Significantly Lower Entropy Estimates for Natural DNA Sequences

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Abstract
If DNA were a random string over its alphabet {A, C, G, T}, an optimal code would assign two bits to each nucleotide. DNA may be imagined to be a highly ordered, purposeful molecule, and one might therefore reasonably expect statistical models of its string representation to produce much lower entropy estimates. Surprisingly, this has not been the case for many natural DNA sequences, including portions of the human genome. We introduce a new statistical model (compression algorithm), the strongest reported to date, for naturally occurring DNA sequences. Conventional techniques code a nucleotide using only slightly fewer bits (1.90) than one obtains by relying only on the frequency statistics of individual nucleotides (1.95). Our method in some cases increases this gap by more than fivefold (1.66) and may lead to better performance in microbiological pattern recognition applications. One of our main contributions, and the principle source of these improvements, is the formal inclusion of inexact match information in the model. The existence of matches at various distances forms a panel of experts which are then combined into a single prediction. The structure of this combination is novel and its parameters are learned using Expectation Maximization (EM). Experiments are reported using a wide variety of DNA sequences and compared whenever possible with earlier work. Four reasonable notions for the string distance function used to identify near matches, are implemented and experimentally compared. We also report lower entropy estimates for coding regions extracted from a large collection of nonredundant human genes. The conventional estimate is 1.92 bits. Our model produces only slightly better results (1.91 bits) when considering nucleotides, but achieves 1.84-1.87 bits when the prediction problem is divided into two stages: (i) predict the next amino acid-based on inexact polypeptide matches, and (ii) predict the particular codon. Our results suggest that matches at the amino acid level play some role, but a small one, in determining the statistical structure of nonredundant coding sequences.