Thursday, October 8, 2015

A Brief History of Exponents

The Math With Bad Drawing blog has a nice little post explaining very lucidly who the notion of exponents of repeated multiplication was generalized in a way that is pretty much unique to allow for exponents that have values other than whole numbers.

This fact, typically first taught in middle school or high school algebra, has been well known for a long time. Euclid toyed with the idea a little.  Ancient Greek scientist Achimedes first generalized the concept and proved the law of exponents. A fairly efficient form of exponential notation was invented by Nicolas Chuquet in 1484. More than three hundred years ago RenĂ© Descartes established the modern superscript notation for exponents in the late 1600s around the same time that Newton's law of gravity and motion were invented and around the same time that Newton and Leibniz invented calculus (the modern notation used in undergraduate calculus follows the practice of Leibniz and not Newton's much more awkward notation).

There has been one notable elaboration of a similar concept in mathematics since then, called the fractal dimension which was first formally defined using that name by the late Benoit Mandelbrot in 1967 and entered the upper level college mathematics curriculum in the late 1980s and early 1990s, around the time I was an undergraduate math major. This concept was also invented in Newton's day, but then consigned to the dustbin of history as a curiosity until the late 1800s when several mathematicians developed it some more, and then remained out of sight until Mandelbrot, more or less single handedly repopularized the concept in a way that actually stuck and found practical applications.

The fractal dimension generalizes the notion of a dimension in a manner similar to the way that the law of exponents generalizes the notion of repeated multiplication by relating change in detail to change in scale.  For example, the smaller the ruler you use to measure a shoreline, the longer the shore gets in ruler lengths, because the ragged pattern of a shoreline has a high fractal dimension, while a smooth shoreline would have a low fractal dimension and doesn't change in length at all based upon the length of the ruler used to measure it.

I probably wouldn't ordinarily have found any of the blog post on exponents notable at all. But, earlier just this week, I had been thinking about the precise issue of how the generalized notion of an exponent is so much more subtle than the naive repeated multiplication definition, in the context of thinking about Euler's formula and the Euler's number "e", which is equal to approximately 2.71828 and is a transcendental number that cannot be produced from the ratio of any two integers (something called a rational number). It felt remarkable to see in illustrated print found at random on the Internet, almost exactly the same line of thought.

I guess I still belong to the math tribe, even though I'm a lawyer now.

4500 Year Old Ethiopian Ancient DNA

UPDATE: This is the first African autosomal ancient DNA sample that I am aware of, a remarkable technological feat, and it is paradigm shifting.  There is so much data in the whole genome of even a single individual, and the accumulated genomes of various modern and ancient populations is sufficiently significant already, that it is possible to make reliable and powerful inferences even with a sample size of just N=1 from a new population, ancient or modern, as this paper does.

UPDATE 2 (October 9, 2015): More figures at this tweet.

I had originally read this paper to imply that the 4500 year old Southwest Ethiopian male Mota in the sample was Eurasian admixed. Upon a more careful reading, it appears that this individual predates significant recent Eurasian admixture and can be used as a reference point to establish the levels of Eurasian back migration found in other African populations, since he has no measurable Eurasian admixture himself.
Characterizing genetic diversity in Africa is a crucial step for most analyses reconstructing the evolutionary history of anatomically modern humans. However, historic migrations from Eurasia into Africa have affected many contemporary populations, confounding inferences. Here, we present a 12.5x coverage ancient genome of an Ethiopian male (‘Mota’) who lived approximately 4,500 years ago. 
We use this genome to demonstrate that the Eurasian backflow into Africa came from a population closely related to Early Neolithic farmers, who had colonized Europe 4,000 years earlier. The extent of this backflow was much greater than previously reported, reaching all the way to Central, West and Southern Africa, affecting even populations such as Yoruba and Mbuti, previously thought to be relatively unadmixed, who harbor 6-7% Eurasian ancestry.
M. Gallego Llorente et al, "Ancient Ethiopian genome reveals extensive Eurasian admixture throughout the African continent" Science (October 8, 2015) DOI: 10.1126/science.aad2879

Hat tip to Dienekes.

