Abstract
We express the Crow-Kimura and Eigen models of quasispecies theory in a functional integral representation. We formulate the spin coherent state functional integrals using the Schwinger Boson method. In this formulation, we are able to deduce the long-time behavior of these models for arbitrary replication and degradation functions. We discuss the phase transitions that occur in these models as a function of mutation rate. We derive for these models the leading order corrections to the infinite genome length limit.
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References
E. Baake, M. Baake and H. Wagner, Ising quantum chain is equivalent to a model of biological evolution. Phys. Rev. Lett. 78:559–562 (1997); 79:1782.
E. Baake, M. Baake and H. Wagner, Quantum mechanics versus classical probability in biological evolution. Phys. Rev. E 57:1191–1192 (1998).
R. B. Bapat and T. E. S. Raghavan, Nonnegative Matrices and Applications. Cambridge University Press, New York (1997).
R. Bürger, Mathematical properties of mutation-selection models. Genetica 102/103:279–298 (1998).
J. F. Crow and M. Kimura, An Introduction to Population Genetics Theory. Harper and Row, New York (1970).
M. Eigen, Self organization of matter and the evolution of biological macor molecules. Naturwissenschaften 58:465–523 (1971).
M. Eigen, J. McCaskill and P. Schuster, The molecular quasi-species. Adv. Chem. Phys. 75:149–263 (1989).
K. Jain and J. Krug, Adaptation in simple and complex fitness landscapes. In: H. R. U. Bastolla, M. Porto, and M. Vendruscolo (eds.) Structural approaches to sequence evolution: Molecules, networks and populations. Springer Verlag, Berlin (2005). Q-bio.PE/0508008.
I. Leuthäusser, An exact correspondence between Eigen’s evolution model and a two-dimensional ising system. J. Chem. Phys. 84:1884–1885 (1986).
L. Peliti, Quasispecies evolution in general mean-field landscapes. Europhys. Lett. 57:745–751 (2002).
D. B. Saakian and C. K. Hu, Eigen model as a quantum spin chain: Exact dynamics. Phys. Rev. E 69:021–913 (2004).
D. B. Saakian and C. K. Hu, Solvable biological evolution model with a parallel mutation-selection scheme. Phys. Rev. E 69, 046–121 (2004).
D. B. Saakian, C. K. Hu and H. Khachatryan, Solvable biological evolution models with general fitness functions and multiple mutations in parallel mutation-selection scheme. Phys. Rev. E 70:041–908 (2004).
D. B. Saakian, E. Munoz, C. K. Hu and M. W. Deem, Quasispecies theory for multiple-peak fitness landscapes. Phys. Rev. E 73:041–913 (2006).
P. Tarazona, Error thresholds for molecular quasispecies as phase transitions: From simple landscapes to spin-glass models. Phys. Rev. A 45:6038–6050 (1992).
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Park, JM., Deem, M.W. Schwinger Boson Formulation and Solution of the Crow-Kimura and Eigen Models of Quasispecies Theory. J Stat Phys 125, 971–1011 (2006). https://doi.org/10.1007/s10955-006-9190-z
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DOI: https://doi.org/10.1007/s10955-006-9190-z