Approximating the Gaussian as a Sum of Exponentials and Its Applications to the Fast Gauss Transform
We develop efficient and accurate sum-of-exponential (SOE) approximations for the Gaussian using rational approximation of the exponential function on the negative real axis. Six digit accuracy can be obtained with eight terms and ten digit accuracy can be obtained with twelve terms. This representation is of potential interest in approximation theory but we focus here on its use in accelerating the fast Gauss transform (FGT) in one and two dimensions. The one-dimensional scheme is particularly straightforward and easy to implement, requiring only twenty-four lines of MATLAB code. The two-dimensional version requires some care with data structures, but is significantly more efficient than existing FGTs. Following a detailed presentation of the theoretical foundations, we demonstrate the performance of the fast transforms with several numerical experiments.
For the past half-century, structural biologists relied on the notion that similar protein sequences give rise to similar structures and functions. While this assumption has driven research to explore certain parts of the protein universe, it disregards spaces that don’t rely on this assumption. Here we explore areas of the protein universe where similar protein functions can be achieved by different sequences and different structures. We predict ∼200,000 structures for diverse protein sequences from 1,003 representative genomes1 across the microbial tree of life, and annotate them functionally on a per-residue basis. Structure prediction is accomplished using the World Community Grid, a large-scale citizen science initiative. The resulting database of structural models is complementary to the AlphaFold database, with regards to domains of life as well as sequence diversity and sequence length. We identify 161 novel folds and describe examples where we map specific functions to structural motifs. We also show that the structural space is continuous and largely saturated, highlighting the need for shifting the focus from obtaining structures to putting them into context, to transform all branches of biology, including a shift from sequence-based to sequence-structure-function based meta-omics analyses.
Gastrulation movements in all animal embryos start with regulated deformations of patterned epithelial sheets, which are driven by cell divisions, cell shape changes, and cell intercalations. Each of these behaviors has been associated with distinct aspects of gastrulation and has been a subject of intense research using genetic, cell biological, and more recently, biophysical approaches. Most of these studies, however, focus either on cellular processes driving gastrulation or on large-scale tissue deformations. Recent advances in microscopy and image processing create a unique opportunity for integrating these complementary viewpoints. Here, we take a step toward bridging these complementary strategies and deconstruct the early stages of gastrulation in the entire Drosophila embryo. Our approach relies on an integrated computational framework for cell segmentation and tracking and on efficient algorithms for event detection. The detected events are then mapped back onto the blastoderm shell, providing an intuitive visual means to examine complex cellular activity patterns within the context of their initial anatomic domains. By analyzing these maps, we identified that the loss of nearly half of surface cells to invaginations is compensated primarily by transient mitotic rounding. In addition, by analyzing mapped cell intercalation events, we derived direct quantitative relations between intercalation frequency and the rate of axis elongation. This work is setting the stage for systems-level dissection of a pivotal step in animal development.
We introduce a closure model for coarse-grained kinetic theories of polar active fluids. Based on a thermodynamically consistent, quasi-equilibrium approximation of the particle distribution function, the model closely captures important analytical properties of the kinetic theory, including its linear stability and the balance of entropy production and dissipation. Nonlinear simulations show the model reproduces the qualitative behavior and nonequilibrium statistics of the kinetic theory, unlike commonly used closure models. We use the closure model to simulate highly turbulent suspensions in both two and three dimensions in which we observe complex multiscale dynamics, including large concentration fluctuations and a proliferation of polar and nematic defects.
Single nucleus transcriptome and chromatin accessibility of postmortem human pituitaries reveal diverse stem cell regulatory mechanisms
Despite their importance in tissue homeostasis and renewal, human pituitary stem cells (PSCs) are incompletely characterized. We describe a human single nucleus RNA-seq and ATAC-seq resource from pediatric, adult, and aged postmortem pituitaries (snpituitaryatlas.princeton.edu) and characterize cell-type-specific gene expression and chromatin accessibility programs for all major pituitary cell lineages. We identify uncommitted PSCs, committing progenitor cells, and sex differences. Pseudotime trajectory analysis indicates that early-life PSCs are distinct from the other age groups. Linear modeling of same-cell multiome data identifies regulatory domain accessibility sites and transcription factors that are significantly associated with gene expression in PSCs compared with other cell types and within PSCs. We identify distinct deterministic mechanisms that contribute to heterogeneous marker expression within PSCs. These findings characterize human stem cell lineages and reveal diverse mechanisms regulating key PSC genes and cell type identity.
A single flexible filament can be actuated to escape from the scallop theorem and generate net propulsion at low Reynolds number. In this work, we study the dynamics of a simple boundary-driven multi-filament swimmer, a two-arm clamshell actuated at the hinged point, using a nonlocal slender body approximation with full hydrodynamic interactions. We first consider an elastic clamshell consisted of flexible filaments with intrinsic curvature, and then build segmental models consisted of rigid segments connected by different mechanical joints with different forms of response torques. The simplicity of the system allows us to fully explore the effect of various parameters on the swimming performance. Optimal included angles and elastoviscous numbers are identified. The segmental models capture the characteristic dynamics of the elastic clamshell. We further demonstrate how the swimming performance can be significantly enhanced by the asymmetric beating patterns induced by biased torques.
