Derivation of evolutionary entropy in the steady-state thermodynamics of evolutionary dynamics

We investigate the parallel mutation-selection model with varying population size, which is formulated in terms of individuals undergoing the evolution processes of reproduction and mutation, to derive evolutionary entropy. Under the framework of the steady-state thermodynamics for evolutionary dynamics, the excess growth (the difference between the maximum growth rate and the total growth rate) can be interpreted as the evolutionary entropy defined in terms of the probability distributions characteristic of evolutionary dynamics. The Clausius inequality states that the excess growth is always less than or equal to the entropy difference in evolutionary dynamics. Analytically, by using the genome sequence length L=3, we derive the growth after evolutionary dynamics with the finite number of environmental changes and calculate the entropy difference during this evolutionary dynamics, and we verify the Clausius inequality. Furthermore, by taking the infinite limit of the number of environmental changes, we verify that the equality holds for the quasistatic environmental change. By using the derived evolutionary entropy, we propose the thermodynamic relation between the free fitness and evolutionary entropy, where the free fitness is the maximum growth rate possible. Numerically, we use the Gillespie-type simulations, which provides direct realizations of the master equation governing evolutionary dynamics, to verify the Clausius inequality and we find that the simulation results are in good agreement with the analytic results.