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Principles of Evolution pp 297–329Cite as

Bacterial Games

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Abstract

Microbial laboratory communities have become model systems for studying the complex interplay between nonlinear dynamics of evolutionary selection forces, stochastic fluctuations arising from the probabilistic nature of interactions, and spatial organization. Major research goals are to identify and understand mechanisms that ensure viability of microbial colonies by allowing for species diversity, cooperative behavior and other kinds of “social” behavior. A synthesis of evolutionary game theory, nonlinear dynamics, and the theory of stochastic processes provides the mathematical tools and conceptual framework for a deeper understanding of these ecological systems. We give an introduction to the modern formulation of these theories and illustrate their effectiveness, focusing on selected examples of microbial systems. Intrinsic fluctuations, stemming from the discreteness of individuals, are ubiquitous, and can have important impact on the stability of ecosystems. In the absence of speciation, extinction of species is unavoidable, may, however, take very long times. We provide a general concept for defining survival and extinction on ecological time scales. Spatial degrees of freedom come with a certain mobility of individuals. When the latter is sufficiently high, bacterial community structures can be understood through mapping individual-based models, in a continuum approach, onto stochastic partial differential equations. These allow progress using methods of nonlinear dynamics such as bifurcation analysis and invariant manifolds. We conclude with a perspective on the current challenges in quantifying bacterial pattern formation, and how this might have an impact on fundamental research in nonequilibrium physics .

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Notes

  1. 1.

    The two-species Lotka–Volterra equations describe a predator–prey system where the per capita growth rate of the prey decreases linearly with the number of predators present. In the absence of prey, predators die, but there is a positive contribution to their growth which increases linearly with the amount of prey present [22, 23].

  2. 2.

    Note that ε is the fraction of carbon source kept by cooperators solely for themselves and \(x(1-\epsilon)\) is the amount of carbon source shared with the whole community. Hence, the linear growth rates of cooperators and defectors would be \(\epsilon + x(1-\epsilon) -c\) and \(x(1-\epsilon)\), respectively, where c is the metabolic cost for invertase production.

  3. 3.

    You may also want to haven a look at the movies posted on http://www.theorie.physik.uni-muenchen.de/lsfrey/research/fields/biological_physics/2007_004/. There is also a Wolfram demonstration project that can be downloaded from the internet: http://demonstrations.wolfram.com/BiodiversityInSpatialRockPaperScissorsGames/.

References

  1. L. Hall-Stoodley, J.W. Costerton, P. Stoodley, Nat. Rev. Microbiol. 2, 95 (2004)

    Article  Google Scholar 

  2. T.J. Battin, W.T. Sloan, S. Kjelleberg, H. Daims, I.M. Head, T.P. Curtis, L. Eberl, Nat. Rev. Microbiol. 5, 76 (2007)

    Article  Google Scholar 

  3. G.J. Velicer, Trends Microbiol. 11, 330 (2003)

    Article  Google Scholar 

  4. J.B. Xavier, K.R. Foster, Proc. Natl. Acad. Sci. USA 104, 876 (2007)

    Article  ADS  Google Scholar 

  5. C.D. Nadell, J.B. Xavier, S.A. Levin, K.R. Foster, PLoS Biol. 6, e14 (2008)

    Article  Google Scholar 

  6. B. Kerr, M.A. Riley, M.W. Feldman, B.J.M. Bohannan, Nature 418, 171 (2002)

    Article  ADS  Google Scholar 

  7. J.B. Xavier, E. Martinez-Gracia, K.R. Foster, Am. Nat. 174, 1 (2009)

    Article  Google Scholar 

  8. J. Gore, H. Youk, A. van Oudenaarden, Nature 459, 253 (2009)

    Article  ADS  Google Scholar 

  9. J.S. Chuang, O. Rivoire, S. Leibler, Science 323, 272 (2009)

    Article  Google Scholar 

  10. S.A. Levin, Am. Nat. 108, 207 (1974)

    Article  Google Scholar 

  11. M.P. Hassell, H.N. Comins, R.M. May, Nature 353, 255 (1991)

    Article  ADS  Google Scholar 

  12. R. Durrett, S. Levin, Theor. Popul. Biol. 46, 363 (1994)

    Article  MATH  Google Scholar 

  13. T. Reichenbach, M. Mobilia, E. Frey, Nature 448, 1046 (2007)

    Article  ADS  Google Scholar 

  14. S.A. West, A.S. Griffin, A. Gardner, S.P. Diggle, Nat. Rev. Microbiol. 4, 597 (2006)

    Article  Google Scholar 

  15. R.M. May, W.J. Leonard, SIAM J. Appl. Math. 29, 243 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  16. M.J. Osborne, An Introduction to Game Theory (Oxford University Press, Oxford, 2004)

