Information‐theoretical entropy as a measure of sequence variability

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Abstract
We propose the use of the information-theoretical entropy, S = −Σpi log2 Pi, as a measure of variability at a given position in a set of aligned sequences. pi stands for the fraction of times the i-th type appears at a position. For protein sequences, the sum has up to 20 terms, for nucleotide sequences, up to 4 terms, and for codon sequences, up to 61 terms. We compare S and VS, a related measure, in detail with VK, the traditional measure of immunoglobulin sequence variability, both in the abstract and as applied to the immunoglobulins. We conclude that S has desirable mathematical properties that VK lacks and has intuitive and statistical meanings that accord well with the notion of variability. We find that VK and the S-based measures are highly correlated for the immunoglobulins. We show by analysis of sequence data and by means of a mathematical model that this correlation is due to a strong tendency for the frequency of occurrence of amino acid types at a given position to be log-linear. It is not known whether the immunoglobulins are typical or atypical of protein families in this regard, nor is the origin of the observed rank-frequency distribution obvious, although we discuss several possible etiologies.