Uniparental Haplogroups

Eurogenes notes that "this individual belongs to Y-haplogroup E1b1 and mtDNA haplogroup L3." These uniparental haplogroups, which are disclosed in the supplemental materials to the paper.

The mtDNA haplogroup is more specifically L3x2a which "is restricted to the Horn of Africa and the Nile Valley in modern Ethiopian samples, suggesting a degree of maternal continuity in Ethiopia over the past 4,500 years. . . . Mutation E-P2, present in Mota, represents the most widespread subclade of haplogroup E and has been found at high frequency in modern Ethiopians."

This individual also strengthens the case for an African origin of Y-DNA E, relative to a back migration hypothesis, because trace levels of Neanderthal ancestry found in other Africans can now be firmly attributed to recent Eurasian sources and are not present in this not really very old Y-DNA E African individual without Neolithic era Eurasian admixture.  This suggests that any back migration of Y-DNA E would have happened, if it did happen, prior to any Neanderthal admixture, which is present in all modern non-Africans.

The Source and Context of the Ancient DNA Sample
Mota Cave, situated 1,963 meters above sea level in the Gamo highlands of southwest Ethiopia, overlooks the Kulano River, a tributary of the Deme-Omo River. The cave was found in 2011 in collaboration with local Gamo elders and partially excavated in 2012. It measures 14 meters in width and 9 meters in depth and contains more than 60 centimeters of anthropogenic deposits and substantial rock fall. The cave’s deposits suggest at least seven different human occupations from the middle to late Holocene (c. 5295 BP to c. 300 BP), and contains the only middle Holocene burial known in southwest Ethiopia. This burial consists of a complete but fragmentary male adult skeleton dated via AMS radiocarbon to the fifth millennium BP (OxA-29631: 3997 ± 29 BP; 4524-4418 Cal BP).
This is part of an endorheic basin that flows into Lake Turkana in the Southwestern Ethiopia.

The context isn't well enough established to know if this individual was part of a hunter-gatherer, Neolithic, or metal age material culture, but there are hints that he might have been a hunter-gatherer because the only relics found with the body are "A geode and at least 27 obsidian, chert, and basalt flaked stone tools were found in the grave; such artefacts are characteristic of the Later Stone Age lithic tool assemblage present in much of the cave’s deposits.", and because of his genetic affinity to the Sandawe people, discussed below, who are a click speaking people who were a relict hunter-gatherer population of East Africa until about 150 years ago.

Genetic Affinities Of Linguistic Groups

This is right in the vicinity of the homeland of the Southern Omotic languages like Ari.  Cushitic languages are also spoken in the region, which has a high level of linguistic, religious and ethnic diversity in an area that is 90% rural.

The supplement also notes that "Principal component analysis shows that Ari and Sandawe are the closest contemporary populations to Mota.", and that Mota has no discernable Neanderthal component relative to modern African populations.
Mota was placed close to the Ethiopian samples, in between the clusters formed by the Ari and the Sandawe (but very close to an Ari individual that stands out from the rest of that group). The Ari can be split into two castes, Ari Cultivator and Ari Blacksmith, which share a common origin within the last 4,500 years. Since data on a larger number of SNPs are available for Ethiopian populations, we repeated the PCA using this higher quality dataset, which gave us 484,161 usable SNPs that could be called in Mota. Once again, Mota fell in between the Ari and the Sandawe cluster. . . . The Ari speak a language classified as Omotic, which is the most differentiated branch of the Afro-Asiatic languages. Gumuz, a population member of the Nilo-Saharan family (also an Afro-Asiatic language), also shows a high level of shared drift with Mota, but significantly less than the Ari. Sandawe, which are closer to Mota in the PCAs, do not show high shared drift with Mota in the f3, possibly because they are closer to the Khoisan populations than the other Eastern African populations.
Southwest Ethiopia is about as far from the place where any putative migration across the Gate of Tears would have taken place as one can be in Ethiopia and is in an area where Omotic languages are among the languages currently spoken.  The Sandawe people who also cluster with Mota currently live in central Tanzania but almost surely had a much larger geographic range in the past.  The highly tonal features of the Omotic languages may perhaps reflect a modified click language heritage.

