Soraggi et al. [2] describe HMMploidy, a statistical method that takes DNA sequencing data as input and uses a hidden Markov model to estimate ploidy. The method allows ploidy to vary not only between individuals, but also between and even within chromosomes. This allows the method to detect aneuploidy and also chromosomal regions in which multiple paralogous loci have been mistakenly assembled on top of one another. 

HMMploidy estimates genotypes and ploidy simultaneously, with a separate estimate for each genome. The genome is divided into a series of non-overlapping windows (typically 100), and HMMploidy provides a separate estimate of ploidy within each window of each genome. The method is thus estimating a large number of parameters, and one might assume that this would reduce its accuracy. However, it benefits from large samples of genomes. Large samples increase the accuracy of internal allele frequency estimates, and this improves the accuracy of genotype and ploidy estimates. In large samples of low-coverage genomes, HMMploidy outperforms all other estimators. It does not require a reference genome of known ploidy. The power of the method increases with coverage and sample size but decreases with ploidy. Consequently, high coverage or large samples may be needed if ploidy is high. 

The method is slower than some alternative methods, but run time is not excessive. Run time increases with number of windows but isn't otherwise affected by genome size. It should be feasible even with large genomes, provided that the number of windows is not too large. The authors apply their method and several alternatives to isolates of a pathogenic yeast, Cryptococcus neoformans, obtained from HIV-infected patients. With these data, HMMploidy replicated previous findings of polyploidy and aneuploidy. There were several surprises. For example, HMMploidy estimates the same ploidy in two isolates taken on different days from a single patient, even though sequencing coverage was three times as high on the later day as on the earlier one. These findings were replicated in data that were down-sampled to mimic low coverage. 

Three alternative methods (ploidyNGS [1], nQuire, and nQuire.Den [3]) estimated the highest ploidy considered in all samples from each patient. The present authors suggest that these results are artifactual and reflect the wide variation in allele frequencies. Because of this variation, these methods seem to have preferred the model with the largest number of parameters. HMMploidy represents a new and potentially useful tool for studying variation in ploidy. It will be of most use in studying the genetics of asexual organisms and cancers, where aneuploidy imposes little or no penalty on reproduction. It should also be useful for detecting assembly errors in de novo genome sequences from non-model organisms.

References

[1] Augusto Corrêa dos Santos R, Goldman GH, Riaño-Pachón DM (2017) ploidyNGS: visually exploring ploidy with Next Generation Sequencing data. Bioinformatics, 33, 2575–2576. https://doi.org/10.1093/bioinformatics/btx204

[2] Soraggi S, Rhodes J, Altinkaya I, Tarrant O, Balloux F, Fisher MC, Fumagalli M (2022) HMMploidy: inference of ploidy levels from short-read sequencing data. bioRxiv, 2021.06.29.450340, ver. 6 peer-reviewed and recommended by Peer Community in Mathematical and Computational Biology. https://doi.org/10.1101/2021.06.29.450340

[3] Weiß CL, Pais M, Cano LM, Kamoun S, Burbano HA (2018) nQuire: a statistical framework for ploidy estimation using next generation sequencing. BMC Bioinformatics, 19, 122. https://doi.org/10.1186/s12859-018-2128-z

Antimicrobial resistance (AMR) arises due to two main reasons: pathogens are either intrinsically resistant to the antimicrobials, or they can develop new resistance mechanisms in a continuous fashion over time and space. The latter has been referred to as within-host evolution of antimicrobial resistance and studied in infectious disease settings such as Tuberculosis [1]. During antibiotic treatment for example within-host evolutionary AMR dynamics plays an important role [2] and presents significant challenges in terms of optimizing treatment dosage. The study by Djidjou-Demasse et al. [3] contributes to addressing such challenges by developing a modelling approach that utilizes integro-differential equations to mathematically capture continuity in the space of the bacterial resistance levels.

Given its importance as a major public health concern with enormous societal consequences around the world, the evolution of drug resistance in the context of various pathogens has been extensively studied using population genetics approaches [4]. This problem has been also addressed using mathematical modelling approaches including Ordinary Differential Equations (ODE)-based [5. 6] and more recently Stochastic Differential Equations (SDE)-based models [7]. In [3] the authors propose a model of within-host AMR evolution in the absence and presence of drug treatment. The advantage of the proposed modelling approach is that it allows for AMR to be represented as a continuous quantitative trait, describing the level of resistance of the bacterial population termed quantitative AMR (qAMR) in [3]. Moreover, consistent with recent experimental evidence [2] integro-differential equations take into account both, the dynamics of the bacterial population density, referred to as “bottleneck size” in [2] as well as the evolution of its level of resistance due to drug-induced selection. 

