Protein sequence evolution has resulted in a vast repertoire of molecular functionality crucial to life. Despite the central importance of sequence evolution to biology, our fundamental understanding of how sequence composition affects evolution is incomplete. This report describes the utilization of lattice model simulations of directed evolution, which indicate that, on average, peptide and protein evolvability is strongly dependent on initial sequence composition. The report also discusses two distinct regimes of sequence evolution by point mutation: (a) the "classical" mode where sequences "crawl" over free energy barriers towards acquiring a target fold, and (b) the "quantum" mode where sequences appear to "tunnel" through large energy barriers generally insurmountable by means of a crawl. Finally, the simulations indicate that oily and charged peptides are the most efficient substrates for evolution at the "classical" and "quantum" regimes, respectively, and that their respective response to temperature is commensurate with analogies made to barrier crossing in classical and quantum systems. On the whole, these results show that sequence composition can tune both the evolvability and the optimal mode of evolution of peptides and proteins.