Capacity, Collision Avoidance and Shopping Rate under a Social Distancing Regime.

Capacity restrictions in stores, maintained by mechanisms like spacing customer intake, became familiar features of retailing in the time of the pandemic. Shopping rates in a crowded store under a social distancing regime are prone to considerable slowdown. Inspired by the random particle collision concepts of statistical mechanics, we introduce a dynamical model of the evolution of the shopping rate as a function of a given customer intake rate. The slowdown of each individual customer is incorporated as an additive term to the baseline value of the shopping time, proportionally to the number of other customers in the store. We determine analytically and via simulation the trajectory of the model as it approaches a Little's law equilibrium and identify the point beyond which equilibrium cannot be achieved. By relating the customer shopping rate to the slowdown compared with the baseline, we can calculate the optimal intake rate leading to maximum equilibrium spending. This turns out to be the maximum rate compatible with equilibrium. The slowdown due to the largest possible number of shoppers is more than compensated for by the increased volume of shopping. This macroscopic model is validated by simulation experiments in which avoidance interactions between pairs of shoppers are responsible for shopping delays.

From comparative gene content and gene order to ancestral contigs, chromosomes and karyotypes.

To reconstruct the ancestral genome of a set of phylogenetically related descendant species, we use the RACCROCHE pipeline for organizing a large number of generalized gene adjacencies into contigs and then into chromosomes. Separate reconstructions are carried out for each ancestral node of the phylogenetic tree for focal taxa. The ancestral reconstructions are monoploids; they each contain at most one member of each gene family constructed from descendants, ordered along the chromosomes. We design and implement a new computational technique for solving the problem of estimating the ancestral monoploid number of chromosomes x. This involves a "g-mer" analysis to resolve a bias due long contigs, and gap statistics to estimate x. We find that the monoploid number of all the rosid and asterid orders is [Formula: see text]. We show that this is not an artifact of our method by deriving [Formula: see text] for the metazoan ancestor.

Escape from Parsimony of a Double-Cut-and-Join Genome Evolution Process.

We analyze models of genome evolution based on both restricted and unrestricted double-cut-and-join (DCJ) operations. Not only do our models allow different types of operations generated by DCJs (including reversals, translocations, transpositions, fissions, and fusions) to take different weights during the course of evolution, but they also let these weights fluctuate over time. We compare the number of operations along the evolutionary trajectory with the DCJ distance of the genome from its ancestor at each step, and determine at what point they diverge: the process escapes from parsimony. Adapting the method developed by Berestycki and Durrett, we approximate the number of cycles in the breakpoint graph of a random genome at time t and its ancestral genome by the number of tree components in a random graph (not necessarily an Erdös-Rényi one) constructed from the model of evolution. In both models, the process on a genome of size n is bound to its parsimonious estimate up to t≈n∕2 steps.

Buxus and Tetracentron genomes help resolve eudicot genome history.

Ancient whole-genome duplications (WGDs) characterize many large angiosperm lineages, including angiosperms themselves. Prominently, the core eudicot lineage accommodates 70% of all angiosperms and shares ancestral hexaploidy, termed gamma. Gamma arose via two WGDs that occurred early in eudicot history; however, the relative timing of these is unclear, largely due to the lack of high-quality genomes among early-diverging eudicots. Here, we provide complete genomes for Buxus sinica (Buxales) and Tetracentron sinense (Trochodendrales), representing the lineages most closely related to core eudicots. We show that Buxus and Tetracentron are both characterized by independent WGDs, resolve relationships among early-diverging eudicots and their respective genomes, and use the RACCROCHE pipeline to reconstruct ancestral genome structure at three key phylogenetic nodes of eudicot diversification. Our reconstructions indicate genome structure remained relatively stable during early eudicot diversification, and reject hypotheses of gamma arising via inter-lineage hybridization between ancestral eudicot lineages, involving, instead, only stem lineage core eudicot ancestors.

Two divergent haplotypes from a highly heterozygous lychee genome suggest independent domestication events for early and late-maturing cultivars.

