​​​Determining if genes are orthologous (i.e. homologous genes whose most common ancestor represents a speciation) or paralogous (homologous genes whose most common ancestor represents a duplication) is a foundational problem in bioinformatics. For instance, the input to almost all phylogenetic methods is a sequence alignment of genes assumed to be orthologous.  Understanding if genes are paralogs or orthologs can also be important for assigning function, for example genes that have diverged following duplication may be more likely to have neofunctionalised or subfunctionalised compared to genes that have diverged following speciation, which may be more likely to have continued in a similar role.

This paper by Jones et al (2022) contributes to a wide range of literature addressing the inference of orthology/paralogy relations but takes a different approach to explaining inconsistency between an assumed species phylogeny and a relation graph (a graph where nodes represent genes and edges represent that the two genes are orthologs). Rather than assuming that inconsistencies are the result of incorrect assessment of orthology (i.e. incorrect edges in the relation graph) they ask if the relation graph could be consistent with a species tree combined with some amount of lateral (horizontal) gene transfer.

The two main questions addressed in this paper are (1) if a network N and a relation graph R are consistent, and (2) if – given a species tree S and a relation graph R – transfer arcs can be added to S in such a way that it becomes consistent with R? 

The first question hinges on the concept of a reconciliation between a gene tree and a network (section 2.1) and amounts to asking if a gene tree can be found that can both be reconciled with the network and consistent with the relation graph. The authors show that the problem is NP hard. Furthermore, the related problem of attempting to find a solution using k or fewer transfers is NP-hard, and also W[1] hard implying that it is in a class of problems for which fixed parameter tractable solutions have not been found. The proof of NP hardness is by reduction to the k-multi-coloured clique problem via an intermediate problem dubbed “antichain on trees” (Section 3). The “antichain on trees” construction may be of interest to others working on algorithmic complexity with phylogenetic networks.

In the second question the possible locations of transfers are not specified (or to put it differently any time consistent transfer arc is considered possible) and it is shown that it generally will be possible to add transfer edges to S in such a way that it can be consistent with R. However, the natural extension to this question of asking if it can be done with k or fewer added arcs is also NP hard.

Many of the proofs in the paper are quite technical, but the authors have relegated a lot of this detail to the appendix thus ensuring that the main ideas and results are clear to follow in the main text. I am grateful to both reviewers for their detailed reviews and through checking of the proofs.

References

Jones M, Lafond M, Scornavacca C (2022) Consistency of orthology and paralogy constraints in the presence of gene transfers. arXiv:1705.01240 [cs], ver. 6 peer-reviewed and recommended by Peer Community in Mathematical and Computational Biology. https://arxiv.org/abs/1705.01240

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/

​To study the annual invasion of a tree pathogen (Melampsora larici-populina, a fungal species responsible for the poplar rust disease), Xhaard et al (2012) had conducted a spatiotemporal survey along the Durance River valley in the French Alps over nearly 200 km, measuring sampled leaves and twigs from 40 to 150 trees at 12 evenly spaced study sites at seven-time points. By combining Bayesian genetic assignment and a landscape epidemiology approach, they were able to estimate the genetic origin and annual spread of the plant pathogen during a single epidemic.

The observed temporal variation in the spatial pattern of infection rates allowed Saubin et al (2023) to estimate the key factors that determine the speed of the invasion dynamics. In particular, it is crucial to estimate the probability and extent of long-distance dispersal. The dynamics of the macroscale population density was formulated by the reaction-diffusion (R.D.) model and by the integro-difference (I.D.) model. Both consist of the diffusion/dispersal component and the reaction component. In the I.D. model, the kernel function represents the distribution of the dispersion. The likelihood function was obtained by coupling the mathematical model of the population dynamics and the statistical model of the observational process.

Saubin et al (2023) considered a thin-tailed Gaussian kernel, a heavy-tailed exponential kernel, and a fat-tailed exponential power kernel. The numerical simulation reflecting the above survey confirmed the identifiability of the propagation kernel and the accuracy of the parameter estimation. In particular, the above survey had the high power to identify the model with frequent long-distance dispersal. The data from the survey selected the exponential power kernel with confidence. The mean dispersal distance was estimated to be 2.01 km. The exponential power was 0.24. This parameter value predicts that 5% of the dispersals will have a distance > 14.3 km and 1% will have a distance > 36.0 km. The mechanistic-statistical approach presented here may become a new standard for the field-based studies of invasion dynamics.

References

Saubin, M., Coville, J., Xhaard, C., Frey, P., Soubeyrand, S., Halkett, F., and Fabre, F. (2023). A mechanistic-statistical approach to infer dispersal and demography from invasion dynamics, applied to a plant pathogen. bioRxiv, ver. 5 peer-reviewed and recommended by Peer Community in Mathematical and Computational Biology. https://doi.org/10.1101/2023.03.21.533642

Xhaard, C., Barrès, B., Andrieux, A., Bousset, L., Halkett, F., and Frey, P. (2012). Disentangling the genetic origins of a plant pathogen during disease spread using an original molecular epidemiology approach. Molecular Ecology, 21(10):2383-2398. https://doi.org/10.1111/j.1365-294X.2012.05556.x

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

The article by Bez et al [1] addresses an important issue for statisticians and ecological modellers: the impact of modelling choices when considering state-space models to represent time series with hidden regimes.

The authors present an empirical study of the impact of model misspecification for models in the HMM and HSMM family. The misspecification can be at the level of the hidden chain (Markovian or semi-Markovian assumption) or at the level of the observed chain (AR0 or AR1 assumption). 

The study uses data on the movements of fishing vessels. Vessels can exert pressure on fish stocks when they are fishing, and the aim is to identify the periods during which fishing vessels are fishing or not fishing, based on GPS tracking data. Two sets of data are available, from two vessels with contrasting fishing behaviour. The empirical study combines experiments on the two real datasets and on data simulated from models whose parameters are estimated on the real datasets. In both cases, the actual sequence of activities is available. The impact of a model misspecification is mainly evaluated on the restored hidden chain (decoding task), which is very relevant since in many applications we are more interested in the quality of decoding than in the accuracy of parameters estimation. Results on parameter estimation are also presented and metrics are developed to help interpret the results. The study is conducted in a rigorous manner and extensive experiments are carried out, making the results robust.

The main conclusion of the study is that choosing the wrong AR model at the observed sequence level has more impact than choosing the wrong model at the hidden chain level.

The article ends with an interesting discussion of this finding, in particular the impact of resolution on the quality of the decoding results. As the authors point out in this discussion, the results of this study are not limited to the application of GPS data to the activities of fishing vessels Beyond ecology, H(S)MMs are also widely used epidemiology, seismology, speech recognition, human activity recognition ... The conclusion of this study will therefore be useful in a wide range of applications. It is a warning that should encourage modellers to design their hidden Markov models carefully or to interpret their results cautiously.

References

[1] Nicolas Bez, Pierre Gloaguen, Marie-Pierre Etienne, Rocio Joo, Sophie Lanco, Etienne Rivot, Emily Walker, Mathieu Woillez, Stéphanie Mahévas (2024) Proper account of auto-correlations improves decoding performances of state-space (semi) Markov models. HAL, ver.3 peer-reviewed and recommended by PCI Math Comp Biol https://hal.science/hal-04547315v3