Distribution Of Branch Lengths Research Articles

Quantification of airway structures by persistent homology.

We propose two types of novel morphological metrics for quantifying the geometry of tubular structures on computed tomography (CT) images. We apply our metrics to identify irregularities in the airway of patients with chronic obstructive pulmonary disease (COPD) and demonstrate that they provide complementary information to the conventional metrics used to assess COPD, such as the tissue density distribution in lung parenchyma and the wall area ratio of the segmented airway. The three-dimensional shape of the airway and its abstraction as a rooted tree with the root at the trachea carina are automatically extracted from a lung CT volume, and the two metrics are computed based on a mathematical tool called persistent homology; treeH0 quantifies the distribution of branch lengths to assess the complexity of the tree-like structure and radialH0 quantifies the irregularities in the luminal radius along the airway. We show our metrics are associated with clinical outcomes.

Transmission dynamics variability of lineage 2 Mycobacterium tuberculosis strains in Kobe, Japan, determined using population-based whole-genome sequencing analysis.

Currently, tuberculosis (TB) in Japan is highly prevalent among elderly patients who were born during a time when TB was highly prevalent. Mycobacterium tuberculosis (Mtb) lineage 2 (L2) is the predominant strain in the country. Moreover, the proportion of foreign-born patients with TB has been increasing. This epidemiological situation in Japan motivated us to explore the heterogeneity in transmission dynamics among the sublineages of Mtb L2 within this aging population. For this purpose, we conducted a population-based whole genome sequencing analysis of 550 Mtb strains in Kobe, Japan, and employed pairwise single nucleotide polymorphism (SNP) distance clustering and terminal branch length (TBL) distribution analysis to assess Mtb transmission. The genomic clustering rate with a threshold of ≤5 SNPs was significantly lower in elderly patients aged 70years or higher than in non-elderly patients. The elderly patient group showed significantly longer TBL than the non-elderly group. These results supported the notion that reactivation of distant infection is a major driving force for the high incidence of TB in elderly individuals. The age group distribution and frequency of lineages/sublineages were found to significantly differ between foreign-born and Japan-born patients. The increased proportion of foreign-born patients might have resulted in more strain diversity in Japan. The L2.2.A sublineage demonstrated a significant association with elderly patients and exhibited lower transmission rates, which indicate to be prone to reactivate from long-term latency. In contrast, L2.2.Modern, showed a strong association with younger and foreign-born patients. This sublineage showed a high genomic cluster rate, suggesting its high transmissibility. The other three major sublineages, namely L2.2.AA2, L2.2.AA3.1, and L2.2.AA3.2, exhibited a consistent increase in cluster rates across varying SNP thresholds, indicating their relatively recent emergence as endemic sublineages in Japan. In conclusion, this study highlights distinct differences in the transmission dynamics of L2 sublineages within an aging society.

Distribution of External Branch Lengths in Yule Histories

The Yule branching process is a classical model for the random generation of gene tree topologies in population genetics. It generates binary ranked trees -also called histories- with a finite number $n$ of leaves. We study the lengths $\ell_1 > \ell_2 > \cdots > \ell_k > \cdots$ of the external branches of a Yule generated random history of size $n$, where the length of an external branch is defined as the rank of its parent node. When $n \rightarrow \infty$, we show that the random variable $\ell_k$, once rescaled as $\frac{n-\ell_k}{\sqrt{n/2}}$, follows a $\chi$-distribution with $2k$ degrees of freedom, with mean $\mathbb E(\ell_k) \sim n$ and variance $\mathbb V(\ell_k) \sim n \big(k-\frac{\pi k^2}{16^k} \binom{2k}{k}^2\big)$. Our results contribute to the study of the combinatorial features of Yule generated gene trees, in which external branches are associated with singleton mutations affecting individual gene copies.

