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

In “Estimating dates of origin and end of COVID-19 epidemics”, Bénéteau et al. develop and apply a mathematical modeling approach to estimate the date of the origin of the SARS-CoV-2 epidemic in France. They also assess how long strict control measures need to last to ensure that the prevalence of the virus remains below key public health thresholds. This problem is challenging because the numbers of infected individuals in both tails of the epidemic are low, which can lead to errors when deterministic models are used. To achieve their goals, the authors developed a discrete stochastic model. The model is non-Markovian, meaning that individual infection histories influence the dynamics. The model also accounts for heterogeneity in the timing between infection and transmission and includes stochasticity as well as consideration of superspreader events. By comparing the outputs of their model with several alternative models, Bénéteau et al. were able to assess the importance of stochasticity, individual heterogeneity, and non-Markovian effects on the estimates of the dates of origin and end of the epidemic, using France as a test case. Some limitations of the study, which the authors acknowledge, are that the time from infection to death remains largely unknown, a lack of data on the heterogeneity of transmission among individuals, and the assumption that only a single infected individual caused the epidemic. Despite the acknowledged limitations of the work, the results suggest that cases may be detected long before the detection of an epidemic wave. Also, the approach may be helpful for informing public health decisions such as the necessary duration of strict lockdowns and for assessing the risks of epidemic rebound as restrictions are lifted. In particular, the authors found that estimates of the end of the epidemic following lockdowns are more sensitive to the assumptions of the models used than estimates of its beginning. In summary, this model adds to a valuable suite of tools to support decision-making in response to disease epidemics.

References

Bénéteau T, Elie B, Sofonea MT, Alizon S (2021) Estimating dates of origin and end of COVID-19 epidemics. medRxiv, 2021.01.19.21250080, ver. 3 peer-reviewed and recommended by Peer Community in Mathematical and Computational Biology. https://doi.org/10.1101/2021.01.19.21250080

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.

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

​​​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