G-family polynomials

We introduce two notions of quandle polynomials for [Formula: see text]-families of quandles: the quandle polynomial of the associated quandle and a [Formula: see text]-family polynomial with coefficients in the group ring of [Formula: see text]. As an application we define image subquandle polynomial enhancements of the [Formula: see text]-family counting invariant for trivalent spatial graphs and handlebody-links. We provide examples to show that the new enhancements are proper.

Sarc-Graph: Automated segmentation, tracking, and analysis of sarcomeres in hiPSC-derived cardiomyocytes

A better fundamental understanding of human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs) has the potential to advance applications ranging from drug discovery to cardiac repair. Automated quantitative analysis of beating hiPSC-CMs is an important and fast developing component of the hiPSC-CM research pipeline. Here we introduce “Sarc-Graph,” a computational framework to segment, track, and analyze sarcomeres in fluorescently tagged hiPSC-CMs. Our framework includes functions to segment z-discs and sarcomeres, track z-discs and sarcomeres in beating cells, and perform automated spatiotemporal analysis and data visualization. In addition to reporting good performance for sarcomere segmentation and tracking with little to no parameter tuning and a short runtime, we introduce two novel analysis approaches. First, we construct spatial graphs where z-discs correspond to nodes and sarcomeres correspond to edges. This makes measuring the network distance between each sarcomere (i.e., the number of connecting sarcomeres separating each sarcomere pair) straightforward. Second, we treat tracked and segmented components as fiducial markers and use them to compute the approximate deformation gradient of the entire tracked population. This represents a new quantitative descriptor of hiPSC-CM function. We showcase and validate our approach with both synthetic and experimental movies of beating hiPSC-CMs. By publishing Sarc-Graph, we aim to make automated quantitative analysis of hiPSC-CM behavior more accessible to the broader research community.

Learning cell communication from spatial graphs of cells

Tissue niches are sources of cellular variation and key to understanding both single-cell and tissue phenotypes. The interaction of a cell with its niche can be described through cell communication events. These events cannot be directly observed in molecular profiling assays of single cells and have to be inferred. However, computational models of cell communication and variance attribution defined on data from dissociated tissues suffer from multiple limitations with respect to their ability to define and to identify communication events. We address these limitations using spatial molecular profiling data with node-centric expression modeling (NCEM), a computational method based on graph neural networks which reconciles variance attribution and communication modeling in a single model of tissue niches. We use these models in varying complexity across spatial assays, such as immunohistochemistry and MERFISH, and biological systems to demonstrate that the statistical cell-cell dependencies discovered by NCEM are plausible signatures of known molecular processes underlying cell communication. We identify principles of tissue organisation as cell communication events across multiple datasets using interpretation mechanisms. In the primary motor cortex, we found gene expression variation that is due to niche composition variation across cortical depth. Using the same approach, we also identified niche-dependent cell state variation in CD8 T cells from inflamed colon and colorectal cancer. Finally, we show that NCEMs can be extended to mixed models of explicit cell communication events and latent intrinsic sources of variation in conditional variational autoencoders to yield holistic models of cellular variation in spatial molecular profiling data. Altogether, this graphical model of cellular niches is a step towards understanding emergent tissue phenotypes.

Topology and habitat assortativity drive neutral and adaptive diversification in spatial graphs

Biodiversity results from differentiation mechanisms developing within biological populations. Such mechanisms are influenced by the properties of the landscape over which individuals interact, disperse and evolve. Notably, landscape connectivity and habitat heterogeneity constrain the movement and survival of individuals, thereby promoting differentiation through drift and local adaptation. Nevertheless, the complexity of landscape features can blur our understanding of how they drive differentiation. Here, we formulate a stochastic, eco-evolutionary model where individuals are structured over a graph that captures complex connectivity patterns and accounts for habitat heterogeneity. Individuals possess neutral and adaptive traits, whose divergence results in differentiation at the population level. The modelling framework enables an analytical underpinning of emerging macroscopic properties, which we complement with numerical simulations to investigate how the graph topology and the spatial habitat distribution affect differentiation. We show that in the absence of selection, graphs with high characteristic length and high heterogeneity in degree promote neutral differentiation. Habitat assortativity, a metric that captures habitat spatial auto-correlation in graphs, additionally drives differentiation patterns under habitat-dependent selection. While assortativity systematically amplifies adaptive differentiation, it can foster or depress neutral differentiation depending on the migration regime. By formalising the eco-evolutionary and spatial dynamics of biological populations in complex landscapes, our study establishes the link between landscape features and the emergence of diversification, contributing to a fundamental understanding of the origin of biodiversity gradients.

Alexander polynomial and spanning trees

Inspired by the combinatorial constructions in earlier work of the authors that generalized the classical Alexander polynomial to a large class of spatial graphs with a balanced weight on edges, we show that the value of the Alexander polynomial evaluated at [Formula: see text] gives the weighted number of the spanning trees of the graph.

Palettes of Dehn colorings for spatial graphs and the classification of vertex conditions

In this paper, we study Dehn colorings of spatial graph diagrams, and classify the vertex conditions, equivalently the palettes. We give some example of spatial graphs which can be distinguished by the number of Dehn colorings with selecting an appropriate palette. Furthermore, we also discuss the generalized version of palettes, which is defined for knot-theoretic ternary-quasigroups and region colorings of spatial graph diagrams.