Transfer of polarity information via diffusion of Wnt ligands in C. elegans embryos.

Different signaling mechanisms concur to ensure robust tissue patterning and cell fate instruction during animal development. Most of these mechanisms rely on signaling proteins that are produced, transported, and detected. The spatiotemporal dynamics of signaling molecules are largely unknown, yet they determine signal activity's spatial range and time frame. Here, we use the Caenorhabditis elegans embryo to study how Wnt ligands, an evolutionarily conserved family of signaling proteins, dynamically organize to establish cell polarity in a developing tissue. We identify how Wnt ligands, produced in the posterior half of the embryos, spread extracellularly to transmit information to distant target cells in the anterior half. With quantitative live imaging and fluorescence correlation spectroscopy, we show that Wnt ligands diffuse through the embryo over a timescale shorter than the cell cycle, in the intercellular space, and outside the tissue below the eggshell. We extracted diffusion coefficients of Wnt ligands and their receptor Frizzled and characterized their co-localization. Integrating our different measurements and observations in a simple computational framework, we show how fast diffusion in the embryo can polarize individual cells through a time integration of the arrival of the ligands at the target cells. The polarity established at the tissue level by a posterior Wnt source can be transferred to the cellular level. Our results support a diffusion-based long-range Wnt signaling, which is consistent with the dynamics of developing processes.

Curvature Heterogeneities Act as Singular Perturbations to Smooth Laplacian Fields: A Fluid Mechanics Demonstration.

In this Letter, we use a model fluid mechanics experiment to elucidate the impact of curvature heterogeneities on two-dimensional fields deriving from harmonic potential functions. This result is directly relevant to explain the smooth stationary structures in physical systems as diverse as curved liquid crystal and magnetic films, heat and Ohmic transport in wrinkled two-dimensional materials, and flows in confined channels. Combining microfluidic experiments and theory, we explain how curvature heterogeneities shape confined viscous flows. We show that isotropic bumps induce local distortions to Darcy's flows, whereas anisotropic curvature heterogeneities disturb them algebraically over system-spanning scales. Thanks to an electrostatic analogy, we gain insight into this singular geometric perturbation, and quantitatively explain it using both conformal mapping and numerical simulations. Altogether, our findings establish the robustness of our experimental observations and their broad relevance to all Laplacian problems beyond the specifics of our fluid mechanics experiment.

Polyhedral Bubble Vibrations.

Underwater bubbles are extremely good acoustic resonators, but are freely evolving and dissolving. Recently it was found that bubbles can be stabilized in frames, but the influence of the frame shape is still undocumented. Here we first explore the vibration of polyhedral bubbles with a low number of faces, shaped as the five Platonic solids. Their resonance frequency is well approximated by the formula for spherical bubbles with the same volume. Then we extend these results to shapes with a larger number of faces using fullerenes, paving the way to obtain arbitrary large resonant bubbles.