How are three-dimensional tissue structures generated during development, and how are conserved organ forms maintained despite substantial variation in embryonic size and developmental tempo across species? Addressing these questions requires both theoretical frameworks for describing tissue mechanics and comparative analyses of morphogenesis in diverse organisms. In this talk, I will discuss the development of a 3D vertex model for analyzing the mechanics of epithelial tissues in three dimensions, with emphasis on the representation of individual cell shapes, force balance, and tissue dynamics in 3D during morphogenesis. I will also present comparative studies of vertebrate inner ear development, which examine how differences in morphogenetic mechanics among species can nevertheless lead to conserved morphological outcomes.

The symbiotic exchange of nutrients between arbuscular mycorrhizal (AM) fungi and their plant hosts underlies growth of terrestrial vegetation that makes up ~85% of the Earth’s biomass, and funnels several Gt of carbon into soil environments each year. Yet despite their importance for ecosystem function and carbon cycling at the planetary scale, little is known about the mechanisms that enable these fungal trade and transport behaviors. AM fungi grow as networks of hyphal filaments. Each of these hyphal tubes are open conduits for fluid flows carrying nutrients both toward and away from host roots. We have been building bespoke imaging robots to map the full network topology of growing fungal networks, as well as internal flow dynamics. I will describe some of the insights gained from these efforts so far – and highlight open questions – about how an organism that functions as a ‘connective tissue’ for much of life on Earth efficiently explores, moves and exchanges nutrients to support a vast underground market for biological resource trade.

Many living and artificial systems improve their fitness or performance by adapting to changing environments or diverse training data. However, it remains unclear how environmental variation shapes adaptation, what is learned, and when memory of past conditions is retained. I will discuss recent results that focus on trainable particle packings that can be inverse-designed to control various material properties on demand. Specifically, I will show that cyclic training, e.g. repeatedly designing the Poisson’s ratio between two values, leads to a return point memory that encodes the range over which the system was trained, reminiscent of the return point memory observed in cyclically sheared systems. Importantly, we are able to establish the precise learning mechanism by which memory is achieved, which we call Gradient Discontinuity Learning. This mechanism is broadly applicable with minimal requirements, and I will speculate on its potential to explain and control learning behaviors in a range of systems, from evolution to machine learning to the brain.

Our bodies and organs are thoroughly covered by epithelial sheets, which exhibit a layered structure, either monolayer or multilayer, corresponding to their function. For example, the intestine has monolayer structure to secrete body fluids, the skin has multilayer structure to protect the body from the outside. These multilayered structures are typically formed from a monolayer to multilayer through a stratification. Although many key molecules and cell behaviors have been elucidated, it remains unclear what common principles govern the layered structure of various epithelia. To unveil these mechanisms, we performed numerical simulations using a three-dimensional vertex model. As a result, epithelial structures were maintained as a monolayer or transitioned to a multilayer depending on mechanical behavior of cells. To understand this transition, we developed a theoretical model by focusing on the deformations of a single cell and its adjacent cells within a planar monolayer. These analyses show that mechanical instability inherent in the foam-like epithelial geometry causes cell delamination. The accumulation of cell delaminations leads to the structural transition from monolayer to multilayer. In this presentation, we will discuss mechanisms that regulate epithelial layer structures.

Epithelial folding is a fundamental process in organogenesis, progressing the sequential transformation of epithelial sheets into complex 3D organ structures. A key feature of this folding is its irreversibility which ensures the stabilization of tissue structure. However, the mechanism underlying this folding irreversibility remains poorly understood. Here, we identify a mechanical property of epithelia that determines folding irreversibility. Using a mechanical indentation assay, we quantitatively characterized the elastoplastic responses of epithelial tissues. We find that short-term or low-curvature folding induces an elastic, shape-restoring response, while long-term and high-curvature folding leads to plastic, irreversible deformation. This elastic-to-plastic transition occurs in a switch-like manner with critical thresholds of both curvature and duration time. This elastoplastic behavior of epithelial tissues may provide a mechanism to stabilize appropriate tissue morphology during embryogenesis. Our finding offers new insights into the fundamental mechanics underlying morphogenesis.

We consider the problem of approximating the Langevin dynamics of inertial particles being transported by a background flow. In particular, we study an acceleration corrected advection-diffusion approximation to the Langevin dynamics under the averaging parametric regime. We prove local time error estimates in the strong and weak sense, whose optimality is checked against computational experiments. Furthermore, our numerical evidence suggests that this approximation also captures the long-time behavior of the Langevin dynamics from which non-uniform patterns emerge.

