Embedding of Markov matrices for d ⩽ 4.

The embedding problem of Markov matrices in Markov semigroups is a classic problem that regained a lot of impetus and activities through recent needs in phylogeny and population genetics. Here, we give an account for dimensions d ⩽ 4 , including a complete and simplified treatment of the case d = 3 , and derive the results in a systematic fashion, with an eye on the potential applications. Further, we reconsider the setup of the corresponding problem for time-inhomogeneous Markov chains, which is needed for real-world applications because transition rates need not be constant over time. Additional cases of this more general embedding occur for any d ⩾ 3 . We review the known case of d = 3 and describe the setting for future work on d = 4 .

How Trustworthy Is Your Tree? Bayesian Phylogenetic Effective Sample Size Through the Lens of Monte Carlo Error.

Bayesian inference is a popular and widely-used approach to infer phylogenies (evolutionary trees). However, despite decades of widespread application, it remains difficult to judge how well a given Bayesian Markov chain Monte Carlo (MCMC) run explores the space of phylogenetic trees. In this paper, we investigate the Monte Carlo error of phylogenies, focusing on high-dimensional summaries of the posterior distribution, including variability in estimated edge/branch (known in phylogenetics as "split") probabilities and tree probabilities, and variability in the estimated summary tree. Specifically, we ask if there is any measure of effective sample size (ESS) applicable to phylogenetic trees which is capable of capturing the Monte Carlo error of these three summary measures. We find that there are some ESS measures capable of capturing the error inherent in using MCMC samples to approximate the posterior distributions on phylogenies. We term these tree ESS measures, and identify a set of three which are useful in practice for assessing the Monte Carlo error. Lastly, we present visualization tools that can improve comparisons between multiple independent MCMC runs by accounting for the Monte Carlo error present in each chain. Our results indicate that common post-MCMC workflows are insufficient to capture the inherent Monte Carlo error of the tree, and highlight the need for both within-chain mixing and between-chain convergence assessments.

Variation in leaf carbon economics, energy balance, and heat tolerance traits highlights differing timescales of adaptation and acclimation.

Multivariate leaf trait correlations are hypothesized to originate from natural selection on carbon economics traits that control lifetime leaf carbon gain, and energy balance traits governing leaf temperatures, physiological rates, and heat injury. However, it is unclear whether macroevolution of leaf traits primarily reflects selection for lifetime carbon gain or energy balance, and whether photosynthetic heat tolerance is coordinated along these axes. To evaluate these hypotheses, we measured carbon economics, energy balance, and photosynthetic heat tolerance traits for 177 species (157 families) in a common garden that minimizes co-variation of taxa and climate. We observed wide variation in carbon economics, energy balance, and heat tolerance traits. Carbon economics and energy balance (but not heat tolerance) traits were phylogenetically structured, suggesting macroevolution of leaf mass per area and leaf dry matter content reflects selection on carbon gain rather than energy balance. Carbon economics and energy balance traits varied along a common axis orthogonal to heat tolerance traits. Our results highlight a fundamental mismatch in the timescales over which morphological and heat tolerance traits respond to environmental variation. Whereas carbon economics and energy balance traits are constrained by species' evolutionary histories, photosynthetic heat tolerance traits are not and can acclimate readily to leaf microclimates.

Scalable Bayesian Divergence Time Estimation With Ratio Transformations.

Divergence time estimation is crucial to provide temporal signals for dating biologically important events from species divergence to viral transmissions in space and time. With the advent of high-throughput sequencing, recent Bayesian phylogenetic studies have analyzed hundreds to thousands of sequences. Such large-scale analyses challenge divergence time reconstruction by requiring inference on highly correlated internal node heights that often become computationally infeasible. To overcome this limitation, we explore a ratio transformation that maps the original $N-1$ internal node heights into a space of one height parameter and $N-2$ ratio parameters. To make the analyses scalable, we develop a collection of linear-time algorithms to compute the gradient and Jacobian-associated terms of the log-likelihood with respect to these ratios. We then apply Hamiltonian Monte Carlo sampling with the ratio transform in a Bayesian framework to learn the divergence times in 4 pathogenic viruses (West Nile virus, rabies virus, Lassa virus, and Ebola virus) and the coralline red algae. Our method both resolves a mixing issue in the West Nile virus example and improves inference efficiency by at least 5-fold for the Lassa and rabies virus examples as well as for the algae example. Our method now also makes it computationally feasible to incorporate mixed-effects molecular clock models for the Ebola virus example, confirms the findings from the original study, and reveals clearer multimodal distributions of the divergence times of some clades of interest.

