How fast and how deep do animals breathe, and how does this depend on how active they are? To answer this question, one needs to dig deeply into how breathing works and what biophysical processes it involves. And one needs to think about body size.

It is impressive how nature adapts the same body plan – e.g. the skeletal structure of mammals – to various shapes and sizes. From mice to whales, also the functioning of most organs remains the same; they are just differently scaled. Scaling does not just mean “making bigger or smaller”. As already noted by Galilei, body shapes change as they are adapted to body dimensions, and the same holds for physiological variables. Many such variables, for instance, heartbeat rates, follow scaling laws of the form y~x^a, where x denotes body mass and the exponent a is typically a multiple of ¼ [1]. These unusual exponents – instead of multiples of ⅓, which would be expected from simple geometrical scaling – are why these laws are called “allometric”. Kleiber’s law for metabolic rates, with a scaling exponent of ¾, is a classic example [2]. As shown by G. West, allometric laws can be explained through a few simple steps [1]. In his models, he focused on network-like organs such as the vascular system and assumed that these systems show a self-similar structure, with a fixed minimal unit (for instance, capillaries) but varying numbers of hierarchy levels depending on body size. To determine the flow through such networks, he employed biophysical models and optimality principles (for instance, assuming that oxygen must be transported at a minimal mechanical effort), and showed that the solutions – and the physiological variables – respect the known scaling relations.

The paper “The origin of the allometric scaling of lung ventilation in mammals“ by Noël et al. [3], applies this thinking to the depth and rate of breathing in mammals. Scaling laws describing breathing in resting animals have been known since the 1950s [4], with exponents of 1 (for tidal volume) and -¼ (for breathing frequency). Equipped with a detailed biophysical model, Noël et al. revisit this question, extending these laws to other metabolic regimes. Their starting point is a model of the human lung, developed previously by two of the authors [5], which assumes that we meet our oxygen demand with minimal lung movements. To state this as an optimization problem, the model combines two submodels: a mechanical model describing the energetic effort of ventilation and a highly detailed model of convection and diffusion in self-similar lung geometries. Breathing depths and rates are computed by numerical optimization, and to obtain results for mammals of any size many of the model parameters are described by known scaling laws. As expected, the depth of breathing (measured by tidal volume) scales almost proportionally with body mass and increases with metabolic demand, while the breathing rate decreases with body mass, with an exponent of about -¼. However, the laws for the breathing rate hold only for basal activity; at higher metabolic rates, which are modeled here for the first time, the exponent deviates strongly from this value, in line with empirical data.

Why is this paper important? The authors present a highly complex model of lung physiology that integrates a wide range of biophysical details and passes a difficult test: the successful prediction of unexplained scaling exponents. These scaling relations may help us transfer insights from animal models to humans and in reverse: data for breathing during exercise, which are easy to measure in humans, can be extrapolated to other species. Aside from the scaling laws, the model also reveals physiological mechanisms. In the larger lung branches, oxygen is transported mainly by air movement (convection), while in smaller branches air flow is slow and oxygen moves by diffusion. The transition between these regimes can occur at different depths in the lung: as the authors state, “the localization of this transition determines how ventilation should be controlled to minimize its energetic cost at any metabolic regime”. In the model, the optimal location for the transition depends on oxygen demand [5, 6]: the transition occurs deeper in the lung in exercise regimes than at rest, allowing for more oxygen to be taken up. However, the effects of this shift depend on body size: while small mammals generally use the entire exchange surface of their lungs, large mammals keep a reserve for higher activities, which becomes accessible as their transition zone moves at high metabolic rates. Hence, scaling can entail qualitative differences between species!

Altogether, the paper shows how the dynamics of ventilation depend on lung morphology. But this may also play out in the other direction: if energy-efficient ventilation depends on body activity, and therefore on ecological niches, a niche may put evolutionary pressures on lung geometry. Hence, by understanding how deep and fast animals breathe, we may also learn about how behavior, physiology, and anatomy co-evolve.

References

[1] West GB, Brown JH, Enquist BJ (1997) A General Model for the Origin of Allometric Scaling Laws in Biology. Science 276 (5309), 122–126. https://doi.org/10.1126/science.276.5309.122

[2] Kleiber M (1947) Body size and metabolic rate. Physiological Reviews, 27, 511–541. https://doi.org/10.1152/physrev.1947.27.4.511

[3] Noël F., Karamaoun C., Dempsey J. A. and Mauroy B. (2021) The origin of the allometric scaling of lung's ventilation in mammals. arXiv, 2005.12362, ver. 6 peer-reviewed and recommended by Peer community in Mathematical and Computational Biology. https://arxiv.org/abs/2005.12362

[4] Otis AB, Fenn WO, Rahn H (1950) Mechanics of Breathing in Man. Journal of Applied Physiology, 2, 592–607. https://doi.org/10.1152/jappl.1950.2.11.592

[5] Noël F, Mauroy B (2019) Interplay Between Optimal Ventilation and Gas Transport in a Model of the Human Lung. Frontiers in Physiology, 10, 488. https://doi.org/10.3389/fphys.2019.00488

[6] Sapoval B, Filoche M, Weibel ER (2002) Smaller is better—but not too small: A physical scale for the design of the mammalian pulmonary acinus. Proceedings of the National Academy of Sciences, 99, 10411–10416. https://doi.org/10.1073/pnas.122352499

As part of many bioinformatics tools, one encodes a k-mer, which is a string, into an integer. The natural encoding uses a bijective function to map the k-mers onto the interval [0, s^k - ], where s is the alphabet size. This encoding is minimal, in the sense that the encoded integer ranges from 0 to the number of represented k-mers minus 1. 

However, often one is only interested in encoding canonical k-mers. One common definition is that a k-mer is canonical if it is lexicographically not larger than its reverse complement. In this case, only about half the k-mers from the universe of k-mers are canonical, and the natural encoding is no longer minimal. For the special case of a DNA alphabet and odd k, there exists a "parity-based" encoding for canonical k-mers which is minimal. 

In [1], the author presents a minimal encoding for canonical k-mers that works for general alphabets and both odd and even k. They also give an efficient bit-based representation for the DNA alphabet. 

This paper fills a theoretically interesting and often overlooked gap in how to encode k-mers as integers. It is not yet clear what practical applications this encoding will have, as the author readily acknowledges in the manuscript. Neither the author nor the reviewers are aware of any practical situations where the lack of a minimal encoding "leads to serious limitations." However, even in an applied field like bioinformatics, it would be short-sighted to only value theoretical work that has an immediate application; often, the application is several hops away and not apparent at the time of the original work. 

In fact, I would speculate that there may be significant benefits reaped if there was more theoretical attention paid to the fact that k-mers are often restricted to be canonical. Many papers in the field sweep under the rug the fact that k-mers are made canonical, leaving it as an implementation detail. This may indicate that the theory to describe and analyze this situation is underdeveloped. This paper makes a step forward to develop this theory, and I am hopeful that it may lead to substantial practical impact in the future. 

References

[1] Roland Wittler (2023) "General encoding of canonical k-mers. bioRxiv, ver.2, peer-reviewed and recommended by Peer Community in Mathematical and Computational Biology https://doi.org/10.1101/2023.03.09.531845

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/

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

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