Formation of Multispheres and Myelin Based on Multiple Solutions of Membrane Shape Equation

In this work, we construct a multiple solutions theory based on a membrane shape equation. The membrane shape of a vesicle or a red blood cell is determined using the Zhongcan-Helfrich shape equation. These spherical solutions, which have an identical radius rs but different center positions, can be described by the same equation: ϕ-ρ/rs=0. A degeneracy for the spherical solutions exists, leading to multisphere solutions with the same radius. Therefore, there can be multiple solutions for the sphere equilibrium shape equation, and these need to satisfy a quadratic equation. The quadratic equation has a maximum of two roots. We also find that the multiple solutions should be in a line to undergo rotational symmetry. We use the quadratic equation to compute the sphere radius, together with a membrane surface constraint condition, to obtain the number of small spheres. We ensure matching with the energy constraint condition to determine the stability of the full solutions. The method is then extended into the myelin formation of red blood cells. Our numerical calculations show excellent agreement with the experimental results and enable the comprehensive investigation of cell fission and fusion phenomena. Additionally, we have predicted the existence of the bifurcation phenomenon in membrane growth and proposed a control strategy.