Lychee is an exotic tropical fruit with a distinct flavor. The genome of cultivar 'Feizixiao' was assembled into 15 pseudochromosomes, totaling ~470 Mb. High heterozygosity (2.27%) resulted in two complete haplotypic assemblies. A total of 13,517 allelic genes (42.4%) were differentially expressed in diverse tissues. Analyses of 72 resequenced lychee accessions revealed two independent domestication events. The extremely early maturing cultivars preferentially aligned to one haplotype were domesticated from a wild population in Yunnan, whereas the late-maturing cultivars that mapped mostly to the second haplotype were domesticated independently from a wild population in Hainan. Early maturing cultivars were probably developed in Guangdong via hybridization between extremely early maturing cultivar and late-maturing cultivar individuals. Variable deletions of a 3.7 kb region encompassed by a pair of CONSTANS-like genes probably regulate fruit maturation differences among lychee cultivars. These genomic resources provide insights into the natural history of lychee domestication and will accelerate the improvement of lychee and related crops.

Peripheral structures in unlabelled trees and the accumulation of subgenomes in the evolution of polyploids.

Many angiosperms have undergone some series of polyploidization events over the course of their evolutionary history. In these genomes, especially those resulting from multiple autopolyploidization, it may be relatively easy to recognize all the ξ sets of n homeologous chromosomes, but it is much harder, if not impossible, to partition these chromosomes into n subgenomes, each representing one distinct genomic component of ξ chromosomes making up the original polyploid. Thus, if we wish to infer the polyploidization history of the genome, we could make use of all the gene trees inferred from the genes in one set of homeologous chromosomes to construct a consensus tree, but there is no evident way of combining the trees from the ξ different sets, because we have no labelling of the chromosomes that is known to be consistent across these sets. We suggest here that lacking a consistent leaf-labelling, the topological structure of the trees may display sufficient resemblance so that a higher level consensus could be revealing of evolutionary history. This would be especially true of the peripheral structures of the tree, likely representing events that occurred more recently and have thus been less obscured by subsequent evolutionary processes. Here, we present a statistical test to assess whether the subgenomes in a polyploid genome could have been added one at a time. The null hypothesis is that the accumulation of chromosomes follows a stochastic process in which transition from one generation to the next is through randomly choosing an edge, and then subdividing this edge in order to link the new internal vertex to a new external vertex. We analyze the probability distributions of a number of peripheral tree substructures, namely leaf- or terminal-pairs, triples and quadruples, arising from this stochastic process, in terms of some exact recurrences. We propose some conjectures regarding the asymptotic behaviours of these distributions. Applying our analysis to a sugarcane genome, we demonstrate that it is unlikely that the accumulation of subgenomes has occurred one at a time.

Ancestral Flowering Plant Chromosomes and Gene Orders Based on Generalized Adjacencies and Chromosomal Gene Co-Occurrences.

Recurrent whole genome duplication and the ensuing loss of redundant genes-fractionation-complicate efforts to reconstruct the gene orders and chromosomes of the ancestors associated with the nodes of a phylogeny. Loss of genes disrupts the gene adjacencies key to current techniques. With our RACCROCHE pipeline, instead of starting with the inference of short ancestral segments, we suggest delaying the choice of gene adjacencies while we accumulate many more syntenically validated generalized (gapped) adjacencies. We obtain longer ancestral contigs using maximum weight matching (MWM). Similarly, we do not construct chromosomes by successively piecing together contigs into larger segments, but rather compile counts of pairwise contig co-occurrences on the set of extant genomes and use these to cluster the contigs. Chromosome-level contig assemblies for a monoploid genome emerge naturally at each node of the phylogeny and the contigs then can be ordered along the chromosome. Sampling alternative MWM solutions, visualizing heat maps, and applying gap statistics allow us to estimate the number of chromosomes in the reconstruction. We introduce several measures of quality: length of contigs, continuity of contig structure on successive ancestors, coverage of the extant genome by the reconstruction, and rearrangement relations among the inferred chromosomes. The reconstructed ancestors are visualized by painting the ancestral projections on the descendant genomes. We submit genomes drawn from a broad range of monocot orders to our pipeline, confirming the tetraploidization event "tau" in the stem lineage between the alismatids and the lilioids. We show additional applications to the Solanaceae and to four Brassica genomes, producing evidence about the monoploid ancestor in each case.

The monoploid chromosome complement of reconstructed ancestral genomes in a phylogeny.

Using RACCROCHE, a method for reconstructing gene content and order of ancestral chromosomes from a phylogeny of extant genomes represented by the gene orders on their chromosomes, we study the evolution of three orders of woody plants. The method retrieves the monoploid complement of each Ancestor in a phylogeny, consisting a complete set of distinct chromosomes, despite some of the extant genomes being recently or historically polyploidized. The three orders are the Sapindales, the Fagales and the Malvales. All of these are independently estimated to have ancestral monoploid number [Formula: see text].

