Published PSMC estimates of Neanderthal effective population size (𝑁e) show an approximately five-fold decline over the past 20,000 years [1]. This observation may be attributed to a true decline in Neanderthal 𝑁e, statistical error that is notorious with PSMC estimation, or geographic subdivision and gene flow that has been hypothesized to occur within the Neanderthal population. Determining which of these factors contributes to the observed decline in Neanderthal 𝑁e is an important question that can provide insight into human evolutionary history.

Though it is widely believed that the decline in Neanderthal 𝑁e is due to geographic subdivision and gene flow, no prior studies have theoretically examined whether these evolutionary processes can yield the observed pattern. In this paper [2], Rogers tackles this problem by employing two mathematical models to explore the roles of geographic subdivision and gene flow in the Neanderthal population. Results from both models show that geographic subdivision and gene flow can indeed result in a decline in 𝑁e that mirrors the observed decline estimated from empirical data. In contrast, Rogers argues that neither statistical error in PSMC estimates nor a true decline in 𝑁e are expected to produce the consistent decline in estimated 𝑁e observed across three distinct Neanderthal fossils. Statistical error would likely result in variation among these curves, whereas a true decline in 𝑁e would produce shifted curves due to the different ages of the three Neanderthal fossils.

In summary, Rogers provides convincing evidence that the most reasonable explanation for the observed decline in Neanderthal 𝑁e is geographic subdivision and gene flow. Rogers also provides a basis for understanding this observation, suggesting that 𝑁e declines over time because coalescence times are shorter between more recent ancestors, as they are more likely to be geographic neighbors. Hence, Rogers’ theoretical findings shed light on an interesting aspect of human evolutionary history.

References

[1] Fabrizio Mafessoni, Steffi Grote, Cesare de Filippo, Svante Pääbo (2020) “A high-coverage Neandertal genome from Chagyrskaya Cave”. Proceedings of the National Academy of Sciences USA 117: 15132- 15136. https://doi.org/10.1073/pnas.2004944117

[2] Alan Rogers (2024) “Genetic evidence for geographic structure within the Neanderthal population”. bioRxiv, version 4 peer-reviewed and recommended by Peer Community in Mathematical and Computational Biology. https://doi.org/10.1101/2023.07.28.551046

When sequencing more and more genomes of a species (or a group of closely related species), a natural question to ask is how quickly the total number of distinct sequences grows as a function of the total number of sequenced genomes. A similar question can be asked about the number of distinct genes or the number of distinct k-mers (length-k subsequences).
 
The paper “Revisiting pangenome openness with k-mers” [1] describes a general mathematical framework that can be applied to each of these versions. A genome is abstractly seen as a set of “items” and a species as a set of genomes. The question then is how fast the function f_tot, the average size of the union of m genomes of the species, grows as a function of m. Basically, the faster the growth the more “open” the species is. More precisely, the function f_tot can be described by a power law plus a constant and the openness $\alpha$ refers to one minus the exponent $\gamma$ of the power law.
 
With these definitions one can make a distinction between “open” genomes ($\alpha < 1$​) where the total size f_tot tends to infinity and “closed” genomes  ($\alpha > 1$)​ where the total size f_tot tends to a constant. However, performing this classification is difficult in practice and the relevance of such a disjunction is debatable. Hence, the authors of the current paper focus on estimating the openness parameter $\alpha$.
 
The definition of openness given in the paper was suggested by one of the reviewers and fixes a problem with a previous definition (in which it was mathematically impossible for a pangenome to be closed).
 
While the framework is very general, the authors apply it by using k-mers to estimate pangenome openness. This is an innovative approach because, even though k-mers are used frequently in pangenomics, they had not been used before to estimate openness. One major advantage of using k-mers is that it can be applied directly to data consisting of sequencing reads, without the need for preprocessing. In addition, k-mers also cover non-coding regions of the genomes which is in particular relevant when studying openness of eukaryotic species.
 
The method is evaluated on 12 bacterial pangenomes with impressive results. The estimated openness is very close to the results of several gene-based tools (Roary, Pantools and BPGA) but the running time is much better: it is one to three orders of magnitude faster than the other methods.
 
Another appealing aspect of the method is that it computes the function f_tot exactly using a method that was known in the ecology literature but had not been noticed in the pangenomics field. The openness is then estimated by fitting a power law function.
 
Finally, the paper [1] offers a clear presentation of the problem, the approach and the results, with nice examples using real data.