Permeability model of fracture network based on branch length distribution and topological connectivity

Fracture networks are of significance in the production of coalbed methane from unconventional reservoirs. However, the complex distribution and geometry of fracture networks make effective predictions of their permeability difficult. This study obtains the shape of a natural fracture network in coal based on a stereomicroscopy experiment and analyzes the structural characteristics of the fracture network using graph theory. The fractal scaling law for the branch length distribution of the fractures and the relationships among the fractal dimensions of the branch length distribution, fracture area, porosity, connectivity, and ratio of maximum to minimum branch length are established. A new permeability model for a complex fracture network is developed based on fractal theory, and two important fracture characteristics, namely, tortuosity and connectivity, are considered. The model is verified using the results of previous studies and seepage tests, and the influence of the fracture network characteristic parameters on the permeability is analyzed. The results show that the permeability increases with increasing porosity, fractal dimension, proportionality coefficient, maximum fracture branch length, and connectivity and decreases with increasing tortuosity fractal dimension and dip angle.

Analysis of the topological motifs of the cellular structure of the tri-spine horseshoe crab (Tachypleus tridentatus) and its associated mechanical properties

Topological motifs in pore architecture can profoundly influence the structural properties of that architecture, such as its mass, porosity, modulus, strength, and surface permeability. Taking the irregular cellular structure of the tri-spine horseshoe crab as a research model, we present a new approach to the quantitative description and analysis of structure-property-function relationships. We employ a robust skeletonization method to construct a curve-skeleton that relies on high-resolution 3D tomographic data. The topological motifs and mechanical properties of the long-range cellular structure were investigated using the Grasshopper plugin and uniaxial compression test to identify the variation gradient. Finite element analysis was conducted for the sub-volumes to obtain the variation in effective modulus along the three principal directions. The results show that the branch length and node distribution density varied from the tip to the base of the sharp corner. These node types formed a low-connectivity network, in which the node types 3-N and 4-N tended to follow the motifs of ideal planar triangle and tetrahedral configurations, respectively, with the highest proportion of inter-branch angles in the angle ranges of 115–120° and 105–110°. In addition, mapping the mechanical gradients to topological properties indicated that narrower profiles with a given branch length gradient, preferred branch orientation, and network connectedness degree are the main factors that affect the mechanical properties. These factors suggest significant potential for designing a controllable, irregularly cellular structure in terms of both morphology and function.

Graph-based algorithms for Laplace transformed coalescence time distributions.

Extracting information on the selective and demographic past of populations that is contained in samples of genome sequences requires a description of the distribution of the underlying genealogies. Using the Laplace transform, this distribution can be generated with a simple recursive procedure, regardless of model complexity. Assuming an infinite-sites mutation model, the probability of observing specific configurations of linked variants within small haplotype blocks can be recovered from the Laplace transform of the joint distribution of branch lengths. However, the repeated differentiation required to compute these probabilities has proven to be a serious computational bottleneck in earlier implementations. Here, I show that the state space diagram can be turned into a computational graph, allowing efficient evaluation of the Laplace transform by means of a graph traversal algorithm. This general algorithm can, for example, be applied to tabulate the likelihoods of mutational configurations in non-recombining blocks. This work provides a crucial speed up for existing composite likelihood approaches that rely on the joint distribution of branch lengths to fit isolation with migration models and estimate the parameters of selective sweeps. The associated software is available as an open-source Python library, agemo.

Characterizing interface topology in multiphase flows using skeletons

The unsteady motion of a gas–liquid interface, such as during splashing or atomization, often results in complex liquid structures embedded in the ambient fluid. Here, we explore the use of skeletonization to identify the minimum amount of information needed to describe their geometry. We skeletonize a periodic liquid jet by a modification of a recently introduced approach to coarsen multiphase flows while retaining a sharp interface. The process consists of diffusing an index function and at the same time moving the interfaces with it, until they “collapse” into each other and form skeletons. The skeleton represents the basic topology of the jet and we also keep track of how much the interface is moved (or how much volume is “accumulated”) during the process, which can be used to approximately reconstruct the jet. We explore various quantitative measures to characterize and distinguish the skeletons. These include standard morphometrics such as branch length distribution, after segmenting the skeletons into branches, and a more sophisticated representation of the skeleton structures called topology morphology descriptor, to obtain an “equivalent” description of the skeletons by retaining information about the topology in a compact way.