The date is also a bit early for any Eurasian admixture to have an Ethio-Semitic source.  And, other recent studies have suggested that Eurasian admixture in non-Ethio-Semitic populations of Ethiopia (presumably arriving via the Blue Nile) took place at about the same time as Ethio-Semitic admixture.

Mota is not particularly close to Nilotic, Cushitic or Ethio-Semitic populations genetically.  Nor was Mota close to the Hadza people, another relict population of Paleo-Africans in East Africa.  Only the Sandawe and Omotic populations were reasonably close to Mota in the PCA analysis.  The study also tends to show that the Sandawe and Ari people of the Owo Valley cluster together rather closely genetically relative to other African populations, possibly shedding some light on the linguistic position of the Omotic language.  Both Cushitic and Ethio-Semitic populations deviate from the cluster that includes Mota in the same direction, while Nilotic and Hadza populations are essentially orthogonal to the Afro-Asiatic populations in the PCA.  The Omotic people, Sandawe people, and Mota are clustered together midway between the Afro-Asiatic, Nilotic and Hadza spokes at about 120 degree angles from each other.

Implications For Eurasian Ancestry In Other Africans
We used f4 ratio analysis to formally assess the extent of back-migration to Africa by West Eurasians . . . . Mota does not show any evidence of a West Eurasian component. . . . This contrasts in particular with the Ari, their closest contemporary relatives, which show large West Eurasian components (17.8%±1.0% and 14.9%±1.2% for Ari Cultivator and Ari Blacksmith, respectively). We confirmed that such a difference is not due to a comparison of a single individual to population estimates by recomputing the f4 ratio for each individual belonging to an Ethiopian population in our dataset. 
The absence of a West Eurasian component in Mota supports the dating of the backflow into Africa, which, at ~3.5kya, is younger than our ancient genome (dated to 4.5 kya). Given that Mota predates the backflow, it potentially provides a better unadmixed African reference than contemporary Yoruba. Thus, we recomputed the extent of the West Eurasian component in contemporary African populations using Mota . . . instead of Yoruba in our f4 ratio. By using this better reference, we estimated West Eurasian admixture to be significantly larger than previously estimated, with an additional 6-9% of the genome of contemporary African populations being of Eurasian origin. Importantly, this analysis shows that the West Eurasian component can be found also in West Africa, albeit at lower levels than in Eastern Africa. Importantly, a sizeable West Eurasian component is also found in the Yoruba and Mbuti, which are often used a representative of an unadmixed African population.
Ethiopians have more Eurasian admixture than other Africans, but essentially all modern Africans have significant levels of Eurasian admixture relative to Mota.

With respect to Neanderthal and Denisovan ancestry:
Given that Mota is our best example of an unadmixed African population, we used it as a reference to assess the affinity of a number of contemporary genomes with Neanderthals. We also investigated the effect of using Mota as a reference when estimating Denisovan introgression. We performed this analysis using the complete genomes (rather than a subset of SNPs as in earlier analyses), since a large number of SNPs is needed to obtain accurate estimates. . . . Both Yoruba and Mbuti were shown to have a small Neanderthal component, in line with their West Eurasian ancestry. As expected, estimates for French and Han were higher than for either of the two contemporary African genomes (from 0.21% in Mbuti to 2.96% in Han).
No evidence of any Denisovan ancestry was found in Mota or any of the other African samples tested with him as a reference for an unadmixed African genome.