    Google Scholar 

  17. J.F. Nash, Proc. Natl. Acad. Sci. USA 36, 48 (1950)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. R.M. Dawes, Annu. Rev. Psychol. 31, 169 (1980)

    Article  Google Scholar 

  19. R. Axelrod, W.D. Hamilton, Science 211, 1390 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  20. J. Maynard Smith, G.R. Price, Nature 246, 15 (1973)

    Article  ADS  Google Scholar 

  21. J. Maynard Smith, Evolution and the Theory of Games (Cambridge University Press, Cambridge, 1982)

    MATH  Google Scholar 

  22. A.J. Lotka, J. Am. Chem. Soc. 42, 1595 (1920)

    Article  Google Scholar 

  23. V. Volterra, Mem. Accad. Lincei 2, 31 (1926)

    Google Scholar 

  24. J. Hofbauer, K. Sigmund, Evolutionary Games and Population Dynamics (Cambridge University Press, Cambridge, 1998)

    MATH  Google Scholar 

  25. M.E. Hibbing, C. Fuqua, M.R. Parsek, S.B. Peterson, Nat. Rev. Microviol. 8, 15 (2010)

    Article  Google Scholar 

  26. J. Roughgarden, Ecology 3, 453 (1971)

    Article  Google Scholar 

  27. A. Melbinger, J. Cremer, E. Frey, Evolutionary game theory in growing populations. Phys. Rev. Lett. 105, 178101 (2010)

    Article  ADS  Google Scholar 

  28. S. Wright, Proc. Natl. Acad. Sci. USA 31, 382 (1945)

    Article  MATH  ADS  Google Scholar 

  29. S. Wright, Evolution and the Genetics of Populations (Chicago University Press, Chicago, IL, 1969)

    Google Scholar 

  30. W.J. Ewens, Mathematical Population Genetics, 2nd edn. (Springer, New York, NY, 2004)

    MATH  Google Scholar 

  31. P.A. Moran, The Statistical Processes of Evolutionary Theory (Clarendon, Oxford, 1964)

    Google Scholar 

  32. S.H. Strogatz, Nonlinear Dynamics and Chaos (Westview, Boulder, CO, 1994)

    Google Scholar 

  33. D. Greig, M. Travisano, Proc. R. Soc. Lond. B 271, S25 (2004)

    Article  Google Scholar 

  34. A. Buckling, F. Harrison, M. Vos, M.A. Brockhurst, A. Gardner, S.A. West, A. Griffin, FEMS Microbiol. Ecol. 62, 135 (2007)

    Article  Google Scholar 

  35. R.L. Trivers, Q. Rev. Biol. 46, 35 (1971)

    Article  Google Scholar 

  36. C.M. Waters, B.L. Bassle, Annu. Rev. Cell Dev. Biol. 21, 319 (2005)

    Article  Google Scholar 

  37. R. Durrett, S. Levin, J. Theor. Biol. 185, 165 (1997)

    Article  Google Scholar 

  38. R. Durrett, S. Levin, Theor. Popul. Biol. 53, 30 (1998)

    Article  MATH  Google Scholar 

  39. T.L. Czárán, R.F. Hoekstra, L. Pagie, Proc. Natl. Acad. Sci. USA 99, 786 (2002)

    Article  ADS  Google Scholar 

  40. R. Axelrod, The Evolution of Cooperation (Basic Books, New York, NY, 1984)

    Google Scholar 

  41. M.A. Nowak, Science 314, 1560 (2006)

    Article  ADS  Google Scholar 

  42. T. Yamagishi, J. Pers. Soc. Psychol. 51, 110 (1986)

    Article  Google Scholar 

  43. W.D. Hamilton, Narrow Roads of Gene Land: Evolution of Social Behaviour (Oxford University Press, Oxford, 1996)

    Google Scholar 

  44. J.F. Crow, M. Kimura, An Introduction to Population Genetics (Blackburn Press, Caldwell, NJ, 2009)

    Google Scholar 

  45. J. Cremer, T. Reichenbach, E. Frey, New J. Phys. 11, 093029 (2009)

    Article  ADS  Google Scholar 

  46. E. Ben-Jacob, I. Cohen, H. Levine, Adv. Phys. 49, 395 (2000)

    Article  ADS  Google Scholar 

  47. O. Hallatschek, P. Hersen, S. Ramanathan, D.R. Nelson, Proc. Natl. Acad. Sci. USA 104, 19926 (2007)

    Article  ADS  Google Scholar 

  48. C.J. Ingham, E.B. Jacob, BMC Microbiol. 8, 36 (2008)

    Article  Google Scholar 

  49. A.T. Henrici, The Biology of Bacteria: The Bacillaceae, 3rd edn. (Heath, Lexington, MA 1948)

    Google Scholar 

  50. E. Ben-Jacob, I. Cohen, I. Golding, D.L. Gutnick, M. Tcherpakov, D. Helbing, I.G. Ron Open, Physica A 282, 247 (2000)

