Metabolic Pathway Analysis as Part of Systems Biology

Stefan SchusterDept. of Bioinformatics

Friedrich Schiller University Jena, Germany

Friedrich Schiller(1759-1805)

Famous people at Jena University:

Ernst Haeckel(1834-1919, Biogenetic rule)

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Introduction

Analysis of metabolic systems requires theoretical methods due to high complexity

Systems theoretical approaches used in this field for a long time, for example:

•1960‘s Dynamic simulation of biochemical systems by David Garfinkel

•Metabolic Control Analysis (1973 H. Kacser/J. Burns, R. Heinrich/T.A. Rapoport, 1980‘s Hans Westerhoff)

•„Biochemical Systems Theory“ (1970‘s Michael Savageau)

Since end 1990‘s: New quality due to high throughput experiments and on-line databases (e.g. KEGG, ExPASy, BRENDA)

„Metabolomics“ is a growing field besides „genomics“, „proteomics“...

Major challenge: clarify relationship between structure and function in complex intracellular networks

Motivation for the modelling of metabolism

Functional genomics – assignment of gene functions can be improved by consideration of interplay between gene products

Study of robustness to enzyme deficiencies and knock-out mutations is of high medical and biotechnological relevance

Increase of rate and yield of bioprocesses important in biotechnology

The evolutionary aspect

Metabolic pathways result from biological evolution

Evolution is actually co-evolution because various species interact

Each species tends to optimize its properties; the outcome depends also on the properties of the other species

Features often studied in Systems Biology:

RobustnessFlexibilityFragilityOptimalityModularity

Theoretical Methods

Dynamic SimulationStability and bifurcation analysesMetabolic Control Analysis (MCA)Metabolic Pathway AnalysisMetabolic Flux Analysis (MFA)Optimizationand others

Theoretical Methods

Dynamic SimulationStability and bifurcation analysesMetabolic Control Analysis (MCA)Metabolic Pathway AnalysisMetabolic Flux Analysis (MFA)Optimizationand others

Metabolic Pathway Analysis (or Metabolic Network Analysis)

Decomposition of the network into the smallest functional entities (metabolic pathways)

Does not require knowledge of kinetic parameters!!

Uses stoichiometric coefficients and reversibility/irreversibility of reactions

History of pathway analysis

„Direct mechanisms“ in chemistry (Milner 1964, Happel & Sellers 1982)

Clarke 1980 „extreme currents“ Seressiotis & Bailey 1986 „biochemical pathways“ Leiser & Blum 1987 „fundamental modes“ Mavrovouniotis et al. 1990 „biochemical pathways“ Fell (1990) „linearly independent basis vectors“ Schuster & Hilgetag 1994 „elementary flux modes“ Liao et al. 1996 „basic reaction modes“ Schilling, Letscher and Palsson 2000 „extreme

pathways“

S. Schuster und C. Hilgetag: J. Biol. Syst. 2 (1994) 165-182

non-elementary flux mode

elementary flux modes

An elementary mode is a minimal set of enzymes thatcan operate at steady state with all irreversible reactions used in the appropriate direction

Related concept: Extreme pathway (C.H. Schilling, D. Letscher and B.O. Palsson, J. theor. Biol. 203 (2000) 229) - distinction between internal and exchange reactions, all internal reversible reactions are split up into forward and reverse steps

All flux distributions in the living cell are non-negative linear combinations of elementary modes

Mathematical background

Steady-state condition NV = 0Sign restriction for irreversible fluxes: Virr 0

This represents a linear equation/inequality system.

Solution is a convex region.

All edges correspond to elementary modes.

In addition, there may be elementary modes in the interior.

Geometrical interpretation

Elementary modes correspond to generating vectors (edges) of a convex polyhedral cone (= pyramid) in flux space (if all modes are irreversible)

