Abstract
In this work first and second moments for a many species Moran model are calculated. The model describes by means of a time-continuous birth- and death process the evolution of an ensemble of N macromolecules out of n possible species. The molecules may replicate (correct or erroneous, in the latter case producing mutants) and may undergo elimination. Replication and elimination will be coupled in order to keep population size constant. In the case of arbitrary replication rates an expansion of the moments in powers of 1/N is found. For equal replication rates exact calculation of the moments is possible. In the case of a v-cube model (binary macromolecules) the second moments may be used to find a simple expression for the mean Hamming distance in the system. This quantity provides a measure for the localization of the ensemble.
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Berman, A., Plemmons, R. J.: Nonnegative matrices in the mathematical sciences. New York London: Academic Press 1979
Demetrius, L., Schuster, P., Sigmund, K.: Polynucleotide evolution and branching processes. Bull. Math. Biol. 47, 239–262 (1985)
Demetrius, L.: An extremal principle of macromolecular evolution. Phys. Scripta 35, 63–71 (1987)
Ebeling, W., Engel, A., Esser, B., Feistel, A.: Diffusion and reaction in random media and models of evolution processes. J. Stat. Phys. 37, 369–384 (1984)
Ebeling, W., Sonntag, I.: A stochastic description of evolutionary processes in underoccupied systems. BioSystems 19, 91–100 (1986)
Ebeling, W., Engel, A.: Models of evolutionary systems and their applications to optimization problems. Syst. Anal. Model. Simul. 3, 3–11 (1986)
Eigen, M.: Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften 58, 465–523 (1971)
Eigen, M., McCaskill, J. S., Schuster, P.: The molecular quasispecies. Adv. Chem. Phys. (in press)
Fontana, W., Schuster, P.: A computer model of evolutionary optimization. Biophys. Chem. 26, 123–147 (1987)
Gardiner, C. W.: Handbook of stochastic methods, 2nd edn. Berlin Heidelberg New York: Springer 1985
Golding, G. B., Stobeck, C.:The distribution of nucleotide site differences between two finite sequences. Theor. Popul. Biol. 22, 96–107 (1982)
Jones, B. L., Leung, H. K.: Stochastic analysis of a nonlinear model for selection of biological macromolecules. Bull. Math. Biol. 43, 665–680 (1981)
Leung, H. K.: Stability analysis of a stochastic model for biomolecular selection. Bull. Math. Biol. 46, 399–406 (1984)
Leuthäusser, I.: An exact correspondence between Eigens evolution model and a two-dimensional Ising model. J. Chem. Phys. 84, 1884 (1986)
McCaskill, J. S.: A stochastic theory of macromolecular evolution. Biol. Cybern. 50, 63–73 (1984)
McCaskill, J. S.: A localization threshold for macromolecular quasispecies from continuously distributed replication rates. J. Chem. Phys. 80, 5194–5202 (1984)
Moran, P. A. P.: The effect of selection in a haploid genetic population. Proc. Camb. Phil. Soc. 54, 463–474 (1958)
Rumschitzky, D. S.: Spectral properties of Eigen evolution matrices. J. Math. Biol. 24, 667–680 (1987)
Schuster, P., Swetina, J.: Stationary mutant distributions and evolutionary optimization. Bull. Math. Biol. 50, 635–660 (1987)
Schuster, P., Sigmund, K.: Fixation probabilities for advantageous mutants. Math. Biosci. (to appear)
Swetina J., Schuster, P.: Selfreplication with errors a model for polynucleotide replication. Biophys. Chem. 16, 329–345 (1982)
Watterson, G. A.: On the number of segregating sites in genetical models without recombination. Theor. Popul. Biol. 7, 256–276 (1975)
Weinberger, E. D.: A stochastic generalization of Eigen's model of natural selection. Thesis, New York University 1987
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Swetina, J. First and second moments and the mean hamming distance in a stochastic replication-mutation model for biological macromolecules. J. Math. Biology 27, 463–483 (1989). https://doi.org/10.1007/BF00290640
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DOI: https://doi.org/10.1007/BF00290640