Quasispecies theory for horizontal gene transfer and recombination

Abstract

We introduce a generalization of the parallel, or Crow-Kimura, and Eigen models of molecular evolution to represent the exchange of genetic information between individuals in a population. We study the effect of different schemes of genetic recombination on the steady-state mean fitness and distribution of individuals in the population, through an analytic field theoretic mapping. We investigate both horizontal gene transfer from a population and recombination between pairs of individuals. Somewhat surprisingly, these nonlinear generalizations of quasi-species theory to modern biology are analytically solvable. For two-parent recombination, we find two selected phases, one of which is spectrally rigid. We present exact analytical formulas for the equilibrium mean fitness of the population, in terms of a maximum principle, which are generally applicable to any permutation invariant replication rate function. For smooth fitness landscapes, we show that when positive epistatic interactions are present, recombination or horizontal gene transfer introduces a mild load against selection. Conversely, if the fitness landscape exhibits negative epistasis, horizontal gene transfer or recombination introduce an advantage by enhancing selection towards the fittest genotypes. These results prove that the mutational deterministic hypothesis holds for quasi-species models. For the discontinuous single sharp peak fitness landscape, we show that horizontal gene transfer has no effect on the fitness, while recombination decreases the fitness, for both the parallel and the Eigen models. We present numerical and analytical results as well as phase diagrams for the different cases.

Quasispecies theory for finite populations

“…The master equation can be exactly projected onto the ξ coordinate and defines the rates at which the sequences of individuals change with time due to replication, mutation, and horizontal gene transfer. We define (1 + u )/2 to be the probability of a wild type letter in the sequence, ρ ± = (1 ± u )/2 is the probability of inserting a wild-type or non-wild-type letter by horizontal gene transfer [27, 34], and $$u=\frac{1}{N}\sum _{\xi =0}^L(2\xi ∕L-1)n\xi$$ is the ‘average base composition,’ where n ξ is the number of individuals at coordinate ξ .…”

“…This phase transition occurs when the mutation rate exceeds a critical value, which depends on the nature of the fitness function [25, 27]. The phase transition is usually of first order for binary alphabets [25, 27], but it is of higher order for smooth fitness functions in larger alphabets [28]. The quasispecies is composed by a collection of nearly neutral mutants, that is, a cloud of closely related individuals sharing similar fitness values, rather than by a single sequence type.…”

Finite population size effects in quasispecies models with single-peak fitness landscape

“…The convergence of evolutionary dynamics with population size depends on the mutation rate and the fitness landscape. In the infinite population limit the evolution equations are deterministic, and, for molecular evolution models [3]- [8], there are many exact results [8]- [19]. It is possible even to find exact solutions for the steady state and dynamics [15,16,18].…”

“…The methods used include quantum mechanics [8,9], statistical mechanics [10]- [12], quantum field theory [10]- [14], [19], Hamilton-Jacobi equation (HJE) [15][16][17]. Such approach has given many exact results for evolution models [3]- [20], solved a paradox of the origin of life [21], and produced exact finite genome length corrections for the mean fitness and gene probabilities in some evolution models [22].…”

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

“…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].…”