Kinetic evaluation of the starch molecular behavior under extrusion-based or laser powder bed fusion 3D printing systems: A systematic structural and biological comparison

Supramolecular polymers are the most used materials for 3D printing applications because of their ability to improve the flow behavior of inks, and consequently enhance the mechanical strength of printed architectures. However, their interactions are assessed without considering the fact that their formed structures are affected by the printing process. Here, the effects of printer temperature of an extrusion-based printing system, or surface temperature of a laser powder bed fusion printer, on the molecular behavior of starch were investigated. Starch, being a highly branched polymer, was selected as the tested biopolymer since it provides numerous noteworthy benefits for an investigation of the depolymerization/crosslinking mechanisms upon printing, which could shed light on the impact of 3D printing on non-degradable materials. Completely different behaviors of starch functional properties are found between the two printing systems, where the extrusion-based printer promoted a significant degradation of the starch chains, while the laser powder bed fusion offered a typical behavior associated with crosslinking/branching. Extrusion 3D printing induced a reduction in starch molecular size and therefore in the structural strength of networks formed from it. However, this was also found to increase the starch digestibility rate/extent. In contrast, laser powder bed fusion increased the molecular size, inducing a viscoelastic gel-like character. At the same time, it decreased the starch digestion rate/extent. The obtained data offers information that can support the mechanistic interpretation of the depolymerization/crosslinking kinetics on the non-degradable materials, where it may be much harder to obtain the branch-length distribution. Therefore, the mechanistic information provided from starch can also be useful in understanding the degradation/chain branching of synthetic branched polymers undergoing the same printing processes.

Bias induced drift and trapping on random combs and the Bethe lattice: Fluctuation regime and first order phase transitions

We study the competition between field-induced transport and trapping in a disordered medium by studying biased random walks on random combs and the bond-diluted Bethe lattice above the percolation threshold. While it is known that the drift velocity vanishes above a critical threshold, here our focus is on fluctuations, characterized by the variance of the transit times. On the random comb, the variance is calculated exactly for a given realization of disorder using a ‘forward transport’ limit which prohibits backward movement along the backbone but allows an arbitrary number of excursions into random-length branches. The disorder-averaged variance diverges at an earlier threshold of the bias, implying a regime of anomalous fluctuations, although the velocity is nonzero. Our results are verified numerically using a Monte Carlo procedure that is adapted to account for ultra-slow returns from long branches. On the Bethe lattice, we derive an upper bound for the critical threshold bias for anomalous fluctuations of the mean transit time averaged over disorder realizations. Finally, as for the passage to the vanishing velocity regime, it is shown that the transition to the anomalous fluctuation regime can change from continuous to first order depending on the distribution of branch lengths.

Population structure, biogeography and transmissibility of Mycobacterium tuberculosis

Mycobacterium tuberculosis is a clonal pathogen proposed to have co-evolved with its human host for millennia, yet our understanding of its genomic diversity and biogeography remains incomplete. Here we use a combination of phylogenetics and dimensionality reduction to reevaluate the population structure of M. tuberculosis, providing an in-depth analysis of the ancient Indo-Oceanic Lineage 1 and the modern Central Asian Lineage 3, and expanding our understanding of Lineages 2 and 4. We assess sub-lineages using genomic sequences from 4939 pan-susceptible strains, and find 30 new genetically distinct clades that we validate in a dataset of 4645 independent isolates. We find a consistent geographically restricted or unrestricted pattern for 20 groups, including three groups of Lineage 1. The distribution of terminal branch lengths across the M. tuberculosis phylogeny supports the hypothesis of a higher transmissibility of Lineages 2 and 4, in comparison with Lineages 3 and 1, on a global scale. We define an expanded barcode of 95 single nucleotide substitutions that allows rapid identification of 69 M. tuberculosis sub-lineages and 26 additional internal groups. Our results paint a higher resolution picture of the M. tuberculosis phylogeny and biogeography.

