Bulletin of Mathematical Biology

, Volume 56, Issue 1, pp 1–64 | Cite as

“The arrival of the fittest”: Toward a theory of biological organization

  • Walter Fontana
  • Leo W. Buss


The formal structure of evolutionary theory is based upon the dynamics of alleles, individuals and populations. As such, the theory must assume the prior existence of these entities. This existence problem was recognized nearly a century ago, when DeVries (1904,Species and Varieties: Their Origin by Mutation) stated. “Natural selection may explain the survival of the fittest, but it cannot explain the arrival of the fittest.” At the heart of the existence problem is determining how biological organizations arise in ontogeny and in phylogeny.

We develop a minimal theory of biological organization based on two abstractions from chemistry. The theory is formulated using λ-calculus, which provides a natural framework capturing (i) the constructive feature of chemistry, that the collision of molecules generates specific new molecules, and (ii) chemistry's diversity of equivalence classes, that many different reactants can yield the same stable product. We employ a well-stirred and constrained stochastic flow reactor to explore the generic behavior of large numbers of applicatively interacting λ-expressions. This constructive dynamical system generates fixed systems of transformation characterized by syntactical and functional invariances.

Organizations are recognized and defined by these syntactical and functional regularities. Objects retained within an organization realize and algebraic structure and possess a grammar which is invariant under the interaction between objects. An organization is self-maintaining, and is characterized by (i) boundaries established by the invariances, (ii) strong self-repair capabilities responsible for a robustness to perturbation, and (iii) a center, defined as the smallest kinetically persistent and self-maintaining generator set of the algebra.

Imposition of different boundary conditions on the stochastic flow reactor generates different levels of organization, and a diversity of organizations within each level. Level 0 is defined by selfcopying objects or simple ensembles of copying objects. Level 1 denotes a new object class, whose objects are self-maintaining organizations made of Level 0 objects, and Level 2 is defined by self-maintaining metaorganizations composed of Level 1 organizations.

These results invite analogy to the history of life, that is, to the progression from self-replication to self-maintaining procaryotic organizations to ultimately yield self-maintaining eucaryotic organizations. In our system self-maintaining organizations arise as a generic consequence of two features of chemistry, without appeal to natural selection. We hold these findings as calling for increased attention to the structural basis of biological order.


