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Introduction to Steady State Metabolic Modeling

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## Introduction to Steady State Metabolic Modeling

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**Introduction to Steady State Metabolic Modeling**Concepts Flux Balance Analysis Applications Predicting knockout phenotypes Quantitative Flux Prediction Lab Practical Flux Balance Analysis of E. coli central metabolism Predicting knockout phenotypes Modeling 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**Streptococcus pneumoniae • Explore capabilities of global network • How do we go from a pretty picture to a model we can manipulate?**Metabolic Pathways**hexokinase Metabolites glucose Enzymes phosphofructokinase Reactions & Stoichiometry 1 F6P => 1 FBP Kinetics Regulation gene regulation metabolite regulation phosphoglucoisomerase phosphofructokinase aldolase triosephosphate isomerase G3P dehydrogenase phosphoglycerate kinase phosphoglycerate mutase enolase pyruvate kinase**Steady state**Steady State Assumptions • Dynamics are transient • At appropriate time-scales and conditions, metabolism is in steady state uptake conversion secretion A B uptake Conversion t t Two key implications • Fluxes are roughly constant • Internal metabolite concentrations are constant**Metabolic Flux**Input fluxes Volume of pool of water = metabolite concentration Output fluxes 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 A B C D E F G H I -1 0 0 -2 0 0 1 0 1 0 -1 0 -1 0 0 0 1 0 Reactions Metabolites**B**v1 v2 A C vin vout v4 D v3 v5 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. v1 v2 v3 v4 v5 vin vout A B C D -1 1 0 0 0 -1 1 0 -1 0 0 1 0 0 1 -1 1 0 0 0 0 0 -1 0 0 0 -1 1 Exchange Reactions**B**0 0 v1 v2 A C vin vout 0 v4 0 1 D v3 v5 0 0 • 0 v1 • 0 v2 • 0 v3 • 0 v4 • 0 v5 • vin • 0 vout 1 0 0 0 v1 v2 v3 v4 v5 vin vout A B C D -1 1 0 0 0 -1 1 0 -1 0 0 1 0 0 1 -1 1 0 0 0 0 0 -1 0 0 0 -1 1 Given these fluxes These are the changes in metabolite concentration Calculating changes in concentration What happens if vin is 1 “unit” per second We can calculate this with S dA/dt dB/dt dC/dt dD/dt A grows by 1 “unit” per “second” =**The Stoichiometric Matrix**R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 = A B C D E F G H I -1 0 0 -2 0 0 1 0 1 0 -1 0 -1 0 0 0 1 0 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**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 reactions Depends 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**There are several online databases with curated and/or automated EC number assignments for sequenced genomes Pathlogic/BioCyc Kegg**From Genomes to the S Matrix**Examples • 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 Gene D Gene E Gene E’ Enzyme A Enzyme B/C Enyzme D Enzyme E Enzyme E’ R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 -1 0 0 -2 0 0 1 0 1 0 -1 0 -1 0 0 0 1 0 A B C D E F G H I 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**1**1 3 v2 v1 v3 flux v3 1 1 3 v1 v2 v3 A C D B V Flux Configuration V Flux Configuration Rate of change of C to D per unit time 4 3 2 1 1 2 3 flux v1 1 flux v2 Flux Configuration, V Imagine we have another simple system: We want to know what region of 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**flux v3**? flux v1 flux v2 Positive Flux Constraint * All steady state flux vectors, V, must satisfy these constraints What region do these V live in? The solution through convex analysis *recall that reversible reactions are represented by two unidirectional fluxes**flux v3**p3 p2 p4 At steady state, the organism “lives in” here v p1 flux v1 flux v2 The Flux Cone Solution is a convexflux cone • Every steady state flux vector is inside this cone • Edges of the cone are circumscribed by Extreme Pathways**Extreme Pathways**Extreme pathways are “fundamental modes” of the metabolic system at steady state They are steady state flux vectors Allother steady state flux vectors arenon-negative linear combinations flux v3 p3 p2 p4 v p1 flux v1 flux v2**B**A C D b2 b1 1 1 1 1 0 0 1 1 0 0 1 1 1 1 v1 v2 v3 v4 vin vout B 1 1 v1 v2 b1 vin vout A C b2 v3 v4 D 1 1 1 1 Example Extreme Patways v1 v2 v3 v4 vin vout v1 v2 A B C D -1 1 0 0 0 -1 1 0 -1 0 0 1 1 0 0 0 0 0 -1 0 0 0 1 -1 vin vout v3 v4 All steady state fluxes configurations are combinations of these extreme pathways**flux v3**p2 p3 p4 max flux v3 p1 flux v1 flux v2 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 flux v3 p3 p2 p4 p1 flux v1 flux v2**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) flux v3 p2 p3 p4 p1 flux v1 flux v2 The Constrained Flux Cone**At any one point in time, organisms have a single flux**configuration How do we select one flux configuration? flux v3 p2 p3 p4 p1 flux v1 flux v2 Selecting One Flux Distribution We will assume organisms are trying to maximize a “fitness” function that is a function of fluxes, F(v)?**7**5 If we choose F(v) to be a linear functionof V: The optimizing flux will always lie on vertex or edge of the cone Linear Programming Optimizing A Fitness Function 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 p2 p3 p4 p1 NADH->NADPH (v1) AMP->ADP (v2)**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 flux v3 p2 p3 p4 p1 flux v1 flux v2 Flux Balance Analysis (all i) (some j)**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 Optimizing E. coli Growth 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**Stoichiometric Matrix Gene annotation Enzyme and reaction catalog Feasible Space S*v=0 Add constraints: vi>0 ai>vi>bi Optimal Flux Growth 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**B**B v1 v2 v3 v4 v5 vin vout v1 v1 v2 v2 A A C C A B C D -1 1 0 0 0 -1 1 0 -1 0 0 1 0 0 1 -1 1 0 0 0 0 0 -1 0 0 0 -1 1 vin vin vout vout v4 v4 D D v3 v3 v5 v5 v1 v2 v3 v4 v5 vin vout A B C D -1 1 0 0 0 -1 1 0 -1 0 0 1 0 0 1 -1 1 0 0 0 0 0 -1 0 0 0 -1 1 in silico Deletion Analysis “wild-type” “mutant” 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 metabolism 436 metabolites 720 reactions Simulated mutants in glycolysis, pentose phosphate, TCA, electron transport Edward & Palsson (2000) PNAS**“reduced growth”**“lethal” E. coli KO simulation results If Zmutant/Z =0, mutant is no growth (-) growth (+) otherwise Compare to experiment (in vivo / prediction) 86% agree Condition specific prediction Used FBA to predict optimal growth of mutants (Zmutant) versus non-mutant (Z) Simulated growth on glucose, galactose, succinate, and acetate Edward & Palsson (2000) PNAS**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**Can models quantitatively predict fluxes and/or growth rate? • Measure some fluxes, say A & B • Controlled uptake rates • Predict optimal flux of A given B as input to model**Growth vs Uptake Fluxes**Predict relationship between - growth rate - oxygen uptake - acetate or succinate uptake Compare to experiments from batch reactors Oxygen Uptake Oxygen Uptake Acetate Uptake Succinate Uptake Edwards et al. (2001) Nat. Biotech.**Specify glucose uptake**Predict - growth - oxygen uptake - acetate secretion Compare to chemostat experiment Uptake vs Growth 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**Expression to Flux?**Interpreting Array Data in Metabolic Context Clustering, GSEA Kegg, PathwayExplorer**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)**Tomorrow’s Lab**• Tools • CellNetAnalyzer • Flux Balance Analysis • Applications • Predict effects of gene knockouts • Central metabolism • TB mycolic acid biosynthesis