Information‐theoretical entropy as a measure of sequence variability

We propose the use of the information‐theoretical entropy, S = −Σp_i log₂ P_i, as a measure of variability at a given position in a set of aligned sequences. p_i 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 V_S, a related measure, in detail with V_K, 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 V_K lacks and has intuitive and statistical meanings that accord well with the notion of variability. We find that V_K 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.