Hox gene clusters are crucial in Embryogenesis. It was observed that some Hox genes are located in order along the telomeric to centromeric direction of the DNA sequence: Hox1, Hox2, Hox3…. These genes are expressed in the same order in the ontogenetic units of the Drosophila embryo along the Anterior-Posterior axis. The two entities (genome and embryo) differ significantly in linear size and in-between distance. This strange phenomenon was named Spatial Collinearity (SP). Later, it was observed that, particularly in the Vertebrates, a Temporal Collinearity (TC) coexists: first is Hox1 expressed, later Hox2 and even later Hox3,…,. According to a Biophysical Model (BM), pulling forces act at the anterior end of the cluster while a cluster fastening applies at the posterior end. Hox clusters are irreversibly elongated along the force direction. During Evolution, the elongated Hox clusters are broken at variable lengths thus split clusters may be created. An Empirical Rule was formulated distinguishing development due to a complete Hox cluster from development due to split Hox clusters. BM can ‘explain’ this Empirical Rule. In a spontaneous mutation where the cluster fastening is dismantled, a minimal pulling force can automatically shift the cluster inside the Hox activation domain. This cluster translocation can probably explain the absence of Temporal Collinearity in Drosophila.

Abstract: Hox gene clusters are crucial in Embryogenesis. It was observed that some Hox genes were located in order along the telomeric to centromeric direction of the DNA sequence: Hox1, Hox2, Hox3…. These genes were expressed in the same order in the ontogenetic units of the Drosophila embryo along the Anterior-Posterior axis. The two entities (genome and embryo) differ significantly in linear size and in-between distance. This strange phenomenon was named Spatial Collinearity (SP). Later, it was observed that, particularly in the Vertebrates, a Temporal Collinearity (TC) coexists: first is Hox1 expressed, later Hox2 and even later Hox3,…,. Hox clusters are irreversibly elongated along the force direction. According to a Biophysical Model (BM), pulling forces act at the anterior end of the cluster while a cluster fastening applies at the posterior end. During Evolution, the elongated Hox clusters are broken at variable lengths thus split clusters may be created. An Empirical Rule was formulated distinguishing development due to a complete Hox cluster from development due to split Hox clusters. BM can explain this Empirical Rule. In an accidental mutation where the cluster fastening is dismantled, a minimal pulling force can automatically shift the cluster inside the Hox activation domain. This cluster translocation can probably explain the absence of temporal collinearity in Drosophila.

It was observed that a cluster of ordered genes (Hox1, Hox2, Hox3,…) in the genome are activated in the ontogenetic units (1, 2, 3,…) of an embryo along the Anterior/Posterior axis following the same order of the Hox genes. This Spatial Collinearity (SC) is very strange since it correlates events of very different spatial dimensions. It was later observed in vertebrates, that, in the above ordering, first is Hox1expressed in ontogenetic unit 1, followed later by Hox2 in unit 2, and even later Hox3 in unit 3….This temporal collinearity (TC) is an enigma and even to-day is explored in depth. In 1999 T. Kondo and D. Duboule, after posterior upstream extended DNA excisions , concluded that the Hox cluster behaves ‘as if’ TC disappears. Here the consideration of TC really disappearing is taken face value and its repercussions are analyzed. Furthermore, an experiment is proposed to test TC disappearance. An outcome of this experiment could be the reappearance (partial or total) of TC.

Hox gene collinearity (HGC) is a multiscalar property of many animal phyla particularly important in embryogenesis. It relates entities and events occurring in Hox clusters inside the chromosome DNA and in embryonic tissues. These two entities differ in linear size by more than four orders of magnitude. HGC is observed as spatial collinearity (SC) where the Hox genes are located in the order (Hox1, Hox2, Hox3 …) along the 3’ to 5’ direction of DNA in the genome and a corresponding sequence of ontogenetic units (E1, E2, E3, …) located along the Anterior – Posterior axis of the embyo. Expression of Hox1 occurs in E1. Hox2 in E2, Hox3 in E3… Besides SC, a temporal collinearity (TC) has been also observed in many vertebrates. According to TC first is Hox1 expressed in E1, later is Hox2 expressed in E2, followed by Hox3 in E3,… Lately doubt has been raised whether TC really exists. A biophysical model (BM) was formulated and tested during the last twenty years. According to BM, physical forces are created which pull the Hox genes one after the other driving them to a transcription factory domain where they are transcribed. The existing experiments support this BM description. Symmetry is a physical-mathematical property of Matter that was explored in depth by Noether who formulated a ground-breaking theory that applies to all sizes of Matter. This theory applied to Biology can explain the origin of HGC as applied not only to animals developing along the A/P axis but also to animals with circular symmetry.

Hox gene collinearity (HGC) is a multiscalar property of many animal phyla particularly important during embryogenesis. It relates events occurring in Hox clusters inside the chromosome DNA and embryonic tissues. These two entities differ in size by more than four orders of magnitude. HGC is observed as spatial collinearity (SC) where the Hox genes are located in the order H1, H2, H3 … along the 3’ to 5’ direction of the DNA sequence. The corresponding embryonic tissues (E1, E2, E3, …) are activated along the Anterior – Posterior axis in the same order. Besides this collinearity a temporal collinearity (TC) has been also observed in many vertebrates. According to TC first is H1 expressed in E1, later is H2 in E2, followed by H3,… Lately doubt has been raised whether TC really exists. A biophysical model (BM) has been formulated and tested in the last twenty years. According to BM, physical forces are created which pull the Hox genes one after the other driving them to a transcription factory domain where they are transcribed. The existing experiments support this BM description. In the present work two equivalent realizations of BM are presented which explain the recent findings on TC as observed in the vertebrates.

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.