The Nature of the Neolithic Eurasian Population Migrating To Africa

The fact that the inferred Eurasian component in other Africans determined with reference to Mota is similar to early Neolithic farmers in Europe is also notable.
Since we have in Mota an unadmixed African population, we can look for the origin of the West Eurasian backflow by modelling contemporary Ari as a mixture of Mota and possible source populations. 
We do this by using the admixture f3-statistics . . . from our global panel or a Eurasian ancient genome. For the latter, we used a representative of Mesolithic hunter-gatherers (Loschbour), and one of the Early Neolithic farmers (LBK, also known as Stuttgart); these two genomes were chosen for their high coverage, allowing us to use most of the SNPs available for contemporary populations and Mota.... 
LBK (an early Neolithic farmer) and Sardinians are the two most likely sources (showing the most negative admixture f3 values) for the Eurasian admixture in the Ari. A number of other analyses have shown Sardinians to be the closest contemporary population to early Neolithic farmers that came into Europe from the Near East, as contemporary populations from that region have been affected by large-scale populations movements in the last few millennia. Thus, the West Eurasian backflow originated from the direct descendants of the same early farmers who brought agriculture into Europe. Given that we have a putative source for the West Eurasian component, we can re-estimate its extent by using LBK as its source in our estimation of the f4 ratio . . . . without having to worry about West African ancestry in the source.

We next tested whether the West Eurasian component found in Yoruba, which had been previously suggested to be older than Mota [dated to 9.6k±1.8k yrs ago . . .], comes from the same source found for the Ari. We use the D statistics . . . . from our global panel or a Eurasian ancient genome. Sardinians and LBK were again found to be the most likely source of the West Eurasian component (giving the strongest positive values that indicate excess affinity between X and Yoruba compared to Mota). This result suggests that there was a single source for the West Eurasian component found throughout Africa.
The Copper and Bronze Age steppe and indigenous European hunter-gatherer population sourced admixtures that transformed the gene pool of early Neolithic Europe did not, by and large, extend to Africa.  But, given recent ancient DNA results from Neolithic Western Anatolia and the shifts that Near Eastern populations have seen genetically since then, I'm inclined to call a population that is similar to LBK and Sardinian individuals, Western Anatolian rather than Near Eastern.

This is consistent with previous studies showing that Eurasian admixture in East African and Khoisan people was more similar to Levantine people than to South Arabians and (except among highly West Eurasian Ethio-Semitic individuals with some South Arabian affinities) was of a uniform character throughout Africa similar in proportion and type to modern Omotic people. It would be interesting to see, however, if the Chadic people who live mostly in the Sahel between North Africa and Sub-Saharan Africa, have different autosomal Eurasian affinities to match their unique Y-DNA Eurasian affinities.

As long as the people with Early European Farmer type genetic began their migration that culminated in sub-Saharan Africa before the influx of Steppe-like people into Europe, this doesn't pose a paradox. As summarized in a blockbuster paper earlier this year:
By ~6,000-5,000 years ago, a resurgence of hunter-gatherer ancestry had occurred throughout much of Europe, but in Russia, the Yamnaya steppe herders of this time were descended not only from the preceding eastern European hunter-gatherers, but from a population of Near Eastern ancestry. Western and Eastern Europe came into contact ~4,500 years ago, as the Late Neolithic Corded Ware people from Germany traced ~3/4 of their ancestry to the Yamnaya, documenting a massive migration into the heartland of Europe from its eastern periphery. This steppe ancestry persisted in all sampled central Europeans until at least ~3,000 years ago, and is ubiquitous in present-day Europeans.
Allowing at least 500-1,500 years for a group of Early European Farmer-like people to migrate from Western Anatolia to Ethiopia before the resurgence of hunter-gatherer ancestry or the steppe ancestry had changed the European gene pool is not an unreasonable scenario. This trip involves a march of about 1200 miles more or less due South (although, obvious, the route would not be as the crow flies).

This is comparable to the time needed for Early European farmers to advance that far (i.e. to the Northern coast of Continental Europe and Southern Scandinavia) and with that much of a change in latitude in Europe during the first wave of the Neolithic revolution in Europe.

The time depth and distribution of Y-DNA T (which is present at relatively high levels on Omotic and Cushitic populations relative to Ethio-Semitic populations) suggests that this may have been an important Y-DNA haplogroup of the EEF-like Neolithic farmers whose autosomal DNA contributed to Africa's gene pool via the Levant, possibly with Y-DNA J mixed in (although the multiple possible historical events that could have spread Y-DNA J complicate the analysis).  But, Y-DNA T is too young to be a plausible candidate accompanying the spread of mtDNA M1 (as has been suggested by some) and U6 in their migrations back from Eurasia ca. 30,000 years ago, and is a poor fit to mtDNA clades that probably arrived in Africa via Iberia and then spread across North Africa to East Africa.

