Advertisement

Journal of Mathematical Biology

, Volume 78, Issue 3, pp 837–878 | Cite as

Generalized quasispecies model on finite metric spaces: isometry groups and spectral properties of evolutionary matrices

  • Yuri S. Semenov
  • Artem S. NovozhilovEmail author
Article
  • 48 Downloads

Abstract

The quasispecies model introduced by Eigen in 1971 has close connections with the isometry group of the space of binary sequences relative to the Hamming distance metric. Generalizing this observation we introduce an abstract quasispecies model on a finite metric space X together with a group of isometries \(\Gamma \) acting transitively on X. We show that if the domain of the fitness function has a natural decomposition into the union of tG-orbits, G being a subgroup of \(\Gamma \), then the dominant eigenvalue of the evolutionary matrix satisfies an algebraic equation of degree at most \(t\cdot \mathrm{rk}_{\mathbf {Z}} R\), where R is the orbital ring that is defined in the text. The general theory is illustrated by three detailed examples. In the first two of them the space X is taken to be the metric space of vertices of a regular polytope with the natural “edge” metric, these are the cases of a regular m-gon and of a hyperoctahedron; the final example takes as X the quotient rings \(\mathbf {Z}/p^n\mathbf {Z}\) with p-adic metric.

Keywords

Quasispecies model Finite metric space Dominant eigenvalue Mean population fitness Isometry group Regular polytope 

