Publications

We introduce a fast direct solver for variable-coefficient elliptic PDEs on surfaces based on the hierarchical Poincaré–Steklov method. The method takes as input an unstructured, high-order quadrilateral mesh of a surface and discretizes surface differential operators on each element using a high-order spectral collocation scheme. Elemental solution operators and Dirichlet-to-Neumann maps tangent to the surface are precomputed and merged in a pairwise fashion to yield a hierarchy of solution operators that may be applied in \(\mathcal{O}(N \log N)\) operations for a mesh with \(N\) degrees of freedom. The resulting fast direct solver may be used to accelerate high-order implicit time-stepping schemes, as the precomputed operators can be reused for fast elliptic solves on surfaces. On a standard laptop, precomputation for a 12th-order surface mesh with over 1 million degrees of freedom takes 10 seconds, while subsequent solves take only 0.25 seconds. We apply the method to a range of problems on both smooth surfaces and surfaces with sharp corners and edges, including the static Laplace–Beltrami problem, the Hodge decomposition of a tangential vector field, and some time-dependent nonlinear reaction-diffusion systems. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: code and data available”, as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/danfortunato/surface-hps-sisc.

Few techniques are available for studying the nature of forces that drive subcellular dynamics. Here we develop two complementary ones. The first is femtosecond stereotactic laser ablation, which rapidly creates complex cuts of subcellular structures and enables precise dissection of when, where and in what direction forces are generated. The second is an assessment of subcellular fluid flows by comparison of direct flow measurements using microinjected fluorescent nanodiamonds with large-scale fluid-structure simulations of different force transduction models. We apply these techniques to study spindle and centrosome positioning in early Caenorhabditis elegans embryos and to probe the contributions of microtubule pushing, cytoplasmic pulling and cortical pulling upon centrosomal microtubules. Based on our results, we construct a biophysical model to explain the dynamics of centrosomes. We demonstrate that cortical pulling forces provide a general explanation for many behaviours mediated by centrosomes, including pronuclear migration and centration, rotation, metaphase spindle positioning, asymmetric spindle elongation and spindle oscillations. This work establishes methodologies for disentangling the forces responsible for cell biological phenomena.

Contaminants and other agents are often present at the interface between two fluids, giving rise to rheological properties such as surface shear and dilatational viscosities. The dynamics of viscous drops with interfacial viscosities has attracted greater interest in recent years, due to the influence of surface rheology on deformation and the surrounding flows. We investigate the effects of shear and dilatational viscosities on the electro-deformation of a viscous drop using the Taylor–Melcher leaky dielectric model. We use a large deformation analysis to derive an ordinary differential equation for the drop shape. Our model elucidates the contributions of each force to the overall deformation of the drop and reveals a rich range of dynamic behaviors that show the effects of surface viscosities and their dependence on rheological and electrical properties of the system. We also examine the physical mechanisms underlying the observed behaviors by analyzing the surface dilatation and surface deformation.

The efficient transport of fluids through disordered media requires a thorough understanding of how the driving rate affects two-phase interface propagation. Despite our understanding of front dynamics in homogeneous environments, as well as how medium heterogeneities shape fluid interfaces at rest, little is known about the effects of localized topographical variations on large-scale interface dynamics. To gain physical insights into this problem, we study here oil-air displacements through an “imperfect” Hele-Shaw cell. Combining experiments, numerical simulations, and theory, we show that the flow rate dramatically alters the interface response to a porous constriction as one approaches the Saffman-Taylor instability, strictly under stable conditions. This gives rise to asymmetric imbibition–drainage hysteresis cycles that feature divergent extensions and nonlocal effects, all of which are aptly captured and explained by a minimal free boundary model.

Finely-tuned enzymatic pathways control cellular processes, and their dysregulation can lead to disease. Creating predictive and interpretable models for these pathways is challenging because of the complexity of the pathways and of the cellular and genomic contexts. Here we introduce Elektrum, a deep learning framework which addresses these challenges with data-driven and biophysically interpretable models for determining the kinetics of biochemical systems. First, it uses in vitro kinetic assays to rapidly hypothesize an ensemble of high-quality Kinetically Interpretable Neural Networks (KINNs) that predict reaction rates. It then employs a novel transfer learning step, where the KINNs are inserted as intermediary layers into deeper convolutional neural networks, fine-tuning the predictions for reaction-dependent in vivo outcomes. Elektrum makes effective use of the limited, but clean in vitro data and the complex, yet plentiful in vivo data that captures cellular context. We apply Elektrum to predict CRISPR-Cas9 off-target editing probabilities and demonstrate that Elektrum achieves state-of-the-art performance, regularizes neural network architectures, and maintains physical interpretability

