Diploid biological evolution models with general smooth fitness landscapes and recombination

Abstract

Using a Hamilton-Jacobi equation approach, we obtain analytic equations for steady-state population distributions and mean fitness functions for Crow-Kimura and Eigen-type diploid biological evolution models with general smooth hypergeometric fitness landscapes. Our numerical solutions of diploid biological evolution models confirm the analytic equations obtained. We also study the parallel diploid model for the simple case of recombination and calculate the variance of distribution, which is consistent with numerical results.

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

“…(1) in [23]. There we derived the analytic solution for the diploid many allele biological evolution models [7][8][9] with general fitness landscapes to the first order using the HamiltonJacobi equation approach.…”

“…In [25] we calculated the finite size correction for the recombination model with single-peak fitness. In the present paper we calculate the finite genome length corrections for a diploid model with symmetric * Electronic address: saakian@phys.sinica.edu.tw † Electronic address: hu@phys.sinica.edu.tw landscape [23] as well as for a haploid model with a simple horizontal gene transfer (HGT) [16,20] for a general symmetric fitness landscape. The method may be applied to rather general cases of nonlinear probabilistic models [29].…”

“…The infinite genome length limit has been solved fora more complicated (more nonlinear) evolution model: the horizontal gene transfer model [16,20,23,25]. This model has been solved both for the haploid [16,20], and the hyper-geometric diploid case [7][8][9]23].…”

“…After the development of molecular biology, the concepts and methods in statistical physics have been widely used to study molecular models of biological evolution [1][2][3][4][5][6][7][8][9][10][11][12][13], especially in recent years [14][15][16][17][18][19][20][22][23][24][25]. 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].…”

“…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]. This model has been solved both for the haploid [16,20], and the hyper-geometric diploid case [7][8][9]23].…”