Quantum mechanics formalism for biological evolution

Highlights: ► Biological evolution is an off-equilibrium process described by path integrals over phylogenies. ► The phylogenies are sums of linear lineages for asexual populations. ► For sexual populations, each lineage is a tree and the path integral is given by a sum over these trees. ► Quantum statistics describe the stationary state of biological populations in simple cases. - Abstract: We study the evolution of sexual and asexual populations in fitness landscapes compatible with epistatic interactions. We find intriguing relations between the mathematics of biological evolution and quantum mechanics formalism. We give the general structure of the evolution of sexual and asexual populations which is in general an off-equilibrium process that can be expressed by path integrals over phylogenies. These phylogenies are the sum of linear lineages for asexual populations. For sexual populations, instead, each lineage is a tree of branching ratio two and the path integral describing the evolving population is given by a sum over these trees. Finally we show that the Bose–Einstein and the Fermi–Dirac distributions describe the stationary state of biological populations in simple cases.