in Words
Decide the flux through each reaction in the network ofmetabolism reactions so that the production of biomass ismaximized subject to the following constraints:
Conservation of mass (stoichiometric coefficients).
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Linear Programming Formulation for FBA
Variables:vj = flux through reactionj
max∑
j∈M
cjvj = biomass/ATP/amino acid/growth
subject to∑
j∈MSijvj = bi ∀ i ∈ N
vj ≥ 0 ∀ j ∈ M
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Linear Programming Formulation for FBA
Variables:vj = flux through reactionj
max∑
j∈M
cjvj = biomass/ATP/amino acid/growth
subject to∑
j∈MSijvj = bi ∀ i ∈ N
vj ≥ 0 ∀ j ∈ M
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Linear Programming Formulation for FBA
Variables:vj = flux through reactionj
max∑
j∈M
cjvj = biomass/ATP/amino acid/growth
subject to∑
j∈MSijvj = bi ∀ i ∈ N
vj ≥ 0 ∀ j ∈ M
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Linear Programming Formulation for FBA
Variables:vj = flux through reactionj
max∑
j∈M
cjvj = biomass/ATP/amino acid/growth
subject to∑
j∈MSijvj = bi ∀ i ∈ N
vj ≥ 0 ∀ j ∈ M
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Stoichiometric Constraints
A + 2B → CC → A + D
B + D → 2C
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Stoichiometric Constraints
A + 2Bv1→ C
Cv2→ A + D
B + Dv3→ 2C
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Stoichiometric Constraints
A + 2Bv1→ C
Cv2→ A + D
B + Dv3→ 2C
− v1 + v2 = bA
− 2v1 − v3 = bB
v1 − v2 + 2v3 = bCv2 − v3 = bD
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Stoichiometric Constraints
A + 2Bv1→ C
Cv2→ A + D
B + Dv3→ 2C
− v1 + v2 = bA
− 2v1 − v3 = bB
v1 − v2 + 2v3 = bCv2 − v3 = bD
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Stoichiometric Constraints
A + 2Bv1→ C
Cv2→ A + D
B + Dv3→ 2C
− v1 + v2 = bA
− 2v1 − v3 = bB
v1 − v2 + 2v3 = bCv2 − v3 = bD
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Stoichiometric Constraints
A + 2Bv1→ C
Cv2→ A + D
B + Dv3→ 2C
− v1 + v2 = bA
− 2v1 − v3 = bB
v1 − v2 + 2v3 = bCv2 − v3 = bD
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Outline
1 Introduction to FBAAssumptions of FBAUses of FBA
2 Math Programming and FBAMaximization of Biomass ProductionGene Addition/Deletion Studies
3 Strengths/Weaknesses of FBA
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Objective in Words
Decide the flux through each reaction in the network ofmetabolism reactions, which genes are added, and whichgenes are deleted so that the number of genes activated isminimized subject to the following constraints:
Conservation of mass (stoichiometric coefficients).
Biomass production is at least P% of theoretical maximum.
If flux through a reaction is non-zero, then the genesencoding the enzymes of that reaction must be functional.
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Objective in Words
Decide the flux through each reaction in the network ofmetabolism reactions, which genes are added, and whichgenes are deleted so that the number of genes activated isminimized subject to the following constraints:
Conservation of mass (stoichiometric coefficients).
Biomass production is at least P% of theoretical maximum.
If flux through a reaction is non-zero, then the genesencoding the enzymes of that reaction must be functional.
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Objective in Words
Decide the flux through each reaction in the network ofmetabolism reactions, which genes are added, and whichgenes are deleted so that the number of genes activated isminimized subject to the following constraints:
Conservation of mass (stoichiometric coefficients).
Biomass production is at least P% of theoretical maximum.
If flux through a reaction is non-zero, then the genesencoding the enzymes of that reaction must be functional.
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Objective in Words
Decide the flux through each reaction in the network ofmetabolism reactions, which genes are added, and whichgenes are deleted so that the number of genes activated isminimized subject to the following constraints:
Conservation of mass (stoichiometric coefficients).