The model proposed in [3] has been extensively and rigorously analysed to address various scenarios including the significance of host immune response in drug efficiency, treatment failure and preventive strategies. The drug treatment chosen to be investigated in this study, namely chemotherapy, has been characterised in terms of the level of evolved resistance by the bacterial population in presence of antimicrobial pressure at equilibrium.

Furthermore, the minimal duration of drug administration on bacterial growth and the emergence of AMR has been probed in the model by changing the initial population size and average resistance levels. A potential limitation of the proposed model is the assumption that mutations occur frequently (i.e. during growth), which may not be necessarily the case in certain experimental and/or clinical situations.

References

[1] Castro RAD, Borrell S, Gagneux S (2021) The within-host evolution of antimicrobial resistance in Mycobacterium tuberculosis. FEMS Microbiology Reviews, 45, fuaa071. https://doi.org/10.1093/femsre/fuaa071

[2] Mahrt N, Tietze A, Künzel S, Franzenburg S, Barbosa C, Jansen G, Schulenburg H (2021) Bottleneck size and selection level reproducibly impact evolution of antibiotic resistance. Nature Ecology & Evolution, 5, 1233–1242. https://doi.org/10.1038/s41559-021-01511-2

[3] Djidjou-Demasse R, Sofonea MT, Choisy M, Alizon S (2021) Within-host evolutionary dynamics of antimicrobial quantitative resistance. HAL, hal-03194023, ver. 4 peer-reviewed and recommended by Peer Community in Mathematical and Computational Biology. https://hal.archives-ouvertes.fr/hal-03194023

[4] Wilson BA, Garud NR, Feder AF, Assaf ZJ, Pennings PS (2016) The population genetics of drug resistance evolution in natural populations of viral, bacterial and eukaryotic pathogens. Molecular Ecology, 25, 42–66. https://doi.org/10.1111/mec.13474

[5] Blanquart F, Lehtinen S, Lipsitch M, Fraser C (2018) The evolution of antibiotic resistance in a structured host population. Journal of The Royal Society Interface, 15, 20180040. https://doi.org/10.1098/rsif.2018.0040

[6] Jacopin E, Lehtinen S, Débarre F, Blanquart F (2020) Factors favouring the evolution of multidrug resistance in bacteria. Journal of The Royal Society Interface, 17, 20200105. https://doi.org/10.1098/rsif.2020.0105

[7] Igler C, Rolff J, Regoes R (2021) Multi-step vs. single-step resistance evolution under different drugs, pharmacokinetics, and treatment regimens (BS Cooper, PJ Wittkopp, Eds,). eLife, 10, e64116. https://doi.org/10.7554/eLife.64116

In standard mutation-selection models for describing the dynamics of phenotypically structured populations, it is often assumed that the mutation rate is constant across the phenotypes. In particular, this assumption leads to a constant diffusion coefficient for diffusion approximation models (Perthame, 2007 and references therein).   

Patout et al (2021) study the dependence of the mutation rate on the birth rate, by introducing some diffusion approximations at the population level, derived from the large population limit of a stochastic, individual-based model. The reaction-diffusion model in this article is of the “cross-diffusion” type: The form of “cross-diffusion” also appeared in ecological literature as a type of biased movement behaviors for organisms (Shigesada et al., 1979). The key underlying assumption for “cross-diffusion” is that the transition probability at the individual level depends solely upon the condition at the departure point. Patout et al (2021) envision that a higher birth rate yields more mutations per unit of time. One of their motivations is that during cancer development, the mutation rates of cancer cells at the population level could be correlated with reproduction success.   

The reaction-diffusion approximation model derived in this article illustrates several interesting phenomena: For the time evolution situation, their model predicts different solution trajectories under various assumptions on the fitness function, e.g. the trajectory could initially move towards the birth optimum but eventually end up at the survival optimum. Their model also predicts that the mean fitness could be flat for some period of time, which might provide another alternative to explain observed data. At the steady-state level, their model suggests that the populations are more concentrated around the survival optimum, which agrees with the evolution of the time-dependent solution trajectories.   

Perhaps one of the most interesting contributions of the study of Patout et al (2021) is to give us a new perspective to model the mutation rate in phenotypically structured populations and subsequently, and to help us better understand the connection between mutation and selection. More broadly, this article offers some new insights into the evolutionary dynamics of phenotypically structured populations, along with potential implications in empirical studies.   

