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The authors have declared that no competing interests exist.

Analyzed the data: AGMN MS. Wrote the paper: AGMN MS. Conceived the idea to build a model for the interbreeding of Neanderthals and the African Ancestors of Eurasians and performed computer simulations on that model for this theoretical paper: AGMN MS.

Considering the recent experimental discovery of Green

The question of whether all of us, living humans, descend exclusively from a recent (i.e. 100,000–200,000 years old) anatomically modern African population which completely replaced archaic populations in other continents, or if Africans could have interbred with these local hominids has been the subject of a long lasting and interesting debate. The first of these possibilities, known as Out of Africa model, is based mainly on genetic evidence

The decision of which model correctly describes the origin of

Until recently, the majority of the scientific community seemed to favour the Out of Africa model, but things changed radically in 2010 with a paper by Green

Based on their findings that all non-Africans have a similar proportion of Neanderthal genes, and also on archaeological evidence

We will describe by a simple and realistic model the dynamics of two

Our work should be compared with the results of a recent paper

We would like to point out that the present research is coherent with our results

As there seems to be no sign of Denisovan

Consider a population of constant size equal to

We also suppose that the subpopulations had lived isolated from each other for a long time before they met. We will start generations count at the instant

Starting with the initial condition just stated, our model can be fully simulated at the computer. At each generation, the model consists of the following three stochastic steps:

We assume that the total population size is constant and equal to

The above described stochastic process permitting fluctuation of the subpopulation sizes is exactly the same as in the well-known

The number of generations until extinction of one subpopulation in the Wright-Fisher process is random, as well as which of the two subpopulations becomes extinct. If

At any time

We assume that at each generation a number

As will be seen ahead, the typical values of

Remind that the sizes

In this process, individuals which came to some subpopulation at generation

At any time

The model, as formulated up to now, is fully stochastic and can be simulated in the computer. Although simple, such a simulation for a large population and for a large number of generations is very expensive in terms of computer time. We have produced some small computer simulations of the model for the sake of comparison with better results, which we obtain as we now describe.

It is possible to derive equations relating the mean allelic fractions at generation

Of course the above assumption of exchanged individuals all having the mean allelic fractions in their subpopulations is a very strong one and it is not strictly true. Nonetheless, it is indeed a very good

In the mean field approximation, the mean allelic fraction

The above equations, after taking the

The above equations describe deterministic gene flow between two subpopulations with stochastically varying sizes, total population size being constant. Although the interbreeding model is fully stochastic, its description through Eqs. (2) maintains only partially the stochastic character, as a consequence of the mean field approximation. Below we will compare the outcome of Eqs. (2) with simulations of the fully stochastic version of the model, showing that the mean field approximation is indeed a good one.

We stress here that we think of

Most of the results we will exhibit are based on performing a large number of times only step 1 in our three-step description of the model. For each Wright-Fisher path

By introducing the auxiliary functions

and

In general,

As the integrand in the exponent of (3) is positive, this formula shows that

Eq. (4) on the other hand shows that introgression of genes from one subpopulation into the other is generally not symmetric. In fact,

An important special case in which the integrals in (3) and (4) can be exactly evaluated is when

and

The importance of this special case is that it provides some useful approximations. For example, if

With the purpose of illustrating the qualitative behaviour of the solutions of (2), we show in

For two different histories

It can be seen that all qualitative features explained above are present. It should also be noticed that the final values of

The final values of

In

For a single Wright-Fisher path

If we complete simulation of the model by performing the last two steps in the model description, we will obtain what we call the the

On the other hand, we may use the same realization of

The left graph in

Indeed, we believe that the randomness in the sexual reproduction process accounts for the largest part of the difference between theoretical and simulated values. In fact, as shown in

On the other hand, if

It should also be noticed that agreement between theoretical and simulated values is worse for

The right graph in

Although the agreement between theoretical and simulated values is not complete, results shown in

We show here the probability density that the final value of

We know that Neanderthals were extinct and, according to

As we do not know the composition of the total population at the time the two subpopulations met, we will take the initial fraction

As can be seen in

The inset in

The above information shows that the experimental data are better explained by values of

We also see that the probability density for

A technical detail in producing

First we observe that if

The free mating situation, in which subpopulations interact as if there were no differences among their members, is a particular case of this large

The conclusion is that if

If we take instead values of

We have produced a large set of Wright-Fisher paths with random

The right plot in

Mitochondrial DNA and Y chromosome are both inherited in a haploid way. The former is inherited by maternal line and the latter is exclusive to male individuals and, thus, inherited by paternal line. Furthermore mtDNA is not subject to recombination and recombination seems to be negligible for the Y chromosome. It is also believed that large portions of both are selectively neutral. These facts allow an easier mathematical treatment of their statistical properties. From the experimental point of view, mtDNA

More recently

Both authors of this paper have separately claimed that the above evidences favouring the Out of Africa model are in fact compatible with anatomically modern Africans and Neanderthals being part of a single interbreeding population at the times they coexisted. In

These facts imply that the large genealogical distances between living humans and Neanderthals, as seen in mtDNA, are not uncommon in an interbreeding population. On the contrary, they turn out to be very likely if the correct statistics is used. Furthermore, typical distances between individuals in the population formed by Neanderthals and anatomically modern Africans may have been much larger at the time of Neanderthals’ extinction than they are nowadays. They also imply that extinction of Neanderthals’ mtDNA is compatible with the survival of their nuclear DNA.

