The sequences, or primary structures, of existing biopolymers--in particular, proteins--are believed to be a product of evolution. Are the sequences random? If not, what is the character of this nonrandomness? To explore the statistics of protein sequences, we use the idea of mapping the sequence onto the trajectory of a random walk, originally proposed by Peng et al. [Peng, C.-K., Buldyrev, S. V., Goldberger, A. L., Havlin, S., Sciortino, F., Simons, M. & Stanley, H. E. (1992) Nature (London) 356, 168-170] in their analysis of DNA sequences. Using three different mappings, corresponding to three basic physical interactions between amino acids, we found pronounced deviations from pure randomness, and these deviations seem directed toward minimization of the energy of the three-dimensional structure. We consider this result as evidence for a physically driven stage of evolution.