sentences:  ""From K = 5-13, all Nilo-Saharan speaking populations from southern Sudan, and Chad cluster with west-central Afroasiatic Chadic-speaking populations (Fig.S15).These results are consistent with linguistic and archeological data, suggesting a possible common ancestry of Nilo-Saharan speaking populations from an eastern Sudanese homeland within the past ~10,500 years, with subsequent bi-directional migration westward to Lake Chad and southward into modern day southern Sudan, and more recent migration eastward into Kenya and Tanzania ~3,000 ya (giving rise to Southern Nilotic speakers) and westward into Chad ~2,500 ya (giving rise to Central Sudanic speakers) (S62, S65, S67, S74).