Exactly the same reasoning can be applied to the mitochondrial and nuclear DNAs of the fossil bones found in Siberia

In

In

As explained before, instead of simulating only Wright-Fisher paths (step 1 in the model description), as we did in the results of

Bar-Yosef

By taking random values for

In the right part of

Our model assumes, for simplicity sake, symmetry in the number of individuals migrating from one subpopulation to the other at each generation. Of course, this does not imply symmetry in the gene introgression of one subpopulation into the other. The reason is that, neutrality assumed, a single individual may change radically the gene pool of a small subpopulation, whereas a single individual in a large subpopulation will probably not alter too much the gene pool of that subpopulation.

This effect is quantified in (4) and explained just after this equation. Moreover, it is clearly visible in

We use the same sample of 790 histories used in Fig. 4 for obtaining the probability density for the final value of

On the other hand, Green

In the framework of the model proposed in this article we could infer that the 1 to 4% fraction

We also estimated the mean number of generations for Neanderthal extinction in the Middle East to be approximately

Moreover, our model is compatible with the lack of introgression of Neanderthals into mtDNA and Y chromosome of living humans. We estimated in only

Neanderthals are implicitly considered in this work as a group within the

Although we do not intend to back up any kind of superiority for Neanderthals, our neutrality hypothesis is at least supported by recent results

In another work

In building the model presented here, one of the strongest concerns of the authors was simplicity. As a consequence of this concern, our model contains a single parameter to be estimated, Eqs. (2) are exactly solvable, and qualitative properties of their solutions (3) and (4) are well understood. Computer simulation was only moderately employed. The use of the Wright-Fisher process as a model for extinction under neutrality of one subpopulation seemed to us more natural than if we had assumed some arbitrary population model with other parameters.

After a first version of this work had been published on line at arxiv.org and reviewed by New Scientist

Whereas we think of neutrality and no biological barriers, with social barriers preventing free mating, they consider a model of African range expansion, free mating with Neanderthals, but strong reproductive isolation between the “ `two species”, probably due to avoidance of interspecific matings, a low fitness of hybrids, or both. They consider several scenarios for the location of Neanderthals during interbreeding, including the possibility of interbreeding restricted to the Middle East, as we did. Nonetheless, the scenario they consider the more probable is the one in which Neanderthals were scattered through a large geographical area including Middle East, Europe and Central Asia. In particular, in the more probable scenario, their model forecasts interbreeding hotspots also in Europe and Central Asia, very far from the Middle East.

We identify two important differences between our assumptions and those of Currat and Excoffier, which could possibly be resolved by future experimental tests. The first difference is that although both models predict asymmetric introgression among interbreeding subpopulations, ours predicts larger introgression of Africans among Middle Eastern Neanderthals. Theirs, on the contrary, predicts smaller introgression of Africans into any Neanderthals. Their prediction is in agreement with the results of Green

Our prediction also agrees with this fact, if we suppose that a fraction of the Neanderthal population never left Europe and did not participate in the postulated Middle Eastern interbreeding. The descendants of these European Neanderthals did not interbreed later with Africans when they came into Europe, or this interbreeding was very small, possibly due to small population densities of the Neanderthals when they were close to definitive extinction. DNA sequencing of late Middle Eastern Neanderthal fossils and comparison with European Neanderthals would be a good test for helping discriminate between the two models.

The second difference is that our model is simpler in that it does not take into account the spatial distribution of the subpopulations. Along with

Current knowledge about Denisovans’ morphology and life style is much less than what we know about Neanderthals. In particular we do not know whether Denisovans lived only in Siberia, where up to now the only known fossils have been found, or elsewhere. Where and when this people made contact with the African ancestors of present day Melanesians and Australians is still a conjecture

As we now know of our Neanderthal and Denisovan inheritances, it is time to ask whether they were the only hominids that Africans mated. We believe that the future may still uncover lots of surprises when Denisovans will be better studied and nuclear DNA of many more Neanderthal and other hominid fossils will become available. In particular, we expect that in a near future experimental tests and archaeological or palaeontological discoveries may discriminate where, when and for how long Africans interbred with Neanderthals and other hominids, and prepare the way for finer theories.

AGMN dedicates this work to the memory of his father, recently deceased. He taught him everything fathers usually teach their children, but also Biology and Mathematics.