The Euler characteristic: A general topological descriptor for complex data

Topological Data Analysis for Particulate Gels

Soft gels, formed via the self-assembly of particulate materials, exhibit intricate multiscale structures that provide them with flexibility and resilience when subjected to external stresses. This work combines particle simulations and topological data analysis (TDA) to characterize the complex multiscale structure of soft gels. Our TDA analysis focuses on the use of the Euler characteristic, which is an interpretable and computationally scalable topological descriptor that is combined with filtration operations to obtain information on the geometric (local) and topological (global) structure of soft gels. We reduce the topological information obtained with TDA using principal component analysis (PCA) and show that this provides an informative low-dimensional representation of the gel structure. We use the proposed computational framework to investigate the influence of gel preparation (e.g., quench rate, volume fraction) on soft gel structure and to explore dynamic deformations that emerge under oscillatory shear in various response regimes (linear, nonlinear, and flow). Our analysis provides evidence of the existence of hierarchical structures in soft gels, which are not easily identifiable otherwise. Moreover, our analysis reveals direct correlations between topological changes of the gel structure under deformation and mechanical phenomena distinctive of gel materials, such as stiffening and yielding. In summary, we show that TDA facilitates the mathematical representation, quantification, and analysis of soft gel structures, extending traditional network analysis methods to capture both local and global organization.

DeepIDA-GRU: a deep learning pipeline for integrative discriminant analysis of cross-sectional and longitudinal multiview data with applications to inflammatory bowel disease classification

Biomedical research now commonly integrates diverse data types or views from the same individuals to better understand the pathobiology of complex diseases, but the challenge lies in meaningfully integrating these diverse views. Existing methods often require the same type of data from all views (cross-sectional data only or longitudinal data only) or do not consider any class outcome in the integration method, which presents limitations. To overcome these limitations, we have developed a pipeline that harnesses the power of statistical and deep learning methods to integrate cross-sectional and longitudinal data from multiple sources. In addition, it identifies key variables that contribute to the association between views and the separation between classes, providing deeper biological insights. This pipeline includes variable selection/ranking using linear and nonlinear methods, feature extraction using functional principal component analysis and Euler characteristics, and joint integration and classification using dense feed-forward networks for cross-sectional data and recurrent neural networks for longitudinal data. We applied this pipeline to cross-sectional and longitudinal multiomics data (metagenomics, transcriptomics and metabolomics) from an inflammatory bowel disease (IBD) study and identified microbial pathways, metabolites and genes that discriminate by IBD status, providing information on the etiology of IBD. We conducted simulations to compare the two feature extraction methods.

Design space determination of pharmaceutical processes: Effects of control strategies and uncertainty

The identification of process Design Space (DS) is of high interest in highly regulated industrial sectors, such as pharmaceutical industry, where assurance of manufacturability and product quality is key for process development and decision-making. If the process can be controlled by a set of manipulated variables, the DS can be expanded in comparison to an open-loop scenario, where there are no controls in place. Determining the benefits of control strategies may be challenging, particularly when the available model is complex and computationally expensive - which is typically the case of pharmaceutical manufacturing. In this study, we exploit surrogate-based feasibility analysis to determine whether the process satisfies all process constraints by manipulating the process inputs and reduce the effect of uncertainty. The proposed approach is successfully tested on two simulated pharmaceutical case studies of increasing complexity, i.e., considering (i) a single pharmaceutical unit operation, and (ii) a pharmaceutical manufacturing line comprised of a sequence of connected unit operations. Results demonstrate that different control actions can be effectively exploited to operate the process in a wider range of inputs and mitigate uncertainty.

Decoding Optical Responses of Contact-Printed Arrays of Thermotropic Liquid Crystals Using Machine Learning: Detection and Reporting of Aqueous Amphiphiles with Enhanced Sensitivity and Selectivity

Surfactants and other amphiphilic molecules are used extensively in household products, industrial processes, and biological applications and are also common environmental contaminants; as such, methods that can detect, sense, or quantify them are of great practical relevance. Aqueous emulsions of thermotropic liquid crystals (LCs) can exhibit distinctive optical responses in the presence of surfactants and have thus emerged as sensitive, rapid, and inexpensive sensors or reporters of environmental amphiphiles. However, many existing LC-in-water emulsions require the use of complicated or expensive instrumentation for quantitative characterization owing to variations in optical responses among individual LC droplets. In many cases, the responses of LC droplets are also analyzed by human inspection, which can miss subtle color or topological changes encoded in LC birefringence patterns. Here, we report an LC-based surfactant sensing platform that takes a step toward addressing several of these issues and can reliably predict concentrations and types of surfactants in aqueous solutions. Our approach uses surface-immobilized, microcontact-printed arrays of micrometer-scale droplets of thermotropic LCs and hierarchical convolutional neural networks (CNNs) to automatically extract and decode rich information about topological defects and color patterns available in optical micrographs of LC droplets to classify and quantify adsorbed surfactants. In addition, we report computational capabilities to determine relevant optical features extracted by the CNN from LC micrographs, which can provide insights into surfactant adsorption phenomena at LC-water interfaces. Overall, the combination of microcontact-printed LC arrays and machine learning provides a convenient and robust platform that could prove useful for developing high-throughput sensors for on-site testing of environmentally or biologically relevant amphiphiles.

