Description of cellular patterns by Dirichlet domains: the two-dimensional case

The membrane domains concentrate specific lipids and proteins at one place within the membrane, which helps in cell signaling, adhesion, and other critical cellular processes. These domains can differ in size, composition, function, and lifespan.
Protein Domains
The membrane comprises a group of distinct proteins responsible for carrying out a cell's specific function. For example, the plasma membrane of the human sperm, or a single germ cell, contains a unique set of proteins in the anterior...

Different physical properties of lipids and proteins allow them to localize and form distinct islands or domains in the membrane. Some membrane domains are formed due to protein-protein interactions, whereas others are formed due to the presence of specific lipids such as sphingolipids and sterols—for example, large proteins, such as bacteriorhodopsin, aggregate and create distinct domains.
Another mechanism for membrane domain formation involves membrane proteins interacting with cytoskeletal...

The simplest case of a surface charge distribution is the uniformly charged disk. Calculating its electric field also helps us calculate the electric field of a large plane of charge.
The system's symmetry is in the cylindrical directions across the plane of the charge. As a result, the electric fields created by various surface charge elements nullify each other in the direction parallel to the surface. Thereby, the resulting electric field is perpendicular to the plane. Since the disk is...

Quadric surfaces are three-dimensional surfaces characterized by second-degree equations in the variables x, y, and z. These surfaces are smooth and continuous, and specific combinations of squared and linear terms define their shapes. The main types of quadric surfaces include ellipsoids, cones, paraboloids, and hyperboloids. Each type exhibits distinct geometric features depending on how the variables are arranged and related within the equation.Ellipsoids are closed surfaces formed when all...

Double integrals are often used to measure quantities distributed across two-dimensional regions, such as rainfall over a lake, heat across a metal plate, or population density over land. In many practical situations, the region of interest does not have straight boundaries and cannot be described conveniently as a rectangle. Instead, the region may have curved or irregular edges. To evaluate integrals over such domains, the region is embedded inside a larger rectangular region where...

In vector calculus, flux measures the total flow of a vector field through a surface. For a closed surface in three-dimensional space, this means measuring how much of the field passes outward through every point on the boundary. Directly calculating this flux can be difficult when the surface has a complicated or irregular shape. The Divergence Theorem provides a powerful alternative by relating surface flux to behavior inside the enclosed region.The Divergence Theorem states that the outward...