"shooting Method" Axisymmetric Membrane Bubble Shape Research

Study on structure and function of biomembrane is becoming one of the hot spots in comdensedmatter physics.The basic structure of biomembrane is a bilayer membrane composed amphiphiliclipids molecules. Now the object of study about biomembrane is the lipid-bilayer that is formed onlyfrom phospholipids without considering the protein cytoskeleton.The phospholipid molecules mayaggregate into bilayer membranes and closed vesicles at a certain critical concentration in aqueoussolution. Physical research shows that lipid-bilayer membrane is the liquid crystal state in the normalphysiological conditions.The shape of red blood cells depends entirely on the membrane force balance because of nonuclear.but clinical trials found the shape diversity and complexity about RBC. In order to betterexplain the old physiological problem why red blood cells are circular biconcave discoidal shape,in the previous studies Helfrich proposed the spontaneous curvature model based on liquid crystalcharacteristics of bilayer membrane. Helfrich simplified the biomembrane into the lipid-bilayermembrane and neglected the thickness of bilayer then he derived the curvature flexible formulaexpressed with the two major surface curvatures C₁, C₂ and the spontaneous curvature C_o. Thesize of the total area and enclosed volume maintain two constants considering the mobility ofvesicles and the incompressibility of the liquid in closed vesicles,in addition the curvature energy ofvesicles ought to be minimum because of the Hamiltonian principle,so the issue on shapes oflipid-bilayer vesicles will become a variational problem of minimum bending energy under theconstraints of area and volume.According to the axisymmetric shape equation derived from SCmodel, Helfrich and Deuling obtained the biconcave shape of red blool cell ,also got the stomatocyteshape etc with numerical calculation. In 1987 Ouyang and Helfrich deduced the general shapeequation of vesicles without any restrictions in order to deal with non-symmetric vesicles,thenOuyang with other partners tested to solve the equation and found three important analyticalsolutions including one may describe well the biconcave shape of erythrocyte and other one wasnamed Clifford anchor ring that was theoretical predicted at first then was experimental verifiedsoon.On the other hand experimental study on vesicle shapes mainly used various lipid or lipidmixtures, especially the phospholipid to synthesize membrane vesicles.Based on the spontaneouscurvature (SC) model, other two curvature models such as the bilayer couple (BC) model and thearea difference elasticity (ADE) model have been proposed to obtain various vesicle shapes andbetter explain some shape transitions versus experimental outcome.For axisymmetric vesicles through introducing five variables such asΨ,U,γ,X, Z andindependent variable of arc length S, one may simplify the high order nonlinear partial differential equat ions describing shapes of vesicle into ordinary differential equations, then analyse boundaryconditions fulfilled charateristics of the shape. After these steps one can solve numerically theboundary value problem of differential equations with shooting method to draw the contour ofminimal solution in the coordinates X and Z. It is major work of this paper that study on theprograms of shooting method to numerically solve the closed axisymmetric shapes of vesicles,main results can be summarized as follows:(1) According to the case of the parameter (?) value, we define a length unit (?) and rescale allfive variables and other two parameters (?) to obtain the simplified differential equations II.(2) After we have obtained the function values of the five variables on the basis of the boundarycondition of a closed axisymmetric vesicle with the spherical topology and the five derivative valuesfrom the differential equations, we expand Taylor one order series for the five variables to eliminatethe problem of singular point because of the variable X - 0 in the south and north poles.(3) We program in C language and control the calculated error to integrate numerically thedifferential equations II with the Bulirsch-Stoer algorithm, then we verify the analytical expressionsof balance condition about spherical and cylindrical vesicles through numerical integral.(4) After again introducing a variable of the arc length Y and a new independent variable /, weobtain a new differential equations III including six variables through variable substitution, thenprepare and debug the program of shooting method to solve the new equations III.(5) Under the condition of (?),we gain the estimate values by extrapolationand the true values through shooting method with regard to U(0) and S₁. By using these true valuesof U(0),> S₁ , we integrate the equations II and obtain six kinds of vesicle shapes in whichoblates> spheres and prolates are stable however other three shapes are metastable.(6) After assigning the values of(?) and finding the estimates of U(0) > Sl from somereferences.we use the shooting method and make numerical integration of the equations II in order toget other three stable vesicle shapes with spherical topology.They are shapes of biconcave >stomatocytes,pears.(7) We integrate the differential equations I through manual searching for the values of (?)and U(0) then gain two categories of cyclical vesicles shapes .On the other hand we find sometubular vesicles which have beads from three to eleven in the SC model by the software of SurfaceEvolver.