John (Conklin-705) and Ananias Conklin (Conklin-72) were glassmakers who emigrated to New England in the 1630s. They both had some association with Nottinghamshire in England but were probably not born there.
I got interested in the families purely because of the association with my home county and they were flagged with various errors and I've put together a resume of what new research has been done, including an extensive ydna study. It's at Space:Conklin_Families_of_America, please add to it if you have more information.
The y-DNA study shows conclusively that John and Ananias were very closely related (note: not that they were brothers, just that they were very close relatives).
It also shows that they were in no way related to a third Conklin emigrant (Conklin-242) though he is widely said to have been the son of Ananias.
Putting together the yDNA results with the parish register entries, the known history of glassmaking in England and the social context, suggests that John and Ananias were perhaps first cousins, both originating from one migrant family from mainland Europe who settled in the West Midlands and then moved to the Nottinghamshire area when glass making was expanded there in the early 1600s.
I'd like to suggest that John Conklin was most probably the son of John Conklin (Conklin-2758) and that Ananias was likely the son of Francis Conklin (Conklin-2725).
Linking them like this is entirely consistent with all the evidence, but would deserve an "unverified parent" tag at the same time. The PGM project manages both profiles.