Unified framework for quasispecies evolution and stochastic quantization

In this paper we provide a unified framework for quasispecies evolution and stochastic quantization. We map the biological evolution described by the quasispecies equation to the stochastic dynamics of an ensemble of particles undergoing a creation-annihilation process. We show that this mapping identifies a natural decomposition of the probability that an individual has a certain genotype into eigenfunctions of the evolutionary operator. This alternative approach to study the quasispecies equation allows for a generalization of the Fisher theorem equivalent to the Price equation. According to this relation the average fitness of an asexual population increases with time proportional to the variance of the eigenvalues of the evolutionary operator. Moreover, from the present alternative formulation of stochastic quantization a novel scenario emerges to be compared with existing approaches. The evolution of an ensemble of particles undergoing diffusion and a creation-annihilation process is parametrized by a variable beta that we call the inverse temperature of the stochastic dynamics. We find that the evolution equation at high temperatures is simply related to the Schrodinger equation, but at low temperature it strongly deviates from it. In the presence of additional noise in scattering processes between the particles, the evolution reaches a steady state described by the Bose-Einstein statistics.