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

@article{Bratus2019RigorousMA,
  title={Rigorous Mathematical Analysis of the Quasispecies Model: From Manfred Eigen to the Recent Developments},
  author={A. S. Bratus and Artem S. Novozhilov and Yuri S Semenov},
  journal={STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics \& Health},
  year={2019}
}
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… 
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References

SHOWING 1-10 OF 41 REFERENCES
The quasispecies distribution
TLDR
The quasispecies model is presented and it is focused on its stationary solutions, which have a surprisingly rich combinatorial structure, involving for instance the Eulerian and Stirling numbers, as well as the up--down coefficients of permutations.
On Eigen’s Quasispecies Model, Two-Valued Fitness Landscapes, and Isometry Groups Acting on Finite Metric Spaces
TLDR
An abstract generalization of Eigen’s model is introduced such that the sequences are identified with the points of a finite metric space X together with a group of isometries acting transitively on X.
Exact solution of the Eigen model with general fitness functions and degradation rates.
  • D. SaakianChin-Kun Hu
  • Computer Science
    Proceedings of the National Academy of Sciences of the United States of America
  • 2006
We present an exact solution of Eigen's quasispecies model with a general degradation rate and fitness functions, including a square root decrease of fitness with increasing Hamming distance from the
Asymptotic Behavior of Eigen’s Quasispecies Model
TLDR
It is proved convergence of trajectories, as well as convergence of the equilibrium solutions, for a discrete-time version of Eigen’s model, which coincides with a model proposed by Moran.
Quasispecies: From Theory to Experimental Systems
TLDR
This paper presents mathematical models of quasispecies theory and exact results for the dynamics of viral population dynamics, which show that viral evolutionary strategies based on lethal mutagenesis and error threshold based on fidelity variants and RNA quasipecies are viable.
On the behavior of the leading eigenvalue of Eigen's evolutionary matrices.
Ising quantum chain is equivalent to a model of biological evolution
TLDR
A problem of biochemical physics that may be mapped exactly onto a quantum chain is presented that can be solved without approximation and describes mutation and selection as going on in parallel.
Mutation-selection balance: ancestry, load, and maximum principle.
TLDR
The results are applied to threshold phenomena caused by the interplay of selection and mutation (known as error thresholds) and lead to a clarification of concepts, as well as criteria for the existence of error thresholds.
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