flux1

flux2

flux3

generating vectors

NADP

NADPH

NADP

NADPH

NADHNAD

ADP

ATP

ADP

ATP

CO2

ATP ADP

G6P

X5P

Ru5P

R5P

S7P

GAP

GAP

6PG

GO6P

F6P FP2

F6P

DHAP

1.3BPG

3PG

2PG

PEP

E4P

Part of monosaccharide metabolism

Red: external metabolites

Pyr

NADHNAD

ADP

ATP

ADP

ATP

ATP ADP

G6P GAPF6P FP2

DHAP

1.3BPG

3PG

2PG

PEP

1st elementary mode: glycolysis

Pyr

2nd elementary mode: fructose-bisphosphate cycle

ATP ADP

F6P FP2

4 out of 7 elementary modes in glycolysis-pentose-phosphate system

NADP

NADPH

NADP

NADPH

NADHNAD

ADP

ATP

ADP

ATP

CO2

ATP ADP

G6P

X5P

Ru5P

R5P

S7P

GAP

GAP

6PG

GO6P

F6P FP2

F6P

DHAP

1.3BPG

3PG

2PG

PEP

E4P

S. Schuster, D.A. Fell, T. Dandekar:Nature Biotechnol. 18 (2000) 326-332

Pyr

Software for computing elementary modes

ELMO (in Turbo-Pascal) - C. Hilgetag

EMPATH (in SmallTalk) - J. Woods

METATOOL (in C) - Th. Pfeiffer, F. Moldenhauer, A. von Kamp, M. Pachkov

Included in GEPASI - P. Mendesand JARNAC - H. Sauro

part of METAFLUX (in MAPLE) - K. Mauch

part of FluxAnalyzer (in MATLAB) - S. Klamt

part of ScrumPy (in Python) - M. Poolman

On-line computation:

pHpMetatool - H. Höpfner, M. Lange

http://pgrc-03.ipk-gatersleben.de/tools/phpMetatool/index.php

NADP

NADPH

NADP

NADPH

NADHNAD

ADP

ATP

ADP

ATP

CO2

ATP ADP

G6P

X5P

Ru5P

R5P

S7P

GAP

GAP

6PG

GO6P

F6P FP2

F6P

DHAP

1.3BPG

3PG

2PG

PEP

E4P

Optimization: Maximizing molar yields

ATP:G6P yield = 3 ATP:G6P yield = 2

Pyr

Maximization of tryptophan:glucose yield

Model of 65 reactions in the central metabolism of E. coli.26 elementary modes. 2 modes with highest tryptophan:glucose yield: 0.451.

Glc

G6P

233

Anthr

Trp105

PEPPyr

3PGGAP

PrpP

S. Schuster, T. Dandekar, D.A. Fell,Trends Biotechnol. 17 (1999) 53

Optimality of metabolism Example of theoretical prediction: Maximization of

pathway flux subject to constant total enzyme concentration (Waley, 1964; Heinrich, Schuster

and Holzhütter, 1987)

1

1

r

jrtot

jq

qqEE

Position in the chain

Optimalenzymeconcn.

1 2 3 4

(q: equilibrium constant)

However, there are more objective functions besides maximization of pathway flux

Maximum stability and other criteria have been suggested (Savageau, Heinrich, Schuster, …)

Optimality criterion for a particular species need not coincide with optimality criterion for a community of species

Optimization theory needs to be extended to cope with this problem Game theory

Maximum flux vs. maximum molar yield

Example: Fermentation has a low yield

(2 moles ATP per mole of glucose) but high ATP production rate (cf. striated muscle); respiration has a high yield (>30 moles ATP per mole of glucose) but low ATP production rate

Two possible strategies

ADP ATP ADP ATPATP ADP

G6P F6P Pyr

ATP ADP

Gluc Ac.ald.

CO2

EtOH

Fermentation

The two cells (strains, species) have two strategies.The outcome for each of them depends on their own strategy as well as on that of the competitor.

Respiration can be considered as a cooperative strategy because it uses the resource more efficiently. By contrast, fermentation is a competitive strategy.

Switch between high yield and high rate has been shownfor bacterium Holophaga foetida growing on methoxylatedaromatic compounds (Kappler et al., 1997).

Game-theoretical problem

How to define the payoff?

We propose taking the steady-state population density as thepayoff. Particular meaningful in spatially distributed systemsbecause spreading of strain depends on population density.

Dependence of the payoff on the strategy of the other species via the steady-state substrate level. This may also be used asa source of information about the strategy of the other species.

Population payoffs and resource level

T. Frick, S. Schuster: An example of the prisoner's dilemma in biochemistry. Naturwissenschaften 90 (2003) 327-331.

Payoff matrix of the „game“of two species feeding on the same resource

Cooperative strategy Competitive strategy

Cooperative 3.2 0.0 strategy larger than

in Nash equilibr.

Competitive 5.5 2.7strategy

This is equivalent to the „Prisoner‘s dilemma“

We take the steady-state population density as the payoff. Values calculated with parameter values from model in Pfeiffer et al. (2001).

Nash equilibrium

Prisoner‘s dilemma

If prisoner A reveals the plan of escape to the jail director, while prisoner B does not, A is set free and gets a reward of 1000 ₤. B is kept in prison for 10 years.

The same vice versa. If none of them betrays, both can escape. If both betray, they are kept in prison for 5 years.They are allowed to know what the other one does.

