Natural Selection in the Wake of Catastrophe

Natural selection favors traits that improve fitness: the ability to survive and reproduce. But how many traits matter? We show that recovery after catastrophe can reveal the answer. When selection acts on abundant existing variation in a population, the resulting replicator dynamics predict that the gap between average and highest fitness shrinks inversely with time. Interestingly, the prefactor of this decay is proportional to the number of traits under selection. This law is a dynamical analog of the equipartition principle in statistical mechanics, with time playing the role of inverse temperature and the relevant trait axes acting as effective degrees of freedom. It therefore allows us to count these axes in two large experimental fitness landscapes measured in E. coli under antibiotic stress. We further show that, rather than following gradient ascent on a fitness landscape, adaptation follows a curvature-sensitive optimization flow corresponding to the Levenberg-Marquardt method. Near fitness peaks, these trajectories become increasingly biased against greediness: from an optimization perspective, post-catastrophic selection becomes progressively optimistic over time. The pace and path of recovery from catastrophe thus quantitatively probe the underlying fitness landscape geometry.