Abstract
Mutation is introduced into autocatalytic reaction networks. The differential equations obtained are neither of repliator-type nor can they be transformed straightway into a linear equation. Examples of low dimensional dynamical systems —n=2, 3 and 4 — are discussed and complete qualitative analysis is presented. Error thresholds known from simple replication-mutation kinetics with frequency independent replication rates occur here as well. Instead of cooperative transitions or higher order phase transitions the thresholds appear here as supercritical or subcritical bifurcations being analogous to first-order phase transitions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature
Andrade, M. A., J. C. Nuño, F. Morán, F. Montero and G. Mptisos. 1993. Complex dynamics of a catalytic network having faulty replication into error-species.Physica D 63, 21–40.
Cech, T. R. 1986. RNA as an enzyme.Sci. Am. 255 (5), 76–84.
Doudna, J. A. and J. W. Szostak. 1989. RNA-catalyzed synthesis of complementary-strand RNA.Nature 339, 519–522.
Eigen, M. 1971. Selforganization of matter and the evolution of biological macromolecules.Die Naturwissenschaften 58, 465–523.
Eigen, M. and P. Schuster. 1977. The hypercycle. A principle of natural self-organization. Part A: Emergence of the hypercycle.Die Naturwissenschaften 64, 541–565.
Eigen, M. and P. Schuster. 1978. The hypercycle. A principle of natural self-organization. Part B: The abstract hypercycle.Die Naturwissenschaften 65, 7–41.
Eigen, M. and P. Schuster. 1979.The Hypercycle. A Principle of Natural Self-organization. Berlin: Springer-Verlag.
Eigen, M., J. McCaskill and P. Schuster. 1988. The molecular quasi-species.J. Phys. Chem. 92, 6881–6891.
Eigen, M., J. McCaskill and P. Schuster. 1989. The molecular quasi-species.Adv. Chem. Phys. 75, 149–263.
Eigen, M., C. K. Biebricher, M. Gebinoga and W. C. Gardiner. 1991. The hypercycle. Coupling of RNA and protein biosynthesis in the infection cycle of an RNA bacteriophage.Biochemistry,30, 11,005–11,018.
Hadeler, P. K. 1981. Stable polymorphisms in a selection model with mutation.SIAM J. Appl. Math. 41, 1–7.
Hamming, R. W. 1950. Error detecting and error correcting codes.Bell Syst. Tech. J. 29, 147–160.
Hofbauer, J. 1981. On the occurrence of limit cycles in the Volterra-Lotka equation.Nonlinear Analysis 5, 1003–1007.
Hofbauer, J. 1985. The selection mutation equation.J. Math. Biol. 23, 41–53.
Hofbauer, J. and K. Sigmund. 1988.The Theory of Evolution and Dynamical Systems. Cambridge: Cambridge University Press.
Hofbauer, J., J. Mallet-Paret and H. L. Smith. 1991. Stable periodic solutions for the hypercycle system.J. Dynam. Diff. Eqns. 3, 423–236.
Leuthäusser, I. 1987. Statistical mechanics of Eigen's evolution model.J. Statist. Phys.,48, 343–360.
Nowak, M. and P. Schuster. 1989. Error thresholds of replication in finite populations. Mutation frequencies and the onset of Muller's ratchet.J. theor. Biol. 137, 375–395.
Nuño, J. C., M. A. Andrade, F. Morán and F. Montero. 1993a. A model of an autocatalytic network formed by error-prone self-replicative species.Bull. Math. Biol. 55, 385–415.
Nuño, J. C., M. A. Andrade and F. Montero. 1993b. Non-uniformities and superimposed competition in a model of an autocatalytic network formed by error-prone self-replicative species.Bull. Math. Biol. 55, 417–449.
Rumschitzky, D. 1987. Spectral properties of Eigen evolution matrices.J. Math. Biol. 24, 667–680.
Schlögl, F. 1972. Chemical reaction models for non-equilibrium phase transitions.Z.f. Physik 253, 147–161.
Schnabl, W., P. F. Stadler, C. Forst and P. Schuster. 1991. Full characterization of a strange attractor. Chaotic dynamics in low-dimensional replicator systems.Physica D 48, 65–90.
Schuster, P. 1986. Dynamics of molecular evolution.Physica D 22, 100–119.
Schuster, P., K. Sigmund and R. Wolff. 1978. Dynamical systems under constant organization I. Topological analysis of a family of non-linear differential equations—a model for catalytic hypercycles.Bull. Math. Biol. 40, 743–769.
Schuster, P. and K. Sigmund 1983. Replicator dynamics.J. theor. Biol. 100, 533–538.
Schuster, P. and K. Sigmund. 1985. Dynamics of evolutionary optimization.Ber. Bunsenges. Phys. Chem. 89, 668–682.
Schuster, P. 1986. Dynamics of molecular evolution.Physica D 22, 100–119.
Schuster, P. and J. Swetina. 1988. Stationary mutant distribution and evolutionary optimization.Bull. Math. Biol. 50, 635–660.
Stadler, P. F. 1991. Complementary replication.Math. Biosc. 107, 83–109.
Stadler, P. F. and P. Schuster. 1992. Mutation in autocatalytic reaction networks. An analysis based on perturbation theory.J. Math. Biol. 30, 597–632.
Stadler, P. F. and J. C. Nuño. 1994. The influence of mutation on autocatalytic reaction networks.Math. Biosc. (in press).
Swetina, J. and P. Schuster. 1982. Self-replication with errors. A model for polynucleotide replication.Biophys. Chem. 16, 329–345.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Stadler, P.F., Schnabl, W., Forst, C.V. et al. Dynamics of small autocatalytic reaction networks—II. Replication, mutation and catalysis. Bltn Mathcal Biology 57, 21–61 (1995). https://doi.org/10.1007/BF02458315
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02458315