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  1. Article ; Online: Generalized quasispecies model on finite metric spaces: isometry groups and spectral properties of evolutionary matrices.

    Semenov, Yuri S / Novozhilov, Artem S

    Journal of mathematical biology

    2018  Volume 78, Issue 3, Page(s) 837–878

    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 ... ...

    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 [Formula: see text] 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 [Formula: see text], then the dominant eigenvalue of the evolutionary matrix satisfies an algebraic equation of degree at most [Formula: see text], 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 [Formula: see text] with p-adic metric.
    MeSH term(s) Biological Evolution ; Genetic Fitness ; Mathematical Concepts ; Models, Biological ; Mutation ; Species Specificity ; Systems Biology
    Language English
    Publishing date 2018-09-05
    Publishing country Germany
    Document type Journal Article
    ZDB-ID 187101-8
    ISSN 1432-1416 ; 0303-6812
    ISSN (online) 1432-1416
    ISSN 0303-6812
    DOI 10.1007/s00285-018-1294-y
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  2. Article ; Online: On Eigen's Quasispecies Model, Two-Valued Fitness Landscapes, and Isometry Groups Acting on Finite Metric Spaces.

    Semenov, Yuri S / Novozhilov, Artem S

    Bulletin of mathematical biology

    2016  Volume 78, Issue 5, Page(s) 991–1038

    Abstract: A two-valued fitness landscape is introduced for the classical Eigen's quasispecies model. This fitness landscape can be considered as a direct generalization of the so-called single- or sharply peaked landscape. A general, non-permutation invariant ... ...

    Abstract A two-valued fitness landscape is introduced for the classical Eigen's quasispecies model. This fitness landscape can be considered as a direct generalization of the so-called single- or sharply peaked landscape. A general, non-permutation invariant quasispecies model is studied, and therefore the dimension of the problem is [Formula: see text], where N is the sequence length. It is shown that if the fitness function is equal to [Formula: see text] on a G-orbit A and is equal to w elsewhere, then the mean population fitness can be found as the largest root of an algebraic equation of degree at most [Formula: see text]. Here G is an arbitrary isometry group acting on the metric space of sequences of zeroes and ones of the length N with the Hamming distance. An explicit form of this exact algebraic equation is given in terms of the spherical growth function of the G-orbit A. Motivated by the analysis of the two-valued fitness landscapes, 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. In particular, a simplicial analog of the original quasispecies model is discussed, which can be considered as a mathematical model of the switching of the antigenic variants for some bacteria.
    MeSH term(s) Genetic Fitness ; Mathematical Concepts ; Models, Genetic ; Mutation ; Selection, Genetic
    Language English
    Publishing date 2016-05
    Publishing country United States
    Document type Journal Article
    ZDB-ID 184905-0
    ISSN 1522-9602 ; 0007-4985 ; 0092-8240
    ISSN (online) 1522-9602
    ISSN 0007-4985 ; 0092-8240
    DOI 10.1007/s11538-016-0172-2
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  3. Article ; Online: Exact solutions for the selection-mutation equilibrium in the Crow-Kimura evolutionary model.

    Semenov, Yuri S / Novozhilov, Artem S

    Mathematical biosciences

    2015  Volume 266, Page(s) 1–9

    Abstract: We reformulate the eigenvalue problem for the selection-mutation equilibrium distribution in the case of a haploid asexually reproduced population in the form of an equation for an unknown probability generating function of this distribution. The special ...

