Physical and mathematical approaches to the deformation of vesicles---from contact to adhesion, fusion and budding

2019 Fiscal Year Final Research Report

• Project/Area Number: 16KT0128
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 特設分野
• Research Field: Mathematical Sciences in Search of New Cooperation
• Research Institution: Tohoku University
• Principal Investigator: Takagi Izumi 東北大学, 理学研究科, 名誉教授 (40154744)
• Co-Investigator(Kenkyū-buntansha): 今井 正幸 東北大学, 理学研究科, 教授 (60251485)
• Project Period (FY): 2016-07-19 – 2020-03-31
• Keywords: ベシクルの形状 / 幾何学的変分問題 / 面積差弾性エネルギー / 反応拡散方程式

Outline of Final Research Achievements

A vesicle is a lipid bilayer forming a spherical or toroidal shell. Its shape is determined as a closed surface which minimizes the sum of the bending energy and area-difference elastic energy, among surfaces having the prescribed surface area and enclosing volume. We found that the Euler-Lagrange equation is an integro-differential equation, which makes mathematical analysis of this model quite difficult. The rigorous analysis of the shape equation for axially symmetric surfaces remains open due to this difficulty. In addition, we considered reaction-diffusion equations with variable coefficients arising from models of two-phase vesicles as well as mechano-chemical models of biological pattern formation. We proved the existence and stability of steady-state solutions for some of such equations; however, we could not study the interaction between the (visco-)elastic deformation of membrane and the chemical patterning within the research period.

Free Research Field

数学，非線形偏微分方程式論，非線形解析学

Academic Significance and Societal Importance of the Research Achievements

脂質の二重膜がつくる小胞がベシクルであるが，生体膜で囲まれた細胞の形状を決めるメカニズムは，ベシクルの形状を支配するものと本質的には変わらないことが知られており，ベシクルに関する知見は細胞の構造を理解する上で有益である．特に，多成分ベシクルでは，相分離が弾性エネルギー係数に影響を及ぼし形状を変化させる．これは生物の形態形成の仕組みと共通する部分があると考えられる．