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Rigorous Mathematical Analysis of the Quasispecies Model: From Manfred Eigen to the Recent Developments

  • Alexander S. Bratus
  • Artem S. NovozhilovEmail author
  • Yuri S. Semenov
Chapter
Part of the STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health book series (STEAM)

Abstract

We review the major progress in the rigorous analysis of the classical quasispecies model that usually comes in two related but different forms: the Eigen model and the Crow–Kimura model. The model itself was formulated almost 50 years ago, and in its stationary form represents an easy to formulate eigenvalue problem. Notwithstanding the simplicity of the problem statement, we still lack full understanding of the behavior of the mean population fitness and the quasispecies distribution for an arbitrary fitness landscape. Our main goal in this review is twofold: first, to highlight a number of impressive mathematical results, including some of the recent ones, which pertain to the mathematical development of the quasispecies theory. Second, to emphasize that, despite these 50 years of vigorous research, there are still very natural both biological and mathematical questions that remain to be addressed within the quasispecies framework. Our hope is that at least some of the approaches we review in this text can be of help for anyone embarking on further analysis of the quasispecies model.

Keywords

The quasispecies Crow–Kimura model Error threshold Mean population fitness 

AMS Subject Classification

15A18 92D15 92D25 

Notes

Acknowledgment

ASB’s research is supported in part by Russian Science Foundation grant #19-11-00008.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alexander S. Bratus
    • 1
    • 2
  • Artem S. Novozhilov
    • 3
    Email author
  • Yuri S. Semenov
    • 2
  1. 1.Faculty of Computational Mathematics and CyberneticsLomonosov Moscow State UniversityMoscowRussia
  2. 2.Applied Mathematics–1Moscow State University of Railway EngineeringMoscowRussia
  3. 3.Department of MathematicsNorth Dakota State UniversityFargoUSA

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