    Article  ADS  Google Scholar 

  51. M. Matsushita, H. Fujikawa, Physica A 168, 498 (1990)

    Article  ADS  Google Scholar 

  52. R. Rudner, O. Martsinkevich, W. Leung, E.D. Jarvis, Mol. Microbiol. 27, 687 (1998)

    Article  Google Scholar 

  53. M. Eisenbach, Mol. Microbiol. 4, 161 (1990)

    Article  Google Scholar 

  54. J. Henrichsen, Acta Pathol. Microbiol. Scand. B 80, 623 (1972)

    Google Scholar 

  55. J. Henrichsen, L.O. Froholm, K. Bovre, Acta Pathol. Microbiol. Scand. B 80, 445 (1972)

    Google Scholar 

  56. S. Park, P.M. Wolanin, E.A. Yuzbashyan, H. Lin, N.C. Darnton, J.B. Stock, P. Silberzan, R. Austin, Proc. Natl. Acad. Sci. USA 100, 13910 (2003)

    Article  ADS  Google Scholar 

  57. O. Hallatschek, D.R. Nelson, Theor. Popul. Biol. 73, 158 (2008)

    Article  MATH  Google Scholar 

  58. H.H. McAdams, A. Arkin, Trends Genet. 15, 65 (1999)

    Article  Google Scholar 

  59. M. Kaern, T.C. Elston, W.J. Blake, J.J. Collins, Nat. Rev. Microbiol. 6, 451 (2005)

    Article  Google Scholar 

  60. J.W. Veening, W.K. Smits, O.P. Kuipers, Annu. Rev. Microbiol. 62, 193 (2008)

    Article  Google Scholar 

  61. W.K. Smits, O.P. Kuipers, J.W. Veening, Nat. Rev. Microbiol. 4, 259 (2006)

    Article  Google Scholar 

  62. D. Dubnau, R. Losick, Mol. Microbiol. 61, 564 (2006)

    Article  Google Scholar 

  63. M. Leisner, K. Stingl, E. Frey, B. Maier, Curr. Opin. Microbiol. 11, 553 (2008)

    Article  Google Scholar 

  64. M. Delbrück, J. Chem. Phys. 8, 120 (1940)

    Article  ADS  Google Scholar 

  65. A. Traulsen, J.C. Claussen, C. Hauert, Phys. Rev. Lett. 95, 238701 (2005)

    Article  ADS  Google Scholar 

  66. T. Reichenbach, M. Mobilia, E. Frey, Phys. Rev. E 74, 051907 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  67. A. Traulsen, J.C. Claussen, C. Hauert, Phys. Rev. E 74, 011901 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  68. J. Cremer, T. Reichenbach, E. Frey, Eur. Phys. J. B 63, 373 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  69. N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981)

    MATH  Google Scholar 

  70. C.W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, Heidelberg, 2007)

    Google Scholar 

  71. T. Antal, I. Scheuring, Bull. Math. Biol. 68, 1923 (2006)

    Article  MathSciNet  Google Scholar 

  72. C. Taylor, Y. Iwasa, M.A. Nowak, J. Theor. Biol. 243, 245 (2006)

    Article  MathSciNet  Google Scholar 

  73. M. Ifti, B. Bergersen, Eur. Phys. J. E 10, 241 (2003)

    Article  Google Scholar 

  74. M. Ifti, B. Bergersen, Eur. Phys. J. B 37, 101 (2004)

    Article  ADS  Google Scholar 

  75. A. Dobrinevski, E. Frey, Extinction in neutrally stable stochastic Lotka-Volterra models, Submitted [arXiv:1001.5235]

