Gauging genetic odds after repeated inbreeding

Cross pollination of two individual plants produces an offspring with two sets of chromosomes. The figure shows three genes, each having a pair of chromosome segments called alleles. One set of alleles, labeled a₁, a₂, and a₃, comes from the mother plant, and the other, labeled A₁, A₂, and A₃, comes from the father. Self pollinate the offspring (F₁) and the next generation’s (F₂’s) sets mix up the a and A alleles. Repeat the procedure for many generations and you end up with a recombinant inbred line (RIL), a plant with identical sets of alleles. Such plants are widely used in agriculture to locate the genes responsible for particular traits. In 1931 biologists John Haldane and Conrad Waddington worked out the probabilities for producing RILs for two and three genes. By reaching into their bag of physics tools, Areejit Samal (now at the Institute of Mathematical Sciences in India) and Olivier Martin (French National Institute for Agricultural Research) have found the long-awaited solution for any number of genes. They described allelic pairs as spins using a formula introduced by Roy Glauber for the probabilities of spin configurations in an Ising magnet, a theoretical model of ferromagnets that restricts spins to two states: up or down. Then they realized that the main unknown in their expression for the RIL probability, the expectation values of spin configurations involving four or more spins, could be solved by turning to quantum field theoretical equations developed by Freeman Dyson and Julian Schwinger. Samal and Martin say their solution may be useful for gene mapping, which relies on comparing the relative likelihood of different genotypes. (A. Samal, O. Martin, Phys. Rev. Lett. 114, 238101, 2015, doi:10.1103/PhysRevLett.114.238101 .)