Global phylogenomic assessment of Leptoseris and Agaricia reveals substantial undescribed diversity at mesophotic depths.

BACKGROUND: Mesophotic coral communities are increasingly gaining attention for the unique biological diversity they host, exemplified by the numerous mesophotic fish species that continue to be discovered. In contrast, many of the photosynthetic scleractinian corals observed at mesophotic depths are assumed to be depth-generalists, with very few species characterised as mesophotic-specialists. This presumed lack of a specialised community remains largely untested, as phylogenetic studies on corals have rarely included mesophotic samples and have long suffered from resolution issues associated with traditional sequence markers. RESULTS: Here, we used reduced-representation genome sequencing to conduct a phylogenomic assessment of the two dominant mesophotic genera of plating corals in the Indo-Pacific and Western Atlantic, respectively, Leptoseris and Agaricia. While these genome-wide phylogenies broadly corroborated the morphological taxonomy, they also exposed deep divergences within the two genera and undescribed diversity across the current taxonomic species. Five of the eight focal species consisted of at least two sympatric and genetically distinct lineages, which were consistently detected across different methods. CONCLUSIONS: The repeated observation of genetically divergent lineages associated with mesophotic depths highlights that there may be many more mesophotic-specialist coral species than currently acknowledged and that an urgent assessment of this largely unstudied biological diversity is warranted.

Postcranial disparity of galeaspids and the evolution of swimming speeds in stem-gnathostomes.

Galeaspids are extinct jawless relatives of living jawed vertebrates whose contribution to understanding the evolutionary assembly of the gnathostome bodyplan has been limited by absence of postcranial remains. Here, we describe Foxaspis novemura gen. et sp. nov., based on complete articulated remains from a newly discovered Konservat-Lagerstätte in the Early Devonian (Pragian, ∼410 Ma) of Guangxi, South China. F. novemura had a broad, circular dorso-ventrally compressed headshield, slender trunk and strongly asymmetrical hypochordal tail fin comprised of nine ray-like scale-covered digitations. This tail morphology contrasts with the symmetrical hypochordal tail fin of Tujiaaspis vividus, evidencing disparity in galeaspid postcranial anatomy. Analysis of swimming speed reveals galeaspids as moderately fast swimmers, capable of achieving greater cruising swimming speeds than their more derived jawless and jawed relatives. Our analyses reject the hypothesis of a driven trend towards increasingly active food acquisition which has been invoked to characterize early vertebrate evolution.

CrusTome: a transcriptome database resource for large-scale analyses across Crustacea.

Transcriptomes from nontraditional model organisms often harbor a wealth of unexplored data. Examining these data sets can lead to clarity and novel insights in traditional systems, as well as to discoveries across a multitude of fields. Despite significant advances in DNA sequencing technologies and in their adoption, access to genomic and transcriptomic resources for nontraditional model organisms remains limited. Crustaceans, for example, being among the most numerous, diverse, and widely distributed taxa on the planet, often serve as excellent systems to address ecological, evolutionary, and organismal questions. While they are ubiquitously present across environments, and of economic and food security importance, they remain severely underrepresented in publicly available sequence databases. Here, we present CrusTome, a multispecies, multitissue, transcriptome database of 201 assembled mRNA transcriptomes (189 crustaceans, 30 of which were previously unpublished, and 12 ecdysozoans for phylogenetic context) as an evolving and publicly available resource. This database is suitable for evolutionary, ecological, and functional studies that employ genomic/transcriptomic techniques and data sets. CrusTome is presented in BLAST and DIAMOND formats, providing robust data sets for sequence similarity searches, orthology assignments, phylogenetic inference, etc. and thus allowing for straightforward incorporation into existing custom pipelines for high-throughput analyses. In addition, to illustrate the use and potential of CrusTome, we conducted phylogenetic analyses elucidating the identity and evolution of the cryptochrome/photolyase family of proteins across crustaceans.

Scalable Bayesian phylogenetics.