References

[1] Parmigiani L., Wittler, R. and Stoye, J. (2024) "Revisiting pangenome openness with k-mers". bioRxiv, ver. 4 peer-reviewed and recommended by Peer Community In Mathematical and Computational Biology. https://doi.org/10.1101/2022.11.15.516472

Galtier [1] introduces “Aphid,” a new statistical method that estimates the contributions of gene flow (GF) and incomplete lineage sorting (ILS) to phylogenetic conflict.  Aphid is based on the observation that GF tends to make gene genealogies shorter, whereas ILS makes them longer.  Rather than fitting the full likelihood, it models the distribution of gene genealogies as a mixture of several canonical gene genealogies in which coalescence times are set equal to their expectations under different models. This simplification makes Aphid far faster than competing methods. In addition, it deals gracefully with bidirectional gene flow—an impossibility under competing models. Because of these advantages, Aphid represents an important addition to the toolkit of evolutionary genetics.

In the interest of speed, Aphid makes several simplifying assumptions. Yet even when these were violated, Aphid did well at estimating parameters from simulated data. It seems to be reasonably robust.

Aphid studies phylogenetic conflict, which occurs when some loci imply one phylogenetic tree and other loci imply another. This happens when the interval between successive speciation events is fairly short. If this interval is too short,  however,  Aphid’s approximations break down, and its estimates are biased. Galtier suggests caution when the fraction of discordant phylogenetic trees exceeds 50–55%. Thus, Aphids will be most useful when the interval between speciation events is short, but not too short.

Galtier applies the new method to three sets of primate data. In two of these data sets  (baboons and African apes), Aphid detects gene flow that would likely be missed by competing methods. These competing methods are primarily sensitive to gene flow that is asymmetric in two senses: (1) greater flow in one direction than the other, and (2) unequal gene flow connecting an outgroup to two sister species.  Aphid finds evidence of symmetric gene flow in the ancestry of baboons and also in that of African apes. The data suggest that ancestral humans and chimpanzees both interbred with ancestral gorillas, and at about the same rate.  Aphid’s ability to detect this signature sets it apart from competing methods.

References

[1]   Nicolas Galtier (2023) “An approximate likelihood method reveals ancient gene flow between human, chimpanzee and gorilla”. bioRxiv, ver. 3 peer-reviewed and recommended by Peer Community in Mathematical and Computational Biology.  https://doi.org/10.1101/2023.07.06.547897

Reconstructing a phylogenetic network for a set of conflicting phylogenetic trees on the same set of leaves remains an active strand of research in mathematical and computational phylogenetic since 2005, when Baroni et al. [1] showed that the minimum number of reticulations h(T,T') needed to simultaneously embed two rooted binary phylogenetic trees T and T' into a rooted binary phylogenetic network is one less than the size of a maximum acyclic agreement forest for T and T'. In the same paper, the authors showed that h(T,T') is bounded from above by n-2, where n is the number of leaves of T and T' and that this bound is sharp. That is, for a fixed n, there exist two rooted binary phylogenetic trees T and T' such that h(T,T')=n-2.

Since 2005, many papers have been published that develop exact algorithms and heuristics to solve the above NP-hard minimisation problem in practice, which is often referred to as Minimum Hybridisation in the literature, and that further investigate the mathematical underpinnings of Minimum Hybridisation and related problems. However, many such studies are restricted to two trees and much less is known about Minimum Hybridisation for when the input consists of more than two phylogenetic trees, which is the more relevant cases from a biological point of view. 

In [2], van Iersel, Jones, and Weller establish the first lower bound for the minimum reticulation number for more than two rooted binary phylogenetic trees, with a focus on exactly three trees. The above-mentioned connection between the minimum number of reticulations and maximum acyclic agreement forests does not extend to three (or more) trees. Instead, to establish their result, the authors use multi-labelled trees as an intermediate structure between phylogenetic trees and phylogenetic networks to show that, for each ε>0, there exist three caterpillar trees on n leaves such that any phylogenetic network that simultaneously embeds these three trees has at least (3/2 - ε)n reticulations. Perhaps unsurprising, caterpillar trees were also used by Baroni et al. [1] to establish that their upper bound on h(T,T') is sharp. Structurally, these trees have the property that each internal vertex is adjacent to a leaf. Each caterpillar tree can therefore be viewed as a sequence of characters, and it is exactly this viewpoint that is heavily used in [2]. More specifically, sequences with short common subsequences correspond to caterpillar trees that need many reticulations when embedded in a phylogenetic network. It would consequently be interesting to further investigate connections between caterpillar trees and certain types of sequences. Can they be used to shed more light on bounds for the minimum reticulation number?