    Abstract We reformulate the eigenvalue problem for the selection-mutation equilibrium distribution in the case of a haploid asexually reproduced population in the form of an equation for an unknown probability generating function of this distribution. The special form of this equation in the infinite sequence limit allows us to obtain analytically the steady state distributions for a number of particular cases of the fitness landscape. The general approach is illustrated by examples; theoretical findings are compared with numerical calculations.
    MeSH term(s) Biological Evolution ; Gene Frequency ; Models, Biological ; Mutation ; Selection, Genetic
    Language English
    Publishing date 2015-08
    Publishing country United States
    Document type Journal Article ; Research Support, U.S. Gov't, Non-P.H.S.
    ZDB-ID 1126-5
    ISSN 1879-3134 ; 0025-5564
    ISSN (online) 1879-3134
    ISSN 0025-5564
    DOI 10.1016/j.mbs.2015.05.002
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  4. Book ; Online: On Eigen's quasispecies model, two-valued fitness landscapes, and isometry groups acting on finite metric spaces

    Semenov, Yuri S. / Novozhilov, Artem S.

    2015  

    Abstract: A two-valued fitness landscape is introduced for the classical Eigen's quasispecies model. This fitness landscape can be considered as a direct generalization of the so-called single or sharply peaked landscape. A general, non permutation invariant ... ...

    Abstract A two-valued fitness landscape is introduced for the classical Eigen's quasispecies model. This fitness landscape can be considered as a direct generalization of the so-called single or sharply peaked landscape. A general, non permutation invariant quasispecies model is studied, therefore the dimension of the problem is $2^N\times 2^N$, where $N$ is the sequence length. It is shown that if the fitness function is equal to $w+s$ on a $G$-orbit $A$ and is equal to $w$ elsewhere, then the mean population fitness can be found as the largest root of an algebraic equation of degree at most $N+1$. Here $G$ is an arbitrary isometry group acting on the metric space of sequences of zeroes and ones of the length $N$ with the Hamming distance. An explicit form of this exact algebraic equation is given in terms of the spherical growth function of the $G$-orbit $A$. Sufficient conditions for the so-called error threshold for sequences of orbits are given. Motivated by the analysis of the two-valued fitness landscapes 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 $X$. In particular, a simplicial analogue of the original quasispecies model is discussed, which can be considered as a mathematical model of the switching of the antigenic variants for some bacteria.

    Comment: 38 pages, 7 figures, several typos were fixed
    Keywords Quantitative Biology - Populations and Evolution ; 92D15 ; 92D25
    Subject code 612
    Publishing date 2015-03-11
    Publishing country us
    Document type Book ; Online
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  5. Article ; Online: Linear algebra of the permutation invariant Crow-Kimura model of prebiotic evolution.

    Bratus, Alexander S / Novozhilov, Artem S / Semenov, Yuri S

    Mathematical biosciences

    2014  Volume 256, Page(s) 42–57

    Abstract: A particular case of the famous quasispecies model - the Crow-Kimura model with a permutation invariant fitness landscape - is investigated. Using the fact that the mutation matrix in the case of a permutation invariant fitness landscape has a special ... ...

    Abstract A particular case of the famous quasispecies model - the Crow-Kimura model with a permutation invariant fitness landscape - is investigated. Using the fact that the mutation matrix in the case of a permutation invariant fitness landscape has a special tridiagonal form, a change of the basis is suggested such that in the new coordinates a number of analytical results can be obtained. In particular, using the eigenvectors of the mutation matrix as the new basis, we show that the quasispecies distribution approaches a binomial one and give simple estimates for the speed of convergence. Another consequence of the suggested approach is a parametric solution to the system of equations determining the quasispecies. Using this parametric solution we show that our approach leads to exact asymptotic results in some cases, which are not covered by the existing methods. In particular, we are able to present not only the limit behavior of the leading eigenvalue (mean population fitness), but also the exact formulas for the limit quasispecies eigenvector for special cases. For instance, this eigenvector has a geometric distribution in the case of the classical single peaked fitness landscape. On the biological side, we propose a mathematical definition, based on the closeness of the quasispecies to the binomial distribution, which can be used as an operational definition of the notorious error threshold. Using this definition, we suggest two approximate formulas to estimate the critical mutation rate after which the quasispecies delocalization occurs.
    MeSH term(s) Biological Evolution ; Models, Theoretical
    Language English
    Publishing date 2014-10
    Publishing country United States
    Document type Journal Article ; Research Support, Non-U.S. Gov't ; Research Support, U.S. Gov't, Non-P.H.S.
    ZDB-ID 1126-5
    ISSN 1879-3134 ; 0025-5564
    ISSN (online) 1879-3134
    ISSN 0025-5564
    DOI 10.1016/j.mbs.2014.08.006
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  6. Article ; Online: On the behavior of the leading eigenvalue of Eigen's evolutionary matrices.

