Solids dispersed in a drying drop will migrate to the edge of the drop and form a solid ring. This phenomenon produces ringlike stains and occurs for a wide range of surfaces, solvents, and solutes. Here we show that the migration is caused by an outward flow within the drop that is driven by the loss of solvent by evaporation and geometrical constraint that the drop maintain an equilibrium droplet shape with a fixed boundary. We describe a theory that predicts the flow velocity, the rate of growth of the ring, and the distribution of solute within the drop. These predictions are compared with our experimental results.

Thin cylindrical tethers are common lipid bilayer membrane structures, arising in situations ranging from micromanipulation experiments on artificial vesicles to the dynamic structure of the Golgi apparatus. We study the shape and formation of a tether in terms of the classical soap-film problem, which is applied to the case of a membrane disk under tension subject to a point force. A tether forms from the elastic boundary layer near the point of application of the force, for sufficiently large displacement. Analytic results for various aspects of the membrane shape are given.

The endoplasmic reticulum (ER) often forms stacked membrane sheets, an arrangement that is likely required to accommodate a maximum of membrane-bound polysomes for secretory protein synthesis. How sheets are stacked is unknown. Here, we used novel staining and automated ultra-thin sectioning electron microscopy methods to analyze stacked ER sheets in neuronal cells and secretory salivary gland cells of mice. Our results show that stacked ER sheets form a continuous membrane system in which the sheets are connected by twisted membrane surfaces with helical edges of left- or right-handedness. The three-dimensional structure of tightly stacked ER sheets resembles a parking garage, in which the different levels are connected by helicoidal ramps. A theoretical model explains the experimental observations and indicates that the structure corresponds to a minimum of elastic energy of sheet edges and surfaces. The structure allows the dense packing of ER sheets in the restricted space of a cell.

We extend elasticity theory of filaments to encompass systems, such as bacterial flagella, that display competition between two helical structures of opposite chirality. A general, fully intrinsic formulation of the dynamics of bend and twist degrees of freedom is developed using the natural frame of space curves, spanning from the inviscid limit to the viscously overdamped regime applicable to cellular biology. Aspects of front propagation found in flagella are discussed.