Constraint-Based Modeling of Metabolic Networks based on:

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Constraint-Based Modeling of Metabolic Networks
based on Genome-scale models of microbial
cellsEvaluating the consequences of
constraints, Price, et. al (2004)
Tomer Shlomi School of Computer Science, Tel-Aviv
University, Tel-Aviv, Israel January, 2006
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Outline

• Metabolism and metabolic networks
• Kinetic models vs. constraints-based modeling
• Flux Balance Analysis
• Exploring the solution space
• Altering phenotypic potential gene knockouts

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Cellular Metabolism

• The essence of life..
• Catabolism and anabolism
• The metabolic core production of energy
  anaerobic and aerobic metabolism
• Probably the best understood of all cellular
  networks metabolic, PPI, regulatory, signaling
• Tremendous importance in Medicine antibiotics,
  metabolic disorders, liver disorders, heart
  disorders
• Bioengineering efficient production of
  biological products.

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Metabolites and Biochemical Reactions

• Metabolite an organic substance, e.g. glucose,
  oxygen
• Biochemical reaction the process in which two or
  more molecules (reactants) interact, usually with
  the help of an enzyme, and produce a product

Glucose ATP
Glucokinase Glucose-6-Phosphate ADP
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Kinetic Models

• Dynamics of metabolic behavior over time
• Metabolite concentrations
• Enzyme concentrations
• Enzyme activity rate depends on enzyme
  concentrations and metabolite concentrations
• Solved using a set of differential equations
• Impossible to model large-scale networks
• Requires specific enzyme rates data
• Too complicated

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Constraint Based Modeling

• Provides a steady-state description of metabolic
  behavior
• A single, constant flux rate for each reaction
• Ignores metabolite concentrations
• Independent of enzyme activity rates
• Assume a set of constraints on reaction fluxes
• Genome scale models

Flux rate µ-mol / (mg h)
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Constraint Based Modeling

• Find a steady-state flux distribution through
  all
• biochemical reactions

• Under the constraints
• Mass balance metabolite production and
  consumption rates are equal
• Thermodynamic irreversibility of reactions
• Enzymatic capacity bounds on enzyme rates
• Availability of nutrients

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Metabolic Networks
Biochemistry
Cell Physiology
Genome Annotation
Inferred Reactions
Network Reconstruction
Analytical Methods
Metabolic Network
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Mathematical Representation

• Stoichiometric matrix network topology with
  stoichiometry of biochemical reactions

Glucokinase
Glucose ATP
Glucokinase Glucose-6-Phosphate ADP
Glucose -1
ATP -1
G-6-P 1
ADP 1
Mass balance Sv 0 Subspace of R
Thermodynamic vi gt 0 Convex cone
Capacity vi lt vmax Bounded convex cone
n
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Growth Medium Constraints

• Exchange reactions enable the uptake of nutrients
  from the media and the secretion of waste products

Lower bound Upper bound
Glucose 0 2.5
Oxygen 0
Inf
CO2 -Inf
0
G-Ex O-Ex Co2-Ex
Glucose 1
Oxygen 1
CO2 1
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Determination of Likely Physiological States

• How to identify plausible physiological states?
• Optimization methods
• Maximal biomass production rate
• Minimal ATP production rate
• Minimal nutrient uptake rate
• Exploring the solution space
• Extreme pathways
• Elementary modes

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Outline Optimization Methods

• Predicting the metabolic state of a wild-type
  strain
• Flux Balance Analysis (FBA)
• Predicting the metabolic state after a gene
  knockout
• Minimization Of Metabolic Adjustment
• Regulatory On/Off Minimization

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Biomass Production Optimization

• Metabolic demands of precursors and cofactors
  required for 1g of biomass of E. coli
• Classes of macromolecules
• Amino Acids, Carbohydrates
• Ribonucleotides, Deoxyribonucleotides
• Lipids, Phospholipids
• Sterol, Fatty acids
• These precursors are removed from the
• metabolic network in the corresponding ratios
• We define a growth reaction
• Z 41.2570 VATP - 3.547VNADH18.225VNADPH .

