Phenogram enumeration: The number of regular genotype-phenotype correspondences in genetic systems

A phenotype system is a mapping of a set of genotypes into a set of phenotypes. Two phenotype systems are permutationally equivalent if some permutation or combination of permutations of gene symbols or locus symbols makes them identical. A set of permutationally equivalent phenotype systems is called a phenogram . Two mappings of a set D into a set R are said to be equivalent and to belong to the same pattern if the first mapping performed on some permutation of D is identical to the second performed in conjunction with some permutation of R . Thus the number of phenotype systems which are identical under some permutation of gene or locus symbols is precisely the number of patterns of phenotype systems. De Bruijn's extension of Polya's fundamental counting theorem, which enumerates the number of patterns, is introduced in terms sufficiently general to allow its application to other biological enumeration problems. The theorem is used to solve the general phenogram enumeration problem, i.e. given any genetic system whatever, how many phenograms are there with exactly φ phenotypes? The results of calculations for several basic genetic systems are presented, along with some observations on these. A nomenclature which simplifies the presentation of two locus systems showing epistasis is suggested. Several applications of phenogram analysis are discussed.