Phylogenetic sampling affects evolutionary patterns of morphological disparity

AbstractCladistic character matrices are routinely repurposed in analyses of morphological disparity. Unfortunately, the sampling of taxa and characters within such datasets reflects their intended application (to resolve phylogeny, rather than distinguish between phenotypes) resulting in tree shapes that often misrepresent broader taxonomic and morphological diversity. Here we use tree shape as a proxy to explore how sampling can affect perceptions of evolving morphological disparity. Through analyses of simulated and empirical data, we demonstrate that sampling can introduce biases in morphospace occupation between clades that are predicted by differences in tree symmetry and branch length distribution. Symmetrical trees with relatively long internal branches predict more expansive patterns of morphospace occupation. Conversely, asymmetrical trees with relatively short internal branches predict more compact distributions. Additionally, we find that long external branches predict greater phenotypic divergence by peripheral morphotypes. Taken together, our results caution against the uncritical repurposing of cladistic datasets in disparity analyses. However, they also demonstrate that when morphological diversity is proportionately sampled, differences in tree shape between clades can speak to genuine differences in morphospace occupation. While cladistic datasets may serve as a useful starting point, disparity datasets must attempt to achieve uniformity in lineage sampling across time and topology. Only when all potential sources of bias are accounted for can evolutionary phenomena be distinguished from artefactual signals. It must be accepted that the non‐uniformity of the fossil record may preclude representative sampling and, therefore, a faithful characterization of the evolution of morphological disparity.

Sweeps in time: leveraging the joint distribution of branch lengths.

Current methods of identifying positively selected regions in the genome are limited in two key ways: the underlying models cannot account for the timing of adaptive events and the comparison between models of selective sweeps and sequence data is generally made via simple summaries of genetic diversity. Here, we develop a tractable method of describing the effect of positive selection on the genealogical histories in the surrounding genome, explicitly modeling both the timing and context of an adaptive event. In addition, our framework allows us to go beyond analyzing polymorphism data via the site frequency spectrum or summaries thereof and instead leverage information contained in patterns of linked variants. Tests on both simulations and a human data example, as well as a comparison to SweepFinder2, show that even with very small sample sizes, our analytic framework has higher power to identify old selective sweeps and to correctly infer both the time and strength of selection. Finally, we derived the marginal distribution of genealogical branch lengths at a locus affected by selection acting at a linked site. This provides a much-needed link between our analytic understanding of the effects of sweeps on sequence variation and recent advances in simulation and heuristic inference procedures that allow researchers to examine the sequence of genealogical histories along the genome.

Estimating effective permeability using connectivity and branch length distribution of fracture network

In this study, a new approach to estimate the effective permeability of the fracture network is proposed. New functional expressions are derived considering laminar fluid flow through fracture networks. Effective permeability of the fracture network is expressed as a function of topological connectivity, statistical branch segment length distributions, maximum, and minimum branch segment lengths. It is observed that the statistical cumulative branch segment length distribution of fracture networks can follow both fractal and multi-fractal patterns. Hence, analytical expressions of effective permeability are proposed considering both fractal and multi-fractal distribution of branch segment lengths. Obtained expressions are validated using numerical simulation results from past studies. It is observed from derived expressions that, effective permeability of fracture networks increases non-linearly with increasing topological connectivity and fractal dimensions of statistical branch segment length distributions. However, the non-linearity in the relationship is mainly governed by the scaling exponent and scaling coefficient of the fracture network. With increasing scaling exponent and scaling coefficient, non-linearity in the relation between effective permeability and topological connectivity increases. It is observed from the derived relation that, effective permeability of a fracture network is more sensitive to longer branch segments compared to the shorter branch segments. Effective permeability of fracture networks increases rapidly with increase in length of the branch segments. Hence it is interpreted, longer branch segments in fracture networks play a critical role in determining the effective permeability of the fractured reservoir.

Dissecting Incongruence between Concatenation- and Quartet-Based Approaches in Phylogenomic Data.