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Biomass Composition Issues

• Varies across different organisms
• Depends on the growth medium
• Depends on the growth rate
• The optimum does not change much with changes in
  composition within a class of macromolecules
• The optimum does change if the relative
  composition of the major macromolecules changes

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Flux Balance Analysis (FBA)

• Successfully predicts
• Growth rates
• Nutrient uptake rates
• Byproduct secretion rates
• Solved using Linear Programming (LP)

• Finds flux distribution with maximal growth rate

Max vgro, - maximize growth s.t Sv
0, - mass balance constraints vmin ? v ?
vmax - capacity constraints
Fell, et al (1986), Varma and Palsson (1993)
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FBA Example (1)
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FBA Example (2)
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FBA Example (2)
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Linear Programming Basics (1)
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Linear Programming Basics (2)
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Linear Programming Basics (3)
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Linear Programming Types of Solutions (1)
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Linear Programming Types of Solutions (2)
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Linear Programming Algorithms

• Simplex
• Used in practice
• Does not guarantee polynomial running time
• Interior point
• Worse case running time is polynomial

growth
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Phenotype Predictions Evolving Growth Rate
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Exploring the Convex Solution Space
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Alternative Optima

• The optimal FBA solution is not unique

One solution Optimal
solutions Near-optimal solutions

• Basic solutions enumeration MILP (Lee, et. al,
  2000)
• Flux variability analysis (Mahadevan, et. al.
  2003)
• Hit and run sampling (Almaas, et. al, 2004)
• Uniform random sampling (Wiback, et. al, 2004)

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What Do Multiple Solutions Represent ?

• Some of the solutions probably do not represent
  biologically meaningful metabolic behaviors as
  there are missing constraints
• Previous studies tackled this problem by
• Incorporating additional constraints regulatory
  constraints (Covert, et. al., 2004)
• Looking for reactions for which new constraints
  may significantly reduce the solution space
  (Wiback, et. al., 2004)

FBA solution space
Meaningful solutions
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Interpretations of Metabolic Space

• Effect of exogenous factors the metabolic space
  corresponds to growth in a medium under various
  external conditions that are beyond the models
  scope such as stress or temperature
• Heterogeneity within a population - the metabolic
  space represents heterogenous metabolic behaviors
  by individuals within a cell population
  (Mahadevan, et. al., 2003, Price, et. al., 2004)
• Alternative evolutionary paths the metabolic
  space represents different metabolic states
  attainable through different evolutionary paths
  (Mahadevan, et. al., 2003, Fong, et. al., 2004)
• The three interpretations are obviously not
  mutually exclusive

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Alternative Optima Basic Solutions Enumeration

• Lee, et. al, 2000
• Basic solutions metabolic states with minimal
  number of non-zero fluxes
• Different solutions differ in at least a single
  zero flux
• Use Mixed Integer Linear Programming
• Formulate optimization as to identify new
  solutions that are different from the previous
  ones
• Applicable only to small scale models

growth
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Alternative Optima Flux Variability Analysis

• Mahadevan, et. al. 2003
• Find metabolic states with extreme values of
  fluxes
• Use linear programming to minimize and maximize
  the flux through each reaction while satisfying
  all constraints

Max / Min vi, - maximize growth s.t Sv
0, - mass balance constraints vmin ? v ?
vmax - capacity constraints Vgro Vopt
- set maximal growth rate
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Alternative Optima Hit and Run Sampling

• Almaas, et. al, 2004
• Based on a random walk inside the solution space
  polytope
• Choose an arbitrary solution
• Iteratively make a step in a random direction
• Bounce off the walls of the polytope in random
  directions

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Alternative Optima Uniform Random Sampling

• Wiback, et. al, 2004
• The problem of uniform sampling a
  high-dimensional polytope is NP-Hard
• Find a tight parallelepiped object that binds the
  polytope
• Randomly sample solutions from the parallelepiped
• Can be used to estimate the volume of the
  polytope

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Topological Methods

• Not biased by a statement of an objective

• Network based pathways
• Extreme Pathways (Schilling, et. al., 1999)
• Elementary Flux Modes (Schuster, el. al., 1999)
• Decomposing flux distribution into extreme
  pathways
• Extreme pathways defining phenotypic phase planes
• Uniform random sampling

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Extreme Pathways andElementary Flux Modes

• Unique set of vectors that spans a solution space
• Consists of minimum number of reactions
• Extreme Pathways are systematically independent
  (convex basis vectors)

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Extreme Pathways andElementary Flux Modes

