Spherical Topology, The Bubble Shape And Its Stability In The Budding And Three Concave Membrane

Amphiphilic molecules composed of a polar hyfrophilic head and two non-polar hydrophobic tail chains may aggregate spontaneously into the bilayer membrane,which will form closed vesicles in low concentration because of the hydrophobic interaction.Study on the lipid vesicles as a simplified model of the biomembrane provide valuable information to understand the complex biological system.In 1973 German physicist W.Helfrich firstly proposed the spontaneous curvature model(SC) based on the similarities of the lipid membrane and the nematic phase liquid crystal,thus proved the existence of the biconcave shape.At the same time,taking into account impact of the energy from the area difference between the bilayers in 1989 S.Svetian et al advanced the bilayer couple model(BC) and then U.Seffet also proposed other area difference elasticity model(ADE) with Miao Ling.After many years of effort by some people,we have a very clear understanding about phase diagrams of the three models,still don't clearly know some regions.It is showed the stomatocytes vesicles will transform the limit shape which have a spherical mother vesicle and a inverted spherical daughter vesicle with a infinitesimal neck,which is expressed with L^sto in the left corner of phase diagram of the BC model,but stable shapes are not clear beyond the region.However in experiments S.Svetina and others observed many interesting vesicle shapes named inside n-budding,the region known of the phase diagram don't show these shapes.The current theoretical research on n-budding shapes deal with the approximate shapes composed of one big ideal mother sphere linking n small inverted perfcet daughter sphere,then give some budding shapes with the reduced volume v=0.85.we need lucubrate such problems as what these actual budding shape are,whether these shape are stable,what region they are in.The clinical triconcave shape of the morbid red blood cell caused a lot of people's interest,so considering the protein skeleton Gerald Lin et al obtained the triconcave and some other abnormal structure of red blood cell by molecular dynamics simulation method.It is a matter of concern whether the triconcave shape can be stable only taking account of the curvature energe.We make an in-depth study on the inside n-budding vesicles and the triconcave of red blood cell, main results can be summarized as follows:(1) We research the region in which the inside n-budding vesicles exist and find the existing region is under the curve of L^sto that is unknown region in the phase diagram of SC and BC model.(2) In practice we calculate the energy with two methods of stn and stp in order to solve the difficulties from the thin neck.The algorithms of stn may keep the neck from repture,the stp may have faster convergence speed and is effective to analysis the stability of vesicle shapes.At the same time we make full use of the relationship between the SC and BC models,and their respective calculational characteristics.It is proved that inside two-budding vesicles are unstable in SC model and the shape will transform into stomatocytes.However two-budding vesicles are stable in BC model,so the example proves the BC model is closer to the actual situation. (3) In order to gain inside 2-budding vesicle,we need to calculate in the BC model.For advancing calculation speed,firstly we calculate in SC model,secondly switch to BC model.We find that a nearly spherical mother vesicle linking inverted dumbbell by one infinitesimal neck with reduced volume v=0.8 and reduced area differenceâ–³a=0.485.In past,The shape is simpled the approximate situation under which the big mother and all the small daughter vesicles are ideal spheres in the past theoretical research.(4) Through the further evolution of the shape,we find that the neck linking spherial mother vesicle and the inverted daughter vesicle constantly become thinner and thinner in the budding process,at last the shape split into one sphere and inverted dumbbell,rather than what the usual literature showed the shape directly split into a spherical mother vesicle and two inverted spherical daughter vesicles.(5) We observe that inside three-budding vesicle will transform into inside two-budding vesicle during the study on three-budding vesicle with reduced volume v=0.8 and reduced area differenceâ–³a=0.485,so one ought to calculate in a smaller reduced area difference to obtain the inside three-budding shape.(6) U.Seifert et al studied symmetric shape of a sphere enclosing above and below two smaller spheres,and thought the shape is unstable and it will transit stomatocyte discovering through research, it is broken from top to bottom symmetry,but only in more reduced area difference it transformed into stomatocyte.In littler reduced area difference it transformed symmetric shape of a sphere enclosing above and below two smaller spheres into asymmetric shape.(7) Zhang yong et al claimed that triconcave is given with Surface-Evolver in SC model,we do their works again in different initial shapes,and find that triconcave is got with stn energy long evolveing in parameters that they used.However,the shape is only a saddle,if the shape is evolved in stp energy,it is faster switched to stomatocyte.Because many triconcave are stable shapes in BC modle but in SC modle,we study futher in BC modle,and find that triconcave is unstable switching to oblate ellipsoidal vesicle.The study show that triconcave is not unstable only given bilayer membranaceous curvature energy,the protein skeleton is vital for triconcave.