Topological conflict or incongruence is widespread in phylogenomic data. Concatenation- and coalescent-based approaches often result in incongruent topologies, but the causes of this conflict can be difficult to characterize. We examined incongruence stemming from conflict the between likelihood-based signal (quantified by the difference in gene-wise log-likelihood score or $\Delta $GLS) and quartet-based topological signal (quantified by the difference in gene-wise quartet score or $\Delta $GQS) for every gene in three phylogenomic studies in animals, fungi, and plants, which were chosen because their concatenation-based IQ-TREE (T1) and quartet-based ASTRAL (T2) phylogenies are known to produce eight conflicting internal branches (bipartitions). By comparing the types of phylogenetic signal for all genes in these three data matrices, we found that 30-36% of genes in each data matrix are inconsistent, that is, each of these genes has a higher log-likelihood score for T1 versus T2 (i.e., $\Delta $GLS $>$0) whereas its T1 topology has lower quartet score than its T2 topology (i.e., $\Delta $GQS $<$0) or vice versa. Comparison of inconsistent and consistent genes using a variety of metrics (e.g., evolutionary rate, gene tree topology, distribution of branch lengths, hidden paralogy, and gene tree discordance) showed that inconsistent genes are more likely to recover neither T1 nor T2 and have higher levels of gene tree discordance than consistent genes. Simulation analyses demonstrate that the removal of inconsistent genes from data sets with low levels of incomplete lineage sorting (ILS) and low and medium levels of gene tree estimation error (GTEE) reduced incongruence and increased accuracy. In contrast, removal of inconsistent genes from data sets with medium and high ILS levels and high GTEE levels eliminated or extensively reduced incongruence, but the resulting congruent species phylogenies were not always topologically identical to the true species trees.[Conflict; gene tree; phylogenetic signal; phylogenetics; phylogenomics; Tree of Life.].

Mathematical Reconstruction of Patient-Specific Vascular Networks Based on Clinical Images and Global Optimization

Cancer is a major cause of death worldwide and becomes particularly threatening once it begins to metastasize. During metastasis, the blood vessels serve as pathways for cancerous cell transportation and hence are crucial for understanding cancer growth. Existing medical imaging modalities can provide 3-D contrast images of the vascular tissues but with limited quality and detailedness. A much-needed tool for cancer research is thus one that can reconstruct vascular networks from low-quality clinical images. To this end, we developed a computational framework that takes 3-D medical images as input and reconstructs complete, patient-specific vascular network models using a mathematical optimization procedure. Our framework extracts major vessels from the images and uses the organ geometry to select vessel termination points. Then, it generates the remainder network based on physiological optimality principles. Using the framework, we obtained a set of network models with over 3000 terminal segments from a brain MRA scan. We analyzed the Strahler order, vessel radius, and branch length distributions of the models, which match with actual human data. We also performed fluid dynamics simulation inside the reconstructed vessels and showed that the pressure and shear stress distributions agree with existing in vivo measurements. The qualitative and quantitative agreements in vessel morphometry and hemodynamics demonstrate the effectiveness of the framework. Our method bridges the gap between image-based vessel models, accuracy of which is limited by the resolution of clinical images, and hypothetical models.

Cole–Cole relation for long-chain branching from general rigid bead–rod theory

Empirically, we find that the parametric plots of the imaginary vs real parts of the complex viscosity may depend neither on temperature nor on average molecular weight. Moreover, for a fixed polydispersity, these viscosity Cole–Cole curves amplify both rightward and upward with long-chain branching content. In this paper, we find that general rigid bead–rod theory [O. Hassager, “Kinetic theory and rheology of bead–rod models for macromolecular solutions. II. Linear unsteady flow properties,” J. Chem. Phys. 60(10), 4001–4008 (1974)] can explain these rightward and upward amplifications. We explore the effects of branching along a straight chain in small-amplitude oscillatory shear flow. Specifically, we explore the number of branches, branch length, branch position, and branch distribution.

Multi-scale fracture network characterisation on carbonate platforms

Characterisation of fracture networks at different scales is challenging and important to many fields of geoscience, especially when access to multiple resolution datasets is limited. Here, we develop an integrated analysis of fracture networks on carbonate platforms using three scales of observation: small (outcrop), intermediate (airborne LiDAR) and large (3D seismic). Statistical analyses and ternary diagrams of geometrical and topological data from Cariatiz (South East Spain) and Pernambuco (East Brazil) are used to understand the relationships and distribution of fracture networks between multi-scale datasets. A variety of fracture types at each scale of observation reveal how complex fracture networks are on carbonate platforms. Our results demonstrate that fracture network properties behave differently depending on the fracture size, and that transitional scale gaps between datasets constrain fracture characterisation. Airborne LiDAR maps show that intermediate-sized fractures appear to have a better controlled orientation and a lower connectivity than smaller fractures from the same area in Cariatiz. Fracture branch length distributions fit a negative exponential or log-normal distribution for massive non-stratabound units. This work is important as it demonstrates that the use of outcrop data is a good approach to understand fracture complexity of carbonate platforms. Understanding sub-seismic fracture networks is therefore critical in quantifying fluid flow and permeability in carbonate reservoirs.

