Mutation-selection models, sequence space and statistical physics

The evolution of populations under the joint action of mutation and selection is, in the framework of classical population genetics, described by systems of ordinary differential equations. These equations carry over to molecular evolution if alleles are identified with sequences, and a suitable mutation model is specified. The reulting systems are, however, very large and hard to treat.

Matters are simplified by a connection to statistical physics. It may be shown that the mutation-reproduction matrix of the evolution model is exactly equivalent to the Hamiltonian of an Ising quantum chain. Here, the mutation rate corresponds to the temperature, and the fitness of a sequence may be identified with the interaction energy of the spins within the chain. Hence, the methods of statistical physics may be used to diagonalize the mutation-reproduction matrix, and thus solve the evolution model exactly. However, the quantum-mechanical states do not translate directly into the probabilities of the evolution model, since they rely on the quantum-mechanical (as opposed to classical) probability concept; here, the methods require some modification.