Y-DNA F* is basically absent from Africa, and Y-DNA I, while old enough, has a distribution that is to thin and patchy to be a very strong candidate for a companion to mtDNA M1 and U6.

Y-DNA J has about the right geographic spread in Africa to match mtDNA M1 and U6 as part of the same back migration, but it is hard to know how much of Y-DNA J is due to Semitic migration to Africa (Ethio-Semitic and Phoenician first, and then Arab later) in the last 4,000 years, how much is due to earlier Neolithic and Paleolithic migrations.  Another possibility is that a Y-DNA E population migrated to Iberia early in the Upper Paleolithic era (where it left genetic traces) and then back migrated to NW Africa ca. 30,000 years ago with mtDNA M1 and U6 women from Europe.

The Origins of African Herding And Farming

The apparent timing of the Eurasian Neolithic admixture in almost all modern Africans is also relevant to determining what role, if any, migrant people from food producing societies played in the conversion of wild African plants into domesticated crops that sustained early African-style farming.

Well dated plant remains can determine when domestication happened, and this can be compared to the apparent dates of admixture of people who resembled Early European Farmers genetically (probably about 1500 BCE - 2500 BCE with the Ethio-Semites arriving closer to 1500 BCE and the best guess for other farmers via the Nile closer to 2000 BCE).

Some of the data on the switch to food production is here. Cattle reached Egypt in the earliest part of the Neolithic revolution in Africa around 7000 BCE, the donkey was locally domesticated around 6000 BCE, and sheep and goats appear around 5000 BCE.

Fertile Crescent crops were mostly unsuited to sub-Saharan climates, so farming came much later to African than herding. Dillon (2007) argues that "The domestication of sorghum has its origins in Ethiopia and surrounding countries, commencing around 4000–3000 BC." Pearl millet cultivation became ca. 3200-2700 BCE in Africa and started in the West with a transfer to the East and to India by 1700 BCE. An early type of locally developed farming in Ethiopia was originally conducted primarily by Omotic people and was flourishing when the Ethio-Semites arrived, but was largely displaced and set aside when Ethio-Semites brought their more developed farming techniques to the Ethiopia ca. 1500 BCE.

The balance of the evidence, therefore, favors the development of African domesticated plants after Fertile Crescent pastoralist populations arrive in Africa, but before a major demic contribution of people with Early European Farmer genetics.

Of course, Mota, because he lived in such an isolated area, could have been one of the last unadmixed people of Africa when he died.  There is a decent chance that there would have been some Omotic farmers within a couple hundred miles or so of the place he died at the time.

Functional Traits

Functional traits discerned from Mota's genome include the following:
Skin colour could not be determined although Mota did not have common European variants associated with light skin colour (rs16891982 and rs1426654). Mota was determined to have had brown eyes (p-value = 0.997) and dark (p-value = 0.996), probably black (p-value = 0.843) hair. . . . Mota did not have any of the major alleles known to cause lactase persistence. . . . Mota . . . lived at high altitude and was . . . likely adapted to hypoxia.
Thus, Mota was probably black, brown eyed and dark haired, lacked lactase persistence associated with many herding and farming populations in Africa, and was genetically adapted to high altitudes.

Wednesday, October 7, 2015

Strict New Limits On BSM Physics

Increasingly, complex theories of particle physics and cosmology are disfavored by experiments.

* There are new combined limits on dark matter product from the ATLAS and CMS experiments at the Large Hadron Collider (LHC) based upon complete Run I data.  No dark matter signal has been observed at the LHC.