Payoff matrixfor the Prisoner‘s Dilemma

Cooperate Defect

Nash equilibrium

Cooperate

Defect

Escape/Escape

A B

10 years prison/Escape + Reward

Escape + Reward/10 years prison

5 years prison/5 years prison

Pareto optimum

)()( SJNSJNS SFF

SRR

RR

ATPRR

dNNScJN

FF

ATPFF

dNNScJN

Substrate level:

Population densities:

v, constant substrate input rate; JS, resource uptake rates;JATP, ATP production rates; d, death rate.

System equations

T. Pfeiffer, S. Schuster, S. Bonhoeffer: Cooperation and Competition in the Evolution of ATP Producing Pathways. Science 292 (2001) 504-507.

Michaelis-Menten rate laws

SK

SVSJ

Mi

iSi

max

Sii

ATPi

JyJ

(yi = ATP:glucose yield of pathway i)

Do we need anthropomorphic concepts?

…such as „strategy“, „cooperation“, „altruism“NO!! They are auxiliary means to understand co-

evolution more easilyThe game-theoretical problem can alternatively be

described by differential equation systems. Nash equilibrium is asymptotically stable steady state

A paradoxical situation:

Both species tend to maximize their population densities.

However, the resultant effect of these two tendencies is that their population densities decrease.

The whole can be worse then the sum of its parts!

n-Player games

„Tragedy of the commons“ - Generalization of the prisoner‘s dilemma to n players

Commons: common possession such as the pasture of avillage or fish stock in the ocean. Each of n users of thecommons may think s/he could over-use it withoutdamaging the others too much. However, when all of them think so…

Biological examples

S. cerevisiae and Lactobacilli use fermentation even under aerobiosis, if sufficient glucose is available. They behave „egotistically“.

Other micro-organisms, such as Kluyverymyces, use respiration.

Multicellular organismsFor multicellular organisms, it would be disadvantageous if

their cells competed against each other.

In fact, most cell types in multicellular organisms use respiration.

Exception: cancer cells. Perhaps, their „egotistic“ behaviour is one of the causes of their pathological effects.

„Healthy“ exceptions:

Cells using fermentation in multicellular organisms

Erythrocytes -small volume prevents mitochondria.

Striated muscle during heavy exercise - diffusion of oxygen not fast enough.

Astrocytes – division oflabour with neurons,which degrade lactate tocarbon dioxide and water.

How did cooperation evolve?

Deterministic system equations: fermenters always win.

However, they can only sustain low population densities. Susceptible to stochastic extinction.

Further effects in spatially distributed systems. Cooperating cells can form aggregates.

Possible way out of the dilemma: Evolution in a 2D (or 3D) habitat with stochastic effects

At low cell diffusion rates and low substrate input,respirators can win in the long run.

Blue: respirators

Red: fermenters

Black: empty sites

Yellow: both

Aggregates of cooperating cells can be seen as an important step towards multicellularity.

T. Pfeiffer, S. Schuster, S. Bonhoeffer: Cooperation and Competition in the Evolution of ATP Producing Pathways. Science 292 (2001) 504-507.

Biotechnological relevance

Communities of different bacteria speciesCompetition for the same substrate or

division of labour so that the product of one bacterium is used as a substrate by another one (crossfeeding, like in astrocytes and neurons)

Pathways operating in microbial communities = „consortium pathways“

From: O. Pelz et al.,Environm. Microb.1 (1999), 167–174

Example: Degradation of 4-chlorosalicylate

Another example: E. coli

E. coli in continuous culture (chemostat) evolves, over many generations, so as to show stable polymorphism (Helling et al., 1987)

One resulting strain degrades glucose to acetate, another degrades acetate to CO2 and water

Example of intra-species crossfeeding

Conclusions

„Analysis“ (Greek) means decompositionScientists tend to analyse: „function of a

gene“, „role of an calcium oscillations“, „impact of an enzyme“

The ability of a steel ship to be afloat

cannot be explained by decomposition

However:

Analytical vs. holistic approaches

Decomposition should not be overdoneExample elementary flux modes: smallest

functional units, rather than decomposition into enzymes

It depends on the question at which level the description should be made

Systems biology motivated by reasoning that the whole is more than the sum of ist parts (sometimes worse than…)

Game theory is one possible holistic approach

•Steffen Klamt, Jörg Stelling, Ernst Dieter Gilles (MPI Magdeburg)•Thomas Dandekar (U Würzburg)•David Fell (Brookes U Oxford)•Thomas Pfeiffer, Sebastian Bonhoeffer (ETH Zürich)•Peer Bork (EMBL Heidelberg)•Reinhart Heinrich, Thomas Höfer (HU Berlin)•I. Zevedei-Oancea (formerly my group, now HU Berlin)•Hans Westerhoff (VU Amsterdam)• and others

•Acknowledgement to DFG for financial support

Cooperations