    Google Scholar 

  76. M. Berr, T. Reichenbach, M. Schottenloher, E. Frey, Phys. Rev. Lett. 102, 048102 (2009)

    Article  ADS  Google Scholar 

  77. R.M. May, Stability and Complexity in Model Ecosystems, 2nd edn. (Princeton University Press, Princeton, NJ, 1974)

    Google Scholar 

  78. J.D. Murray, Mathematical Biology, 3rd edn. (Springer, Berlin, Heidelberg, 2002)

    MATH  Google Scholar 

  79. A.M. Turing, Philos. Trans. R. Soc. Lond. B 237, 37 (1952)

    Article  ADS  Google Scholar 

  80. M.A. Nowak, R.M. May, Nature 359, 826 (1992)

    Article  ADS  Google Scholar 

  81. M.P. Hassell, H.N. Comins, R.M. May, Nature 370, 290 (1994)

    Article  ADS  Google Scholar 

  82. B. Blasius, A. Huppert, L. Stone, Nature 399, 354 (1999)

    Article  ADS  Google Scholar 

  83. A.A. King, A. Hastings, Theor. Popul. Biol. 64, 431 (2003)

    Article  MATH  Google Scholar 

  84. G. Szabo, C. Hauert, Phys. Rev. Lett. 89, 118101 (2002)

    Article  ADS  Google Scholar 

  85. C. Hauert, M. Doebeli, Nature 428, 643 (2004)

    Article  ADS  Google Scholar 

  86. T.M. Scanlon, K.K. Caylor, I. Rodriguez-Iturbe, Nature 449, 209 (2007)

    Article  ADS  Google Scholar 

  87. S. Kefi, M. Rietkerk, C.L. Alados, Y. Pueyo, V.P. Papanastasis, A. ElAich, P.C. de Ruiter, Nature 449, 213 (2007)

    Article  ADS  Google Scholar 

  88. G. Szabó, G. Fáth, Phys. Rep. 446, 97 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  89. M. Perc, A. Szolnoki, G. Szabó, Phys. Rev. E 75, 052102 (2007)

    Article  ADS  Google Scholar 

  90. M.A. Nowak, Evolutionary Dynamics (Belknap Press, Cambridge, MA, 2006)

    MATH  Google Scholar 

  91. O.A. Igoshin, R. Welch, D. Kaiser, G. Oster, Proc. Natl. Acad. Sci. USA 101, 4256 (2004)

    Article  ADS  Google Scholar 

  92. A. McKane, D. Alonso, Bull. Math. Biol. 64, 913 (2002)

    Article  Google Scholar 

  93. E. Liebermann, C. Hauert, M.A. Nowak, Nature 433, 312 (2005)

    Article  ADS  Google Scholar 

  94. M. Mobilia, I.T. Georgiev, U.C. Täuber, Phys. Rev. E 73, 040903(R) (2006)

    Article  ADS  Google Scholar 

  95. M. Mobilia, I.T. Georgiev, U.C. Täuber, J. Stat. Phys. 128, 447 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  96. J.B.C. Jackson, L. Buss, Proc. Natl. Acad. Sci. USA 72, 5160 (1975)

    Article  ADS  Google Scholar 

  97. O. Gilg, I. Hanski, B. Sittler, Science 302, 866 (2001)

    Article  ADS  Google Scholar 

  98. B. Sinervo, C.M. Lively, Nature 380, 240 (1996)

    Article  ADS  Google Scholar 

  99. B.C. Kirkup, M.A. Riley, Nature 428, 412 (2004)

    Article  ADS  Google Scholar 

  100. A.J. Nicholson, Aust. J. Zool. 2, 9 (1954)

    Article  Google Scholar 

  101. T. Reichenbach, M. Mobilia, E. Frey. J. Theor. Biol. 254, 368 (2008)

    Article  Google Scholar 

  102. T. Reichenbach, M. Mobilia, E. Frey. Phys. Rev. Lett. 99, 238105 (2007)

    Article  ADS  Google Scholar 

  103. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, Berlin, Heidelberg, 1990)

    MATH  Google Scholar 

  104. M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993)

    Article  ADS  Google Scholar 

  105. I.S. Aranson, L. Kramer, Rev. Mod. Phys. 74, 99 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  106. W. van Saarloos, Phys. Rep. 386, 29 (2003)

    Article  MATH  ADS  Google Scholar 

  107. K. Tainaka, Phys. Rev. E 50, 3401 (1994)

    Article  ADS  Google Scholar 

  108. K. Sato, N. Konno, T. Yamaguchi, Mem. Muroran Inst. Technol. 47, 109 (1997)

    MathSciNet  Google Scholar 

  109. G. Szabó, A. Szolnoki, R. Izsak, J. Phys. A Math. Gen. 37, 2599 (2004)

    Article  MATH  ADS  Google Scholar 

  110. M. Peltomaki, M. Alava, Phys. Rev. E 78, 031906 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  111. T. Reichenbach, E. Frey, Phys. Rev. Lett. 101, 058102 (2008)

