On the application of statistical physics to evolutionary biology

There is a close analogy between statistical thermodynamics and the evolution of allele frequencies under mutation, selection and random drift. Wrightʹs formula for the stationary distribution of allele frequencies is analogous to the Boltzmann distribution in statistical physics. Population size, 2 N , plays the role of the inverse temperature, 1 / kT , and determines the magnitude of random fluctuations. Log mean fitness, log ( W ¯ ) , tends to increase under selection, and is analogous to a (negative) energy; a potential function, U , increases under mutation in a similar way. An entropy, S H , can be defined which measures the deviation from the distribution of allele frequencies expected under random drift alone; the sum G = E [ log ( W ¯ ) + U + S H ] gives a free fitness that increases as the population evolves towards its stationary distribution. Usually, we observe the distribution of a few quantitative traits that depend on the frequencies of very many alleles. The mean and variance of such traits are analogous to observable quantities in statistical thermodynamics. Thus, we can define an entropy, S Ω , which measures the volume of allele frequency space that is consistent with the observed trait distribution. The stationary distribution of the traits is exp [ 2 N ( log ( W ¯ ) + U + S Ω ) ] ; this applies with arbitrary epistasis and dominance. The entropies S Ω , S H are distinct, but converge when there are so many alleles that traits fluctuate close to their expectations. Populations tend to evolve towards states that can be realised in many ways (i.e., large S Ω ), which may lead to a substantial drop below the adaptive peak; we illustrate this point with a simple model of genetic redundancy. This analogy with statistical thermodynamics brings together previous ideas in a general framework, and justifies a maximum entropy approximation to the dynamics of quantitative traits.