Helfrich Theory To Study The Membrane-colloidal Particle Interaction

Biophysics is a rising interdisciplinary field which combines physics and biology,and it is an important branch of the life science. The research on the membrane haslong been the focus of biophysics. Most of the fundamental biochemical functions incells involve membranes at some point, including processes as DNA replication,protein biosynthesis, protein secretion and hormonal responses. Many biologicalprocesses are controlled by the interaction of macromolecules with a cell membrane.One of the most important properties of cell membranes is their abilities to controltransport mechanisms, such as endocytosis and exocytosis. It is now experimentallypossible to extract and stabilize a freely standing lipid-bilayer membrane in manyinteresting shapes, which can be used as test beds for understanding importantbiophysical processes. The Helfrich model provides a theoretical basis of theinteraction between membrane and colloidal particle.Based on the Helfrich model, we study the system of multiple spherical particlesconfined in a soft membrane tube. The free energy of the system is described by thephenomenological parameters (the bending rigidity, surface tension and adsorptionenergy) and geometric parameters. Minimizing the obtained free energy with respect tothe geometric parameters, we can obtain the stable structures of the system.We calculate the stable structures of the membrane-colloidal particle interactionsystem and the phase diagram in terms of basic parameters in the system. The detailedworks are as follows:(1) The basic theory of the Helfrich model is introduced, together with theconsideration of the geometric model and energy model, such as angle-arclengthparameterization, the bending energy, surface tension energy and adsorption energy.Then taking the membrane-colloidal particle interaction system as an example, weshow how to calculate the free energy and the stable structures.(2) By use of the treatment of the canonical equations of motion in Hamiltonian dynamics, the energy minimization is converted to the solution of four first-orderdifferential equations which can be processed by the shooting method.(3) The system of one or two or even more spherical particles confined in thefreely supported soft membrane tube is investigated. A new phase is found in the twospherical particles-membrane interacting system, which was missed in a publishedLetter. The new phase is characterized by the inner contact angle and the relatedstructures and phase transition are also discussed. The multiple sphericalparticles-membrane system is calculated based on the above works. The results showthe phase boundary can change as a function of the particles number. The previouswork mentioned in this paper can be a theoretical reference for the related experiments.This submitted work is divided into four chapters. The first chapter is theintroduction, which briefly discusses the history of the biophysics and the study of themembrane-colloidal particle interacting systems. Chapter two introduces the majortheory of Helfrich model. Chapter three introduces the Helfrich Hamiltonian and thenumerical approach used to calculate the stable structures. In chapter four, the resultsare shown and related structures and phase transition are discussed in detail.