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 Metabolites glucose Enzymes phosphofructokinase Reactions & Stoichiometry 1 F6P => 1 FBP Kinetics Regulation gene regulation metabolite regulation hexokinase phosphoglucoisomerase phosphofructokinase aldolase triosephosphate isomerase G3P dehydrogenase phosphoglycerate kinase phosphoglycerate mutase enolase pyruvate kinase

Metabolic Modeling: The Dream

uptake Steady State Assumptions Dynamics are transient At appropriate time- scales and conditions, metabolism is in steady state A B uptakeconversionsecretion Two key implications 1.Fluxes are roughly constant 2.Internal metabolite concentrations are constant t Conversion t Steady state

Metabolic Flux Output fluxes Input fluxes Volume of pool of water = metabolite concentration 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: [ ] T [ ] T

The Stoichiometric Matrix (S) R1 R2 R3 R4 R5 R6 R7 R8 R9 R ABCDEFGHIABCDEFGHI Reactions Metabolites

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 C vin v1 v3 v2 v5 vout v4 ABCDABCD v1 v2 v3 v4 v5 vin vout Exchange Reactions

Calculating changes in concentration ABCDABCD v1 v2 v3 v4 v5 vin vout A B D C vin v1 v3 v2 v5 vout v4 What happens if v in is 1 “unit” per second v1 0 v2 0 v3 0 v4 0 v5 1 vin 0 vout = dA/dt dB/dt dC/dt dD/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 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 R ABCDEFGHIABCDEFGHI =

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 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 ABCDEFGHIABCDEFGHI R1 R2 R3 R4 R5 R6 R7 R8 R9 R 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 AGene B Gene C Examples Gene DGene E Gene E’ Enzyme AEnzyme B/CEnyzme DEnzyme 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, V Imagine we have another simple system: v1 v2 v3 V Flux Configuration AB v1 CD v3 v2 flux v1 flux v2 flux v Rate of change of C to D per unit time V Flux Configuration 113 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

Positive Flux Constraint *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 v1 flux 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 p1 p2 p2 p3 p3 p4 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 They are steady state flux vectors All other steady state flux vectors are non-negative linear combinations v p1 p1 p2 p2 p3 p3 p4 p4 flux v1 flux v2 flux v3

Example Extreme Patways A B D C vin v1 v3 v2 v4 vout ABCDABCD v1 v2 v3 v4 vin vout b b2 v1 v2 v3 v4 vin vout All steady state fluxes configurations are combinations of these extreme pathways A B D C vin v1 v3 v2 v4 vout b1 b

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 p1 p2 p2 p3 p3 p4 p4 flux v1 flux v2 flux v3 p1 p1 p2 p2 p3 p3 p4 p4 max flux v3 flux v1 flux v2 flux v3

The Constrained Flux Cone p1 p1 p2 p2 p3 p3 p4 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 p1 p2 p2 p3p3 p4 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 p1 p2 p2 p3p3 p4 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 cone Linear Programming

Flux Balance Analysis p1 p1 p2 p2 p3p3 p4 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 (all i) (some j)

Optimizing E. coli Growth Z = v ATP v NADH v NADPH v G6P v F6P v R5P v E4P v T3P v 3PG v PEP v PYR v AcCoA v OAA v AKG Metabolite(mmol) ATP NADH NADPH G6P0.205 F6P R5P E4P0.361 T3P PG1.496 PEP PYR AcCoA OAA AKG 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: v i >0  i >v i >  i 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

in silico Deletion Analysis A B D C vin v1 v3 v2 v5 vout v4 ABCDABCD v1 v2 v3 v4 v5 vin vout “wild-type” “mutant” A B D C vin v1 v3 v2 v5 vout v4 ABCDABCD v1 v2 v3 v4 v5 vin vout 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

E. coli KO simulation results If 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 (Z mutant ) versus non-mutant (Z) Simulated growth on glucose, galactose, succinate, and acetate “lethal” “reduced growth” Edward & Palsson (2000) PNAS in vivo in silico 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 Oxygen Uptake 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 for 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 ( – –Biomodels database ( –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