# 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…

## 4 Citations

Open quasispecies models: Stability, optimization, and distributed extension

- BiologyJournal of Mathematical Analysis and Applications
- 2020

Virus-immune dynamics determined by prey-predator interaction network and epistasis in viral fitness landscape

- Biology
- 2021

It is shown that long term dynamics of viral mutants evolving resistance at distinct epitopes (viral proteins targeted by immune responses) are governed by epistasis in the virus fitness landscape, and how pairwise or multiplicative epistatic interactions shape viral evolution against concurrent immune responses and convergence to the multi-variant steady state predicted by theoretical results.

Modeling the dynamics of within-host viral infection and evolution predicts quasispecies distributions and phase boundaries separating distinct classes of infections

- BiologybioRxiv
- 2021

Analysis of viral of viral mutant populations over a wide range of permissivity and immunity, for large numbers of cells and viruses, reveals phase transitions that separate the three classes of viruses, both in the infection-cycle dynamics and at steady state.

Open quasispecies systems: New approach to evolutionary adaptation

- BiologyChinese Journal of Physics
- 2021

## References

SHOWING 1-10 OF 41 REFERENCES

Linear algebra of the permutation invariant Crow-Kimura model of prebiotic evolution.

- MathematicsMathematical biosciences
- 2014

The quasispecies distribution

- Mathematics
- 2016

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

- MathematicsBulletin of mathematical biology
- 2016

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.

- Computer ScienceProceedings 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

- MathematicsBulletin of mathematical biology
- 2018

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

- BiologyCurrent Topics in Microbiology and Immunology
- 2016

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.

- MathematicsMathematical biosciences
- 2014

Ising quantum chain is equivalent to a model of biological evolution

- Biology
- 1997

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.

Exact solutions for the selection-mutation equilibrium in the Crow-Kimura evolutionary model.

- MathematicsMathematical biosciences
- 2015

Mutation-selection balance: ancestry, load, and maximum principle.

- BiologyTheoretical population biology
- 2002

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.