Hox Gene Collinearity May Be Related to Noether’s Theory on Symmetry and Its Linked Conserved Quantities

Hox Gene Collinearity (HGC) is a fundamental property that determines the development of many animal clades including Vertebrates. In the Hox gene clusters the genes are located in a sequence Hox1, Hox2, Hox3,… along the 3’ to 5’ direction of the cluster in the chromosome. During Hox cluster activation the Hox genes are expressed sequentially in the ontogenetic units D1, D2, D3,… along the anterior (A)- Posterior (P) axis of the early embryo. This collinearity, first observed by E.B. Lewis, is surprising because the spatial extent of these structures (Hox clusters and embryos) differ by about 4 orders of magnitude. Biomolecular mechanisms alone cannot explain this correlation. Long range physical interactions like diffusion or electric attractions should be involved. A biophysical model (BM) has been  formulated which cooperates with the biomolecular processes and describes the data successfully. Hundred years ago E. Noether made a fundamental discovery in Mathematics and Physics. She proved rigorously that a physical system obeying a symmetry law (e.g.rotations or self similarity) is linked to a conserved physical quantity. It is argued here that HGC obeys a ‘primitive’ self similarity symmetry of the genes of a Hox cluster along a finite straight line. In the case of Vertebrates, the associated partially conserved quantity is the ever increasing ‘ratchet’- like gene ordering where some Hox genes are missing. Another application of Noether’s Theory is performed to rotationally symmetric embryos like the sea urchin.