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A Game-Theoretical Approach to the Analysis of Metabolic Pathways

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A Game-Theoretical Approach to the Analysis of Metabolic Pathways

Stefan Schuster

Dept. of Bioinformatics

Friedrich Schiller University Jena, Germany

- Widely used hypothesis: During evolution, metabolic systems have reached (nearly) optimal states (M.A. Savageau, R. Heinrich, E. Meléndez-Hevia, J. Stucki, …). Based on Darwin‘s dogma „Survival of the fittest“
- This hypothesis has been used to predict structural and dynamic properties of metabolic systems

- However: 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
- Optimization theory needs to be extended to cope with this situation Game theory

Initiated in the 1940‘s by

John von Neumann and Oskar Morgenstern.

(1903-1957) (1902 – 1976)

They dealt with non-cooperative (zero-sum) games.

- 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.

B

Cooperate Defect

A

Pareto optimum

10 years prison/

Escape + Reward

Cooperate

Escape/Escape

5 years prison/

5 years prison

Escape + Reward/

10 years prison

Defect

Nash equilibrium

If both of them try to maximize their profit individually,

they both betray. However, then they both lose.

It would be better for them to cooperate, but they cannot be sure

that the other one does not change his mind.

This is a dilemma…

- During evolution, metabolic systems have reached (nearly) optimal states
- Example of theoretical prediction: Maximization of pathway flux subject to constant total enzyme concentration (Waley, 1964; Heinrich et al., 1987)

(q: equilibrium constant)

Optimal

enzyme

concn.

1

2

3

4

Position in the chain

- However, there are more objective functions besides maximization of pathway flux
- Maximum stability and other criteria have been suggested (Savageau, Heinrich, Schuster, …)
- Here, we analyse 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

CO2

Gluc

EtOH

Pyr

Ac.ald.

G6P

F6P

2

2

ATP

ADP

ATP

ADP

ATP

ADP

ADP

ATP

Fermentation

Glucose

Ethanol

(Pyruvate,

Lactate)

Respiration

+ 2 ATP

H2O + CO2

+ 32 ATP

cell A

cell B

respiration or fermentation?

respiration or 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 shown

for bacterium Holophaga foetida growing on methoxylated

aromatic compounds (Kappler et al., 1997).

Substrate level:

Population densities:

v, constant substrate input rate; JS, resource uptake rates;

JATP, ATP production rates; d, death rate.

For J(S), simple Michaelis-Menten rate laws are used.

T. Pfeiffer, S. Schuster, S. Bonhoeffer: Cooperation and

Competition in the Evolution of ATP Producing Pathways.

Science 292 (2001) 504-507.

(yi = ATP:glucose yield of pathway i)

- …such as „strategy“, „cooperation“, „altruism“
- NO!! They are auxiliary means to understand co-evolution more easily
- The game-theoretical problem can alternatively be described by differential equation systems. Nash equilibrium is asymptotically stable steady state

We propose taking the steady-state population density as the

payoff. Particular meaningful in spatially distributed systems

because 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 as

a source of information about the strategy of the other species.

T. Frick, S. Schuster: An example of the prisoner's dilemma in biochemistry.

Naturwissenschaften 90 (2003) 327-331.

We take the steady-state population density as the payoff.

Values calculated with parameter values from model in Pfeiffer et al. (2001).

Cooperative strategy Competitive strategy

Cooperative 3.2 0.0

strategy larger than

in Nash equilibr.

Competitive 5.5 2.7

strategy

This is equivalent to the „Prisoner‘s dilemma“

Nash equilibrium

- 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!

„Tragedy of the commons“ - Generalization of the

prisoner‘s dilemma to n players

Commons: common possession such as the pasture of a

village or fish stock in the ocean. Each of n users of the

commons may think s/he could over-use it without

damaging the others too much. However, when all of

them think so…

- 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.

- For 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.

- Cells using fermentation in multicellular organisms

Striated muscle during heavy

exercise - diffusion of oxygen

not fast enough.

Erythrocytes -

small volume

prevents mitochondria.

Astrocytes -

Job sharing with neurons,

which degrade lactate to

carbon dioxide and water.

- 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

Blue: respirators

Red: fermenters

Yellow: both

Black: empty sites

At low cell diffusion rates and low substrate input,

respirators can win in the long run.

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.

- Communities of different bacteria species
- Competition 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“

Example: Degradation

of 4-chlorosalicylate

From: O. Pelz et al.,

Environm. Microb.

1 (1999), 167–174

- 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

- The prisoner‘s dilemma is relevant in biochemistry.
- In many situations, it would be advantageous for all interacting species to cooperate. However, this strategy is unstable w.r.t invasion by species using the competitive strategy, which gives high growth rates but wastes the resource.
- Stable solution = Pareto optimal solution

- Such dilemmas have to be overcome in the evolution towards cooperation
- Way out of the dilemma may be due to stochastic and spatial effects
- Competition vs. cooperation is relevant in biotechnologically used bacterial communities
- Many questions open…

Cooperations on this project

- Sebastian Bonhoeffer, Thomas Pfeiffer (ETH Zürich)
- Tobias Frick (U Tübingen)
- (starting) Vítor Martins dos Santos (GBF Braunschweig)