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

Predicting knockout phenotypes central metabolism 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 phosphoglucoisomerase phosphofructokinase aldolase triosephosphate isomerase G3P dehydrogenase phosphoglycerate kinase phosphoglycerate mutase enolase

Metabolites

glucose

Enzymes

phosphofructokinase

Reactions & Stoichiometry

1 F6P => 1 FBP

Kinetics

pyruvate kinase

Regulation

gene regulation metabolite regulation

Metabolic Modeling: The Dream

Steady State Assumptions

• Dynamics are transient • At appropriate time scales and conditions,

metabolism is in steady state

Two key implications 1. Fluxes are roughly constant 2. Internal metabolite concentrations are constant

uptake A conversion B secretion t Steady state

uptake

  

dt

constant  0

t

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

Not required for steady

Substrate affinity

state modeling!

Rate constants

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: Reactions: Stoichiometry Vectors: [ A B C D ] T 2A + B -> C [ -2 -1 1 0 ] T C -> D [ 0 0 -1 1 ] T

The Stoichiometric Matrix (S)

F G I H A B C D E Reactions 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 (Very) Simple System

vin A v1 v3 B D v2 v4 v5 C vout A B C D v1 v2 v3 v4 v5 vin vout -1 1 0 0 0 -1 1 0 -1 0 0 1 0 0 -1 1 0 0 1 -1 1 0 0 0 0 0 -1 0 Exchange Reactions

• 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.

Calculating changes in concentration

What happens if v in is 1 “unit” per second We can calculate this with S 0 v1 vin 1 A v3 0 B D 0 v2 0 v4 v5 0 C vout 0 A grows by 1 “unit” per “second” dA/dt dB/dt dC/dt dD/dt 1 0 0 0 = A B C D v1 v2 v3 v4 v5 vin vout -1 1 0 0 0 -1 1 0 -1 0 0 1 0 0 -1 1 0 0 1 -1 1 0 0 0 0 0 -1 0 0 v1 0 v2 0 v3 0 v4 0 v5 1 vin 0 vout Given these fluxes These are the changes in metabolite concentration

The Stoichiometric Matrix

A B C = D E F I G H 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 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

dx dt

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

Kegg Pathlogic/BioCyc

From Genomes to the S Matrix

Examples Columns encode reactions Gene A Gene B Gene C Gene D Gene E Gene E’ Enzyme E Enzyme E’ Relationships btw genes and rxns

-

1 gene 1 rxn

-

1 gene 1+ rxns

-

1+ genes 1 rxn Enzyme A Enzyme B/C Enyzme D The same reaction can be included as multiple roles (paralogs) A B C D I E F G H 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 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: A B C D

Rate of change of C to D per unit time

4

flux v3

Flux Configuration 3 2 1

We want to know what region of this space contains feasible fluxes given our constraints

1

flux v2

1 2 3

flux v1

The Steady State Constraint

• We have

dx dt

• But also recall that at steady state, metabolite concentrations are constant:

dx/dt

=0

dx dt

0

Positive Flux Constraint

0 

dx dt v i

 0,

i

 1..

n

*

flux v3

All steady state flux vectors, V, must satisfy these constraints What region do these V live in?

?

flux v1 flux v2

The solution through

convex analysis

*recall that reversible reactions are represented by two unidirectional fluxes

The Flux Cone

Solution is a

convex flux cone

• Every steady state flux vector is inside this cone • At steady state, the organism “lives in” here

flux v3 p

2

p

3 v

p

4

p

1

flux v1 flux v2

Extreme Pathways

Extreme pathways are “fundamental modes” of the metabolic system at steady state

flux v3

They are steady state flux vectors

All

other steady state flux vectors are

non-negative linear combinations V

 

i



i p i



i

 0

p

2

p

3 v

p

4

p

1

flux v1 flux v2

Example Extreme Patways

vin A v1 v3 B D v2 v4 C vout A B C D v1 v2 v3 v4 vin vout -1 1 0 0 0 -1 1 0 -1 0 0 1 0 0 1 -1 1 0 0 0 0 0 -1 0 1 b1 vin b2 1 A v1 v3 1 B D 1 v2 v4 1 C vout 1 b1 1 1 0 0 1 1 1 1

