Skip to main content
SpringerLink
Log in
Book cover

Quasispecies: From Theory to Experimental Systems pp 121–139Cite as

Mathematical Models of Quasi-Species Theory and Exact Results for the Dynamics

  • Chapter
  • First Online:

Part of the book series: Current Topics in Microbiology and Immunology ((CT MICROBIOLOGY,volume 392))

Abstract

We formulate the Crow–Kimura, discrete-time Eigen model, and continuous-time Eigen model. These models are interrelated and we established an exact mapping between them. We consider the evolutionary dynamics for the single-peak fitness and symmetric smooth fitness. We applied the quantum mechanical methods to find the exact dynamics of the evolution model with a single-peak fitness. For the smooth symmetric fitness landscape, we map exactly the evolution equations into Hamilton–Jacobi equation (HJE). We apply the method to the Crow–Kimura (parallel) and Eigen models. We get simple formulas to calculate the dynamics of the maximum of distribution and the variance. We review the existing mathematical tools of quasi-species theory.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  • Alves D, Fontanari JF (1998) Error threshold in finite populations. Phys Rev E 57:7008

    Article  CAS  Google Scholar 

  • Ancliff M, Park JM (2012) Dynamics of quasi-species models with a complex spin coherent state representation. J Korean Phys Soc 61:1898

    Article  Google Scholar 

  • Ancliff M, Park JM (2010) Optimal mutation rates in dynamic environments: the Eigen model. Phys Rev E 82:021904

    Article  Google Scholar 

  • Avetisyan Zh, Saakian DB (2010) Recombination in one and two dimensional fitness landscapes. Phys Rev E 81:051916

    Article  Google Scholar 

  • Baake E, Baake M, Wagner H (1997) Ising quantum chain is equivalent a model of biological evolution. Phys Rev Lett 78:559

    Article  CAS  Google Scholar 

  • Baake E, Gabriel W (2000) Biological evolution through mutation, selection, and drift: an introductory review. In: Stauffer D (ed) Annual review of computational physics, vol VII. World Scientific, Singapore, pp 203–264

    Google Scholar 

  • Baake E, Wagner H (2001) Mutation-selection models solved exactly with methods of statistical mechanics. Genet Res 78:93

    Article  CAS  PubMed  Google Scholar 

  • Bagnoli F, Bezzi M (1997) Speciation as pattern formation by competition in a smooth fitness landscape. Phys Rev Lett 79:3302

    Article  CAS  Google Scholar 

  • Crow JF, Kimura M (1970) An introduction to population genetics theory. Harper Row, New York

    Google Scholar 

  • Drossel B (2001) Biological evolution and statistical physics. Adv Phys 50:209

    Article  CAS  Google Scholar 

  • Eigen M (1971) Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften 58:465

    Article  CAS  PubMed  Google Scholar 

  • Eigen M, McCaskill JS, Schuster P (1989) The molecular quasispecies. Adv Chem Phys 75:149

    CAS  Google Scholar 

  • Franz S, Peliti L (1997) Error threshold in simple landscapes. J Phys A 30:4481

    Article  Google Scholar 

  • Galstyan V, Saakian DB (2012) Dynamics of the chemical master equation, a strip of chains of equations in d-dimensional space. Phys Rev E 86:011125

    Article  Google Scholar 

  • Gerland U, Hwa T (2002) On the selection and evolution of regulatory binding motifs. J Mol Evol 55:386

    Article  CAS  PubMed  Google Scholar 

  • Hermisson J, Redner O, Wagner H, Baake E (2002) Mutation–selection balance: ancestry, load, and maximum principle. Theor Popul Biol 62:9

    Article  PubMed  Google Scholar 

  • Kirakosyan Z, Saakian DB, Hu C-K (2010) Evolution models with lethal mutations on symmetric or random fitness landscapes. Phys Rev E 82:011904

    Article  Google Scholar 

  • Kirakosyan Z, Saakian DB, Hu C-K (2011) Finite genome length corrections for the mean fitness and gene probabilities in evolution models, vol 144, p 149

    Google Scholar 

  • Kirakosyan Z, Saakian D, Hu C-K (2012a) Biological evolution in a multidimensional fitness landscape. Phys Rev E 86:031920

    Article  Google Scholar 

  • Kirakosyan Z, Saakian DB, Hu C-K (2012b) Eigen model with correlated multiple mutations and solution of error catastrophe paradox in the origin of life. J Phys soc Jpn 81:114801

    Article  Google Scholar 

  • Leuthausser I (1987) Statistical mechanics of Eigen’s model. J Stat Phys 48:343

    Article  Google Scholar 

  • Melikyan A (1998) Generalized characteristics of first order PDEs. Birkhauser, Boston

    Book  Google Scholar 

  • Neves AGM (2010) Detailed analysis of an Eigen quasispecies model in a periodically moving sharp-peak landscape. Phys Rev E 82:031915

    Article  Google Scholar 

  • Park J-M, Deem MW (2006) Schwinger Boson formulation and solution of the Crow-Kimura and Eigen models of quasispecies theory. J Stat Phys 125:975–1015

