Study On The Elasticity Of Lipid Membrane In Small Scale

Biophysics is a newly emerged discipline crossing physics and biology, which usesphysical theories and methods to study biological systems. The cell as the basic unit of lifesystems has been a hot topic in this field. With the continuous development of experimentaltechniques, especially the development of single-molecule experimental techniques, peoplecan obtain more information about cell structure and function. Cell membrane is a selectivebarrier to distinguish between cells and the extracellular environment. It controls thetransmission of materials between cells and maintains a relatively stable environment. Lipidrafts are the micro-domains in the plasma membrane, which are rich in cholesterols andsphingomyelins. These micro-domains have a close relation to many biological functions suchas signal transduction, protein sorting and so on. As a result, the physical mechanism hiddenbehind the properties of lipid rafts aroused researchers’ interests.The material transport process in the cell is also a hot research topic. As one way toachieve the protein sorting, vesicular transport can be described as follows. The intracellularvesicles which play the role of cargo containers carry out targeted transport by attaching tomotor proteins which move along microtubules. During this process, if the vesicles meet actinfilaments, deformation will occur. The deformation of the vesicles is an attractive researchtopic.This thesis consists of two parts.The first part mainly discusses the physical mechanism that leads to the formation oflipid rafts on curved surfaces. Lipid membranes are generally composed of saturated lipids,unsaturated phospholipids and cholesterols. At high temperatures, the three main componentsof the membranes (saturated lipids, unsaturated phospholipids and cholesterols) form ahomogeneous mixture. Below a critical demixing temperature, the three componentssegregate into two coexistent fluid phases, a saturated-lipid-enriched liquid-ordered (Lo)phase and unsaturated-phospholipid-enriched liquid-disordered (Ld) phase. Small Lodomainsforming on cell membranes are also referred to as rafts. In this part, we investigate how auniform externally imposed curvature may influence lipid segregation on a Lo-Ldcoexistent membrane. We show that, with the presence of a bending-modulus contrast, the curvature ofthe membrane induces the lipids to tilt away from the membrane normal and yields a reducedeffective line tension between the lipid domains.The main results obtained in this section are listed as follows. Firstly, there exists acritical value of the externally imposed curvature. When a lipid membrane is subjected to acurvature with value larger than the critical value, lipid segregation leads to the formation ofmultiple Lo/Lddomains (or rafts) of microscopic length scale rather than a completeseparation of the two phases. The critical value of the curvature ranges from one over ten toone over a few hundred nanometers.Secondly, we determine the critical line tension for the multi-domain pattern to occur.We point out that the line tension can be lowered either by increasing the external curvature orthe bending stiffness contrast between Loand Ld.Thirdly, we treat the lipid molecules as unit vectors on a cylindrical surface and use atwo-dimensional surface variational method to calculate the phase separation of the lipidmolecules on the cylindrical surface. We obtain an analytical expression of the total energy ofthe system and then discuss the relationship between the domain size and the line tensionunder different component fractions of Lo/Lddomains. Also, we give an analytical expressionof the Lodomain size by considering two special cases:(i) the component fractions of Lo/Lddomains are about the same, and (ii) one fraction is much larger than the other.Finally，we also consider the case of lipid segregation on a spherically curved membrane.Following a similar process, we obtain an analytical energy expression of the system afterperforming some approximation. Then, we discuss relations between the domain size and thenormalized line tension on a spherically curved membrane.The second part of this thesis investigates the deformation of vesicles transported bymotor proteins along microtubules in living cells. When the vesicles encounter the blocking ofmicrofilaments, deformation occurs in the process.Different from the traditional experiments, vesicles in vivo consist of multiple types oflipids and various types have different shapes. An external force leads to an asymmetricaldistribution of the lipid molecules, and thus the spontaneous curvatures of different parts aredifferent. So we introduce a component-dependent spontaneous curvature into the classic Helfrich energy, and consider also the osmotic pressure and surface tension energy. Then weobtain the relationship between the pulling force and the pulling length. The results show thatby introducing such a spontaneous curvature the pulling force resulting from the modifiedHelfrich energy is smaller than the one from the previous classical Helfrich model and agreeswell with the in vivo experimental results.This thesis is divided into four chapters. Chapter1introduces the history of thebio-membrane research, as well as its important in vivo functions. We describe the mainchemical compositions of the bio-membranes, and also introduce some important propertiesof the two biomaterials----lipid bilayers and liposomes. After showing the single-moleculetechniques in biological experiments, we give a brief review of the researches on elastictheories of membranes.In the second chapter, we discuss elastic theories of lipid membranes. At the beginning,there is a short introduction of differential geometry and variational methods, which will beused in this thesis. Then we introduce the various concepts and energies, such as the bendingenergy, tilt energy, compression energy, surface tension and so on. We derive the shapeequation of lipid membranes on a cylinder by taking into account all the aforementionedenergies of lipid membranes for the first time. Finally, we compare our results to others’.In the third chapter, we investigate how an externally imposed curvature influences lipidsegregation on two-phase-coexistent membranes. We consider two cases---membranes on acylindrical surface and those on a spherical surface.In the fourth chapter, we investigate the deformation of vesicles transported by motorproteins along microtubules in living cells. After introducing a component-dependentspontaneous curvature into the classic Helfrich free energy, we calculate how the pulling forcevaries with the tubular length under the far-field and near-field approximation.