Acute kidney injury (AKI) is a major risk factor for long-term adverse outcomes, including chronic kidney disease. In mouse models of AKI, maladaptive repair of the injured proximal tubule (PT) prevents complete tissue recovery. However, evidence for PT maladaptation and its etiological relationship with complications of AKI is lacking in humans. We performed single-nucleus RNA sequencing of 120,985 nuclei in kidneys from 17 participants with AKI and seven healthy controls from the Kidney Precision Medicine Project. Maladaptive PT cells, which exhibited transcriptomic features of dedifferentiation and enrichment in pro-inflammatory and profibrotic pathways, were present in participants with AKI of diverse etiologies. To develop plasma markers of PT maladaptation, we analyzed the plasma proteome in two independent cohorts of patients undergoing cardiac surgery and a cohort of marathon runners, linked it to the transcriptomic signatures associated with maladaptive PT, and identified nine proteins whose genes were specifically up- or down-regulated by maladaptive PT. After cardiac surgery, both cohorts of patients had increased transforming growth factor–β2 (TGFB2), collagen type XXIII-α1 (COL23A1), and X-linked neuroligin 4 (NLGN4X) and had decreased plasminogen (PLG), ectonucleotide pyrophosphatase/phosphodiesterase 6 (ENPP6), and protein C (PROC). Similar changes were observed in marathon runners with exercise-associated kidney injury. Postoperative changes in these markers were associated with AKI progression in adults after cardiac surgery and post-AKI kidney atrophy in mouse models of ischemia-reperfusion injury and toxic injury. Our results demonstrate the feasibility of a multiomics approach to discovering noninvasive markers and associating PT maladaptation with adverse clinical outcomes.

Soft materials are usually defined as materials made of mesoscopic entities, often self-organised, sensitive to thermal fluctuations and to weak perturbations. Archetypal examples are colloids, polymers, amphiphiles, liquid crystals, foams. The importance of soft materials in everyday commodity products, as well as in technological applications, is enormous, and controlling or improving their properties is the focus of many efforts. From a fundamental perspective, the possibility of manipulating soft material properties, by tuning interactions between constituents and by applying external perturbations, gives rise to an almost unlimited variety in physical properties. Together with the relative ease to observe and characterise them, this renders soft matter systems powerful model systems to investigate statistical physics phenomena, many of them relevant as well to hard condensed matter systems. Understanding the emerging properties from mesoscale constituents still poses enormous challenges, which have stimulated a wealth of new experimental approaches, including the synthesis of new systems with, e.g. tailored self-assembling properties, or novel experimental techniques in imaging, scattering or rheology. Theoretical and numerical methods, and coarse-grained models, have become central to predict physical properties of soft materials, while computational approaches that also use machine learning tools are playing a progressively major role in many investigations. This Roadmap intends to give a broad overview of recent and possible future activities in the field of soft materials, with experts covering various developments and challenges in material synthesis and characterisation, instrumental, simulation and theoretical methods as well as general concepts.

Here we introduce a variation of the trap model of supercooled liquids based on softness, a particle-based variable identified by machine learning that quantifies the local structural environment and energy barrier for the particle to rearrange. As in the trap model, we assume that each particle's softness, and hence energy barrier, evolves independently. We show that our model makes qualitatively reasonable predictions of behaviors such as the dependence of fragility on density in a model supercooled liquid. We also show failures of the model, indicating in some cases signs that softness may be missing important information, and in other cases features that may only be explained by correlations neglected in the trap model.

Fluid-structure interactions between active and passive components are important for many biological systems to function. A particular example is chromatin in the cell nucleus, where ATP-powered processes drive coherent motions of the chromatin fiber over micron lengths. Motivated by this system, we develop a multiscale model of a long flexible polymer immersed in a suspension of active force dipoles as an analog to a chromatin fiber in an active fluid – the nucleoplasm. Linear analysis identifies an orientational instability driven by hydrodynamic and alignment interactions between the fiber and the suspension, and numerical simulations show activity can drive coherent motions and structured conformations. These results demonstrate how active and passive components, connected through fluid-structure interactions, can generate coherent structures and self-organize on large scales.

This work addresses the question of uniqueness of the minimizers of a convex but not strictly convex integral functional with linear growth in a two-dimensional setting. The integrand -- whose precise form derives directly from the theory of perfect plasticity -- behaves quadratically close to the origin and grows linearly once a specific threshold is reached. Thus, in contrast with the only existing literature on uniqueness for functionals with linear growth, that is that which pertains to the generalized least gradient, the integrand is not a norm. We make use of hyperbolic conservation laws hidden in the structure of the problem to tackle uniqueness. Our argument strongly relies on the regularity of a vector field -- the Cauchy stress in the terminology of perfect plasticity -- which allows us to define characteristic lines, and then to employ the method of characteristics. Using the detailed structure of the characteristic landscape evidenced in our preliminary study \cite{BF}, we show that this vector field is actually continuous, save for possibly two points. The different behaviors of the energy density at zero and at infinity imply an inequality constraint on the Cauchy stress. Under a barrier type convexity assumption on the set where the inequality constraint is saturated, we show that uniqueness holds for pure Dirichlet boundary data, a stronger result than that of uniqueness for a given trace on the whole boundary since our minimizers can fail to attain the boundary data.