Normal Form Flow Reactor Biological Organization Existence Problem Random Object 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Anderson, P. W. 1972. More is different.Science 177, 393–396.Google Scholar
  2. Baas, N. A. 1993. Emergence and higher order structures. InProceedings of the Conference on Systems Research and Cybernetics, Baden-Baden. In press.Google Scholar
  3. Bachmann, P. A., P. L. Luisi and J. Lang. 1992. Autocatalytic self-replicating micelles as models for prebiotic structures.Nature 357, 57–59.CrossRefGoogle Scholar
  4. Bagley, R. J. and J. D. Farmer. 1992. Spontaneous emergence of a metabolism. InArtificial Life II, C. G. Langton, C. Taylor, J. D. Farmer and S. Rasmussen (Eds), pp. 93–141. Redwood City: Addison-Wesley.Google Scholar
  5. Bagley, R. J., J. D. Farmer and W. Fontana. 1992. Evolution of a metabolism. InArtificial Life II, C. G. Langton, C. Taylor, J. D. Farmer and S. Rasmussen (Eds), pp. 141–158 Redwood City: Addison-Wesley.Google Scholar
  6. Bagley, R. J., J. D. Farmer, S. A. Kauffman, N. H. Packard, A. S. Perelson and I. M. Stadnyk. 1989. Modeling adaptive biological systems.Biosystems 23, 113–138.CrossRefGoogle Scholar
  7. Banatre, J.-P. and D. Le Metayer. 1986. A new computational model and its discipline of programming. Technical Report. INRIA Report 566.Google Scholar
  8. Barendregt, H. G. 1984.The Lambda Calculus: Its Syntax and Semantics (second edition). Amsterdam: North-Holland.zbMATHGoogle Scholar
  9. Berry, G. and G. Boudol. 1990. The Chemical Abstract Machine. In17th ACM Annual Symposium on Principles of Programming Languages, pp. 81–94. New York: ACM Press.Google Scholar
  10. Bjoerlist, M. C. and P. Hogeweg. 1991. Spiral wave structure in prebiotic evolution: hypercycles stable against parasites.Physica D 48, 17–28.CrossRefGoogle Scholar
  11. Buss, L. W. 1987.The Evolution of Individuality. Princeton: Princeton University Press.Google Scholar
  12. Buss, L. W. 1994. Protocell life cycles. InEarly Life on Earth, S. Bergstrom (Ed.). New York: Columbia University Press, In press.Google Scholar
  13. Church, A. 1932. A set of postulates for the foundation of logic.Annals of Math. (2)33, 346–366 and34, 839–864.zbMATHMathSciNetCrossRefGoogle Scholar
  14. Church, A. 1941.The Calculi of Lambda Conversion. Princeton: Princeton University Press.Google Scholar
  15. Church, A. and J. B. Rosser. 1936. Some properties of conversion.Trans. Amer. Math. Soc. 39, 472–482.zbMATHMathSciNetCrossRefGoogle Scholar
  16. Curry, H. B. and R. Feys. 1958.Combinatory Logic. Volume 1. Amsterdam: North-Holland.Google Scholar
  17. Curry, H. B., J. R. Hindley and J. P. Seldin 1972.Combinatory Logic. Volume 2, Amsterdam: North-Holland.Google Scholar
  18. Dyson, F. 1985.Origins of Life, Cambridge: Cambridge University Press.Google Scholar
  19. Eigen, M. 1971. Self-organization of matter and the evolution of biological macro-molecules.Naturwissenschaften 58, 465–526.CrossRefGoogle Scholar
  20. Eigen, M. and P. Schuster. 1977. The hypercycle. A principle of natural self-organization. A: Emergence of the hypercycle.Naturwissenschaften 64, 541–565.CrossRefGoogle Scholar
  21. Eigen, M. and P. Schuster. 1978a. The hypercycle. A principle of natural self-organization. B: The abstract hypercycle.Naturwissenschaften 65, 7–41.CrossRefGoogle Scholar
  22. Eigen, N. and P. Schuster. 1978b. The hypercycle. A principle of natural self-organization. C: The realistic hypercycle.Naturwissenschaften 65, 341–369.CrossRefGoogle Scholar
  23. Eigen, M. and P. Schuster. 1979.The Hypercycle. Berlin: Springer.Google Scholar
  24. Eigen, M. and P. Schuster. 1982. Stages of emerging life-five principles of early organization.J. Mol. Evol. 19, 47–61.CrossRefGoogle Scholar
  25. Farmer, J. D., S. A. Kauffman and N. H. Packard. 1986. Autocatalytic replication of polymers.Physica D 22, 50–67.MathSciNetCrossRefGoogle Scholar
  26. Fisher, R. A. 1930.The Genetical Theory of Natural Selection. Oxford: Clarendon Press.zbMATHGoogle Scholar