Integrated synteny- and similarity-based inference on the polyploidization-fractionation cycle.

Whole-genome doubling, tripling or replicating to a greater degree, due to fixation of polyploidization events, is attested in almost all lineages of the flowering plants, recurring in the ancestry of some plants two, three or more times in retracing their history to the earliest angiosperm. This major mechanism in plant genome evolution, which generally appears as instantaneous on the evolutionary time scale, sets in operation a compensatory process called fractionation, the loss of duplicate genes, initially rapid, but continuing at a diminishing rate over millions and tens of millions of years. We study this process by statistically comparing the distribution of duplicate gene pairs as a function of their time of creation through polyploidization, as measured by sequence similarity. The stochastic model that accounts for this distribution, though exceedingly simple, still has too many parameters to be estimated based only on the similarity distribution, while the computational procedures for compiling the distribution from annotated genomic data is heavily biased against earlier polyploidization events-syntenic 'crumble'. Other parameters, such as the size of the initial gene complement and the ploidy of the various events giving rise to duplicate gene pairs, are even more inaccessible to estimation. Here, we show how the frequency of unpaired genes, identified via their embedding in stretches of duplicate pairs, together with previously established constraints among some parameters, adds enormously to the range of successive polyploidization events that can be analysed. This also allows us to estimate the initial gene complement and to correct for the bias due to crumble. We explore the applicability of our methodology to four flowering plant genomes covering a range of different polyploidization histories.

The contributions from the progenitor genomes of the mesopolyploid Brassiceae are evolutionarily distinct but functionally compatible.

The members of the tribe Brassiceae share a whole-genome triplication (WGT), and one proposed model for its formation is a two-step pair of hybridizations producing hexaploid descendants. However, evidence for this model is incomplete, and the evolutionary and functional constraints that drove evolution after the hexaploidy are even less understood. Here, we report a new genome sequence of Crambe hispanica, a species sister to most sequenced Brassiceae. Using this new genome and three others that share the hexaploidy, we traced the history of gene loss after the WGT using the Polyploidy Orthology Inference Tool (POInT). We confirm the two-step formation model and infer that there was a significant temporal gap between those two allopolyploidizations, with about a third of the gene losses from the first two subgenomes occurring before the arrival of the third. We also, for the 90,000 individual genes in our study, make parental subgenome assignments, inferring, with measured uncertainty, from which of the progenitor genomes of the allohexaploidy each gene derives. We further show that each subgenome has a statistically distinguishable rate of homoeolog losses. There is little indication of functional distinction between the three subgenomes: the individual subgenomes show no patterns of functional enrichment, no excess of shared protein-protein or metabolic interactions between their members, and no biases in their likelihood of having experienced a recent selective sweep. We propose a "mix and match" model of allopolyploidy, in which subgenome origin drives homoeolog loss propensities but where genes from different subgenomes function together without difficulty.

Genome sequence and evolution of Betula platyphylla.

Betula L. (birch) is a pioneer hardwood tree species with ecological, economic, and evolutionary importance in the Northern Hemisphere. We sequenced the Betula platyphylla genome and assembled the sequences into 14 chromosomes. The Betula genome lacks evidence of recent whole-genome duplication and has the same paleoploidy level as Vitis vinifera and Prunus mume. Phylogenetic analysis of lignin pathway genes coupled with tissue-specific expression patterns provided clues for understanding the formation of higher ratios of syringyl to guaiacyl lignin observed in Betula species. Our transcriptome analysis of leaf tissues under a time-series cold stress experiment revealed the presence of the MEKK1-MKK2-MPK4 cascade and six additional mitogen-activated protein kinases that can be linked to a gene regulatory network involving many transcription factors and cold tolerance genes. Our genomic and transcriptome analyses provide insight into the structures, features, and evolution of the B. platyphylla genome. The chromosome-level genome and gene resources of B. platyphylla obtained in this study will facilitate the identification of important and essential genes governing important traits of trees and genetic improvement of B. platyphylla.

Branching Out to Speciation in a Model of Fractionation: The Malvaceae.