Recent advances in Bayesian phylogenetics offer substantial computational savings to accommodate increased genomic sampling that challenges traditional inference methods. In this review, we begin with a brief summary of the Bayesian phylogenetic framework, and then conceptualize a variety of methods to improve posterior approximations via Markov chain Monte Carlo (MCMC) sampling. Specifically, we discuss methods to improve the speed of likelihood calculations, reduce MCMC burn-in, and generate better MCMC proposals. We apply several of these techniques to study the evolution of HIV virulence along a 1536-tip phylogeny and estimate the internal node heights of a 1000-tip SARS-CoV-2 phylogenetic tree in order to illustrate the speed-up of such analyses using current state-of-the-art approaches. We conclude our review with a discussion of promising alternatives to MCMC that approximate the phylogenetic posterior. This article is part of a discussion meeting issue 'Genomic population structures of microbial pathogens'.

A class of identifiable phylogenetic birth-death models.

In a striking result, Louca and Pennell [S. Louca, M. W. Pennell, Nature 580, 502-505 (2020)] recently proved that a large class of phylogenetic birth-death models is statistically unidentifiable from lineage-through-time (LTT) data: Any pair of sufficiently smooth birth and death rate functions is "congruent" to an infinite collection of other rate functions, all of which have the same likelihood for any LTT vector of any dimension. As Louca and Pennell argue, this fact has distressing implications for the thousands of studies that have utilized birth-death models to study evolution. In this paper, we qualify their finding by proving that an alternative and widely used class of birth-death models is indeed identifiable. Specifically, we show that piecewise constant birth-death models can, in principle, be consistently estimated and distinguished from one another, given a sufficiently large extant timetree and some knowledge of the present-day population. Subject to mild regularity conditions, we further show that any unidentifiable birth-death model class can be arbitrarily closely approximated by a class of identifiable models. The sampling requirements needed for our results to hold are explicit and are expected to be satisfied in many contexts such as the phylodynamic analysis of a global pandemic.

Real-Time and Remote MCMC Trace Inspection with Beastiary.

Bayesian phylogenetics has gained substantial popularity in the last decade, with most implementations relying on Markov chain Monte Carlo (MCMC). The computational demands of MCMC mean that remote servers are increasingly used. We present Beastiary, a package for real-time and remote inspection of log files generated by MCMC analyses. Beastiary is an easily deployed web-app that can be used to summarize and visualize the output of many popular software packages including BEAST, BEAST2, RevBayes, and MrBayes via a web browser. We describe the design and implementation of Beastiary and some typical use-cases, with a focus on real-time remote monitoring.

Large-scale characterization of the macrolide resistome reveals high diversity and several new pathogen-associated genes.

Macrolides are broad-spectrum antibiotics used to treat a range of infections. Resistance to macrolides is often conferred by mobile resistance genes encoding Erm methyltransferases or Mph phosphotransferases. New erm and mph genes keep being discovered in clinical settings but their origins remain unknown, as is the type of macrolide resistance genes that will appear in the future. In this study, we used optimized hidden Markov models to characterize the macrolide resistome. Over 16 terabases of genomic and metagenomic data, representing a large taxonomic diversity (11 030 species) and diverse environments (1944 metagenomic samples), were searched for the presence of erm and mph genes. From this data, we predicted 28 340 macrolide resistance genes encoding 2892 unique protein sequences, which were clustered into 663 gene families (<70 % amino acid identity), of which 619 (94 %) were previously uncharacterized. This included six new resistance gene families, which were located on mobile genetic elements in pathogens. The function of ten predicted new resistance genes were experimentally validated in Escherichia coli using a growth assay. Among the ten tested genes, seven conferred increased resistance to erythromycin, with five genes additionally conferring increased resistance to azithromycin, showing that our models can be used to predict new functional resistance genes. Our analysis also showed that macrolide resistance genes have diverse origins and have transferred horizontally over large phylogenetic distances into human pathogens. This study expands the known macrolide resistome more than ten-fold, provides insights into its evolution, and demonstrates how computational screening can identify new resistance genes before they become a significant clinical problem.

Weeds Harbor Fusarium Species that Cause Malformation Disease of Economically Important Trees in Western Mexico.