    Article  ADS  Google Scholar 

  112. M.R. Parsek, E.P. Greenberg, Trends Microbiol. 13, 27 (2005)

    Article  Google Scholar 

  113. C.T. MacDonald, J.H. Gibbs, A.C. Pipkin, Biopolymers 6, 1 (1968)

    Article  Google Scholar 

  114. G.M. Schütz, Exactly Solvable Models for Many-Body Systems Far from Equilibrium, Vol. 19 of Phase Transitions and Critical Phenomena (Academic, San Diego, CA, 2001), pp. 1–251

    Google Scholar 

  115. M. Mobilia, T. Reichenbach, H. Hinsch, T. Franosch, E. Frey, Banach Center Publ. 80, 101 (2008); arXiv:cond-mat/0612516

    Article  MathSciNet  Google Scholar 

  116. T.E. Harris, Ann. Probab. 2, 969 (1974)

    Article  MATH  Google Scholar 

  117. H. Hinrichsen, Adv. Phys. 49, 815 (2000)

    Article  ADS  Google Scholar 

  118. C. Castellano, S. Fortunato, V. Loreto, Rev. Mod. Phys. 81, 591 (2009)

    Article  ADS  Google Scholar 

  119. P. Clifford, A. Sudbury, Biometrika 60, 581 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  120. R. Holley, T.M. Liggett, Ann. Probab. 6, 198 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  121. T.M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes (Springer, Berlin, Heidelberg, 1999)

    MATH  Google Scholar 

  122. L. Frachebourg, P.L. Krapivsky, E. Ben-Naim, Phys. Rev. E 54, 6186 (1996)

    Article  ADS  Google Scholar 

  123. L. Frachebourg, P.L. Krapivsky, E. Ben-Naim, Phys. Rev. Lett. 77, 2125 (1996)

    Article  ADS  Google Scholar 

  124. S. Rulands, T. Reichenbach, E. Frey, Three-fold way to extinction in populations of cyclically competing species, J. Stat. Mech. L01003 (2011)

    Google Scholar 

  125. A. Winkler, T. Reichenbach, E. Frey, Phys. Rev. E 81, 060901(R) (2010)

    Article  ADS  Google Scholar 

  126. D.-Q. Jiang, M. Qian, M.-P. Qian, Mathematical Theory of Nonequilibrium Steady States (Springer, Berlin, Heidelberg, 2004)

    MATH  Google Scholar 

  127. F. Schlögl, Z. Phys. 198, 559 (1967)

    Article  ADS  Google Scholar 

  128. E.T. Jaynes, Ann. Rev. Phys. Chem. 3, 579 (1980)

    Article  ADS  Google Scholar 

  129. P. Glansdorff, I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations (Wiley-Interscience, New York, NY, 1971)

    MATH  Google Scholar 

  130. S. Goldstein, J.L. Lebowitz, Physica D 193, 53 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  131. U. Seifert, Phys. Rev. Lett. 95, 040602 (2005)

    Article  ADS  Google Scholar 

  132. B. Andrae, J. Cremer, T. Reichenbach, E. Frey, Phys. Rev. Lett. 104, 218102 (2010)

    Article  ADS  Google Scholar 

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Acknowledgements

We are indebted to Benjamin Andrae, Maximilian Berr, Jonas Cremer, Alexander Dobrinevsky, Anna Melbinger, Mauro Mobilia, Steffen Rulands, and Anton Winkler, with whom we had the pleasure to work on game theory. They have contributed with a multitude of creative ideas and through many insightful discussions have shaped our understanding of the topic. Financial support from the German Excellence Initiative via the program “Nanosystems Initiative Munich” and the German Research Foundation via the SFB TR12 “Symmetries and Universalities in Mesoscopic Systems” is gratefully acknowledged.

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Authors and Affiliations

  1. Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, D-80333, München, Germany

    Erwin Frey & Tobias Reichenbach

  2. Howard Hughes Medical Institute and Laboratory of Sensory Neuroscience, The Rockefeller University New York, New York, NY, 10065-6399, USA

    Tobias Reichenbach

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  1. Erwin Frey
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  1. , School of Engineering & Science, Jacobs University Bremen, Campus Ring 8, Bremen, 28759, Germany

    Hildegard Meyer-Ortmanns

  2. , Inst. für Wissenschaft Komplexer Systeme, Medizinische Universität Wien, Währinger Gürtel 18 - 20, Wien, 1090, Austria

    Stefan Thurner

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Frey, E., Reichenbach, T. (2011). Bacterial Games. In: Meyer-Ortmanns, H., Thurner, S. (eds) Principles of Evolution. The Frontiers Collection. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18137-5_13

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