References

Perthame B (2007) Transport Equations in Biology Frontiers in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7842-4_2

Patout F, Forien R, Alfaro M, Papaïx J, Roques L (2021) The emergence of a birth-dependent mutation rate in asexuals: causes and consequences. bioRxiv, 2021.06.11.448026, ver. 3 peer-reviewed and recommended by Peer Community in Mathematical and Computational Biology. https://doi.org/10.1101/2021.06.11.448026

Shigesada N, Kawasaki K, Teramoto E (1979) Spatial segregation of interacting species. Journal of Theoretical Biology, 79, 83–99. https://doi.org/10.1016/0022-5193(79)90258-3

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

The paper [1] presents and applies a new Bayesian inference method of phylogenetic reconstruction for multiple sequence alignments in the case of low sequencing coverage but diverse copy number aberrations (CNA), with applications to single cell sequencing of tumors.

The idea is to take advantage of CNA to reconstruct the topology of the phylogenetic tree of sequenced cells in a first step (the `sitka' method), and in a second step to assign single nucleotide variants (SNV) to tree edges (and then calibrate their lengths) (the `sitka-snv' method).

The data are summarized into a binary-valued CxL matrix Y, where C is the number of cells and L is the number of loci (here, loci are segments of prescribed length called `bins'). The entry of Y at row i and column j is 1 (otherwise 0) iff in the ancestral lineage of cell i, at least one genomic rearrangement has occurred, and more specifically the gain or loss of a segment with at least one endpoint in locus j or in locus j+1. The authors expect the infinite-allele assumption to approximately hold (i.e., that at most one mutation occurs at any given marker and that 0 is the ancestral state). They refer to this assumption as the `perfect phylogeny assumption'. By only recording from CNA events the endpoints at which they occur, the authors lose the information on copy number, but they gain the assumption of independence of the mutational processes occurring at different sites, which approximately holds for CNA endpoints.

The goal of sitka is to produce a posterior distribution on phylogenetic trees conditional on the matrix Y , where here a phylogenetic tree is understood as containing the information on 1) the topology of the tree but not its edge lengths, and 2) for each edge, the identity of markers having undergone a mutation, in the sense of the previous paragraph. 

The results of the method are tested against synthetic datasets simulated under various assumptions, including conditions violating the perfect phylogeny assumption and compared to results obtained under other baseline methods. The method is extended to assign SNV to edges of the tree inferred by sitka. It is also applied to real datasets of single cell genomes of tumors. 

The manuscript is very well-written, with a high degree of detail. The method is novel, scalable, fast and appears to perform favorably compared to other approaches. It has been applied in independent publications, for example to multi-year time-series single-cell whole-genome sequencing of tumors, in order to infer the fitness landscape and its dynamics through time, see [2].

The reviewing process has taken too long, mainly because of other commitments I had during the period and to the difficulty of finding reviewers. Let me apologize to the authors and thank them for their patience as well as for the scientific rigor they brought to their revisions and answers to reviewers, who I also warmly thank for their quality work.

REFERENCES

[1] Sohrab Salehi, Fatemeh Dorri, Kevin Chern, Farhia Kabeer, Nicole Rusk, Tyler Funnell, Marc J Williams, Daniel Lai, Mirela Andronescu, Kieran R. Campbell, Andrew McPherson, Samuel Aparicio, Andrew Roth, Sohrab Shah, and Alexandre Bouchard-Côté. Cancer phylogenetic tree inference at scale from 1000s of single cell genomes (2023). bioRxiv, 2020.05.06.058180, ver. 4 peer-reviewed and recommended by Peer Community in Mathematical and Computational Biology. 
https://doi.org/10.1101/2020.05.06.058180

[2] Sohrab Salehi, Farhia Kabeer, Nicholas Ceglia, Mirela Andronescu, Marc J. Williams, Kieran R. Campbell, Tehmina Masud, Beixi Wang, Justina Biele, Jazmine Brimhall, David Gee, Hakwoo Lee, Jerome Ting, Allen W. Zhang, Hoa Tran, Ciara O’Flanagan, Fatemeh Dorri, Nicole Rusk, Teresa Ruiz de Algara, So Ra Lee, Brian Yu Chieh Cheng, Peter Eirew, Takako Kono, Jenifer Pham, Diljot Grewal, Daniel Lai, Richard Moore, Andrew J. Mungall, Marco A. Marra, IMAXT Consortium, Andrew McPherson, Alexandre Bouchard-Côté, Samuel Aparicio & Sohrab P. Shah. Clonal fitness inferred from time-series modelling of single-cell cancer genomes (2021).  Nature 595, 585–590. https://doi.org/10.1038/s41586-021-03648-3