Mango malformation disease (MMD) caused by Fusarium spp. is an important limiting factor in most production areas worldwide. Fusarium mexicanum and F. pseudocircinatum have been reported as causing MMD in Mexico. These two pathogens also cause a similar disease in Swietenia macrophylla (big-leaf mahogany malformation disease) in central western Mexico, and F. pseudocircinatum was recently reported as causing malformation disease in Tabebuia rosea (rosy trumpet) in the same region. These studies suggest that additional plant species, including weeds, might be hosts of these pathogens. The role that weed hosts might have in the disease cycle is unknown. The objectives of this work were to recover Fusarium isolates from understory vegetation in mango orchards with MMD, identify the Fusarium isolates through DNA sequence data, and determine whether F. mexicanum is capable of inducing disease in the weedy legume Senna uniflora (oneleaf senna). Additional objectives in this work were to compare Fusarium isolates recovered from weeds and mango trees in the same orchards by characterizing their phylogenetic relationships, assessing in vitro production of mycotoxins, and identifying their mating type idiomorph. A total of 59 Fusarium isolates from five species complexes were recovered from apical and lateral buds from four weed species. Two of the species within the F. fujikuroi species complex are known to cause MMD in Mexico. Trichothecene production was detected in five isolates, including F. sulawense and F. irregulare in the F. incarnatum-equiseti species complex and F. boothii in the F. sambucinum species complex. Both mating types were present among mango and weed isolates. This is the first report of herbaceous hosts harboring Fusarium species that cause mango malformation in Mexico. The information provided should prove valuable for further study of the epidemiological role of weeds in MMD and help manage the disease.

The phylogenetic relationships of geoemydid turtles from the Eocene Messel Pit Quarry: a first assessment using methods for continuous and discrete characters.

The geoemydid turtles of the Eocoene Messel Pit Quarry of Hesse, Germany, are part of a rich Western European fossil record of testudinoids. Originally referred to as "Ocadia" kehreri and "Ocadia" messeliana, their systematic relationships remain unclear. A previous study proposed that a majority of the Western European geoemydids, including the Messel geoemydids, are closely related to the Recent European representatives of the clade Mauremys. Another study hypothesised that the Western European geoemydid fauna is more phylogenetically diverse, and that the Messel geoemydids are closely related to the East Asian turtles Orlitia and Malayemys. Here we present the first quantitative analyses to date that investigate this question. We use continuous characters in the form of ratios to estimate the placement of the Messel geoemydids in a reference tree that was estimated from molecular data. We explore the placement error obtained from that data with maximum likelihood and Bayesian methods, as well as linear parsimony in combination with discrete characters. We find good overall performance with Bayesian and parsimony analyses. Parsimony performs even better when we also incorporated discrete characters. Yet, we cannot pin down the position of the Messel geoemydids with high confidence. Depending on how intraspecific variation of the ratio characters is treated, parsimony favours a placement of the Messel fossils sister to Orlitia borneensis or sister to Geoemyda spengleri, with weak bootstrap support. The latter placement is suspect because G. spengleri is a phylogenetically problematic species with molecular and morphological data. There is even less support for placements within the Mauremys clade.

Epidemiology, diversity, and management of bacterial spot of tomato caused by Xanthomonas perforans.

Tomato is an important crop grown worldwide. Various plant diseases cause massive losses in tomato plants due to diverse biotic agents. Bacterial spot of tomato (BST) is a worldwide disease that results in high losses in processed and fresh tomato. Xanthomonas perforans, an aerobic, single-flagellated, rod-shaped, Gram-negative plant pathogenic bacterium, is one of the leading causes of BST. Over the past three decades, X. perforans has increasingly been reported from tomato-growing regions and became a major bacterial disease. X. perforans thrives under high humidity and high temperature, which is commonplace in tropical and subtropical climates. Distinguishing symptoms of BST are necrotic lesions that can coalesce and cause a shot-hole appearance. X. perforans can occasionally cause fruit symptoms depending on disease pressure during fruit development. Short-distance movement in the field is mainly dependent on wind-driven rain, whereas long distance movement occurs through contaminated seed or plant material. X. perforans harbors a suite of effectors that increase pathogen virulence, fitness, and dissemination. BST management mainly relies on copper-based compounds; however, resistance is widespread. Alternative compounds, such as nanomaterials, are currently being evaluated and show high potential for BST management. Resistance breeding remains difficult to attain due to limited resistant germplasm. While the increased genetic diversity and gain and loss of effectors in X. perforans limits the success of single-gene resistance, the adoption of effector-specific transgenes and quantitative resistance may lead to durable host resistance. However, further research that aims to more effectively implement novel management tools is required to curb disease spread. KEY POINTS: • Xanthomonas perforans causes bacterial spot on tomato epidemics through infected seedlings and movement of plant material. • Genetic diversity plays a major role in shaping populations which is evident in loss and gain of effectors. • Management relies on copper sprays, but nanoparticles are a promising alternative to reduce copper toxicity.