Allopatry increases the balance of phylogenetic trees during radiation under neutral speciation

The shape of a phylogenetic tree is defined by the sequence of speciation events, represented by its branching points, and extinctions, represented by branch interruptions. In a neutral scenario of parapatry and isolation by distance, species tend to branch off the original population one after the other, leading to highly unbalanced trees. In this case the degree of imbalance, measured by the normalized Sackin index, grows linearly with species richness. Here we claim that moderate values of imbalance for trees with large number of species can occur if the geographic distribution involves more than one deme (allopatry) and speciation is parapatric within demes. The combined values of balance (normalized Sackin index) and species richness provide an estimate of how many demes were involved in the process if it happened in such neutral scenario. We also show that the spatial division in demes moderately slows down the diversification process, portraying a neutral mechanism for structuring the branch length distribution of phylogenetic trees.

Spatial phylogenetics of the North American flora

AbstractNorth America is a large continent with extensive climatic, geological, soil, and biological diversity. As biota faces threat from habitat destruction and climate change, making a quantitative assessment of biodiversity becomes critically important. Rapid digitization of plant specimen records and accumulation of DNA sequence data enable a much‐needed broad synthesis of species occurrences with phylogenetic data. In this study, the first such synthesis of a flora from such a large and diverse part of the world is attempted, all seed plants from the North American continent (here defined to include Canada, United States, and Mexico), with a focus on examining phylogenetic diversity and endemism. We collected digitized plant specimen records and chose a coarse grain for analysis, recognizing that this grain is currently necessary for reasonable completeness per sampling unit. We found that raw richness and endemism patterns largely support previous hypotheses of biodiversity hotspots. The application of phylogenetic metrics and a randomization test revealed novel results, including a significant phylogenetic clustering across the continent, a striking east–west geographical difference in the distribution of branch lengths, and the discovery of centers of neo‐ and paleoendemism in Mexico, the southwestern USA, and the southeastern USA. Finally, our examination of phylogenetic beta diversity provides a new approach to compare centers of endemism. We discuss the empirical challenges of working at the continental scale and the need for more sampling across large parts of the continent, for both DNA data for terminal taxa and spatial data for poorly understood regions, to confirm and extend these results.

Large-amplitude oscillatory shear flow loops for long-chain branching from general rigid bead-rod theory

General rigid bead-rod theory [O. Hassager, “Kinetic theory and rheology of bead-rod models for macromolecular solutions. II. Linear unsteady flow properties,” J. Chem. Phys. 60, 4001–4008 (1974)] explains polymer viscoelasticity from macromolecular orientation. By means of this theory, we relate the complex viscosity of polymeric liquids to the architecture of axisymmetric branched macromolecules. In this work, we explore how adding long-chain branching to polymers affects the shapes of large-amplitude oscillatory shear (LAOS) flow loops. By loops, we mean plots of the alternant part of the shear stress response vs the cosinusoidal shear rate. We choose LAOS for its ability to amplify subtle differences in small-amplitude oscillatory shear flow at a high Weissenberg number. When non-dimensionalized with the product of the zero-shear viscosity and the shear rate amplitude, the loop shapes depend on the sole dimensionless architectural parameter, the macromolecular lopsidedness of the long-chain branched macromolecule. In this work, in this way, we compare and contrast the loop shapes of macromolecular chains that are straight with those branched. Specifically, we explore symmetric branch multiplicity, branch functionality, branch length, branch position, branch distribution, and multiple branch asymmetry. We find that adding branching collapses and distorts the loops. We then find that so long as branch length, branch position, and branch distribution are held constant and so long as the branching is symmetric about the center of mass, the peak shear stress increases with branch multiplicity. We also find that branch functionality hardly affects the loops. The structural details explored in this paper have yet to be explored in the laboratory.