The Adhesion Of Two Cylindrical Colloids To A Tubular Membrane

As the important component of cells, membranes play a crucial role in bothstructure and function of cells. Most of the fundamental life processes in cells involvemembranes at some point, including processes as protein biosynthesis and secretion,DNA replication, and hormonal responses. Many biological processes are controlled bythe interaction of macromolecules with a cell membrane, such as endocytosis andexocytosis. The research on the membrane has long been the focus of biophysics. It isnow experimentally possible to extract freely standing lipid-bilayer membranes in manyinteresting shapes, which can be used as test beds for understanding importantbiophysical processes. As an idealized model, the interaction of colloids and membraneshas been widely used to provide fundamental understanding of the physical propertiesof biological systems. Thus, the study of the colloids interacting with membranes issignificant, and the Helfrich model provides a theoretical basis for the investigation ofthe interaction between membrane and colloidal particle.Firstly, we introduce the basic theory of the Helfrich model. It includes thegeometric model and energy model, such as angle-arclength parameterization, thebending energy, surface tension energy and adsorption energy. Then taking thecylindrical colloids-flat membrane interaction system as an example, we show how tocalculate the free energy and the stable structures with Helfrich model. For complexsystems, we introduce the treatment of the canonical equations, and HelfrichHamiltonian dynamics can be obtained, then the problem of energy minimization isconverted to the problem of integral minimum. The solution of differential equationscan be processed by the shooting method.With the full treatment of Helfrich model we theoretically study the symmetricaladhesion of two cylindrical colloids to a tubular membrane. The free energy of thesystem is described by the energy parameters such as the surface tension, bending rigidity and adsorption energy, and geometric parameters. Minimizing the obtained freeenergy with respect to the geometric parameters, we can obtain the stable structures ofthe system. The adhesion of the rigid cylinders with different radius to the membranetube surface can produce both shallow wrapping with relatively small wrapping angleand deep wrapping with big wrapping angle. These significant structural behaviors canbe obtained by analyzing the system energy. A second order adhesion transition fromthe desorbed to weakly adhered states is found, and a first order phase transition wherethe cylindrical colloids undergoes an abrupt transition from weakly adhered to stronglyadhered states can be obtained as well.For the asymmetric adhesion of two cylindrical colloids to a tubular membrane, wealso use Helfrich model to study. it is found that this adhesion can produce both shallowwrapping with relatively small wrapping angle and deep wrapping with big wrappingangle. A second order adhesion transition from the desorbed to weakly adhered states isfound, and a first order phase transition where the cylindrical colloids undergoes anabrupt transition from weakly adhered to strongly adhered states can be obtained as well.The different relative position of rigid cylinders and membrane tube will influnce thephase transition and structure of the system.