Mallocybe heimii ectomycorrhizae with Cistus creticus and Pinus halepensis in Mediterranean littoral sand dunes - assessment of phylogenetic relationships to M. arenaria and M. agardhii.

Ectomycorrhizal symbiosis appears extensively in the Northern Hemisphere, where Mediterranean ecosystems constitute an important ecological area of considerable biodiversity value. Littoral sand dunes are among high-risk habitats, and ectomycorrhizal lifestyle contributes significantly to supporting life in such regions. Mallocybe heimii (Bon) Matheny & Esteve-Rav. (Inocybaceae, Basidiomycota) and the very similar M. arenaria (Bon) Matheny & Esteve-Rav. grow in poor, usually sandy soils, in association with angiosperms or gymnosperms. Basidiomata originally identified under these names were collected from littoral sand dunes of Greece, and their morpho-anatomical characteristics were examined in conjunction with material derived from other European regions. Sequences from basidiomata and root tips corresponding to the nuclear rDNA internal transcribed spacer region (ITS) and large subunit (LSU) were obtained and analyzed. Phylogenetic results demonstrated that material identified as M. heimii or M. arenaria form a single well-supported group, while M. agardhii (N. Lund) Matheny & Esteve-Rav. is confirmed to be distinct from M. arenaria (the latter was initially described as a variety of the former, i.e., I. agardhii var. arenaria Bon). A detailed tree of the genus Mallocybe was generated on the basis of concatenated ITS and LSU sequences, and relationships of selected taxa are discussed in the light of morphological and sequence data. In addition, the first morphotype descriptions of M. heimii ectomycorrhizae with Cistus creticus L. and Pinus halepensis Miller are hereby provided. Both morphotypes exhibited the typical characteristics of Inocybe/Mallocybe ectomycorrhizae; however, differences were noted, the most significant being the presence of clamps on mantle hyphae and the type of anastomoses.

Evolution of initiation rites during the Austronesian dispersal.

As adaptive systems, kinship and its accompanying rules have co-evolved with elements of complex societies, including wealth inheritance, subsistence, and power relations. Here we consider an aspect of kinship evolution in the Austronesian dispersal that began from about 5500 BP in Taiwan, reaching Melanesia about 3200 BP, and dispersing into Micronesia by 1500 BP. Previous, foundational work has used phylogenetic comparative methods and ethnolinguistic information to infer matrilocal residence in proto-Austronesian societies. Here we apply Bayesian phylogenetic analyses to a data set on Austronesian societies that combines existing data on marital residence systems with a new set of ethnographic data, introduced here, on initiation rites. Transition likelihoods between cultural-trait combinations were modeled on an ensemble of 1000 possible Austronesian language trees, using Reversible Jump Markov Chain Monte Carlo (RJ-MCMC) simulations. Compared against a baseline phylogenetic model of independent evolution, a phylogenetic model of correlated evolution between female and male initiation rites is substantially more likely (log Bayes factor: 17.9). This indicates, over the generations of Austronesian dispersal, initiation rites were culturally stable when both female and male rites were in the same state (both present or both absent), yet relatively unstable for female-only rites. The results indicate correlated phylogeographic evolution of cultural initiation rites in the prehistoric dispersal of Austronesian societies across the Pacific. Once acquired, male initiation rites were more resilient than female-only rites among Austronesian societies.

Coincident rapid expansion of two SARS-CoV-2 lineages with enhanced infectivity in Nigeria.

The emergence of new SARS-CoV-2 variants with enhanced transmissibility or decreased susceptibility to immune responses is a major threat to global efforts to end the coronavirus disease 2019 (COVID-19) pandemic. Disparities in viral genomic surveillance capabilities and efforts have resulted in gaps in our understanding of the viral population dynamics across the globe. Nigeria, despite having the largest population of any nation in Africa, has had relatively little SARS-CoV-2 sequence data made publicly available. Here we report the whole-genome sequences of 74 SARS-CoV-2 isolates collected from individuals in Oyo State, Nigeria in January 2021. Most isolates belonged to either the B.1.1.7 Alpha "variant of concern" or the B.1.525 Eta lineage, which is currently considered a "variant of interest" containing multiple spike protein mutations previously associated with enhanced transmissibility and possible immune escape. Nigeria has the highest reported frequency of the B.1.525 lineage globally with phylogenetic characteristics consistent with a recent monophyletic origin and rapid expansion. Spike protein from the B.1.525 lineage displayed both increased infectivity and decreased neutralization by convalescent sera compared to Spike proteins from other clades. These results, along with indications that the virus is outpacing the B.1.1.7 lineage in Nigeria, suggest that the B.1.525 lineage represents another "variant of concern" and further underline the importance of genomic surveillance in undersampled regions across the globe.

