Abstract
We analyse the stochastic properties of dynamical systems with finite populations of a few differentreplicator species. Our main interest is to evaluate the typicallifetime, i.e. the time for the extinction of the first species in the network, for different catalytic structures, as a function of the population size.
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Literature
Boerlijst, M. C. and P. Hogeweg. 1991. Spiral wave structure in pre-biotic evolution: hypercycles stable against parasites.Physica D 48, 17–28.
Eigen, M. 1971. Self-organization of matter and the evolution of biological macromolecules.Naturwissenschaften 58, 465–523.
Eigen, M. and P. Schuster. 1979.The Hypercycle—A Principle of Natural Self-organization. Berlin: Springer-Verlag.
Eigen, M., J. McCaskill and P. Schuster. 1989. The molecular quasi-species.Adv. Chem. Phys. 75, 149–263.
Gabriel, W. and R. Bürger. 1992. Survival of small populations under demographic stochasticity.Theor. Pop. Biol. 41, 44–71.
García-Tejedor, A., J. C. Sanz-Nuño, J. Olarrea, F. J. De la Rubia and F. Montero. 1988. Influence of the hypercycle on the error threshold: a stochastic approach.J. Theor. Biol. 134, 431–443.
Gardiner, C. W. 1985.Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Berlin: Springer-Verlag.
Gillespie, D. T. 1976. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions.J. Comp. Phys. 22, 403–434.
Hofbauer, J. and K. Sigmund. 1988.The Theory of Evolution and Dynamical Systems. Cambridge: Cambridge University Press.
Hunter, J. 1983.Mathematical Techniques of Applied Probability. Vol. I and II. New York: Academic Press.
Karlin, S. and Taylor, H. M., 1975.A First Course in Stochastic Processes, 2nd ed. New York: Academic Press.
Leung, H. K. 1984. Metastable states in a nonlinear stochastic model.Phys. Rev. A,30, 2609–2612.
Murray, J. D. 1989.Mathematical Biology. Berlin: Springer-Verlag.
Noller, H. F., V. Hoffarth and L. Zimniak. 1992. Unusual resistance of peptidyl transferase to protein extraction procedures.Science 256, 1416–1419.
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. 1993. A model of autocatalytic network formed by error-prone self-replicative species.Bull. Math. Biol. 55, 385–415.
Piccirilli, J. A., T. S. McConnell, A. J. Zaug, H. F. Noller and T. R. Cech. 1992. Aminoacyl esterase activity of thetetrahymena ribozyme.Science 256, 1420–1424.
Schuster, P., K. Sigmund and R. Wolf. 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.
Stadler, P. F. and J. C. Nuño. 1994. On a class of selection-mutation equations: the influence of mutation on autocatalytic reaction networks.Math. Biosc., in press.
Tarazona, P. 1992. Error thresholds for molecular quasi-species as phase transitions: from simple landscapes to spin-glass models.Phys. Rev. A 45, 6038–6050.
Van Kampen, N. G. 1981.Stochastic Processes in Physics and Chemistry. Amsterdam: North-Holland.
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Nuño, J.C., Tarazona, P. Lifetimes of small catalytic networks. Bltn Mathcal Biology 56, 875–898 (1994). https://doi.org/10.1007/BF02458272
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DOI: https://doi.org/10.1007/BF02458272