What Is Gut Bacteria?
We are symbionts. Every human being needs symbiotic macro-organisms in and on us to survive.
One of our particularly important symbiotic partners are our gut bacteria, which make it possible for us to digest food - something we can't do properly with just our own bodies built to our own DNA specifications. The human appendix, an organ long thought to have no purpose, appears to be an organ that has the purpose of rebooting a person's gut bacteria after it is destroyed by a parasitic infection or some other form of GI distress. Different gut bacteria can result in different kinds of digestive functioning. One of the cutting edge treatments for obesity that is being developed is to do a "gut bacteria transplant," i.e. to kill off dysfunctional gut bacteria that contributes to obesity and then restock the gut with different gut bacteria that doesn't contribute to obesity.
Gut bacteria lineages accumulate mutations over time that make them distinctive, and a passed from parents to children in a manner that is often hereditary, but is not genetic (and generally draws from one's community even when it is not hereditary).
Gut Bacteria Lineages Generally Track Human DNA Lineages
It turns out that the lineages of gut bacteria observed out in the world in modern humans generally tracks the genetic variation found in modern humans. Ethio Helix has a nice post discussing a new open access paper on genetic variation in human gut bacteria (and in particular Helicobacter pylori) showing this correspondence, which largely corroborates the phylogenies developed from human genetics (e.g. Y-DNA, mtDNA, particular autosomal DNA sites, archaic admixture proportions, whole genomes) which shows an African origin for all non-Africans with serial founder events as one population breaks off from another in the process of settling the Earth. The tool also provides an independent potential source from which to date the branching of different peoples in human prehistory from each other. Maju provides a shorter summary of the paper's findings.
Why Aren't There Neanderthal or Denisovian Gut Bacteria Lineages?
All of the human gut bacteria genotyped in the study fits neatly into a unitary phylogeny from a common source. Maju notes that apparently none of the lineages are traceable to archaic hominins (something also true of modern human Y-DNA and mtDNA lineages).
I personally suspect that this is because initial gut bacteria transmission is probably primarily from mother to child, and that almost all of the archaic admixture that survives in modern human populations today derives from archaic hominin men having children modern human women. A hybrid child born into a modern human tribe is also more likely to receive modern human gut bacteria infusions later in life, than that child is to transform the rest of the tribe's gut bacteria.
The reverse scenario (archaic hominin women having children with modern human men) probably took place as well, but those children were probably born into archaic hominin tribes which ultimately went extinct rather than leaving modern descendants.
This kind of pattern is particularly likely if the hybrid pregnancies were generally the result of isolated instances of sexual intercourse, rather than being the product of long term marriage type relationships, which might have been more difficult to sustain on an interspecies basis.
This same pattern would explain why there are no archaic mtDNA lineages even though there is archaic autosomal DNA, since mtDNA is inherited uniparentally from mother to child.
As an aside, the lack of archaic Y-DNA in modern humans requires a separate explanation. I have proposed that this is due to Haldane's law, which states that hybrid children of interspecies unions tend to be homozygotes (in humans women are homozygotes, i.e. XX and men are heterozygotes, i.e. XY). Hence, Haldane's law would provide that hybrid archaic human-modern humans, certainly those that are fertile and leave descendants, would tend overwhelmingly to be female, hence removing archaic Y-DNA lineages in modern human populations.