Biological tissues undergo complex shape changes during embryonic development as a result of both autonomous cell behaviors and mechanical inputs from neighboring tissues. Combinations of in-plane and out-of-plane deformations make a complete and biologically interpretable kinematic description of development challenging. I will describe two problems involving shape symmetry-breaking of biological tissues. First, I will show a quantitative framework to extract and analyze 3D time-dependent strain tensor fields from tracked nuclei in the embryonic fruit fly hindgut primordium, a ring of tissue that rapidly deforms into a complicated 3D structure. We find that the hindgut undergoes spatially heterogeneous and temporally ordered deformations, with adjacent regions exhibiting coordinated out-of-plane and circumferential strain. This approach reveals the hindgut primordium to be a tissue with rich strain patterns and provides a generalizable method for interpreting complex tissue deformations in both in vivo and engineered systems. Second, I will describe how we are using reductionist organoid models to understand how the human kidney collecting ducts undergo repeated bifurcations to yield a branched structure. The organoids begin as spheres in a uniform bath of growth factors and appear to emergently break shape symmetry to yield buds and branches. Together, these problems illustrate the variety of mechanisms driving even simple shape changes of tissues during development.

Lipid bilayer membranes with open boundaries arise in many biophysical processes, including pore formation, vesicle rupture, and parasite egress. In this talk, I present a mathematical and computational framework for simulating their dynamics in viscous Stokes flow. The model couples a three-dimensional bulk fluid, a two-dimensional membrane surface, and a one-dimensional free edge via an energy variational formulation. Under axisymmetry, the system reduces to a one-dimensional mixed dimensional boundary integral problem. We develop a hybrid boundary element–finite element method with local mesh refinement to resolve edge singularities. To better understand the solution structure, we also study a simplified elliptic problem with open boundaries, establishing well posedness and characterizing the edge singularity. Numerical results demonstrate accurate resolution of multiscale fluid–membrane interactions and edge dynamics.

The encapsulation of active particles, such as bacteria or active colloids, inside a droplet gives rise to nontrivial shape dynamics and droplet motility. To understand the fluid-mediated coupling between the deformable interface and the active particle, we derive asymptotic solutions in the regime of small shape deformation for the fluid flow about a deformable Stokes droplet containing a point-singularity model of an active particle. Off-centering of the active particle from the drop’s center leads to the excitation of numerous shape modes and, in some cases, net droplet motion. Flows generated by common singularity representations, such as Stokeslets, rotlets, and stresslets, are computed and compared to results for non-deformable droplets. Additionally, we use the reciprocal theorem to determine the drop velocity for an active particle subject to arbitrary surface rheology and enclosed singularity. Finally, we explore the flow in two-dimensional Stokes drops and Hele-Shaw drops enclosing active particles. These results provide insight into the flows generated by active particles in soft confinement and the conditions under which container motion occurs. Such idealized systems serve as a starting point for applications ranging from the design of targeted drug delivery machines to understanding the collective motion of biological swimmers in confinement.

Many physical systems—such as optical waveguide lattices and dense neuronal or vascular networks—can be modeled by metric graphs, where slender “wires” (edges) support wave or diffusion equations subject to Kirchhoff boundary conditions at the nodes. We propose a continuum-limit framework which replaces edgewise differential equations with a coarse-grained partial differential equation defined on the continuous space occupied by the network. The derivation naturally introduces an edge-conductivity tensor, an edge-capacity function, and a vertex number density to encode how each microscopic patch of the graph contributes to the macroscopic phenomena. These results have interesting similarities and differences with the Riemannian Laplace-Beltrami operator. We calculate all macroscopic parameters from first principles via a systematic discrete-to-continuous local homogenization, finding an anomalous effective embedding dimension resulting from a homogenized diffusivity. Numerical examples—including periodic lattices, random graphs, and quasiperiodic monotiles—demonstrate that each finite model converges to its corresponding PDE (posed on different manifolds like tori, disks, and spheres) in the limit of increasing vertex density. These high-density networks encode emergent material and functional properties. They reflect the ability of many real-world, space-filling networks to function simultaneously at multiple scales using the continuum as a feature.