Biomass production is at least P% of theoretical maximum.
If flux through a reaction is non-zero, then the genesencoding the enzymes of that reaction must be functional.
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Objective in Words
Decide the flux through each reaction in the network ofmetabolism reactions, which genes are added, and whichgenes are deleted so that the number of genes activated isminimized subject to the following constraints:
Conservation of mass (stoichiometric coefficients).
Biomass production is at least P% of theoretical maximum.
If flux through a reaction is non-zero, then the genesencoding the enzymes of that reaction must be functional.
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Objective in Words
Decide the flux through each reaction in the network ofmetabolism reactions, which genes are added, and whichgenes are deleted so that the number of genes activated isminimized subject to the following constraints:
Conservation of mass (stoichiometric coefficients).
Biomass production is at least P% of theoretical maximum.
If flux through a reaction is non-zero, then the genesencoding the enzymes of that reaction must be functional.
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Variables
vj = flux through reactionj
yk =
{
1 if gene encoding enzymek is present0 otherwise
AP Burgard and CD Maranas. Probing the performance limits of the Escherichia coli metabolic network subject to
gene additions or deletions. Biotechnology and Bioengineering 74:364-375, 2001.
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Integer Programming Formulation for GeneAddition/Deletion
min∑
k∈G
yk
subject to∑
j∈MSijvj = bi ∀ i ∈ N
∑
j∈Mcjvj ≥ Pvmax
vj ≤ V∑
k∈Gajkyk ∀ j ∈ M
vj ≥ 0 ∀ j ∈ Myk ∈ {0, 1} ∀k ∈ G
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Integer Programming Formulation for GeneAddition/Deletion
min∑
k∈G
yk
subject to∑
j∈MSijvj = bi ∀ i ∈ N
∑
j∈Mcjvj ≥ Pvmax
vj ≤ V∑
k∈Gajkyk ∀ j ∈ M
vj ≥ 0 ∀ j ∈ Myk ∈ {0, 1} ∀k ∈ G
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Integer Programming Formulation for GeneAddition/Deletion
min∑
k∈G
yk
subject to∑
j∈MSijvj = bi ∀ i ∈ N
∑
j∈Mcjvj ≥ Pvmax
vj ≤ V∑
k∈Gajkyk ∀ j ∈ M
vj ≥ 0 ∀ j ∈ Myk ∈ {0, 1} ∀k ∈ G
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Integer Programming Formulation for GeneAddition/Deletion
min∑
k∈G
yk
subject to∑
j∈MSijvj = bi ∀ i ∈ N
∑
j∈Mcjvj ≥ Pvmax
vj ≤ V∑
k∈Gajkyk ∀ j ∈ M
vj ≥ 0 ∀ j ∈ Myk ∈ {0, 1} ∀k ∈ G
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Integer Programming Formulation for GeneAddition/Deletion
min∑
k∈G
yk
subject to∑
j∈MSijvj = bi ∀ i ∈ N
∑
j∈Mcjvj ≥ Pvmax
vj ≤ V∑
k∈Gajkyk ∀ j ∈ M
vj ≥ 0 ∀ j ∈ Myk ∈ {0, 1} ∀k ∈ G
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Maximization of Biomass ProductionGene Addition/Deletion Studies
Integer Programming Formulation for GeneAddition/Deletion
min∑
k∈G
yk
subject to∑
j∈MSijvj = bi ∀ i ∈ N
∑
j∈Mcjvj ≥ Pvmax
vj ≤ V∑
k∈Gajkyk ∀ j ∈ M
vj ≥ 0 ∀ j ∈ Myk ∈ {0, 1} ∀k ∈ G
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
university-logo
Introduction to FBAMath Programming and FBA
Strengths/Weaknesses of FBA
Strengths and Weaknesses of FBA
Strengths
Conservation of mass is a known, fixed constraint.
FBA gives an upper bound on the production capacity of acell.
Weaknesses
How good is the upper bound?
Free energy constraints are generally not considered.
The linear programs have alternate optimal solutions; howdo we decide between them?
Paul Brooks, Jill Hardin Flux Balance Analysis and Math Programming
