Topology of configuration space model for viewing flow lines of flexible molecules

2021 Fiscal Year Final Research Report

• Project/Area Number: 17K05366
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Kochi University
• Principal Investigator: Komatsu Kazushi 高知大学, 教育研究部自然科学系理工学部門, 教授 (00253336)
• Co-Investigator(Kenkyū-buntansha): 後藤 了 東京理科大学, 薬学部生命創薬科学科, 教授 (50253232) ; 江居 宏美 弘前大学, 理工学研究科, 准教授 (60333051)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Keywords: 数理モデル / 配置空間 / トポロジー

Outline of Final Research Achievements

When n is small, a chain model of n-membered ring hydrocarbon molecules was given and the topology of the model's configuration space was studied. In this research project, we extend to the case where n is large, and obtain the following results. We consider the configuration space of equilateral polygons in 3-dim Euclid space whose bond angles are equal a fixed angle except for two successive ones. We obtain that when the fixed angle is suffiently close to the inner angle of the regular polygon, the configuration space is homeomorphic to the sphere with the corresponding dimension.
To explore the possibility of approximating large macrocyclic molecules with small molecules, we have obtained the following results.we consider a kind of polyhedral surfaces obtained by attaching egular pentagons called a regular pentagon ring. We obtain that if a regular pentagon ring is on a plane, it can be folded in one regular pentagon.

Free Research Field

幾何学的数理モデル

Academic Significance and Societal Importance of the Research Achievements

立体構造の変位を許す分子においては，その立体構造の違いによってもつ化学的な性質が異なる場合があることが知られている．このような変形可能な幾何学的構造をもつ数理モデルを考え，その取り得る形状全体からなる配置空間を考えることで，変形の様子の動線を明らかにすることができる．小さな幾何学的構造で得られた結果を大きな幾何学的構造に拡張することは，分子の場合はタンパク質などの大きな分子への応用が得られることを意味する。