Averaging fuzzy biopolymers

Let d be a metric on the set FP(X) of fuzzy subsets of a finite set X. A midpoint of two fuzzy subsets μ,ν∈FP(X) is any fuzzy subset ξ∈FP(X) such that d(ξ,μ)=d(ξ,ν)=12d(μ,ν). These midpoints can be used to represent “middle ways” or “compromises” between two situations described by the fuzzy subsets μ and ν. Now, the imprecise knowledge of a nucleic acid or protein sequence of length N can be modeled by means of a fuzzy biopolymer, a fuzzy subset of a kN-element set with k the number of bases, 4, in the case of nucleic acids, and of amino acids, 20, in the case of proteins. Thus, a midpoint of two fuzzy biopolymers of the same length can be understood as an average of the knowledge of the sequences represented by them. In this paper we explicitly describe the midpoints of two fuzzy biopolymers with respect to distances obtained by aggregating, through several suitable mappings, metrics on each position of the sequences represented by the fuzzy biopolymers.