On the use of Algebra in Genetics: From Phenotype to Genotype

In this work, we propose a mathematical framework based on linear algebra that connects the observed phenotype frequencies in finite population samples with the underlying genetic information encoded in genotype and allele frequencies, and the allelic interactions that govern dominance in gene expression. In cases where a genotype uniquely determines the observed phenotype, we derive an analytical relationship between phenotype frequency, genotypic frequencies, and allelic interactions. Given known dominant recessive allelic interactions and the observed phenotypic frequencies in a population sample, we express all compatible genotypic frequencies in a closed algebraic form. Furthermore, under the assumption of complete (100%) penetrance, and given both phenotypic and genotypic frequencies in a population sample, we derive a matrix representation of the allelic dominance relationships. Future implementations of the proposed framework could incorporate high throughput genomic data, thereby enhancing our ability to investigate the genotype phenotype relationship a crucial step in understanding complex traits, genetic disorders, and disease susceptibilities.