Systembiologie 4 – Flux Balance Analysis -...

Humboldt-Universität

Zu Berlin

Edda Klipp, Humboldt-Universität zu Berlin

Edda Klipp

Systembiologie4 –

Flux Balance Analysis

Sommersemester 2011


zu

BerlinInstitut

für

BiologieTheoretische

Biophysik


Zu Berlin


Reminder: Stoichiometry

Stoichiometric matrix

Vector of metabolite

concentrations

Vector of reaction

rates

Parameter vector

ijnN ni ...1 rj ...1

T1,..., nSSS

T1 rvv ,...,v T1,..., mppp

Systems equations

in matrix

formddS vt N

00 NKNvIn steady state:

K represents the basis vector for all possible steady state fluxes.


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Constraints

in Biology

•

Evolution: Organisms

exist

in resource-scarce

environment

–

The

more

“fit”

organisms

survive

with

a higher

probability

than the

less

“fit”

–

Fitness requires

satisfying

a myriad

of constraints

which

limit

the range

of available

phenotypes

•

Survival

thus

depends

on best utilization

of resources

to survive & grow, subject

to constraints

•

All expressed

phenotypes

must

satisfy

imposed

constraints

constraints

therefore

enable

us

to eliminate

impossible

cellular behaviors


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Constraints

in Biology•

Physico-chemical constraints

–

Conservation

of mass, energy, & momentum

–

Maximal reaction/transport

rates

–

Thermodynamic

constraints

•

Topobiological constraints

–

Macromolecular

crowding

constrains

possible

interactions

& diffusion of large molecules

–

DNA, e.g., must

be

both

tightly

packed

and yet

easily

accessible

to the

transcriptional

machinery

two

competing

needs

constrain

the

physical

arrangement

of DNA within

the

cell


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Constraints

in Biology•

Environmental constraints

–

Condition-dependent

variable constraints

–

pH, temperature, osmolarity, availability

of electron

receptors, etc.

–

Availability

of carbon, oxygen, sulfur, nitrogen, and phosphate sources

in surrounding

media

•

Regulatory constraints

–

Self-imposed

“restraints”

–

Subject

to evolutionary

change

–

Allow

cells

to eliminate

suboptimal phenotypes

and confine themselves

to behaviors

of increased

fitness


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Metabolic

Networks: Complexity

E. coli


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Metabolic

Networks: Pathways


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The Flux Cone


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The Flux Cone

vi

vj

vk

vi

vj

vk

v1

v2

v3

111

k

011

1k

101

2k

Illustrative case S1 S2 S3

v1 v4v2 v3S4

v5 S1

v1v2

v3


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Example

for

Flux

Balance Analysis

max 32 vvv1

v2

v3

011

1k

101

2k

S1

v1v2

v3

Constraint: 1321 vvv

Objective:

v3

v21

1 Multiple optima

max 322 vv

Constraint: 1321 vvv

Objective:

v3

v21

1 Single optimum

max 322 vv

Constraints: 20

1

3

321.

vvvv

Objective:

v3

v21

1 Single optimum


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Mathematical

Representation

of Constraints

•

Balances– Mass– Energy–

Solvent capacity

• Bounds–

Thermodynamics

–

Enzyme/transporter

capacity

•

Non-linear

P/C phenomena

N · v = 0


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Metabolic

Network: Constraints


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Metabolic

Pathways: Elementary

Flux

Modes

v: Set ofv:

R(v) vi


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Elementary

Flux

Modes: Definitions


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Elementary

Flux

Modes: Example


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Elementary

Flux

Modes: Example

2


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Elementary

Flux

Modes

& Extreme Pathways


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EFM Applications: Example

Network


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EFM Applications

1: Pathway

Lengths


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EFM Applications

2: Optimal Yields


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EFM Application

3: Essential Genes


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Caveat

1: Applicability


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Caveat

2: Combinatorial

Complexity


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Caveat

3: Feasibility

of EFM


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Flux Balance Analysis in Metabolic Networks


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Predicting the E.coli

Optimal Growth

•Ibarra et al. Escherichia coli k-12 undergoes adaptive evolution to achieve in silico predicted optimal growth. Nature 2002.

•Daniel Segre, Dennis Vitkup, and George M. Church. Analysis of optimality in natural and perturbed metabolic networks. PNAS, vol. 99, 2002.

•Edwards et al. Characterizing the metabolic phenotype. A phenotype phase plan. Biotechnology and bioengineering. 2002

•Kenneth et al. Advances in flux balance analysis. Current Opinion in Biotechnology. 2003.

•Schillling

et. Al Combining pathway analysis with flux balance analysis for the comprehensive study of metabolic systems. Biotechnology and bioengineering, 2001.


