Stationary mutant distributions and evolutionary optimization Peter Schuster Jörg Swetina Article

Received: 25 November 1987 Revised: 16 February 1988 DOI :
10.1007/BF02460094

Cite this article as: Schuster, P. & Swetina, J. Bltn Mathcal Biology (1988) 50: 635. doi:10.1007/BF02460094 Abstract Molecular evolution is modelled by erroneous replication of binary sequences. We show how the selection of two species of equal or almost equal selective value is influenced by its nearest neighbours in sequence space. In the case of perfect neutrality and sufficiently small error rates we find that the Hamming distance between the species determines selection. As the error rate increases the fitness parameters of neighbouring species become more and more important. In the case of almost neutral sequences we observe a critical replication accuracy at which a drastic change in the “quasispecies”, in the stationary mutant distribution occurs. Thus, in frequently mutating populations fitness turns out to be an ensemble property rather than an attribute of the individual.

In addition we investigate the time dependence of the mean excess production as a function of initial conditions. Although it is optimized under most conditions, cases can be found which are characterized by decrease or non-monotonous change in mean excess productions.

Literature Anderson, P. W. 1958. “Absence of Diffusion in Certain Random Lattices.”

Phys. Rev.
109 , 1492–1505.

CrossRef Demetrius, L. 1974. “Demographic Parameters and Natural Selection.” Proc. natn. Acad. Sci. U.S.A.

71 , 4645–4647; (1983). “Statistical Mechanics and Population Biology.”

J. stat. Phys.
30 , 729–773.

MATH MathSciNet CrossRef —, P. Schuster and K. Sigmund. 1985. Polynucleotide Evolution and Branching Processes.

Bull. math. Biol.
47 , 239–262.

MATH MathSciNet CrossRef Domingo, E., P. Ahlquist and J. J. Holland (Eds.) In press.RNA-Genetics , Vols I and II. Baton Rouge: CRC-Press.

Eigen, M. 1971. “Selforganization of Matter and the Evolution of Biological Macromolecues”

Naturwissenshcaften
58 , 465–526.

CrossRef — 1985. “Macromolecular Evolution: Dynamical Ordering in Sequence Space.”Ber. Bunsenges. phys. Chem. ,89 , 658–667.

— and P. Schuster. 1979.The Hypercycle—a Principle of Natural Self-Organization . Berlin: Springer.

Eigen, M., J. McCaskill and P. Schuster. In press. “Dynamics of Darwinian Molecular Systems.”J. phys. Chem.

Ewens, W. J. 1979. “Mathematical Population Genetics.” InBiomathematics , Vol. 9. Berlin: Springer.

Feinberg, M. 1977. “Mathematical Aspects of Mass Action Kinetics.” InChemical Reaction Theory: A Review , N. Amundsen and L. Lapidus (Eds), pp. 1–78. New Jersey: Prentice Hall.

Jones, B. L. 1978 (Eds) “Some Principles Governing Selection in Self-Reproducing Macromolecular Systems.”J. math. Biol.
6 , 169–75.

—, R. H. Enns and S. S. Ragnekar. 1976. On the Theory of Selection of coupled Macromolecular Systems.”

Bull. math. Biol.
38 , 12–28.

CrossRef — and K. H. Leung, 1981. “Stochastic Analysis of a Nonlinear Model for Selection of Biological Macromolecules.”

Bull. math. Biol.
43 , 665–680.

MATH MathSciNet Kato, T. 1966. “Perturbation Theory for Linear OperatorsGrundl. Math. Wiss.
132 .

Kimura, M. 1983.The Neutral Theory of Molecular Evolution , Cambridge University Press.

Leuthäusser, I. 1986. “An Exact Correspondence Between Eigens Evolution Model and a Two-Dimensional Ising Model.”

J. chem. Phys.
84 , 1884.

MathSciNet CrossRef McCaskill, J. 1984. “A Localization Threshold for Macromolecular Quasispecies from Continuously Distributed Replication Rates.”

J. chem. Phys.
80 , 5194–5202.

MathSciNet CrossRef — 1984. “A Stochastic Theory of Molecular Evolution.”

Biol. Cybernetics ,

50 , 63–73.

MATH CrossRef Rumschitzky, D. 1987. “Spectral Properties of Eigen Evolution Matrices.”

J. math. Biol. ,

24 , 667–680.

MathSciNet Schuster, P. 1986. “Dynamics of Molecular Evolution.”

Physica ,

16D , 100–119.

MathSciNet — and K. Sigmund. 1985. “Dynamics of Evolutionary Optimization.”Ber. Bunsenges. phys. Chem. ,89 , 668–682.

Stein, D. L. and P. W. Anderson. 1984. A Model for the Origin of Biological Catalysis.

Proc. Natn. Acad. Sci. U.S.A.
81 , 1751–1753.

CrossRef Swetina, J. and P. Schuster. 1982. Selfreplication with Errors. A Model for Polynucleotide Replication.

Biophys. Chem.
16 , 329–353.

CrossRef Thompson, C. J. and J. L. McBride. 1974. “On Eigens Theory of the Selforganization of Molecules and the Evolution of Biological Macromolecules.”

Math. Biosci. ,

21 , 127–142.

MATH MathSciNet CrossRef © Society for Mathematical Biology 1988

Authors and Affiliations Peter Schuster Jörg Swetina 1. Institut für theoretische Chemie und Strahlenchemie der Universität Wien Wien Austria