Theoretical Studies On The Shapes Of Biological Membranes

Biological membrane is not only the basic unit of the cell structure, but also the structural foundation to provide the life activity. Biological membrane is in close related with many life process, such as signal recognisee and transduction, cell growth and differentiation, energy transformation, substance transportation etc.. Because of the complexity of the real biological membrane, the appropriate model is developed. Fortunately, vesicle which is consisted of am-phiphilic is the simple and effective model of the biological membrane. Then, in this thesis, we have studies four parts of the project surrounded in the biological membrane: polymer anchored vesicle, rigid rod/flexible chain anchored the infinite membrane, the complicated shapes of the membrane confined in the outer membrane, phase separation dynamics study of two-component membrane.The first part predicates shapes of polymer anchored vesicles. In biological systems, lipid bilayers or membrane are often "decorated" by a large number of macromolecules, such as protein, cytoskeleton and glycocalix. As the simplified model of such biological membrane, it is important for biology to study the shapes of polymer anchored vesicles. Shape transformations have been extensively predicted in such fluid vesicles/polymer compound system by the method combined self-consistent-field theory (SCFT) for the polymer with Helfrich curvature energy for the vesicle. With the impenetrability of the membrane to the polymer, we obtain the new shape equation of the membrane, taking account of the interaction potential of membrane/polymer and polymer/solvents. The main difference between the ordinary shape equation and polymer anchored vesicle's is that the surface of the vesicle is exerted the inhomogeneous entropic pressure because of the confined available space. The asymmetric shape of polymer anchored vesicle is induced because of the inhomogeneous entropic pressure. With the interaction potential between the membrane and the chain segments, the anchored segments changed not only the exerted pressure, but also the surface tension of the vesicle. However, the extents of the alterative pressure and surface tension are in close related with the interaction parameter between the polymer and the vesicle, the polymer segments distribution on the surface of the vesicle. In repulsive or weak adsorption between the membrane and segments, the anchored polymer forms "mushroom", in which the configuration entropy dominates over the interactive energy. However, when the interaction play a dominated role in the state of strong adsorption, polymer prone to form the "pancake" to broaden much more surface contacts with the membrane, which seriously enhance the inhomogeneous spatial distribution of the polymer on the surface of the vesicle. The term concerned with the interaction between the polymer andsolvents is not explicitly appearing in the shape equation. However, the shape of the vesicle is only influenced with the help of the polymer distribution in the solvent molecules. Since the segment concentrations near the membrane will be not increased in relation to the larger size of chain length anchoring, as well as in the good solvent, shape transformations have much less dependence on chain length of the anchored polymer and good solvent in such systems. At the weak adsorption, the effect of chain length is to some extent augmented because the chain segments' densities have been seriously increased adjacent the membrane surface, which can augment the exerted entropic pressure and stretched the tensile stress on the membrane. Also, larger bending rigidity of the membrane can strongly resist fluctuation arising from both the pressure and tensile stress, that is, suppressing the degree of shape transformation irrespective of short or long-ranged interaction. The shape of the polymer anchored vesicle is transformed based on the global factors, and can not be described by the simple symmetric characteristics. Moreover, extension of this method to more complicated systems, such as chain stiffness, different chain architectures, multi-components and chain systems, is straightforward, which provide one method to study the life science of cell.The second part is the infinite membrane anchored by rigid rod/flexible chain. We investigate the shape deformation of such systems, which the density of rigid rod/flexible chains is calculated with the self-consistent field theory (SCFT) and the shape of the membrane is predicted by the Helfrich membrane elasticity theory. It is found that the membrane bends away from polymer, so that the polymer decrease the entropic reduction due to the restriction of the available space. Close to the anchoring position, the polymer distribution is different between the rod and flexible chain, resulting in different shape behaviors of the membrane. Importantly, an evident gap is found between the membrane and the rigid rod because the membrane's curvature has to be continuous. Also, the interaction between the membrane and the polymer will influence the shapes of the membrane. When the interaction between the membrane and the polymer increase from the weak to the strong adsorption, the distribution of the flexible changes from "mushroom" to "pancake", while that of the rigid rod change from "fan" to "cone" and the radius of the "fan" is equal to the chain length of the rod. Without the interaction between the membrane and the flexible chain, the shape of the membrane is almost independent of the chain length; with the interaction, the shape transformation of the membrane is seriously augmented as the chain length increases. Even without the interaction, the shape of the membrane will much bend away from the membrane as the rod length increase, because the end-to-end distance of the rod and the coverage area of the membrane are in proportion to the rod length. Moreover, the bending rigidity and surface tension of the membrane are also one of the factorsfor the shape transformation of the membrane.The third part studies the shape problem of the double layer vesicle with the discrete-spatial variation method, that is, the complicated shape of the inner vesicle confined in the soft and outer environment. We focus on the factors of the complicated shape: the area ratio between the outer and inner vesicle, the bending rigidity parameter between the outer and inner vesicle, the pressure of different space and the electrostatic interaction among the inner vesicle. When the area of the inner vesicle is larger than that of the outer vesicle, the inner is prone to form the cristae in the confined space of the outer vesicle. When the area of the inner membrane is less than that of the outer vesicle, the shapes of the divided vesicle concern with the pressure of different space: the shape of the vesicle is circle if the pressure of the outer space is much less than that of the inner space for each vesicle. The bending rigidity of the outer and inner vesicle impacts the shapes of the double layer vesicle according to the different space pressure. Inhomogeneous electrostatic interaction among the inner vesicle can cause the cristae uniformly distributed in the confined space. The double vesicle adjusts its complicated shape to minimize its whole energy. These results would provide the significant basis to understand the shapes of the organelle, such as mitochondria.The last part studies phase separation dynamics of two-component membrane by means of Dissipative particle dynamics. In biological membrane, amphiphilic is usually inhomogeneous distributed in the bilayers under certain experiment condition. Many complicated domains, which can laterally diffuse on the surface of the membrane, are extensively found on the surface of the membrane. All these phenomena are related to the cell function, such as protein diffusion. We use the mesoscopic model called dissipative particle dynamics to study the spontaneous aggregation formation of amphiphilic molecules in aqueous solution, for example micelles, rod shapes and vesicles. Also, we study the phase separation dynamics of two-component aggregations. Under appropriate conditions, typical shape transformations of aggregations, such as budding, fission, domain diffusion, are observed. We also wish to develop this method to simulate more complicated systems, such as the floating behavior of the vesicle in the confined tube, the interaction between the cell and the base plane, the interaction between the cell and the cell.