Abstract
Eigen [7] has employed deterministic kinetic-like equations to describe macromolecular replication and mutation leading to selection. The solutions to these equations and their physically interesting properties depend upon the spectrum of a type of matrix appearing in those equations. Below we explicitly solve for the spectrum of a fairly general class of such matrices. These solutions are obtained recursively for equations describing macromolecules of any length v.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, P.: Discussion with P. A. at Winter Seminar, Klosters, Switzerland 1985
Bellman, R.: Introduction to Matrix analysis. New York: McGraw-Hill 1960
Biebricher, C. K.: Darwinian selection of self-replicating RNA molecules. In: Hechet, M., Wallace, B., Prance, G. (eds.) Evolutionary biology, vol. 16, pp. 1–52. New York: Plenum Press 1983
Biebricher, C. K., Eigen M., Gardiner, W. C.: Kinetics of RNA Replication. Biochemistry 22, 2544–2559 (1983) and references cited therein
Dress, A., Kruger, M.: Parsimonious phylogenic trees in metric spaces and simulated annealing. To appear in Adv. Appl. Math. 1986
Dress, A., Rumschitzki, D.: Evolution on sequence space and tensor products of representation spaces. Submitted to Am. Math. Monthly
Eigen, M.: Selforganization of sequence space and tensor products of representation spaces. Naturwissenschaften. 58, 465–523 (1971)
Eigen, M., Schuster, P.: The Hypercycle: A principle of natural selforganization. Naturwissenschaften 65, 7–41 (1978)
Gantmacher, F. R.: Matrix theory, vol. II. New York: Chelsea 1959
Hearn, A. C.: REDUCE 2. University of Utah, Salt Lake City, Utah 84112
Jones, B. L., Enns, R. H., Rangnekar, S. S.: On the theory of selection of coupled macromolecular systems. Bull. Math. Biol. 38, 15–28 (1976)
Jost, W.: Über den Ablauf zusammengesetzter chemischer Reaktionen. Z. Naturforsch. 2A, 159–163 (1947)
Leuthäuser, I.: Submitted to J. Chem. Phys.
McCaskill, J. S.: A localization threshold for macromolecular quasispecies from continuously distributed replication rates. J. Chem. Phys. 80, 5194–5202 (1984)
Swetina, J., Schuster, P.: Self-replication with errors: A model for polynucleotide replication. Biophys. Chem. 16, 329–345 (1982)
Thompson, C. J., McBride, J. L.: On Eigen's theory of selforganization of matter and evolution of biological macromolecules. Math. Biosci. 27, 127–142 (1974)
Wei, J., Prater, C. D.: The structure and analysis of chemical reaction systems. Adv. Catal. 13, 203–392 (1962)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rumschitzki, D.S. Spectral properties of Eigen evolution matrices. J. Math. Biology 24, 667–680 (1987). https://doi.org/10.1007/BF00275509
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00275509