  27. Fontana, W. 1990. Algorithmic chemistry: a model for functional self-organization. Technical Report. SFI90-011, Santa Fe Institute, Santa Fe, and Technical Report LA-UR-90-1959, Los Alamos National Laboratory, Los Alamos.Google Scholar
  28. Fontana, W. 1991. Functional self-organization in complex systems. In1990 Lectures in Complex Systems, L. Nadel and D. Stein (Eds), pp. 407–426. Redwood City: Addison-Wesley.Google Scholar
  29. Fontana, W. 1992. Algorithmic chemistry. InArtificial Life II, C. G. Langton, C. Taylor, J. D. Farmer and S. Rasmussen (Eds), pp. 159–209. Redwood City: Addison-Wesley.Google Scholar
  30. Fontana, W. and L. W. Buss. 1993. What would be conserved ‘if the tape were played twice’Proceedings of the National Academy of Sciences, USA. (In press).Google Scholar
  31. Forrest, S. 1990. Emergent computation: self-organizing, collective cooperative phenomena in natural and artificial computing networks.Physica D 42, 1–11.zbMATHMathSciNetCrossRefGoogle Scholar
  32. Fox, S. W. and K. Dose. 1977.Molecular Evolution and the Origin of Life. New York: Marcel Dekker.Google Scholar
  33. Futrelle, R. P. and A. W. Miller. 1992. Simulation of metabolism and evolution using Quasichemistry. InBiomedical Modeling and Simulation, J. Eisenfeld, D. S. Levine and M. Witten (Eds), pp. 273–280. Amsterdam: Elsevier.Google Scholar
  34. Gould, S. J. and R. C. Lewontin. 1979. The spandrels of San Marco and the Panglossian paradigm: a critique of the adaptionist programme.Proc. R. Soc., London B205, 581–598.CrossRefGoogle Scholar
  35. Haldane, J. B. S. 1932.The Causes of Evolution, New York: Longmans, Green.Google Scholar
  36. Haldane, J. B. S. 1954. The origins of life.New Biology 16, 12–27.Google Scholar
  37. Hindley, J. R. and J. P. Seldin. 1986.Introduction to Combinators and λ-Calculus. Cambridge: Cambridge University Press.Google Scholar
  38. Hofbauer, J. and K. Sigmund. 1988.The Theory of Evolution and Dynamics Systems. Cambridge: Cambridge University Press.Google Scholar
  39. Huet, G. 1992.Constructive Computation Theory. Part I. Notes de Cours. DEa Informatique, Mathématiques et Applications. (Available by anonymous ftp from ftp. inriafr,/INRIA/formel /cct/CCT.dvi.)Google Scholar
  40. Huet, G. and D. C. Oppen. 1980. Equations as rewrite rules. InFormal Languages: Perspectives and Open Problems, R. Book (Ed.), New York: Academic Press.Google Scholar
  41. Jacob, F. (1982).The Possible and the Actual. New York: Pantheon Books.Google Scholar
  42. Kauffman, S. A. 1971. Cellular homeostasis, epigenesis and replication in randomly aggregated macromolecular systems.J. Cybernetics 1, 71–96.Google Scholar
  43. Kauffman, S. A. 1986. Autocatalytic sets of proteins.J. theor. Biol. 119, 1–24.CrossRefGoogle Scholar
  44. von Kiedrowski, G. 1986. A self-replicating hexadeoxynucleotide.Angew. Chem. 98, 932–934.Google Scholar
  45. Knuth, D. E. and P. E. Bendix. 1970. Simple word problems in universal algebra. InComputational Problems in Abstract Algebra, J. Leech (ed.), pp. 263–297. New York: Pergamon Press.Google Scholar
  46. Lakoff, G. 1987.Women, Fire, and Dangerous Things. What Categories Reveal about the Mind. Chicago: University of Chicago Press.Google Scholar
  47. Lane, D. 1993a. Artificial worlds and economics. Part I.J. Evol. Econ. 3, 89–107.CrossRefGoogle Scholar
  48. Lane, D. 1993b. Artificial worlds and economics. Part II.J. Evol. Econ. (in press).Google Scholar
  49. Leifer, E. M. 1991.Actors as Observers. A Theory of Skill in Social Relationships New York: Garland Publishing.Google Scholar
  50. Lewontin, R. C. 1970. The units of selection.Ann. Rev. Ecol. System. 1, 1–18.zbMATHCrossRefGoogle Scholar
  51. Lindgren, K. 1992. Evolutionary phenomena in simple dynamics. InArtificial Life II, C. G. Langton, C. Taylor, J. D. Farmer and S. Rasmussen (Eds), pp. 295–312. Redwood City: Addison-Wesley.Google Scholar
