Constraint-Based Modeling of Metabolic Networks

based on: “Genome-scale models of microbial cells:

Evaluating the consequences of constraints”, Price, et. al (2004)

Tomer ShlomiSchool 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

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

Find a steady-state flux distribution through all biochemical reactions

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Metabolic Networks

Network Reconstruction

GenomeAnnotation

BiochemistryCell

Physiology InferredReactions

Metabolic Network Analytical Methods

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Mathematical Representation Stoichiometric matrix – network topology with stoichiometry of

biochemical reactions

Mass balance

S·v = 0

Subspace of R

Thermodynamicvi > 0Convex cone

Capacityvi < vmax

Bounded convex cone

Glucose + ATP

Glucokinase

Glucose-6-Phosphate + ADP

Glucose -1ATP -1

G-6-P +1ADP +1

Glucokinase

n

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Growth Medium Constraints

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

Oxygen 0 Inf

Glucose 0 2.5

CO2 -Inf 0

Glucose 1Oxygen 1

G-Ex O-Ex Co2-Ex

CO2 1

Lower bound Upper bound

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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 - 3.547VNADH+18.225VNADPH + ….

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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)

Max vgro, - maximize growth

s.t

S∙v = 0, - mass balance constraints

vmin v vmax - capacity constraintsgrowth

Finds flux distribution with maximal growth rate

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

growthgrowth

growth

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 model’s 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

S∙v = 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

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

Not biased by a statement of an objective

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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-type’s state under the Euclidian norm Regulatory On/Off Minimization (ROOM) (Shlomi et. al,

2005) Minimize the number of Boolean flux changes from

the wild-type’s state

w

v

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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)

minutes

gro

wth

generations

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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

w

v

Boolean RegulatoryChange

Boolean FluxChange

Finds flux distribution with minimal number of Boolean flux changes

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ROOM: Implementation

Min yi - minimize changes

s.t

v – y ( vmax - w) w - distance constraints

v – y ( vmin - w) w- distance constraints

S∙v = 0, - mass balance constraints

vj = 0, jG - knockout constraints

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

yi = 1 Flux vi change from wild-type

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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ROOM’s 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

Mn

.

.

Growth ReactionBiomass

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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 predicts metabolic steady-state after adaptationProvides accurate flux predictionsPreserved flux linearityFinds alternative pathwaysPredicts steady-state growth rates

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

ROOM vs. MOMA

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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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