Theoretical Modelling And Simulations Of The Deformation And Phase Separation Of Membranes

The research field of bio-membrane should be the first research field within which people make one great achievement after another after laws of physics and mathematics were introduced into the community of the biological science.From the year 1973 when the Helfrich free energy was proposed to depict the membrane behavior,to about ten years later when Ou-Yang successfully obtained the shape of red blood cell(RBC) analytically,finally to the year 2002 when Wortis and his coworkers reproduced the whole SDE sequence of RBC using Monte-Carlo simulations;from Molecular Dynamics which is capable of simulating the formation of bilayer membrane,to various particle-based models that are well suited for describing the phenomena of fission and fusion process of membranes,and also to various field-based models related with membranes;all of these above achievements indicate that the applications of the knowledges of physics and mathematics in biology are of super success.This success may also indicate the end of this research field.However,it may be not true:we clearly see clouds up in the sky of the field.1.There is a simple but useful model-spherical cap model-for the mechanism of bilayer-membrane formation,which claims that no meta-stable intermediate exists during the disk-vesicle transformation. This assertion is usually supported by the facts that bowl-like membranes are hard to be observed and the cells are always closed up.However,it has been challenged by recent observations of meta-stable vesicle pores in some solution with high viscosity.These observations are telling a new story that intermediate states along the disk-vesicle transition can be meta-stable though they are in some shallow local energy minima.Is that the simple model being completely wrong,or just because of the other unknown reasons like the laws of physics and mathematics cannot fit in here anymore? After vesicle pore was observed for the first time,ten years passed;however,the answer to this very fundamental problem of membranes still remains unknown.2.Most problems about the red blood cell(RBC) shape have been well resolved;however,it is always a huge trouble for biologists to precisely determine the values of the RBC's elastic constants,which play a key role in explaining how the RBCs do their jobs in the blood.Most previous measurements depend on mechanical pulling experiments using optical tweezers and the results were not very accurate.However,this inaccuracy is not because of the experimental techniques they used,but because the RBC models which they used to interpret their force-extension curves are not real enough and,thus,not accurate.Therefore,it should be of top priority for bio-membrane scientists to develop a theoretical model that is capable of simulating the pulling process of RBCs accurately.3.All the following processes will involve drastic deformations of the membrane surface:fission of the cells,deformation coupling with phase separation of the multi-component membrane or deformation coupling with chemical reactions within the cell.Nowadays,the methods that can simulate the drastic deformations of the membrane are mainly referred to the particle-based models,which are unfortunately very time-consuming for computer simulation and cannot give definite answers to many problems.The computations based on triangulated-mesh method are much faster;however,due to the numerical instability of the method,they are not very well suited for big-deformation problems.……Clearing up the annoying clouds above,there may be some fresh starts ahead instead of ends.In this thesis,we will study these problems from three hierarchies: the properties and formation of the membrane(PartⅠ),the deformation of the membrane(PartⅡ),and the phase separation on the membrane(PartⅢ). Specifically,they are as follows:1.PartⅠ,the properties and formation of the membrane,includes Chapter 1, 2 and 3.Chapter 1 is the introduction section of the thesis.In Chapter 2, we study the mechanical properties of diblock bilayer membranes using the self-consistent field theory for grand canonical systems.The dependences of the bending modulus,Gaussian modulus and line tension of the membrane on the architecture of diblocks can be obtained by calculating the free energies of the membranes with different sizes and in different geometrical confinements.In Chapter 3,we reconsider the forming mechanism of the closed membrane or vesicle.We assume that, different from the spherical-cap model,the intermediate state along the disk-vesicle transition is not a perfect spherical cap but some axis-symmetric bowl-like membrane.Based on this assumption,we re-examine the transition using the string method,which was primarily designed to study the phase transitions.And we find that the energy barrier of the transition obtained by us is much smaller than that of spherical-cap model,the critical size of the instable disk is much smaller than that of previous one,and more importantly we observe the meta-stable intermediate states along the transitions.In order to obtain more systematic results,we implement some iterative numerical method to solve the shape equation to further study this transition,and construct various phase diagrams of the transition under different parameter spaces.2.PartⅡ,the deformation of the membrane,includes Chapter 4,5 and 6.In order to resolve the problems that prohibits the triangulated-mesh method from simulating the big deformations of membranes.We propose a novel numerical method-Discrete Spatial Variational Method(DSVM) in Chapter 4 and successfully solve the problems.In Chapter 5 and Chapter 6,we try to set up a model to simulate the pulling of RBC.Wortis's RBC model should be the most realistic one to date in the sense that it has reproduced the whole SDE(Stomatocyte Discocyte Echinocyte) shape sequence of RBC precisely.However,due to the optimizing method-Monte Carlo method-they used,which is stochastic,their model is not very well suited for simulating the pulling experiment of RBC.This thesis improves it using dissipative dynamics in order to simulate the whole pulling process precisely.To check our modified model,we also reproduce SDE sequence in Chapter 5,and in Chapter 6 we simulate two different pulling experiments of RBC.And our simulations can "see" things that can not be "seen" by experiments.For example,in the second experiment experimenter want to make sure whether the lower portion of the RBC has been attached to the bottom substrate but the microscope cannot tell. However,by comparing the force curves obtained by our simulations and the experimental data,we assert that it does collapse down to the substrate.3.PartⅢ,the phase separation on the membrane,includes Chapter 7,8 and 9.In Chapter 7 and Chapter 8,we study the micro-phase separation of block copolymers on the surface of the sphere and on the general curved surface.The real-space self-consistent field theory(SCFT) is a successful method to simulate the phase behaviour of the block copolymers.However, this method is only available for the fiat space because it is super difficult to design alternating direct impiicit scheme(ADI) in curved space,which is indispensable for real-space SCFT.We expand the idea of the ADI greatly.As a result,we are able to design spherical ADI and curved-surface ADI.In other words,we have successfully extended the ordinary SCFT to spherical SCFT and to curved-surface SCFT.Now,all the spatial constraints on SCFT method have been released:You can "simulate" SCFT in whatever space you want.In Chapter 9,the novel method, DSVM,proposed in Chapter 4,has been further used to simulate the deformation coupling with phase separation of multi-component membranes.Specifically,we study the late stage of the dynamics and find that,if the budding extent of the phase domain is measured by the ratio of the area to the squared length of the domain,the budding speed of the domain will accelerate at some point during the late stage.This should be the second nucleation process of this dynamics which is resulted from the competition between the bending energy and the interfacial energy.At the critical point of the nucleation,the shape of budding domain approximates a semi-spherical cap.Besides,the first nucleation occurs at the beginning of the dynamics,in which the mixing energy is competing with interfacial energy.