Multifractals, encoded walks and the ergodicity of protein sequences

A variety of statistical methods have been developed to explore correlations in protein and nucleic acid sequences. Such correlations have important implications for the evolution and stability of these macromolecules. Recently, a number of fractal analyses of sequence data have been developed. These analyses have considerable appeal as they are extremely sensitive to long range correlations and to hierarchical structures. One such analysis decodes sequence information into a random walk and the statistics of the resulting random walk is investigated. Anomalous scaling of such walks has been interpreted as indicative of a fractal structure. Alternatively, a generalized box counting analysis of decoded sequences can be used to establish multifractal properties. In this work, the connection between these two seemingly disparate approaches is established. This connection is exploited to investigate correlations in protein sequences. An ensemble consisting of a comprehensive data set of representative protein sequences is analyzed to establish the ergodicity of protein sequences. The implications of this ergodicity for information theoretical approaches to protein structure prediction is explored.