• Inherent redundancy in metabolic networks (Price,
  et. al., 2002)
• Robustness to gene deletion and changes in gene
  expression (Stelling, et. al., 2002)
• Enzyme subsets (correlated reaction sets) in
  yeast (Papin, et. al., 2002)
• Design strains (Carlson, et. al., 2002)
• Assign functions to genes (Forster, et. al, 2002)

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Altering Phenotypic Potential Gene Knockouts
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Altering Phenotypic Potential Gene Knockouts

• Minimization Of Metabolic Adjustment (MOMA)
  (Segre et. al, 2002)
• The flux distribution after a knockout is close
  to the wild-types state under the Euclidian norm
• Regulatory On/Off Minimization (ROOM) (Shlomi et.
  al, 2005)
• Minimize the number of Boolean flux changes from
  the wild-types state

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Altering Phenotypic Potential

• Explaining gene dispensability (Papp, el. al.,
  2004)
• Only 32 of yeast genes contribute to biomass
  production in rich media
• Considered one arbitrary optimal growth solution
• OptKnock Identify gene deletions that generate
  desired phenotype (Burgard, et. al., 2003)
• OptStrain Identify strains which can generate
  desired phenotypes by adding/deleting genes
  (Pharkya, el., al., 2004)

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Modeling Gene Knockouts

• Gene knockout
• Enzyme knockout
• Reaction knockout

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Cellular Adaptation to Genetic and Environmental
Perturbations

• Transient changes in expression levels in
  hundreds of genes (Gasch 2000, Ideker 2001)
• Convergence to expression steady-state close to
  the wild-type (Gasch 2000, Daran 2004, Braun
  2004)
• Drop in growth rates followed by a gradual
  increase (Fong 2004)

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Regulatory On/Off Minimization (ROOM)

• Predicts the metabolic steady-state following the
  adaptation to the knockout
• Assumes the organism adapts by minimizing the set
  of regulatory changes

Boolean Regulatory Change
Boolean Flux Change

• Finds flux distribution with minimal number of
  Boolean flux changes

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ROOM Implementation

• Solved using Mixed Integer Linear Programming
  (MILP)
• Boolean variable yi

yi 1
Flux vi change from wild-type

• Min ?yi - minimize changes
• s.t
• v y ( vmax - w) ? w - distance constraints
• v y ( vmin - w) ? w - distance constraints
• Sv 0, - mass balance constraints
• vj 0, j?G - knockout constraints

• MILP is NP-Hard
• Relax Boolean constraints - solve using LP
• Relax strict constraint of proximity to
  wild-type

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Example Network
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ROOMs Implicit Growth Rate Maximization

• ROOM implicitly attempts to maintain the maximal
  possible growth rate of the wild-type organism
• A change in growth requires numerous changes in
  fluxes

M1
M2
Growth Reaction
. .
Biomass
Mn
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Intracellular Flux Measurements

• Intracellular fluxes measurements in E. coli
  central carbon metabolism
• Obtained using NMR spectroscopy in C labeling
  experiments
• 5 knockouts pyk, pgi, zwf, gnd, ppc in
  Glycolysis and Pentose Phosphate pathways
• Glucose limited and Ammonia limited medias
• FBA wild-type predictions above 90 accuracy

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• Emmerling, M. et al. (2002), Hua, Q. et al.
  (2003), Jiao, Z et al. (2003), Peng, et. al
  (2004)

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Knockout Flux Predictions

• ROOM flux predictions are significantly more
  accurate than MOMA and FBA in 5 out of 9
  experiments

• ROOM steady-state growth rate predictions are
  significantly more accurate than MOMA

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ROOM vs. MOMA

• ROOM predicts metabolic steady-state after
  adaptation
• Provides accurate flux predictions
• Preserved flux linearity
• Finds alternative pathways
• Predicts steady-state growth rates

• MOMA predicts transient metabolic states
  following the knockout
• Provides more accurate transient growth rates

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Additional Constraints

• Transcriptional regulatory constraints (Covert,
  et. al., 2002)
• Boolean representation of regulatory network
• Used to predict growth, changes in expression
  levels, simulate courses of batch cultures
• Energy balance analysis (Beard, et. al., 2002)
• Loops are not feasible according to thermodynamic
  principles resulting in a non-convex solution
  space

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Additional Constraints Slow Changes in the
Environment

• Timescales of cellular process are shorter than
  those of surrounding environment
• Generate dynamic curves to simulate batch
  experiments (Varma, et. al., 1994)

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• Thank you for listening
• Questions