Fractionation is the genome-wide process of losing one gene per duplicate pair following whole genome doubling (WGD). An important type of evidence for duplicate gene loss is the frequency distribution of similarities between paralogous gene pairs in a genome or orthologous gene pairs in two species. We extend a previous branching process model for fractionation, originally accounting for paralog similarities, to encompass the distribution of ortholog similarities, after multiple rounds of whole genome doubling and fractionation, with the speciation event occurring at any point. We estimate the fractionation rates during all the inter-event periods in each lineage of the plant family Malvaceae. We suggest a major correction of the phylogenetic position of the durian sub-family, and discover a new triplication event in this lineage.

A high-contiguity Brassica nigra genome localizes active centromeres and defines the ancestral Brassica genome.

It is only recently, with the advent of long-read sequencing technologies, that we are beginning to uncover previously uncharted regions of complex and inherently recursive plant genomes. To comprehensively study and exploit the genome of the neglected oilseed Brassica nigra, we generated two high-quality nanopore de novo genome assemblies. The N50 contig lengths for the two assemblies were 17.1 Mb (12 contigs), one of the best among 324 sequenced plant genomes, and 0.29 Mb (424 contigs), respectively, reflecting recent improvements in the technology. Comparison with a de novo short-read assembly corroborated genome integrity and quantified sequence-related error rates (0.2%). The contiguity and coverage allowed unprecedented access to low-complexity regions of the genome. Pericentromeric regions and coincidence of hypomethylation enabled localization of active centromeres and identified centromere-associated ALE family retro-elements that appear to have proliferated through relatively recent nested transposition events (<1 Ma). Genomic distances calculated based on synteny relationships were used to define a post-triplication Brassica-specific ancestral genome, and to calculate the extensive rearrangements that define the evolutionary distance separating B. nigra from its diploid relatives.

Excision Dominates Pseudogenization During Fractionation After Whole Genome Duplication and in Gene Loss After Speciation in Plants.

We take advantage of synteny blocks, the analytical construct enabled at the evolutionary moment of speciation or polyploidization, to follow the independent loss of duplicate genes in two sister species or the loss through fractionation of syntenic paralogs in a doubled genome. By examining how much sequence remains after a contiguous series of genes is deleted, we find that this residue remains at a constant low level independent of how many genes are lost-there are few if any relics of the missing sequence. Pseudogenes are rare or extremely transient in this context. The potential exceptions lie exclusively with a few examples of speciation, where the synteny blocks in some larger genomes tolerate degenerate sequence during genomic divergence of two species, but not after whole genome doubling in the same species where fractionation pressure eliminates virtually all non-coding sequence.

Distinguishing successive ancient polyploidy levels based on genome-internal syntenic alignment.

BACKGROUND: A basic tool for studying the polyploidization history of a genome, especially in plants, is the distribution of duplicate gene similarities in syntenically aligned regions of a genome. This distribution can usually be decomposed into two or more components identifiable by peaks, or local maxima, each representing a different polyploidization event. The distributions may be generated by means of a discrete time branching process, followed by a sequence divergence model. The branching process, as well as the inference of fractionation rates based on it, requires knowledge of the ploidy level of each event, which cannot be directly inferred from the pair similarity distribution. RESULTS: For a sequence of two events of unknown ploidy, either tetraploid, giving rise to whole genome doubling (WGD), or hexaploid, giving rise to whole genome tripling (WGT), we base our analysis on triples of similar genes. We calculate the probability of the four triplet types with origins in one or the other event, or both, and impose a mutational model so that the distribution resembles the original data. Using a ML transition point in the similarities between the two events as a discriminator for the hypothesized origin of each similarity, we calculate the predicted number of triplets of each type for each model combining WGT and/or WGD. This yields a predicted profile of triplet types for each model. We compare the observed and predicted triplet profiles for each model to confirm the polyploidization history of durian, poplar and cabbage. CONCLUSIONS: We have developed a way of inferring the ploidy of up to three successive WGD and/or WGT events by estimating the time of origin of each of the similarities in triples of genes. This may be generalized to a larger number of events and to higher ploidies.

The avocado genome informs deep angiosperm phylogeny, highlights introgressive hybridization, and reveals pathogen-influenced gene space adaptation.

