Introduction to Steady State Metabolic Modeling

ConceptsFlux Balance Analysis

ApplicationsPredicting knockout phenotypesQuantitative Flux Prediction

Lab PracticalFlux Balance Analysis of E. coli central metabolismPredicting knockout phenotypesModeling 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

Streptococcuspneumoniae

• Explore capabilities of global network• How do we go from a pretty picture to

a model we can manipulate?

Metabolic Pathways

Metabolitesglucose

Enzymesphosphofructokinase

Reactions & Stoichiometry1 F6P => 1 FBP

Kinetics

Regulationgene regulation

metabolite regulation

hexokinase

phosphoglucoisomerase

phosphofructokinase

aldolase

triosephosphate isomerase

G3P dehydrogenase

phosphoglycerate kinase

phosphoglycerate mutase

enolase

pyruvate kinase

Metabolic Modeling: The Dream

up

take

Steady State Assumptions

• Dynamics are transient• At appropriate time-

scales and conditions, metabolism is in steady state

A Buptake conversion secretion

Two key implications1. Fluxes are roughly

constant2. Internal metabolite

concentrations are constant

constant

0

uptake

d A

dt

t Co

nv

ers

ion

t

Steady state

Metabolic Flux

Output fluxes

Input fluxes

Volume of pool of water =

metaboliteconcentration

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

-100-200101

0-10-100010

ABCDEFGHI

ReactionsM

etab

oli

tes

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

Cvin

v1

v3

v2

v5

vout

v4

ABCD

v1 v2 v3 v4 v5 vin vout-1001

00-10

1000

-1100

0-110

001-1

00-11

ExchangeReactions

Calculating changes in concentration

ABCD

v1 v2 v3 v4 v5 vin vout-1001

00-10

1000

-1100

0-110

001-1

00-11

A

B

D

Cvin

v1

v3

v2

v5

vout

v4

What happens ifvin is 1 “unit” per

second

1

0

0

0

0

00

0 v10 v20 v30 v40 v51 vin0 vout

=

1000

dA/dtdB/dtdC/dtdD/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

dxS V

dt

v1v2v3v4v5v6v7v8v9

v10

v1v2v3v4v5v6v7v8v9

v10

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 R10

-100-200101

0-10-100010

ABCDEFGHI

dA/dtdB/dtdC/dtdD/dtdE/dtdF/dtdG/dtdH/dtdI/dt

dA/dtdB/dtdC/dtdD/dtdE/dtdF/dtdG/dtdH/dtdI/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 reactionsDepends 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

ABCDEFGHI

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10

-100-200101

0-10-100010

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

Examples

Gene D Gene E Gene E’

Enzyme A Enzyme B/C Enyzme D Enzyme 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, VImagine we have another simple system:

v1v2v3

VFlux

Configuration

A Bv1

C Dv3v2

flux v1

flux v2

flux v3

1

2

3

4

1

1 2 3

Rate of change of C to D per

unit time113

VFlux

Configuration

1 1 3

We want to know what regionof 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

dxS V

dt

0dx

S Vdt

Positive Flux Constraint

0 0, 1..i

dxS V v i n

dt

*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 v1flux 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

p2 p3

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

0i i ii

V p

They are steady state flux vectors

All other steady state flux vectors are non-negative linear combinations

v

p1

p2 p3

p4

flux v1

flux v2

flux v3

Example Extreme Patways

A

B

D

Cvin

v1

v3

v2

v4

vout

ABCD

v1 v2 v3 v4 vin vout-1001

00-10

1000

-1100

0-110

001-1

b1110011

001111

b2v1v2v3v4vinvout

All steady state fluxes configurations are combinations of these extreme pathways

A

B

D

Cvin

v1

v3

v2

v4

voutb1

b2

1

1 1

1

1

1 1

1

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

p2

p3

p4

flux v1

flux v2

flux v3

p1

p2 p3

p4

max flux v3

flux v1

flux v2

flux v3

The Constrained Flux Cone

p1

p2

p3

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

p2

p3

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

p2

p3

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 coneLinear Programming

( ) i ii

F v v

Flux Balance Analysis

p1

p2

p3

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

( ) i ii

F v v

0 0,i j

dxS V v v

dt

(all i) (some j)

Optimizing E. coli Growth

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

Metabolite (mmol)

ATP 41.257

NADH -3.547

NADPH 18.225

G6P 0.205

F6P 0.0709

R5P 0.8977

E4P 0.361

T3P 0.129

3PG 1.496

PEP 0.5191

PYR 2.8328

AcCoA 3.7478

OAA 1.7867

AKG 1.0789

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

p1

p2

p3

p4

flux v1

flux v2

flux v3

p1

p2

p3

p4

flux v1

flux v2

flux v3

p1

p2

p3

p4

flux v1

flux v2

flux v3

p1

p2

p3

p4

flux v1

flux v2

flux v3

Stoichiometric MatrixGene annotation

Enzyme and reaction catalog

Feasible SpaceS*v=0

Add constraints:vi>0

i>vi>i

Optimal FluxGrowth 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

Cvin

v1

v3

v2

v5

vout

v4

ABCD

v1 v2 v3 v4 v5 vin vout-1001

00-10

1000

-1100

0-110

001-1

00-11

“wild-type”

“mutant”

A

B

D

Cvin

v1

v3

v2

v5

vout

v4

ABCD

v1 v2 v3 v4 v5 vin vout-1001

00-10

1000

-1100

0-110

001-1

00-11

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 metabolism436 metabolites720 reactions

Simulated mutants in glycolysis, pentose phosphate, TCA, electron transport

Edward & Palsson (2000) PNAS

E. coli KO simulation resultsIf 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 (Zmutant) versus non-mutant (Z)

Simulated growth on glucose, galactose, succinate, and acetate

“lethal”

“reduced growth”

Edward & Palsson (2000) PNAS

in vivo

in silico

+ -

+ 36 2

- 9 32

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

Oxy

gen

Up

take

Oxy

gen

Up

take

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

http://gcrg.ucsd.edufor 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 (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