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E. Coli

Network: 53 metabolites, 78 internal

fluxes, 8 exchange

fluxes


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(I) Narrowing the Steady State Flux Cone

•

The steady state flux cone contains infinite flux distributions!

•

Only a small portion of them is physiologically feasible.

•

More constraints

on the external fluxes.

These depend on factors as:–

Organism–

Environment and accessibility substrates –

maximum rates of diffusion mediated transport–

Etc…


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(II) Calculating Optimal Flux Distribution

•

The constrained flux cone in E.coli

contains ~10^6 EFM(Schilling 2001)

•

How can we identify a “biologically meaningful”

flux?

Assumption…

the metabolic network is optimized with respect to a certain objective function Z.

Z will be a linear function. Later, we will deal with how exactly to choose Z


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Minimize/Maximize S.T + inequality constraints

What we want to do is find the vector v in the flux cone which maximizes Z.

This optimization problem is a classical linear programming (LP) problem that can be solved using the simplex algorithm. W. Wiechert

. Journal of Biotechnology(2002)

j jjZ c v

(II) Calculating Optimal Flux Distribution

…this can be can formulated as an optimization problem:

vN0


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(III) How to Choose the Objective Function Z

We want to choose a Z that is biologically meaningful. Reasonable options could be:1.

Z: Cellular growth

(maximization)

2.

Z: Particular metabolite engineering

(maximization)3.

Z: Energy consumption

(minimization)

We want a v that:(A) Resides in side the cone.(B) maximizes Z=B+D+2E.

Example:cellular growth is correlated with the production of B,D and 2E.


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1. “It has been shown that under rich growth conditions (i.e. no lack of phosphate and nitrogen), E. Coli grows in a stoichiometrically optimal manner.”

(Schilling 2001, Edwards 1994)

We shall use Z which reflects:

Cellular Growth

(III) How to Choose the Objective Function Z

2. “It is reasonable to hypothesize that unicellular organisms have evolved toward maximal growth performance.”

(Segre, 2002.)


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(IV) Phenotype phase planes -

PPPPredicting Cellular Growth

X axis –

Succinate

uptake rate

Observations:

Schilling 2001

Y axis –

Oxygene

uptake rate

Z axis -

Growth rate

(maximal value of the objective function as function of succinate and oxygen uptake)

•Metabolic network is unable to utilize succinate as sole carbon source in anaerobic conditions.

•Region 1: oxygen excess –

this region is wasteful – (less carbon is available for biomass production since

it is oxidized to eliminate the excess oxygen.)

•Line of optimality

Succinate

Oxygene

Gro

wth

rat

e

1 23

4


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(IV) Phenotype phase planes -

PPPPredicting Cellular Growth

X axis –

Succinate

uptake rate

Observations:

Schilling 2001

Y axis –

Oxygene

uptake rateZ axis –

Growth rate

(maximal value of the

objective function as function of succinate and oxygen uptake)

•Region 2 –

limitation on both oxygen and succinate

•Region 3 –

the uptake of additional succinate has a negative effect. Cellular resources are required to eliminate excessive succinate.

Succinate

Oxygene

Gro

wth

rat

e

1 23

4


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Robustness

Flux dependencies. Centre: phenotypic phase plane (PhPP). It shows the maximum biomass production that is achievable at every possible combination of O2 and succinate

uptake rates. The line of optimality corresponds to the conditions that are necessary for maximal biomass yield (g DW cell mmol–1 carbon source, where DW is dry weight). Robustness analysis of the two uptake rates is shown in the two side panels. Left: effect on growth rate of varying O2

uptake at a fixed succinate

uptake rate. Right: the effect on biomass generation of varying the succinate

uptake rate at a fixed O2

uptake rate.


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Model vs. biological experiments


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Does E. coli behave according to optimal behavior predictions?

•

E. coli was grown with malate as sole carbon source. •

A range of substrate concentrations and temperatures were used in order to vary the malate uptake rate (MUR).

•

Oxygen uptake rate (OUR) and growth rate were measured

.

.

.


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Malate/oxygen PPP

Ibarra et al., Nature 2002

1- The experimentally determined growth rates were on the line of optimality of the PPP !


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Malate/oxygen PPP

Ibarra et al., Nature 2002

Is the optimal performance on malate stable over prolonged periods of time?

Evolution of E. coli on malate was studied for 500 generations in a single condition…

2- An adaptive evolution was observed with an increase of 19% in growth rate!

3- Same adaptive evolution was observed for succinate and Malate!


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•Why does this adaptive evolution occur?

•In other words why is the starting point at the bottom of the hill?


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Same experiments were made using glycerol as sole carbon source

Day 0 –

Sub optimal growth

Day 1-40 –

evolution toward optimal growth

Day 40 –optimal growth

Day 60 –optimal growth

(no change)

Why?