Inspired by the recent realization of a 2D chiral fluid as an active monolayer droplet moving atop a 3D Stokesian fluid, we formulate mathematically its free-boundary dynamics. The surface droplet is described as a general 2D linear, incompressible, and isotropic fluid, having a viscous shear stress, an active chiral driving stress, and a Hall stress allowed by the lack of time-reversal symmetry. The droplet interacts with itself through its driven internal mechanics and by driving flows in the underlying 3D Stokes phase. We pose the dynamics as the solution to a singular integral-differential equation, over the droplet surface, using the mapping from surface stress to surface velocity for the 3D Stokes equations. Specializing to the case of axisymmetric droplets, exact representations for the chiral surface flow are given in terms of solutions to a singular integral equation, solved using both analytical and numerical techniques. For a disc-shaped monolayer, we additionally employ a semi-analytical solution that hinges on an orthogonal basis of Bessel functions and allows for efficient computation of the monolayer velocity field, which ranges from a nearly solid-body rotation to a unidirectional edge current depending on the subphase depth and the Saffman-Delbruck length. Except in the near-wall limit, these solutions have divergent surface shear stresses at droplet boundaries, a signature of systems with codimension one domains embedded in a three-dimensional medium. We further investigate the effect of a Hall viscosity, which couples radial and transverse surface velocity components, on the dynamics of a closing cavity
Autism is a highly heritable neurodevelopmental disorder characterized by heterogeneous cognitive, behavioral and communication impairments. Disruption of the gut-brain axis (GBA) has been implicated in autism, with dozens of cross-sectional microbiome and other omic studies revealing autism-specific profiles along the GBA albeit with little agreement in composition or magnitude. To explore the functional architecture of autism, we developed an age and sex-matched Bayesian differential ranking algorithm that identified autism-specific profiles across 10 cross-sectional microbiome datasets and 15 other omic datasets, including dietary patterns, metabolomics, cytokine profiles, and human brain expression profiles. The analysis uncovered a highly significant, functional architecture along the GBA that encapsulated the overall heterogeneity of autism phenotypes. This architecture was determined by autism-specific amino acid, carbohydrate and lipid metabolism profiles predominantly encoded by microbial species in the genera Prevotella, Enterococcus, Bifidobacterium, and Desulfovibrio, and was mirrored in brain-associated gene expression profiles and restrictive dietary patterns in individuals with autism. Pro-inflammatory cytokine profiling and virome association analysis further supported the existence of an autism-specific architecture associated with particular microbial genera. Re-analysis of a longitudinal intervention study in autism recapitulated the cross-sectional profiles, and showed a strong association between temporal changes in microbiome composition and autism symptoms. Further elucidation of the functional architecture of autism, including of the role the microbiome plays in it, will require deep, multi-omic longitudinal intervention studies on well-defined stratified cohorts to support causal and mechanistic inference.
The large-scale organization of the genome inside the cell nucleus is critical for the cell’s function. Chromatin – the functional form of DNA in cells – serves as a substrate for active nuclear processes such as transcription, replication and DNA repair. Chromatin’s spatial organization directly affects its accessibility by ATP-powered enzymes, e.g., RNA polymerase II in the case of transcription. In differentiated cells, chromatin is spatially segregated into compartments – euchromatin and heterochromatin – the former being largely transcriptionally active and loosely packed, the latter containing mostly silent genes and densely compacted. The euchromatin/heterochromatin segregation is crucial for proper genomic function, yet the physical principles behind it are far from understood. Here, we model the nucleus as filled with hydrodynamically interacting active Zimm chains – chromosomes – and investigate how large heterochromatic regions form and segregate from euchromatin through their complex interactions. Each chromosome presents a block copolymer composed of heterochromatic blocks, capable of crosslinking that increases chromatin’s local compaction, and euchromatic blocks, subjected to stochastic force dipoles that capture the microscopic stresses exerted by nuclear ATPases. These active stresses lead to a dynamic self-organization of the genome, with its coherent motions driving the mixing of chromosome territories as well as large-scale heterochromatic segregation through crosslinking of distant genomic regions. We study the stresses and flows that arise in the nucleus during the heterochromatic segregation, and identify their signatures in Hi-C proximity maps. Our results reveal the fundamental role of active mechanical processes and hydrodynamic interactions in the kinetics of chromatin compartmentalization and in the emergent large-scale organization of the nucleus.
Prolonged and Pervasive Perturbations in the Composition of the Southern Hemisphere Midlatitude Lower Stratosphere From the Australian New Year’s Fires
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