The avocado, Persea americana, is a fruit crop of immense importance to Mexican agriculture with an increasing demand worldwide. Avocado lies in the anciently diverged magnoliid clade of angiosperms, which has a controversial phylogenetic position relative to eudicots and monocots. We sequenced the nuclear genomes of the Mexican avocado race, P. americana var. drymifolia, and the most commercially popular hybrid cultivar, Hass, and anchored the latter to chromosomes using a genetic map. Resequencing of Guatemalan and West Indian varieties revealed that ∼39% of the Hass genome represents Guatemalan source regions introgressed into a Mexican race background. Some introgressed blocks are extremely large, consistent with the recent origin of the cultivar. The avocado lineage experienced 2 lineage-specific polyploidy events during its evolutionary history. Although gene-tree/species-tree phylogenomic results are inconclusive, syntenic ortholog distances to other species place avocado as sister to the enormous monocot and eudicot lineages combined. Duplicate genes descending from polyploidy augmented the transcription factor diversity of avocado, while tandem duplicates enhanced the secondary metabolism of the species. Phenylpropanoid biosynthesis, known to be elicited by Colletotrichum (anthracnose) pathogen infection in avocado, is one enriched function among tandems. Furthermore, transcriptome data show that tandem duplicates are significantly up- and down-regulated in response to anthracnose infection, whereas polyploid duplicates are not, supporting the general view that collections of tandem duplicates contribute evolutionarily recent "tuning knobs" in the genome adaptive landscapes of given species.

A branching process for homology distribution-based inference of polyploidy, speciation and loss.

BACKGROUND: The statistical distribution of the similarity or difference between pairs of paralogous genes, created by whole genome doubling, or between pairs of orthologous genes in two related species is an important source of information about genomic evolution, especially in plants. METHODS: We derive the mixture of distributions of sequence similarity for duplicate gene pairs generated by repeated episodes of whole gene doubling. This involves integrating sequence divergence and gene pair loss through fractionation, using a branching process and a mutational model. We account not only for the timing of these events in terms of local modes, but also the amplitude and variance of the component distributions. This model is then extended to orthologous gene pairs. RESULTS: We apply the model and inference procedures to the evolution of the Solanaceae, focusing on the genomes of economically important crops. We assess how consistent or variable fractionation rates are from species to species and over time.

Models for Similarity Distributions of Syntenic Homologs and Applications to Phylogenomics.

We outline an integrated approach to speciation and whole genome duplication (WGD) to resolve the occurrence of these events in phylogenetic analysis. We propose a more principled way of estimating the parameters of gene divergence and fractionation than the standard mixture of normals analysis. We formulate an algorithm for resolving data on local peaks in the distributions of duplicate gene similarities for a number of related genomes. We illustrate with a comprehensive analysis of WGD-origin duplicate gene data from the family Brassicaceae.

Accurate prediction of orthologs in the presence of divergence after duplication.

Motivation: When gene duplication occurs, one of the copies may become free of selective pressure and evolve at an accelerated pace. This has important consequences on the prediction of orthology relationships, since two orthologous genes separated by divergence after duplication may differ in both sequence and function. In this work, we make the distinction between the primary orthologs, which have not been affected by accelerated mutation rates on their evolutionary path, and the secondary orthologs, which have. Similarity-based prediction methods will tend to miss secondary orthologs, whereas phylogeny-based methods cannot separate primary and secondary orthologs. However, both types of orthology have applications in important areas such as gene function prediction and phylogenetic reconstruction, motivating the need for methods that can distinguish the two types. Results: We formalize the notion of divergence after duplication and provide a theoretical basis for the inference of primary and secondary orthologs. We then put these ideas to practice with the Hybrid Prediction of Paralogs and Orthologs (HyPPO) framework, which combines ideas from both similarity and phylogeny approaches. We apply our method to simulated and empirical datasets and show that we achieve superior accuracy in predicting primary orthologs, secondary orthologs and paralogs. Availability and implementation: HyPPO is a modular framework with a core developed in Python and is provided with a variety of C++ modules. The source code is available at https://github.com/manuellafond/HyPPO. Supplementary information: Supplementary data are available at Bioinformatics online.

Resolution effects in reconstructing ancestral genomes.

BACKGROUND: The reconstruction of ancestral genomes must deal with the problem of resolution, necessarily involving a trade-off between trying to identify genomic details and being overwhelmed by noise at higher resolutions. RESULTS: We use the median reconstruction at the synteny block level, of the ancestral genome of the order Gentianales, based on coffee, Rhazya stricta and grape, to exemplify the effects of resolution (granularity) on comparative genomic analyses. CONCLUSIONS: We show how decreased resolution blurs the differences between evolving genomes, with respect to rate, mutational process and other characteristics.