# A Game-Theoretical Approach to the Analysis of Metabolic Pathways Stefan Schuster Dept. of Bioinformatics Friedrich Schiller University Jena, Germany.

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A Game-Theoretical Approach to the Analysis of Metabolic Pathways Stefan Schuster Dept. of Bioinformatics Friedrich Schiller University Jena, Germany

Introduction §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

Introduction (2) §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

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.

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 matrix for 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

Can the other one be trusted? 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…

What has this to do with biochemistry?

Optimality of metabolism §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) Position in the chain Optimal enzyme concn. 1234 (q: equilibrium constant)

Optimality of metabolism (2) §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

Two possible strategies

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). Game-theoretical problem

Substrate level: Population densities: v, constant substrate input rate; J S, resource uptake rates; J ATP, ATP production rates; d, death rate. System equations 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.

Michaelis-Menten rate laws (y i = 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 easily §The game-theoretical problem can alternatively be described by differential equation systems. Nash equilibrium is asymptotically stable steady state

How to define the payoff? 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.

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 2.7 strategy 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

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 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…

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 organisms §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.

„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 - Job sharing with neurons, which degrade lactate to carbon 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 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“

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 CO 2 and water §Example of intra-species crossfeeding

Conclusions §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

Conclusions (2) §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…

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

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