Rabies control in Liberia: Joint efforts towards zero by 30.

Despite declaration as a national priority disease, dog rabies remains endemic in Liberia, with surveillance systems and disease control activities still developing. The objective of these initial efforts was to establish animal rabies diagnostics, foster collaboration between all rabies control stakeholders, and develop a short-term action plan with estimated costs for rabies control and elimination in Liberia. Four rabies diagnostic tests, the direct fluorescent antibody (DFA) test, the direct immunohistochemical test (dRIT), the reverse transcriptase polymerase chain reaction (RT-PCR) assay and the rapid immunochromatographic diagnostic test (RIDT), were implemented at the Central Veterinary Laboratory (CVL) in Monrovia between July 2017 and February 2018. Seven samples (n=7) out of eight suspected animals were confirmed positive for rabies lyssavirus, and molecular analyses revealed that all isolates belonged to the Africa 2 lineage, subgroup H. During a comprehensive in-country One Health rabies stakeholder meeting in 2018, a practical workplan, a short-term action plan and an accurately costed mass dog vaccination strategy were developed. Liberia is currently at stage 1.5/5 of the Stepwise Approach towards Rabies Elimination (SARE) tool, which corresponds with countries that are scaling up local-level interventions (e.g. dog vaccination campaigns) to the national level. Overall an estimated 5.3 - 8 million USD invested over 13 years is needed to eliminate rabies in Liberia by 2030. Liberia still has a long road to become free from dog-rabies. However, the dialogue between all relevant stakeholders took place, and disease surveillance considerably improved through implementing rabies diagnosis at the CVL. The joint efforts of diverse national and international stakeholders laid important foundations to achieve the goal of zero dog-mediated human rabies deaths by 2030.

How Well Does Evolution Explain Endogenous Retroviruses?-A Lakatosian Assessment.

One of the most sophisticated philosophies of science is the methodology of scientific research programmes (MSRP), developed by Imre Lakatos. According to MSRP, scientists are working within so-called research programmes, consisting of a hard core of fixed convictions and a flexible protective belt of auxiliary hypotheses. Anomalies are accommodated by changes to the protective belt that do not affect the hard core. Under MSRP, research programmes are appraised as 'progressive' if they successfully predict novel facts but are judged as 'degenerative' if they merely offer ad hoc solutions to anomalies. This paper applies these criteria to the evolutionary research programme as it has performed during half a century of ERV research. It describes the early history of the field and the emergence of the endogenization-amplification theory on the origins of retroviral-like sequences. It then discusses various predictions and postdictions that were generated by the programme, regarding orthologous ERVs in different species, the presence of target site duplications and the divergence of long terminal repeats, and appraises how the programme has dealt with data that did not conform to initial expectations. It is concluded that the evolutionary research programme has been progressive with regard to the issues here examined.

A Model of Indel Evolution by Finite-State, Continuous-Time Machines.

We introduce a systematic method of approximating finite-time transition probabilities for continuous-time insertion-deletion models on sequences. The method uses automata theory to describe the action of an infinitesimal evolutionary generator on a probability distribution over alignments, where both the generator and the alignment distribution can be represented by pair hidden Markov models (HMMs). In general, combining HMMs in this way induces a multiplication of their state spaces; to control this, we introduce a coarse-graining operation to keep the state space at a constant size. This leads naturally to ordinary differential equations for the evolution of the transition probabilities of the approximating pair HMM. The TKF91 model emerges as an exact solution to these equations for the special case of single-residue indels. For the more general case of multiple-residue indels, the equations can be solved by numerical integration. Using simulated data, we show that the resulting distribution over alignments, when compared to previous approximations, is a better fit over a broader range of parameters. We also propose a related approach to develop differential equations for sufficient statistics to estimate the underlying instantaneous indel rates by expectation maximization. Our code and data are available at https://github.com/ihh/trajectory-likelihood.