Moving Boundary Problems (MBPs) are ubiquitous in modeling complex phenomena across physics, materials science, and mathematical biology, such as grain boundary coarsening, solid-state dewetting, and the shape transformations of lipid bilayer biomembranes. Mathematically, these processes are formulated as geometric gradient flows that minimize specific energy functionals (e.g., perimeter or bending elasticity) with respect to physical metrics like $L^2$ or $H^{-1}$, often subject to global non-linear constraints like volume preservation. However, solving these high-order, severely non-linear geometric PDEs using Parametric Finite Element Methods (PFEM) presents a profound computational dilemma. While explicit parametric tracking offers superior geometric accuracy, it inevitably suffers from severe mesh distortion during large deformations. Traditional ad-hoc mesh regularization techniques fundamentally destroy the exact energy dissipation laws, leading to numerical instability. In this talk, we introduce a novel Unified Stabilized Dual-SAV (Scalar Auxiliary Variable) Framework to definitively resolve this dilemma. By conceptually elevating the artificial mesh regularization to an independent “thermodynamic” energy, we construct a generalized system that orthogonally decouples the physical normal flow from the tangential mesh optimization. We rigorously prove that this system unconditionally dissipates both energies simultaneously and independently (Simultaneous Orthogonal Dissipation). Furthermore, we present a Universal Algebraic Block Reduction strategy. We demonstrate how to strictly isolate massive non-linear global constraints into a microscopic scalar root-finding problem, reducing the computational overhead to virtually zero. The framework will be validated through challenging numerical experiments, including the 4th-order Curve Diffusion flow and the fully constrained Helfrich flow.

The intracellular environment is highly crowded, with macromolecules occupying up to 20–30% of the available volume, rendering passive diffusion inefficient for long-range transport. Cells overcome this limitation through cytoplasmic streaming, in which molecular motors moving along cytoskeletal filaments generate forces that drive large-scale fluid flow.

We develop a numerical framework for a coupled system in which motors undergo advection–diffusion in the bulk and attachment–detachment dynamics at the cell boundary. Using this model, we investigate the emergence of metastable flow states and demonstrate how perturbations in the concentration of attached motors can trigger transitions between them.

Physical computing exploits unconventional physical substrates to overcome limitations such as the high energy consumption inherent in digital computation. However, intrinsic noise and temporal fluctuations (e.g., oscillations) generally deteriorate computational performance. Here, we propose ensemble reservoir computing (ERC), a novel framework that employs ensemble averaging of spatially multiplexed systems to achieve robust information processing despite noise and temporal fluctuations. First, we prove that ensemble averaging in ERC eliminates temporal fluctuations and noise from dynamical states under certain conditions, thereby restoring computational performance to its noise-free level. Next, we show that ERC not only removes the noise and fluctuations but also actively exploits the computational capabilities that conventional reservoir computing (RC) leaves unutilized. This computational enhancement is demonstrated across diverse dynamical systems (e.g., periodic, chaotic, and strange- nonchaotic systems), in which ERC outperforms conventional RC. Finally, using energy-efficient spin-torque oscillators (STOs), we demonstrate that ERC maintains high performance even under realistic conditions, in which noise and temporal fluctuations coexist: STOs with ERC achieved 99% accuracy on an error detection test, where conventional STO reservoir with linear regression only shows a chance level performance, highlighting ERC’s robustness and performance gains for physical systems.

Simulations of stochastic neuron potential models, which describe the voltage potential along the length of a neuron’s axon and incorporate ion channel noise as Gaussian fluctuations, have shown that channel noise can induce complex phenomena such as jitters and splitting of action potentials [1] and place constraints on the miniaturization of axons [2]. To develop a robust analytic framework for understanding stochastic effects of channel noise on action potential propagation in a neuron, we need to begin by investigating how many, independent and spatially distributed ion channels can collectively yield deterministic behavior. We start with an electrophysiological derivation of a simple discrete model and contrast this with a common, yet less physically accurate approach where the law of large numbers and the central limit theorem are more easily applied. Our model couples a spatially discretized diffusive PDE for the voltage with continuous-time Markov processes that govern the behavior of the ion channels. We will then outline an argument using homogenization theory to estimate the rate of strong convergence to the typical deterministic PDE as the spacing between ion channels approaches zero. Finally, we present a numerical technique for simulating our model and discuss the challenges involved in increasing computational efficiency of simulations.

[1] Faisal AA, Laughlin SB. Stochastic simulations on the reliability of action potential propagation in thin axons. PLoS Comput Biol. 2007 May;3(5):e79. doi: 10.1371/journal.pcbi.0030079. PMID: 17480115; PMCID: PMC1864994.

[2] Faisal AA, White JA, Laughlin SB. Ion-channel noise places limits on the miniaturization of the brain’s wiring. Curr Biol. 2005 Jun 21;15(12):1143-9. doi: 10.1016/j.cub.2005.05.056. PMID: 15964281.

Adaptive moving mesh methods are highly effective for achieving high-accuracy numerical computations. However, their error analysis is nontrivial, since the movement of the mesh induces time dependence in the interpolation operator, which requires careful treatment when interchanging differentiation in time and integration in space. In this talk, we present a framework for Lagrangian finite element schemes with moving meshes for convection–diffusion problems. We introduce a concrete moving mesh scheme for the one-dimensional case and discuss its fundamental properties together with rigorous optimal error estimates. We also briefly address extensions to multidimensional problems. Numerical experiments demonstrate that the proposed approach can accurately capture sharp spike-type solutions.