A graph-based modeling framework for tracing hydrological pollutant transport in surface waters

Anthropogenic pollution of hydrological systems affects diverse communities and ecosystems around the world. Data analytics and modeling tools play a key role in fighting this challenge, as they can help identify key sources as well as trace transport and quantify impact within complex hydrological systems. Several tools exist for simulating and tracing pollutant transport throughout surface waters using detailed physical models; these tools are powerful, but can be computationally intensive, require significant amounts of data to be developed, and require expert knowledge for their use (ultimately limiting application scope). In this work, we present a graph modeling framework - which we call HydroGraphs - for understanding pollutant transport and fate across waterbodies, rivers, and watersheds. This framework uses a simplified representation of hydrological systems that can be constructed based purely on open-source data (National Hydrography Dataset and Watershed Boundary Dataset). The graph representation provides a flexible intuitive approach for capturing connectivity and for identifying upstream pollutant sources and for tracing downstream impacts within small and large hydrological systems. Moreover, the graph representation can facilitate the use of advanced algorithms and tools of graph theory, topology, optimization, and machine learning to aid data analytics and decision-making. We demonstrate the capabilities of our framework by using case studies in the State of Wisconsin; here, we aim to identify upstream nutrient pollutant sources that arise from agricultural practices and trace downstream impacts to waterbodies, rivers, and streams. Our tool ultimately seeks to help stakeholders design effective pollution prevention/mitigation practices and evaluate how surface waters respond to such practices.

Topological transformations of a nematic drop

Morphogenesis of living systems involves topological shape transformations which are highly unusual in the inanimate world. Here, we demonstrate that a droplet of a nematic liquid crystal changes its equilibrium shape from a simply connected tactoid, which is topologically equivalent to a sphere, to a torus, which is not simply connected. The topological shape transformation is caused by the interplay of nematic elastic constants, which facilitates splay and bend of molecular orientations in tactoids but hinders splay in the toroids. The elastic anisotropy mechanism might be helpful in understanding topology transformations in morphogenesis and paves the way to control and transform shapes of droplets of liquid crystals and related soft materials.

Topological Analysis of Molecular Dynamics Simulations using the Euler Characteristic

Molecular dynamics (MD) simulations are used in diverse scientific and engineering fields such as drug discovery, materials design, separations, biological systems, and reaction engineering. These simulations generate highly complex data sets that capture the 3D spatial positions, dynamics, and interactions of thousands of molecules. Analyzing MD data sets is key for understanding and predicting emergent phenomena and in identifying key drivers and tuning design knobs of such phenomena. In this work, we show that the Euler characteristic (EC) provides an effective topological descriptor that facilitates MD analysis. The EC is a versatile, low-dimensional, and easy-to-interpret descriptor that can be used to reduce, analyze, and quantify complex data objects that are represented as graphs/networks, manifolds/functions, and point clouds. Specifically, we show that the EC is an informative descriptor that can be used for machine learning and data analysis tasks such as classification, visualization, and regression. We demonstrate the benefits of the proposed approach through case studies that aim to understand and predict the hydrophobicity of self-assembled monolayers and the reactivity of complex solvent environments.

Strain-Driven Mixed-Phase Domain Architectures and Topological Transitions in Pb1- x Srx TiO3 Thin Films

The potential for creating hierarchical domain structures, or mixtures of energetically degenerate phases with distinct patterns that can be modified continually, in ferroelectric thin films offers a pathway to control their mesoscale structure beyond lattice-mismatch strain with a substrate. Here, it is demonstrated that varying the strontium content provides deterministic strain-driven control of hierarchical domain structures in Pb_{1- x} Sr_x TiO₃  solid-solution thin films wherein two types, c/a and a₁ /a₂ , of nanodomains can coexist. Combining phase-field simulations, epitaxial thin-film growth, detailed structural, domain, and physical-property characterization, it is observed that the system undergoes a gradual transformation (with increasing strontium content) from droplet-like a₁ /a₂  domains in a c/a domain matrix, to a connected-labyrinth geometry of c/a domains, to a disconnected labyrinth structure of the same, and, finally, to droplet-like c/a domains in an a₁ /a₂  domain matrix. A relationship between the different mixed-phase modulation patterns and its topological nature is established. Annealing the connected-labyrinth structure leads to domain coarsening forming distinctive regions of parallel c/a and a₁ /a₂  domain stripes, offering additional design flexibility. Finally, it is found that the connected-labyrinth domain patterns exhibit the highest dielectric permittivity.