    Semenov, Yuri S / Bratus, Alexander S / Novozhilov, Artem S

    Mathematical biosciences

    2014  Volume 258, Page(s) 134–147

    Abstract: We study general properties of the leading eigenvalue w¯(q) of Eigen's evolutionary matrices depending on the replication fidelity q. This is a linear algebra problem that has various applications in theoretical biology, including such diverse fields as ... ...

    Abstract We study general properties of the leading eigenvalue w¯(q) of Eigen's evolutionary matrices depending on the replication fidelity q. This is a linear algebra problem that has various applications in theoretical biology, including such diverse fields as the origin of life, evolution of cancer progression, and virus evolution. We present the exact expressions for w¯(q),w¯(')(q),w¯('')(q) for q = 0, 0.5, 1 and prove that the absolute minimum of w¯(q), which always exists, belongs to the interval (0, 0.5]. For the specific case of a single peaked landscape we also find lower and upper bounds on w¯(q), which are used to estimate the critical mutation rate, after which the distribution of the types of individuals in the population becomes almost uniform. This estimate is used as a starting point to conjecture another estimate, valid for any fitness landscape, and which is checked by numerical calculations. The last estimate stresses the fact that the inverse dependence of the critical mutation rate on the sequence length is not a generally valid fact.
    MeSH term(s) Biological Evolution ; Models, Theoretical
    Language English
    Publishing date 2014-12
    Publishing country United States
    Document type Journal Article ; Research Support, Non-U.S. Gov't ; Research Support, U.S. Gov't, Non-P.H.S.
    ZDB-ID 1126-5
    ISSN 1879-3134 ; 0025-5564
    ISSN (online) 1879-3134
    ISSN 0025-5564
    DOI 10.1016/j.mbs.2014.10.004
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  7. Book ; Online: Linear algebra of the permutation invariant Crow-Kimura model of prebiotic evolution

    Bratus, Alexander S. / Novozhilov, Artem S. / Semenov, Yuri S.

    2013  

    Abstract: A particular case of the famous quasispecies model - the Crow-Kimura model with a permutation invariant fitness landscape - is investigated. Using the fact that the mutation matrix in the case of a permutation invariant fitness landscape has a special ... ...

    Abstract A particular case of the famous quasispecies model - the Crow-Kimura model with a permutation invariant fitness landscape - is investigated. Using the fact that the mutation matrix in the case of a permutation invariant fitness landscape has a special tridiagonal form, a change of the basis is suggested such that in the new coordinates a number of analytical results can be obtained. In particular, using the eigenvectors of the mutation matrix as the new basis, we show that the quasispecies distribution approaches a binomial one and give simple estimates for the speed of convergence. Another consequence of the suggested approach is a parametric solution to the system of equations determining the quasispecies. Using this parametric solution we show that our approach leads to exact asymptotic results in some cases, which are not covered by the existing methods. In particular, we are able to present not only the limit behavior of the leading eigenvalue (mean population fitness), but also the exact formulas for the limit quasispecies eigenvector for special cases. For instance, this eigenvector has a geometric distribution in the case of the classical single peaked fitness landscape. On the biological side, we propose a mathematical definition, based on the closeness of the quasispecies to the binomial distribution, which can be used as an operational definition of the notorious error threshold. Using this definition, we suggest two approximate formulas to estimate the critical mutation rate after which the quasispecies delocalization occurs.

    Comment: 35 pages. Several changes are incorporated and one reference is added
    Keywords Quantitative Biology - Populations and Evolution
    Subject code 612
    Publishing date 2013-06-01
    Publishing country us
    Document type Book ; Online
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