Two recent papers on Neanderthals may have some relevance regarding the peopling of America. Although neither one of them deal with Amerindians (studies on Modern Human - Nenderthal gene comparisons tend to ignore Native Americans and only focus on Africans, Asians and Europeans!), they do shed some light on them too:
On Admixture between us and them
The first paper (Lohse and Frantz, 2014) , looks into the admixture issue, which some (Eriksson & Manica, 2012) , have put in doubt by considering that the signal of archaic genes found in Neanderthals and Modern Humans does not mean an actual physical introgression, instead it is just the genes that both groups inherited from a common ancestor in Africa, and are therefored shared by both groups without any "intercourse" necessary between them.
Lohse and Frantz prove that admixture took place out of Africa, after the modern Humans began their expansion out of Africa.
They estimate divergence between Neanderthals and H. sapiens as having taken place between 329 and 349 kya, and the split between humans that remained in Africa and those that left at 122 -141 kya.
Interbreeding took place out of Africa and resulted in a slightly higher proportion (fraction) of Neanderthal admixture in modern East Asians (Han Chinese) in comparison to Europeans:
- 5.9% admixture in CHB (Chinese Han in Beijing) [3.9% to 7.9%].
- 5.3% admixture in CEU (Europeans) [3.4% to 7.3%].
This figure of between 3.4 and 7.9% admixture is higher than those disclosed by previous studies (1 - 6% such as Green et al., 2010) .
Also, and this is interesting, the Time of admixture is estimated as having taken place between 37 and 86 kya., with the following values:
- 75.8 ky for CHB (Asians)
- 55.1 ky for CEU (Europeans)
This means that the Neanderthals encountered the Asians first, mixed with them, leaving a higher proportion of their genes in them. And later mixed with humans in Europe in a lower proportion.
The admixture figures are ratified by another paper (Khrameeva et al., 2014) , which looked into lipid catabolism in modern humans and its Neanderthal origin as well as its predominance in Europeans.
They calculated the D statistic (fraction of Neanderthal-like sites, or NLS), which showed a higher admixutre in Asians compared to Europeans:
"reflect[s] relative similarity between the Neanderthal and the modern human genomes tested, they do not provide a quantitative estimate of Neanderthal ancestry, that is, a 5% D-statistic value reflects a higher similarity between population A and Neanderthal compared with that for population B, but does not signify a 5% level of Neanderthal ancestry in the population A genome.
there was no substantial difference in the genome-wide frequencies of NLS between European and Asian populations, with a slight tendency for higher frequencies in Asians: 5.9±0.08 and 6.2±0.06%, respectively." 
They pointed out that the lipid (fat) catabolism has organic implications and mention seven metabolic categories associated with LCP (2-lysophosphatidylcholine) among which is "glucose-dependent insulin secretion [...] These observations support a contribution of Neanderthal genetic variants to the phenotype of contemporary Europeans." .
This ties in with another paper  which shows that a risk gene known as SLC16A11, found at very high frequencies among Native Americans, causes Type 2 diabetes, by changing levels of the protein that the gene encodes which alters the amount of a specific type of fat (lipid) which other studies have linked to the risk of diabetes. This gene is of Neanderthal origin.
According to this paper  :"SLC16A11; it is present at ~50% frequency in Native American samples and ~10% in east Asian, but is rare in European and African samples. Analysis of an archaic genome sequence indicated that the risk haplotype introgressed into modern humans via admixture with Neanderthals."
Its prevalence is five times higher than among Asians! Maybe an indication of a strong Neanderthal introgression in Amerindians.
So we have a link between Amerindian diabetes and fat, caused by a gene of Neanderthal origin and Europeans who also have high levels of fats due to their Neanderthal genes. It is a pity that the recent study on lipids  and the one measuring introgression  did not sample Native Americans. I understand the reason following the orthodox viewpoint:
American Natives descend from Asian stock so they should have diluted Neanderthal genes compared to the Asians. But, what if... Neanderthals reached America first and mixed with H. sapiens there? Wouldn't Amerindians display (as they indeed do - with diabetes), a higher frequency of Neanderthal genetic traits? Could these have spilled back into Eurasia from America? I believe so:
The temporal gradient from East to West of admixture timing seems to imply this: early mixing in Asia, later in Europe. And also the proportion of Neanderthal genes: higher in Asia, closest to the point of admixture (America), lower as you move West, into Europe, where more modern humans watered the Neanderthal genes down.
We will keep on waiting for studies that include American Natives' data.
 Konrad Lohse and Laurent A. F. Frantz, (2014). Neandertal admixture in Eurasia confirmed by Maximum likelihood analysis of three genomes. Genetics: Early Online, published on February 13, 2014, doi: 10.1534/genetics.114.162396
 Eriksson, A. & Manica, A., (2012). Effect of ancient population structure on the degree of polymorphism shared between modern human populations and ancient hominins. PNAS, page doi: 10.1073/pnas.1200567109
 Ekaterina E. Khrameeva et al., (2014). Neanderthal ancestry drives evolution of lipid catabolism in contemporary Europeans. Nature Communications 5, Article number: 3584 doi:10.1038/ncomms4584 April 1, 2014
 Green, R.E., et al., (2010). A draft sequence of the Neanderthal genome. Science, 328(5979), 710–722.
 The SIGMA Type 2 Diabetes Consortium, (2013). Sequence variants in SLC16A11 are a common risk factor for type 2 diabetes in Mexico. Nature, DOI: 10.1038/nature12828
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