The LUX direct dark matter detection experiment still places the most strict bounds on a cross-section of interaction with nucleons for spin independent dark matter (about 10^-45 per cm^2) for dark matter particles of about 10 GeV/c^2 or more of mass.  But, for lighter dark matter particles (certainly below 1 GeV), the maximum cross section of interaction with nucleons is set by CMS at about 10^-40 per cm^2 for spin independent dark matter and about 10^-41 per cm^2.

The cross-section of interaction of a neutrino with a nucleon is on the order of 4*10^-39 to 8*10^-39 per cm^2/GeV.  Thus, the bounds on dark matter cross-sections of interaction from CMS are comparable to those of neutrinos with hundreds of MeV/c^2 of kinetic energy for dark matter particles up to about 10 GeV.  For dark matter particles with masses of 10 GeV or more, exclusion from LUX is comparable to that of neutrinos with less than 10 eV/c^2 of kinetic energy (still relativistic by about three orders of magnitude, but nevertheless a very low energy for a neutrino).

Also, as recently noted, experimental observations of cosmic rays emitted by dwarf galaxies which are dark matter dominated in the dark matter particle theories, place strict bounds on the mean lifetime and dark matter annihilation cross-sections of any potential dark matter particle.  Dark matter must have a mean lifetime much longer than the age of the universe and must very rarely annihilate. But, this limitation is more model dependent than some of the other boundaries.

None of these experiments, of course, can rule out any kind of dark matter particles whose only interactions with ordinary matter are via gravity, a particularly simple kind of dark matter model that is increasingly favored.

* Theories with an additional Higgs doublet predict an additional pseudo-scalar neutral Higgs boson, often called A, which could be light.  The BESIII collaboration has put increasingly tight boundaries on this possibility in the 212 MeV to 3 GeV mass range, where maximum branching fractions can now be not more than 4.7*10^-6 in J/Psi decays, and is about 100 times smaller than that in parts of that mass range.

Previous experiments have excluded it in other mass ranges for the pseudo-scalar neutral Higgs boson called A.  Generally, these experiments rule out light A bosons for masses from about 212 MeV to 9 GeV with significant branching fractions in a quite model independent fashion, and rule out supersymmetric A bosons with masses of less than that of the Z boson (about 90.1 GeV).

There is simply no meaningful experimental evidence to support theories with multiple Higgs doublets, including supersymmetry.

* New, more strict, limits have been set on the maximum magnetic moment of the neutrino.
The scattering of solar neutrinos off electrons in Borexino provides the most stringent restrictions, due to its robust statistics and the low energies observed, below 1 MeV. Our new limit on the effective neutrino magnetic moment which follows from the most recent Borexino data is 3.1 x 10^-11 mu_B at 90% C.L. This corresponds to the individual transition magnetic moment constraints: |Lambda_1| less than 5.6 x10^-11 mu_B, |Lambda_2| less than 4.0 x 10^-11 mu_B, and |Lambda_3| less than 3.1 x 10^-11 mu_B (90% C.L.), irrespective of any complex phase.
The Standard Model expectation with a simple Dirac mass neutrino model is 3*10^-19 mu_B.  This is non-zero mostly because there is a chance that the neutrino will emit a virtual W boson and a virtual charged lepton that emits a photon at the one loop level.  But, it can be much higher (to the point of approaching thresholds of experimental detection) in models where neutrinos have Majorana mass and in supersymmetric models.

Essentially, this is yet more evidence (along with the continuing non-detection of neutrinoless double beta decay) tending to show that violations of baryon number conservation and lepton number conservation are non-existent, or at least virtually non-existent (high energy sphalerons aside) to the point where they are insufficient to account for the baryon asymmetry to the universe, if you assume that the starting point of the universe had matter and antimatter in equal amounts, or was pure energy.

* There are some two sigma tensions between SM predictions and experimental data in the areas of CP violation and the CKM matrix at the LHC, but researchers think that this it is likely that this is due to "penguin pollution" in the Standard Model predicted value (i.e. the impact of often ignored Feynman diagrams that go into the final prediction but are hard to calculate called "penguins" based upon the way that the Feynman diagram that goes into the calculation looks visually). Overall, however, the new data "set strong constraints on models" beyond the Standard model.