Under The Force Of The Lipid Membrane, The Process Of Forming A Thin Tube

Biologic membranes such as the endoplasmic reticulum, the Golgi apparatus and the inner mitochondrial membrane often form highly dynamic tubular network. The formation and transport of membrane tubes are thought to involve motor proteins that are able to grab the membrane and pull on it as they move along the filaments of the cytoskeleton. At present, thin tube of tens of nanometers in diameter can also be pulled out in the laboratory by micropipettes and optical tweezers. Pulling out a thin tube from a flat membrane is a delicate process. An in-depth understanding of the mechanisms of this process will help us to understand the related life processes, and to lay the foundation for further studies.The shape of a tube pulled out from the flat membrane is closely related to that of the catenoid, we first give a detailed study on the catenoid. Starting from the analytic expression of the catenoid surface we conclude that the minimum radius b of the catenoid will continue to increase with the increasing pulling force f. We give a review on the existence conditions of the catenoid surface, which is l≤lmax≈R圆环/0.75444. To discuss the stability of the catenoid under the axial force, we use direct minimization method to calculate the shape. For soap bubbles, the corresponding free energy contains only the surface tension term, and the corresponding solution is catenoid. Increasing the distance between the two rings, we found that catenoid is stable whenl<lmax, which is the same as the existence conditions of the catenoid. When l>lmax the catenoid becomes unstable, which will transform into two circular flat discs. For membranes, the free energy contains both the surface tension and the curvature energy. The behavior is quite different from that of the soap film. It is found that forl≤lmax, the catenoid is a stable solution. For l>lmax, there also exist stable catenoid-like surfaces. With fixed distance between the two rings, the middle part of the catenoid-like membrane will become thinner with increasing λ. At the same time the corresponding area keeps decreasing and the curvature energy keeps increasing, with the total energy increasing.The physical process to pull out a small tube in the center of a circular flat membrane is then discussed in the axisymmetric case. The classical theory of this process is first reviewed in detail. For ψ<<1the approximate analytic solution can be obtained. In this case the linear approximation holds. The length of the tube is proportional to the axial force. When the force is further increased, thin tube will be pulled out from the center of the membrane. In the thin tube section, ψâ†'Ï€/2, the approximate analytical solution can be obtained. The axial force reaches the saturation value at f0. One almost does not need to increase the axial force to elongate the thin tube further. Then the shapes of tubular membrane near the catenoid parameters are calculated numerically. It is found that when other parameters unchanged, the greater Uo, the longer the length of the tubular membrane vesicle, at the same time the minimum radius of the tubular membrane vesicle is greater. While keeping the other parameters constant, the greater V0, the smaller the Minimum radius of the tubular membrane vesicle. The greater the axial force f, the longer the membrane vesicle is pulled out, but at the same time the greater the minimum radius of the tubular membrane vesicle is.To study the stability of the tubular membrane, a calculation is made by the direct minimization method. It is found that stable tubular vesicles can be obtained by the curvature energy and surface tension energy. Many tubes with different lengths and thicknesses are obtained. Our calculations show that:When the length of the tube is constant, the energy increases with the increasing surface tension λ. When λ is constant, the energy increases with the increasing length of the tubular membrane vesicles. With the same length of the tubular membrane vesicles, the greater the surface tension λ, the thinner the tube radius is.The shapes of the undulation membrane vesicles are obtained. At c0=0, the undulation membrane vesicle is unstable. It will gradually evolve into a thin tubular membrane vesicle. At c0≠0, with the spontaneous curvature c0increases, a series of undulation membrane vesicles shapes can be obtained. The total energy is related to the spontaneous curvature c0, the greater the spontaneous curvature c0, the greater the total energy of the undulation membrane vesicle.