Are the transverse septum in sharks and the diaphragm in mammals homologous structures?
I have searched on Google Scholar and Web of Science, but haven't found substantial evidence to prove or falsify the claim.
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A beginning of answer here below, I hope. Please first consider that many structures are involved in the question here, the diaphragm (UBERON:0001103), the diaphragmaticus muscle (UBERON:0036071) and the septum transversum (UBERON:0004161). At Bgee (bgee.org) we aim annotating relations of similarity between anatomical structures, please have a look at our GitHub https://github.com/BgeeDB/anatomical-similarity-annotations
We already annotated 'diaphragm' as a mammalian structure, not homologous in Amniota (please see https://raw.githubusercontent.com/BgeeDB/anatomical-similarity-annotations/master/release/similarity.tsv). In our next release, you will see the annotation for the 'diaphragmaticus muscle' which is an analog organ in Crocodylians (and Turtles) but not homologous to the mammalian diaphragm either. See here for more details about this new Uberon class: https://github.com/obophenotype/uberon/issues/1229.
Based on the comments here above, I would say that currently we can argue that there is no evidence for a homologous relationship between the 'septum transversum' in sharks and the mammalian diaphragm. Please note that UBERON:0004161 septum transversum describes the (mammalian) embryonic structure that will give rise to the central tendon of the diaphragm, while here you are talking about a adult structure closer to a 'diaphragmaticus muscle'-like septum, as far as I understand. But anyway thank you for your interesting question that points out a very exciting and rapidly evolving evo-devo field, as this recent paper also suggests https://www.ncbi.nlm.nih.gov/pubmed/?term=27859325