Biophysical connection between evolutionary dynamics and thermodynamics in in vitro evolution.

We analyzed a mathematical model of in vitro evolution conducted by repetition of mutagenesis and selection processes. The selection process consists of the selective enrichment and subsequent sampling as follows: each mutant with fitness W is amplified by the Boltzmann factor exp(rW/k(B)T(the)), where the fitness W is defined as the negative Gibbs free energy (-ΔG) in a reaction of the phenotypic molecules and r is the round number of the selective enrichment; then, an arbitrary mutant is randomly chosen from the resulting mutant population and it becomes a new parent in the next generation. As a result, we found that the evolutionary dynamics is described in a mathematical framework similar to thermodynamics: the "evolution constant" k(E) and "evolutionary temperature" T(evo) play key roles similar to the Boltzmann constant k(B) and thermodynamic temperature T(the), respectively. In the stationary state of the evolutionary dynamics, the attractor of the fitness is in inverse proportion to k(E)T(evo). Furthermore, beyond the mathematical analogy, we obtained a biophysical connection between evolutionary dynamics and thermodynamics. Particularly, we found that T(evo) and T(the) are connected by k(E)T(evo)≈k(B)T(the)/2r. These results suggest that we can predict the fitness value in the stationary state by the thermodynamic temperature T(the) in the experimental setup.