Molecular phenotypes link genomic information with organismic functions,
fitness, and evolution. Quantitative traits are complex phenotypes that depend
on multiple genomic loci. In this paper, we study the adaptive evolution of a
quantitative trait under time-dependent selection, which arises from
environmental changes or through fitness interactions with other co-evolving
phenotypes. We analyze a model of trait evolution under mutations and genetic
drift in a single-peak fitness seascape. The fitness peak performs a
constrained random walk in the trait amplitude, which determines the
time-dependent trait optimum in a given population. We derive analytical
expressions for the distribution of the time-dependent trait divergence between
populations and of the trait diversity within populations. Based on this
solution, we develop a method to infer adaptive evolution of quantitative
traits. Specifically, we show that the ratio of the average trait divergence
and the diversity is a universal function of evolutionary time, which predicts
the stabilizing strength and the driving rate of the fitness seascape. From an
information-theoretic point of view, this function measures the
macro-evolutionary entropy in a population ensemble, which determines the
predictability of the evolutionary process. Our solution also quantifies two
key characteristics of adapting populations: the cumulative fitness flux, which
measures the total amount of adaptation, and the adaptive load, which is the
fitness cost due to a population's lag behind the fitness peak.Comment: Figures are not optimally displayed in Firefo
