Good question. The green pigment is biliverdin, which is a product of the breakdown of heme. But more specifically I am not quite sure, I'll have a look.

I have found this open access article from 2020:Taboada, Carlos, et al. "Multiple origins of green coloration in frogs mediated by a novel biliverdin-binding serpin." Proceedings of the National Academy of Sciences 117.31 (2020): 18574-18581.

This article goes into more specificity about the protein that binds the pigment and allows it to become concentrated inside of the body and bones. In this article they point out that green is a difficult pigment to make using vertebrate biochemistry. Multiple frogs have evolved in parallel this mechanism to sequester the green biliverdin within the body, develop a translucent skin, and become green this way (for camouflage).

However, the common milk frog is not very green. It could be that the biliverdin-binding protein is a trait that is derived from an ancestor and is now vestigial. In the cited article, near the end, they do include a section "Other BBSs Functions". In this section they point out that some frogs with green bones do not look green, and suggest that the protein could be involved in inflammation pathways and behave as an anti-oxidant. But nothing conclusive.

I quickly skimmed for other recent papers but I can't find anything conclusive.

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