  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.