References

[1] Baroni, M., Grünewald, S., Moulton, V., and Semple, C. (2005) "Bounding the number of hybridisation events for a consistent evolutionary history". J. Math. Biol. 51, 171–182. https://doi.org/10.1007/s00285-005-0315-9
  
[2] van Iersel, L., Jones, M., and Weller, M. (2023) “When three trees go to war”. HAL, ver. 3 peer-reviewed and recommended by Peer Community In Mathematical and Computational Biology. https://hal.science/hal-04013152/

We often use genetic data from a single site, or even a single individual, to estimate the history of effective population size, Ne, over time scales in excess of a million years. Mazet and Noûs [2] emphasize that such estimates may not mean what they seem to mean.  The ups and downs of Ne may reflect changes in gene flow or selection, rather than changes in census population size. In fact, gene flow may cause Ne to decline even if the rate of gene flow has remained constant.

Consider for example the estimates of archaic population size in Fig. 1, which show an apparent decline in population size between roughly 700 kya and 300 kya. It is tempting to interpret this as evidence of a declining number of individuals, but that is not the only plausible interpretation.

Each of these estimates is based on the genome of a single diploid individual. As we trace the ancestry of that individual backwards into the past, the ancestors are likely to remain in the same locale for at least a generation or two. Being neighbors, there’s a chance they will mate. This implies that in the recent past, the ancestors of a sampled individual lived in a population of small effective size.

As we continue backwards into the past, there is more and more time for the ancestors to move around on the landscape. The farther back we go, the less likely they are to be neighbors, and the less likely they are to mate. In this more remote past, the ancestors of our sample lived in a population of larger effective size, even if neither the number of individuals nor the rate of gene flow has changed.

For awhile then, Ne should increase as we move backwards into the past. This process does not continue forever, because eventually the ancestors will be randomly distributed across the population as a whole. We therefore expect Ne to increase towards an asymptote, which represents the effective size of the entire population.

This simple story gets more complex if there is change in either the census size or the rate of gene flow.  Mazet and Noûs [2] have shown that one can mimic real estimates of population history using models in which the rate of gene flow varies, but census size does not. This implies that the curves in Fig. 1 are ambiguous. The observed changes in Ne could reflect changes in census size, gene flow, or both.

For  this  reason,  Mazet  and  Noûs [2]  would  like  to  replace  the  term  “effective  population size” with an alternative, the “inverse instantaneous coalescent rate,” or IIRC. I don’t share this preference, because the same critique could be made of all definitions of Ne. For example, Wright [3, p. 108] showed in 1931 that Ne varies in response to the sex ratio, and this implies that changes in Ne need not involve any change in census size. This is also true when populations are geographically structured, as Mazet and Noûs [2] have emphasized, but this does not seem to require a new vocabulary.

Figure 1: PSMC estimates of the history of population size based on three archaic genomes: two Neanderthals and a Denisovan [1].

Mazet  and  Noûs  [2]  also  show  that  estimates  of  Ne  can  vary  in  response  to  selection.   It is not hard to see why such an effect might exist. In genomic regions affected by directional or purifying selection, heterozygosity is low, and common ancestors tend to be recent. Such regions may contribute to small estimates of recent Ne. In regions under balancing selection, heterozygosity is high, and common ancestors tend to be ancient. Such regions may contribute to large estimates of ancient Ne. The magnitude of this effect presumably depends on the fraction of the genome under selection and the rate of recombination.

In summary, this article describes several processes that can affect estimates of the history of effective population size. This makes existing estimates ambiguous. For example, should we interpret Fig. 1 as evidence of a declining number of archaic individuals, or in terms of gene flow among archaic subpopulations? But these questions also present research opportunities. If the observed decline reflects gene flow, what does this imply about the geographic structure of archaic populations? Can we resolve the ambiguity by integrating samples from different locales, or using archaeological estimates of population density or interregional trade?

REFERENCES

[1] Fabrizio Mafessoni et al. “A high-coverage Neandertal genome from Chagyrskaya Cave”. Proceedings of the National Academy of Sciences, USA  117.26 (2020), pp. 15132–15136. https://doi.org/10.1073/pnas.2004944117.

[2] Olivier Mazet and Camille Noûs. “Population genetics: coalescence rate and demographic parameters inference”. arXiv, ver. 2 peer-reviewed and recommended by Peer Community In Mathematical and Computational Biology (2023). https://doi.org/10.48550/ARXIV.2207.02111.

[3] Sewall Wright. “Evolution in mendelian populations”. Genetics 16 (1931), pp. 97–159. https://doi.org/10.48550/ARXIV.2207.0211110.1093/genetics/16.2.97.