    Google Scholar 

  • Peliti L (1997) Introduction to the statistical theory of Darwinian evolution, cond-mat/9712027

    Google Scholar 

  • Peng W, Gerland U, Hwa T, Levine H (2003) Dynamics of competitive evolution on a smooth landscape. Phys Rev Lett 90:088103

    Article  PubMed  Google Scholar 

  • Rouzine IM, Wakeley J, Coffin JM (2003) The solitary wave of asexual evolution. Proc Natl Acad Sci USA 100:587

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Saakian DB (2007) A new method for the solution of models of biological evolution: derivation of exact steady-state distributions. J Stat Phys 128:781

    Article  Google Scholar 

  • Saakian DB (2008) Evolution models with base substitutions, insertions, deletions and selection. Phys Rev E 78:061920

    Article  CAS  Google Scholar 

  • Saakian DB, Biebricher C, Hu C-K (2011) Lethal mutants and truncated selection together solve a paradox of the origin of life. Plos one 6:e21904

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Saakian DB, Biebricher CK, Hu C-K (2009) Phase diagram for the Eigen quasispecies theory with the truncated fitness landscape. Phys Rev E 79:041905

    Article  Google Scholar 

  • Saakian DB, Fontanari JF (2009) Fontanari, evolutionary dynamics on rugged fitness landscapes: exact dynamics and information theoretical aspects. Phys Rev E 80:041903

    Google Scholar 

  • Saakian DB, Gazaryan M, Hu C-K (2014) The shock waves and positive epistasis in evolution. Phys Rev E 90:022712

    Article  Google Scholar 

  • Saakian DB, Hu C-K (2004a) Eigen model as a quantum spin chain: exact dynamics. Phys Rev E 69:021913

    Article  Google Scholar 

  • Saakian DB, Hu C-K (2004b) Solvable biological evolution model with a parallel mutation-selection scheme. Phys Rev E 69:046121

    Article  Google Scholar 

  • Saakian DB, Hu C-K (2006) Exact solution of the Eigen model with general fitness functions and degradation rates. Proc Natl Acad Sci USA 103:4935

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Saakian DB, Munoz E, Hu C-K, Deem MW (2006) Quasispecies theory for multiple-peak fitness landscapes. Phys Rev E 73:041913

    Article  Google Scholar 

  • Saakian DB, Hu C-K, Khachatryan H (2004) Solvable biological evolution models with general fitness functions and multiple mutations in parallel mutation-selection scheme. Phys Rev E 70:041908

    Article  Google Scholar 

  • Saakian DB, Rozanova O, Akmetzhanov A (2008) Exactly solvable dynamics of the Eigen and the Crow-Kimura model. Phys Rev E 78:041908

    Article  Google Scholar 

  • Sato K, Kaneko K (2007) Evolution equation of phenotype distribution: general formulation and application to error catastrophe. Phys Rev E 75:061909

    Article  Google Scholar 

  • Swetina J, Schuster P (1982) Self-replication with errors. A model for polynucleotide replication. Biophys Chem 16:329

    Article  CAS  PubMed  Google Scholar 

  • Tarazona P (1992) Error threshold for molecular quasispecies as phase transitions: from simple landscapes to spin-glass models. Phys Rev A 45:6038

    Article  CAS  PubMed  Google Scholar 

  • Thompson CJ, McBride JL (1974) On Eigen’s theory of the self-organization of matter and the evolution of biological macromolecules. Math Biosci 21:127

    Article  Google Scholar 

  • Tsimring LS, Levine H, Kessler DA (1996) RNA virus evolution via a fitness-space model. Phys Rev Lett 76:4440

    Article  CAS  PubMed  Google Scholar 

  • Wagner H, Baake E, Gerisch T (1998) J Stat Phys 92:1017

    Google Scholar 

  • Woodcock H, Higgs PG (1996) Population evolution on a multiplicative single-peak fitness landscape. J Theor Biol 179:61

    Article  CAS  PubMed  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, 2 Alikhanian Brothers St., Yerevan, 375036, Armenia

    David B. Saakian

  2. Institute of Physics, Academia Sinica, Nankang, Taipei, 11529, Taiwan

    Chin-Kun Hu

Authors
  1. David B. Saakian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chin-Kun Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David B. Saakian .

Editor information

Editors and Affiliations

  1. Campus de Cantoblanco, Centro de Biología Molecular Severo Ochoa (CSIC-UAM), Madrid, Spain

    Esteban Domingo

  2. The Santa Fe Institute, Santa Fe, NM, USA and Institut f. Theoretische Chemie, Universität Wien, Vienna, Austria

    Peter Schuster

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Saakian, D.B., Hu, CK. (2015). Mathematical Models of Quasi-Species Theory and Exact Results for the Dynamics. In: Domingo, E., Schuster, P. (eds) Quasispecies: From Theory to Experimental Systems. Current Topics in Microbiology and Immunology, vol 392. Springer, Cham. https://doi.org/10.1007/82_2015_471

Download citation

Publish with us

Policies and ethics

Search

Navigation