A game-theoretical approach to the analysis of microbial metabolic pathways

Transcription

1 A game-theoretical approach to the analysis of microbial metabolic pathways Stefan Schuster Dept. of Bioinformatics Friedrich Schiller University Jena, Germany

2 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

3 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

4 Prisoner s dilemma

5 Payoff matrix for the Prisoner s Dilemma A B Cooperate Defect Cooperate Pareto optimum Escape/Escape 10 years prison/ Escape + Reward Defect Escape + Reward/ 10 years prison 5 years prison/ 5 years prison Nash equilibrium

6 What has this to do with biochemistry?

7 Optimality principles in metabolism Two very important principles: maximum flux vs. maximum molar yield In many situations, these two criteria contradict each other 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

8 Fermentation CO 2 Gluc G6P F6P Pyr Ac.ald. EtOH ATP ADP ATP ADP ADP ATP ADP ATP

9 glycolysis Respiration Figures by Dr. Roger Davis et al. San Diego State University

10 Two possible strategies

11 Game-theoretical problem 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).

12 Substrate level: Population densities: System equations S& =ν N& N& R F = = cj cj N R ATP R ATP F J S R S ( S) N J ( S) F ( S) N dn R R F ( S) N dn F F v, constant substrate input rate; J S, resource uptake rates; J ATP, 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)

13 Michaelis-Menten rate laws J S i ( S ) = V K max i M i S + S J = ATP i y i J S i (y i = ATP:glucose yield of pathway i) Due to the saturation effect, respiro-fermentation is better than pure fermentation.

14 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 of population dynamics. Nash equilibrium is asymptotically stable steady state Advantage of game-theoretical description in comparison to population dynamics: fewer parameters needed.

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

16 Payoff matrix of the game of two species feeding on the same resource 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 strategy larger than in Nash equilibr. Competitive strategy This is equivalent to the Prisoner s dilemma Nash equilibrium

17 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!

18 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

19 Evolution is n-player game Not only various microbial species compete for substrate, but also different strains of the same species If only two distinct strategies are considered (respiro-fermentation vs. respiration), the tragedy of the commons can be mapped to the Prisoner s Dilemma two players with varying numbers of individuals Respiro-fermentation is Nash equilibrium and evolutionary stable strategy (ESS)

20 Biological examples S. cerevisiae and Lactobacilli use fermentation even under aerobic conditions, if sufficient glucose is available. They behave in a selfish way. Other micro-organisms, such as Kluyverymyces, use respiration.

21 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 selfish behaviour is one of the causes of their pathological effects.

22 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 of labour with neurons, which degrade lactate to carbon dioxide and water.

23 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.

24 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)

25 Conclusions Game-theoretical concepts can be applied 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

26 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 Further reading: T. Pfeiffer, S. Schuster: Gametheoretical approaches to studying the evolution of biochemical systems. Trends Biochem. Sci. 30 (2005)

27 Conclusions (3) So far, game theory has not yet been intensely used in Systems Biology. However, this may change in the future because game theory is well-suited for analysing situations where the whole is more than the sum of its parts

28 Cooperations on this project Thomas Pfeiffer (Harvard U) Sebastian Bonhoeffer (ETH Zürich) Tobias Frick (U Tübingen) Sergio Rossell (VU Amsterdam) Gunter Neumann (my group at Jena University)

29 Population payoffs and resource level T. Frick, S. Schuster: An example of the prisoner's dilemma in biochemistry. Naturwissenschaften 90 (2003)

30 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) Optimal enzyme concn. E j = E tot ( ) ( r j q 1 q ) q r 1 (q: equilibrium constant) Position in the chain

31 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

32 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

33 Example: Degradation of 4-chlorosalicylate From: O. Pelz et al., Environm. Microb. 1 (1999),