Mathematics Subject Classification

15A18 92D15 92D25 

References

  1. Baake E, Gabriel W (1999) Biological evolution through mutation, selection, and drift: an introductory review. In: Stauffer D (ed) Annual reviews of computational physics VII. World Scientific, Singapore, pp 203–264Google Scholar
  2. Baake E, Georgii H-O (2007) Mutation, selection, and ancestry in branching models: a variational approach. J Math Biol 54(2):257–303MathSciNetzbMATHGoogle Scholar
  3. Baake E, Wagner H (2001) Mutation-selection models solved exactly with methods of statistical mechanics. Genet Res 78(1):93–117Google Scholar
  4. Biggs N (1993) Algebraic graph theory. Cambridge University Press, Cambridge (second edition, 1993) zbMATHGoogle Scholar
  5. Bratus AS, Novozhilov AS, Semenov YS (2014) Linear algebra of the permutation invariant Crow–Kimura model of prebiotic evolution. Math Biosci 256:42–57MathSciNetzbMATHGoogle Scholar
  6. Bratus AS, Novozhilov AS, Semenov YS (2017) Rigorous mathematical analysis of the quasispecies model: from Manfred Eigen to the recent developments. arXiv: 1712.03855
  7. Brouwer AE, Cohen AM, Neumaier A (1989) Distance-regular graphs. Springer, BerlinzbMATHGoogle Scholar
  8. Brown KS (1982) Cohomology of groups, vol 87. Springer, BerlinGoogle Scholar
  9. Cerf R, Dalmau J (2016a) The quasispecies distribution. arXiv preprint arXiv:1609.05738
  10. Cerf R, Dalmau J (2016b) Quasispecies on class-dependent fitness landscapes. Bull Math Biol 78(6):1238–1258MathSciNetzbMATHGoogle Scholar
  11. Coxeter HSM (1973) Regular polytopes. Courier Corporation, ChelmsfordzbMATHGoogle Scholar
  12. de la Harpe P (2000) Topics in geometric group theory. University of Chicago Press, ChicagozbMATHGoogle Scholar
  13. Dress AWM, Rumschitzki DS (1988) Evolution on sequence space and tensor products of representation spaces. Acta Appl Math 11(2):103–115MathSciNetzbMATHGoogle Scholar
  14. Eigen M (1971) Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften 58(10):465–523Google Scholar
  15. Eigen M, McCaskill J, Schuster P (1988) Molecular quasi-species. J Phys Chem 92(24):6881–6891Google Scholar
  16. Feit W (1982) The representation theory of finite groups, vol 2. Elsevier, AmsterdamzbMATHGoogle Scholar
  17. Hermisson J, Redner O, Wagner H, Baake E (2002) Mutation-selection balance: ancestry, load, and maximum principle. Theor Popul Biol 62(1):9–46zbMATHGoogle Scholar
  18. Jain K, Krug J (2007) Adaptation in simple and complex fitness landscapes. In: Bastolla U, Porto M, Eduardo Roman H, Vendruscolo M (eds) Structural approaches to sequence evolution, chap 14. Springer, Berlin, pp 299–339Google Scholar
  19. Kirillov AA (1976) Elements of the theory of representations, vol 145. Springer, BerlinzbMATHGoogle Scholar
  20. Leuthäusser I (1986) An exact correspondence between Eigen’s evolution model and a two-dimensional Ising system. J Chem Phys 84(3):1884–1885MathSciNetGoogle Scholar
  21. Leuthäusser I (1987) Statistical mechanics of Eigen’s evolution model. J Stat Phys 48(1):343–360MathSciNetGoogle Scholar
  22. Onsager L (1944) Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys Rev 65(3–4):117MathSciNetzbMATHGoogle Scholar
  23. Rumschitzki DS (1987) Spectral properties of Eigen evolution matrices. J Math Biol 24(6):667–680MathSciNetzbMATHGoogle Scholar
  24. Saakian DB, Hu CK (2006) Exact solution of the Eigen model with general fitness functions and degradation rates. Proc Natl Acad Sci USA 103(13):4935–4939Google Scholar
  25. Schuster P (2015) Quasispecies on fitness landscapes. In: Domingo E, Schuster P (eds) Quasispecies: from theory to experimental systems, Current topics in microbiology and immunology. Springer, Berlin, pp 61–120Google Scholar
  26. Semenov Y (1994) Rings associated with hyperbolic groups. Commun Algebra 22(15):6323–6347MathSciNetzbMATHGoogle Scholar
  27. Semenov YS, Novozhilov AS (2015) Exact solutions for the selection-mutation equilibrium in the Crow–Kimura evolutionary model. Math Biosci 266:1–9MathSciNetzbMATHGoogle Scholar
  28. Semenov YS, Novozhilov AS (2016) On Eigen’s quasispecies model, two-valued fitness landscapes, and isometry groups acting on finite metric spaces. Bull Math Biol 78(5):991–1038MathSciNetzbMATHGoogle Scholar
  29. Serre J-P (1996) Linear representations of finite groups, vol 42. Springer, BerlinGoogle Scholar
  30. Stadler PF, Happel R (1999) Random field models for fitness landscapes. J Math Biol 38(5):435–478MathSciNetzbMATHGoogle Scholar
  31. Stadler PF, Tinhofer G (1999) Equitable partitions, coherent algebras and random walks: applications to the correlation structure of landscapes. Match 40:215–261MathSciNetzbMATHGoogle Scholar
  32. Swetina J, Schuster P (1982) Self-replication with errors: a model for polvnucleotide replication. Biophys Chem 16(4):329–345Google Scholar
  33. Thompson CJ (1972) Mathematical statistical mechanics. Macmillan, New YorkzbMATHGoogle Scholar
  34. van Dam ER, Koolen JH, Tanaka H (2016) Distance-regular graphs. Electron J Comb. Dynamic Survey #DS22Google Scholar
  35. Wiehe T (1997) Model dependency of error thresholds: the role of fitness functions and contrasts between the finite and infinite sites models. Genet Res 69(02):127–136Google Scholar
  36. Wilke CO (2005) Quasispecies theory in the context of population genetics. BMC Evol Biol 5(1):44Google Scholar
  37. Wolff A, Krug J (2009) Robustness and epistasis in mutation-selection models. Phys Biol 6(3):036007Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Applied Mathematics-1Russian University of TransportMoscowRussia
  2. 2.Department of MathematicsNorth Dakota State UniversityFargoUSA

Personalised recommendations