All steady state fluxes configurations are combinations of these extreme pathways

b2 0 0 1 1 1 1 v1 v2 v3 v4 vin vout

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 p

2

max flux v3

p

3

p

4

p

1

flux v1 flux v2 flux v3 p

2

p

3

p

4

p

1

flux v1 flux v2

The Constrained Flux Cone

• Contains all achievable flux distributions given the constraints : – Stoichiometry – Reversibility – Max and Min Fluxes

flux v3 p

2

p

3

p

4 • Only requires: – Annotation – Stoichiometry – Small number of flux constraints (small relative to number of reactions)

flux v2 p

1

flux v1

Selecting One Flux Distribution

• At any one point in time, organisms have a single flux configuration • How do we select one flux configuration?

flux v3 p

2

p

3

p

4

p

1

flux v1 flux v2

We will assume organisms are trying to maximize a “fitness” function that is a function of fluxes, F(v)?

Optimizing A Fitness Function

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

ADP->ATP (v3)

7 5

p

2

p

3

p

4

p

1

NADH->NADPH (v1) AMP->ADP (v2)

If we choose F(v) to be a linear function of V :  

i

 The optimizing flux will

always

lie on vertex or edge of the cone Linear Programming

Flux Balance Analysis

Start with stoichiometric matrix and constraints: 0 

dx dt v i

 (all i) 0,

v j

  (some j) Choose a linear function of fluxes to optimize:  

i



v i i

Use linear programming we can find a feasible steady state flux configuration that maximizes the F

flux v3 p

2

p

3

p

4

flux v2 p

1

flux v1

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

Metabolite ATP NADH NADPH G6P F6P R5P E4P T3P 3PG PEP PYR AcCoA OAA AKG (mmol) 41.257

-3.547

18.225

0.205

0.0709

0.8977

0.361

0.129

1.496

0.5191

2.8328

3.7478

1.7867

1.0789

Z = 41.257v

ATP +0.8977v

R5P - 3.547v

+ 0.361v

NADH E4P + 18.225v

+ 0.129v

T3P NADPH + 0.205v

+ 1.496v

3PG G6P + 0.0709v

+ 0.5191v

PEP F6P +2.8328v

PYR + 3.7478v

AcCoA + 1.7867v

OAA + 1.0789v

AKG

FBA Overview

Stoichiometric Matrix

Gene annotation Enzyme and reaction catalog

Feasible Space

S*v=0 Add constraints: v i >0



i >v i >



i

p

2

p

3

p

4

p

1

Optimal Flux

Growth objective Z=c*v Solve with linear programming

p

2

Flux solution

p

3

p

4

p

1

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

“wild-type” vin A v1 v3 B D v2 v4 v5 C vout A B C D v1 v2 v3 v4 v5 vin vout -1 1 0 0 0 -1 1 0 -1 0 0 1 0 0 -1 1 0 0 1 -1 1 0 0 0 0 0 -1 0 “mutant” vin A v1 v3 B D v2 v4 v5 C vout A B C D v1 v2 v3 v4 v5 vin vout -1 1 0 0 0 -1 1 0 -1 0 0 1 0 0 -1 1 0 0 1 -1 1 0 0 0 0 0 -1 0 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

Difference of F(V) (i.e growth) between optima is a measure of the KO phenotype No change in optimal flux Calculate new optimal flux

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

+ -

in vivo

+ 36 2 9 32

Used FBA to predict optimal If Zmutant/Z =0, mutant is no growth (-) growth (+) otherwise Compare to experiment (in vivo / prediction) 86% agree Condition specific prediction growth of mutants (Z mutant ) versus non-mutant (Z) “reduced growth” “lethal” Edward & Palsson (2000) PNAS

Simulated growth on glucose, galactose, succinate, and acetate

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

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 Acetate Uptake Succinate Uptake Edwards et al. (2001) Nat. Biotech.

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 http://gcrg.ucsd.edu

for more info

Price, Reed, & Palsson (2004) Nature Rev. Microbiology

Interpreting Array Data in Metabolic Context

Clustering, GSEA Expression to Flux?

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)

CellNetAnalyzer

Tomorrow’s Lab

• Tools – CellNetAnalyzer – Flux Balance Analysis • Applications – Predict effects of gene knockouts – Central metabolism – TB mycolic acid biosynthesis