Non-fiction: An Algebra for Theoretical Genetics

Overview
Claude Shannon’s 1940 paper proposes a compact algebraic language for Mendelian inheritance, replacing verbal rules and Punnett squares with symbolic operations that compute genotype and phenotype distributions. The work shows how mating, segregation, dominance, and linkage can be expressed within a unified algebra over numerical coefficients that represent probabilities, yielding the familiar ratios of elementary genetics as straightforward algebraic expansions. It is an early example of Shannon’s broader style: abstracting a domain into clean operators and equivalence rules so that reasoning becomes calculation.

Purpose and Framework
Shannon’s aim is to formalize the combinatorial steps of inheritance so that general problems, multi-locus crosses, sex-linked traits, and partial linkage, can be handled uniformly. He introduces symbols for alleles and structured products for genotypes and matings. Coefficients record frequencies or probabilities, and the operations are defined to mirror biological processes: bilinear combination for mating, projection for gamete formation, and homomorphic mapping to collapse genotype classes into phenotypes under dominance relations.

Core Constructions
Alleles at a locus are represented by basic elements; genotypes are products of these elements, with rules enforcing that a genotype contains the appropriate number of alleles per locus. Mating is treated as a bilinear product: the offspring distribution is obtained by expanding the product of parental genotype expressions and then applying segregation operators that split heterozygous terms with the correct probabilities. Dominance is captured by a mapping from the genotype algebra to a phenotype algebra that identifies genotypes with the same phenotypic effect, allowing genotype counts to be post-processed into phenotypic frequencies without redoing the combinatorics. Linkage and recombination enter as linear transformations with parameters such as the recombination fraction r, which mix multi-locus terms in proportions dictated by crossing-over.

Worked Implications
The algebra reproduces the canonical Mendelian results as quick computations. A monohybrid cross expands to yield the 1:2:1 genotype proportions, which map to the 3:1 phenotypic ratio under simple dominance via the homomorphism. Dihybrid crosses factor into independent single-locus components in the case of free assortment, producing the 9:3:3:1 phenotypic distribution. When loci are linked, the algebra replaces independence with a recombination operator, cleanly shifting mass from parental combinations to recombinants according to r, and recovering the expected departure from the 9:3:3:1 pattern. Sex linkage is accommodated by assigning sex-specific allele carriers and modifying the segregation operators for X and Y, yielding characteristic differences between male and female offspring distributions.

Scope and Methodological Value
Beyond re-deriving textbook ratios, the system consolidates multi-step reasoning, mating, segregation, dominance, and mapping to phenotypes, into a single pipeline that is easy to extend. Because everything is linear or bilinear with probabilistic coefficients, the computations can be mechanized or translated into matrices, a point that anticipates later algorithmic treatments of genetic prediction.

Significance and Legacy
The paper’s contribution is not new biology but a unifying calculus that clarifies assumptions, prevents combinatorial slips, and scales to more complex crosses. It shows how probability, algebra, and biology can be fused so that qualitative rules become quantitative operators. In retrospect it foreshadows two hallmarks of Shannon’s later work: representing discrete alternatives with algebraic objects, and separating structural mappings (genotype to phenotype) from noise-like mixing processes (segregation and recombination). The algebra provides a crisp foundation for theoretical genetics problems of its era and a template for subsequent formal systems, including later genetic algebras and matrix-based population genetics models.