Schwinger Boson Formulation and Solution of the Crow-Kimura and Eigen Models of Quasispecies Theory

Abstract

We express the Crow-Kimura and Eigen models of quasispecies theory in a functional integral representation. We formulate the spin coherent state functional integrals using the Schwinger Boson method. In this formulation, we are able to deduce the long-time behavior of these models for arbitrary replication and degradation functions. We discuss the phase transitions that occur in these models as a function of mutation rate. We derive for these models the leading order corrections to the infinite genome length limit.

Quasispecies theory for horizontal gene transfer and recombination

“…In the absence of recombination or horizontal gene transfer this variance is 𝒪( N −1 ), which implies correlations along the sequence are 𝒪( N −1 ) [33]. We expect the same scaling of the variance in the presence of recombination or horizontal gene transfer.…”

“…More recently, exact mappings into a quantum spin chain [44–48] or field theoretic representations [33] have been developed. Analytical and numerical studies of these systems, in the large genome limit, are possible when the fitness function is considered to be permutation invariant [32, 33, 44, 45, 49], or depending on the overlap with several peaks in sequence space [34].…”

“…More recently, exact mappings into a quantum spin chain [44–48] or field theoretic representations [33] have been developed. Analytical and numerical studies of these systems, in the large genome limit, are possible when the fitness function is considered to be permutation invariant [32, 33, 44, 45, 49], or depending on the overlap with several peaks in sequence space [34]. The mapping of the quasi-species models into a physical system allows for the application of the powerful mathematical techniques of statistical mechanics, thus obtaining exact analytical solutions which provide significant insight over numerical studies [33, 34, 46].…”

A comparison between the quasi-species evolution and stochastic quantization of fields

“…INTRODUCTION Increasing evidence shows that the Darwinian theory of biological evolution [1,2] described by the quasi-species equation [3] shares surprising similarities with quantum mechanics at the mathematical level [4][5][6][7][8][9][10][11][12][13][14][15][16]. The quasi-species equation describes the evolution of self-replicating macromolecules such as RNA or DNA or asexual organisms.…”

Finite Genome Length Corrections for the Mean Fitness and Gene Probabilities in Evolution Models

“…Analytic solutions for models with the infinite genome length may be obtained via several methods, including the maximum principle [12,13], Trotter-Suzuki method in quantum statistical physics [14,15,17,18], quantum field theory [15,18,19] and the Hamiltonian-Jacobi equation method [21][22][23]25]. The infinite genome length limit has been solved fora more complicated (more nonlinear) evolution model: the horizontal gene transfer model [16,20,23,25].…”