Evolutionary dynamics on rugged fitness landscapes: exact dynamics and information theoretical aspects

TL;DR

This paper provides an exact solution for the evolutionary dynamics on rugged fitness landscapes using quantum mappings, revealing distinct dynamical regimes and phase transitions relevant to understanding evolution and information processing.

Contribution

It introduces an exact analytical approach to evolutionary dynamics on complex landscapes by mapping to quantum Ising models, uncovering new dynamical regimes and phase transitions.

Findings

Identification of three distinct dynamical regimes in REM landscapes

Discontinuous transitions in mean fitness and overlap at regime boundaries

Finite-time relaxation to equilibrium in ferromagnetic REM

Abstract

The parallel mutation-selection evolutionary dynamics, in which mutation and replication are independent events, is solved exactly in the case that the Malthusian fitnesses associated to the genomes are described by the Random Energy Model (REM) and by a ferromagnetic version of the REM. The solution method uses the mapping of the evolutionary dynamics into a quantum Ising chain in a transverse field and the Suzuki-Trotter formalism to calculate the transition probabilities between configurations at different times. We find that in the case of the REM landscape the dynamics can exhibit three distinct regimes: pure diffusion or stasis for short times, depending on the fitness of the initial configuration, and a spin-glass regime for large times. The dynamic transition between these dynamical regimes is marked by discontinuities in the mean-fitness as well as in the overlap with the initial reference sequence. The relaxation to equilibrium is described by an inverse time decay. In the ferromagnetic REM, we find in addition to these three regimes, a ferromagnetic regime where the overlap and the mean-fitness are frozen. In this case, the system relaxes to equilibrium in a finite time. The relevance of our results to information processing aspects of evolution is discussed.