  52. Lotka, A. J. 1925.Elements of Physical Biology. New York: Dover.zbMATHGoogle Scholar
  53. Luisi, P. L. 1993. Defining the transition to life: self-replicating bounded structures and chemical autopoiesis. InThinking About Biology, W. Stein and F. J. Varela (eds), pp. 3–23. Redwood City: Addison-Wesley.Google Scholar
  54. Maturana, H. and F. Varela. 1973.De Máquinas y Seres Vivos: Una teoría de la organizacíon biológica. Santiago de Chile: Editorial Universitaria. (Reprinted in English in Maturana and Varela, 1980).Google Scholar
  55. Maturana, H. and F. Varela. 1980.Autopoiesis and Cognition: The Realization of the Living. Boston: D. Reidel.Google Scholar
  56. May, R. M. (ed.) 1976.Theoretical Ecology: Principles and Applications. Oxford: Blackwell Scientific.Google Scholar
  57. Maynard-Smith, J. 1982.Evolution and the Theory of Games. Cambridge. Cambridge University Press.Google Scholar
  58. Maynard-Smith, J., R. Burian, S. A. Kauffman, P. Alberch, J. Campbell, B. Goodwin, R. Lande, D. Raup and L. Wolpert. 1985. Development constraints and evolution.Q. Rev. Biol. 60, 265–287.CrossRefGoogle Scholar
  59. Miller, S. M. and L. E. Orgel. 1974.The Origins of Life on the Earth. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  60. Morowitz, H. J. 1992.Beginnings of Cellular Life. New Haven: Yale University Press.Google Scholar
  61. Newman, M. H. A. 1941. On theories with a combinatorial definition of “equivalence”.Annals of Math. 43, 223–243.CrossRefGoogle Scholar
  62. Niesert, U., D. Harnasch and C. Bresch. 1981. Origin of life Between Scylla and Charybdis.J. Mol. Evol. 17, 348–353.CrossRefGoogle Scholar
  63. Odifreddi, P. 1989.Classical Recursion Theory. Amsterdam: North-Holland.zbMATHGoogle Scholar
  64. Oparin, A. I. 1924.Proiskhozhdenie zhizni. Moscow: Moskovskij Rabochij.Google Scholar
  65. Padgett, J. F. and C. K. Ansel. 1993. Robust action in the rise of the Medici: 1400–1434.Am. J. Soc. 98, 1259–1319.CrossRefGoogle Scholar
  66. Penrose, R. 1989.The Emperor's New Mind. Oxford: Oxford University Press.Google Scholar
  67. Revesz, G. E. 1988.Lambda-Calculus, Combinators, and Functional Programming. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  68. Rössler, O. 1971. Ein systemtheoretisches Modell zur Biogenese.Zeitschrift für Naturforschung 26b, 741–746.Google Scholar
  69. Stadler, P.F., W. Fontana and J. H. Miller. 1993. Random catalytic reaction networks.Physica D 63, 378–392.zbMATHMathSciNetCrossRefGoogle Scholar
  70. Thürk, M. 1993. Ein Modell zur Selbstorganisation von Automatenalgorithmen zum Studium molekularer Evolution. PhD thesis. Friedrich-Schiller Universität Jena. Germany.Google Scholar
  71. Tjiuikaua, T., P. Ballester and J. Rebek Jr. 1990. A self-replicating system.J. Am. Chem. Soc. 112, 1249–1250.CrossRefGoogle Scholar
  72. Varela, F., A. Coutinho, B. Dupire and N. N. Vaz. 1988. Cognitive networks Immune, neural, and otherwise. InTheoretical Immunology, Part Two, A. S. Perelson (ed.), pp. 359–375. Redwood City: Addison-Wesley.Google Scholar
  73. Varela, F., H. R. Maturana and R. Uribe. 1974. Autopoiesis: the organization of living systems, its characterization and a model.Bio Systems 5, 187–196.Google Scholar
  74. Vrba, E. and S. J. Gould. 1986. Sorting is not selection.Paleobiology 12, 217–228.Google Scholar
  75. DeVries, H. 1904.Species and Varieties: Their Origin by Mutation. Chicago: Open Court.Google Scholar
  76. Winograd, T. and F. Flores. 1986.Understanding Computers and Cognition. Reading: Addison-Wesley.zbMATHGoogle Scholar
  77. Wright, S. 1931. Evolution in Mendelian populations.Genetics 16, 97–159.Google Scholar

Copyright information

© Society for Mathematical Biology 1993

Authors and Affiliations

  • Walter Fontana
    • 1
  • Leo W. Buss
    • 2
  1. 1.Santa Fe InstituteSanta FeUSA
  2. 2.Department of Biology and Department of Geology and GeophysicsYale UniversityNew HavenUSA

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