蝴蝶斑紋之有限元素模擬與分析

Abstract

摘要 涂林系統是用來解釋生物形態發生(Morphogenesis)的一種可行理論，也就是說生物的成長發育或生物斑紋圖案形成可用涂林系統描述。生物形態發生的動力來自形態素的擴散和反應，而造成圖案形成的原因在於使用兩種以上的化學物質，經由濃度擴散以及化學物之間的交互作用與自我催化而形成圖案。本研究藉由涂林系統模擬蝴蝶翅膀上的斑紋，模擬之蝴蝶對象包括蛇目蝶科(Satyrinae)和鳳蝶科(Papilionidae)翅膀上的斑紋，同時也討論翅膀上的翅脈和翅膀邊緣與活化劑的關係，以及涂林系統參數對系統的穩定度與波數關係。我們以2D和3D參數平面來表示穩定範圍及趨勢，另外討論系統的邊界條件對圖案形成的影響，還有以Source-Source、Source-Sink和擴散反應方程式這三種方法模擬翅室內的基本斑紋。本研究主要探討Joakim Linde、眼斑Dilão、G-M、G-S與Schnakenberg五種涂林系統，用以模擬涂林系統的工具為FEMLAB有限元素軟體。我們分別討論這五個系統的收斂速度，利用收斂速度最近似的兩個或兩個以上之系統，配合翅脈與活化劑的關係來模擬與分析蝴蝶斑紋。本研究發現在系統收斂速度上以眼斑Dilão和G-M系統最接近，在同一翅膀上套用這兩個涂林系統模擬出台灣黑蔭蝶(Lethe butleri)和尖尾黛眼蝶(Lethe sinorix)的斑紋，另外利用Dilão、Schnakenberg和G-M系統三個涂林系統模擬出寬帶黛眼蝶(Lethe Helena Leech)的斑紋。本研究發現以翅脈為邊界條件會對斑紋形成造成影響，利用此關係我們可根據模擬的蝴蝶對象選擇適合之邊界條件，另外模擬斑紋時涂林系統的波數參數對斑紋形成樣式也會造成影響，所以模擬蝴蝶斑紋必須重複修正邊界條件和微調波數參數，直到模擬出相似的斑紋。過去前人研究模擬生物斑紋均以矩形或圓形和單一的涂林系統來模擬生物體上某一區域的斑紋，本研究則進一步以完整的蝴蝶翅膀和複合的涂林系統來模擬全區域的斑紋，得到更佳的模擬效果。 關鍵字：涂林系統、生物斑紋形成、擴散反應方程式

ABSTRACT One of the elementary processes in morphogenesis is the formation of a spatial pattern of tissue structures. It has been shown that relatively simple molecular mechanisms based on auto and cross catalysis can account for a primary pattern of morphogens to determine pattern formation of the tissue. This study simulates and analyses butterfly wing pattern by using Turing system. The simulated wing patterns include Satyrinae and Papilionidae butterflies. We focus on the effect of key factors such as parameter values for mode selection, wave number, wing shape and boundary conditions. We express the stability and the tendency by constructing 2D and 3D parameter plane, and we discuss the boundary conditions of the system effect on pattern formation. Elementary patterns of butterfly wings are simulated by the source-sink, source-source and diffusion-reaction methods. This study utilizes five kinds of different Turing systems to simulate butterfly wing patterns. These systems are Joakim Linde Turing system, Dilão Turing system, Gierer-Meinhardt Turing system, Gray-Scott Turing system, and Schnakenberg Turing system. The finite element software FEMLAB was used for the simulation and analyses. Judging from their similar rate of convergence, Dilão and G-M systems were used to simulate wing patterns of Lethe butler and Lethe sinorix butterfly. We also used Dilão, G-M and Schnakenberg systems to simulate wing patterns of Lethe Helena Leech butterfly. In this study we found that wing pattern formation simulation is affected by two major factors of boundary condition and wave number parameters. The parameters are adjusted iteratively until the simulated pattern error is under a threshold. In contrast to simulate wing patterns of simple shape and using single Turing system, this study expands the scope to simulate whole butterfly wing patterns using composite Turing systems. As a result, more realistic butterfly wing patterns can be simulated and visualized. Keywords: Turing system, Biological Pattern Formation, Diffusion Reaction