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Exploitation of ConstraintsA growing toolbox for constraint-based analysis. The two steps that are used to form a solution space —

reconstruction and the imposition of governing constraints —

are illustrated in the centre of the figure. As indicated, several methods are being developed at various laboratories to analyze the solution space.

Price, 2004


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Metabolic functions of duplicate genes inSaccharomyces cerevisiae

„Genome-scale“

metabolisches Modell

672 Reaktionen105 duplizierte Genfamilien

Systematischer Test von

Knock-outs

Gesamt-Genome (Hefe):~6000 Gene1500 duplizierte Gene

(mehrere Kopien)


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13C-basierte metabolische

Flußanalyse


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Funktionen duplizierter Gene

Back-up

für essentielle GeneNetzwerkrobustheit; Nullmutanten ohne Phänotyp32 von 52 Mutanten (einer Kopie) essentieller Gene lebensfähig

Gene dosageWege mit hohen Flüssen: Erhöhen der Aktivität kritischer Enzymebetrifft 19 der duplizierten FamilienBedeutung ist abhängig von Nährstoffquelle

Regulatorische

Funktion duplizierter Gene18 der 105 duplizierten Genfamilien spielen regulatorische

Rolle

z.B. bei lineare Pathways, Aminosäuresynthese

Gene mit neuer Aufgabe


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Veränderte Flußverteilungbei anderen Nährstoffen


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Metabolism

of Mycoplasma

pneumoniae

Judith Wodke, project

together

with

Luis Serrano, BarcelonaYus

et al., 2009

To understand

basic

principles

of bacterial

metabolism

organization

and regulation, but

also the

impact

of genome

size, one

of the

smallest

bacteria, Mycoplasma pneumoniae, was systematically

studied.

A manually

curated

metabolic

network

of 189 reactions

catalyzed

by

129 enzymes

allowed

the

design

of a defined, minimal medium

with

19 essential nutrients.


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Flux

Model of M. pneumoniae


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Detailed

Dynamic

Model of M. pneumoniae Metabolism

-

Dissecting growth on different conditions (rich vs

defined medium)- Regulation on the metabolic and the genetic level- Dynamics of input and output metabolites


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Bibliography[1] Daniel Segre, Dennis Vitkup, and George M. Church. Analysis of optimality in natural and perturbed metabolic

networks. PNAS, vol. 99, 2002.[2] C. H. Schilling, D. Letscher

and Bernhard Palsson. Theory for the Systemic Definition of Metabolic Pathways and their use in Interpreting Metabolic Function from a Pathway-Oriented Perspective. J. theor. Biol. (2000)

[3] Schillling

et. Al Combining pathway analysis with flux balance analysis for the comprehensive study of metabolic systems. Biotechnology and bioengineering, 2001.

[4] Edwards et al. 2002. Characterizing the metabolic phenotype”

A phenotype phase plan. Biotechnology and bioengineering

[5] Kenethh

et al. Advances in flux balance analysis. Current Opinion in Biotechnology. [6] Ibarra et al. Escherichia coli k-12 undergoes adaptive evolution to achiev

in silico

predicted optimal growth. Nature 2002.

[7] W. Wiechert

. Modeling and simulation: tools for metabolic engineering. Journal of biotechnology(2002)[8] Cornish-Bowden. From genome to cellular phenotype-

a role for meatbolic

flux analysis? Nature biotechnology, vol

18, 2000.

[9] Schuster et al. Detection of elelmtary

flux modes in biochemical networks: a promising tool for pathway analysis and metabolic engineering. TIBTECH 1999

[10] J. Papin, Nathan D Price, B. Palsson. Extreme pathway lengths and reaction participation in genome scale metabolic networks. Genome research, 2002.

[11] Stelling

eta l. Metabolic netwrok

structure determines key aspects of functionality and regulation. Nature 2002.[12] A general definition of metabolic pathways useful for systematic organization and analysis of complex metabolic

networks.


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Growth Phenotypes of insilico

deletion strains

The biomass yields are normalized with respect to the results for the full metabolic genotype. The α

and β

value for the constraints on the external fluxes for glucose and oxygen uptake are defined as follows (units-

mmole

g-1 hr-1): Phase 1 -

vglc = 10, voxy = 23; LO -

vgic = 10, voxy = 20.3; Phase 2 -

vglc = 10, voxy = 17; Phase 3 -




vglc = 10, voxy = 0.

Maximal biomass yields on glucose for all possible single gene deletions in central intermediary metabolism. The environmental variables (uptake rate/external metabolic fluxes) are set to correspond to a point within each of the phases of the wild-type PhPP

(figure inset). The maximal yields were calculated using flux-balance analysis with the objective of maximizing the growth flux.

Systembiologie 4 – Flux Balance Analysis -...

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