2018 Fiscal Year Annual Research Report
Theory and applications of motions of anisotropic interfacial networks
| Project/Area Number |
18F18016
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
MOHAMMAD RHUDAINA 京都大学, 理学(系)研究科(研究院), 外国人特別研究員
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| Project Period (FY) |
2018-07-25 – 2020-03-31
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| Keywords | interface dynamics / interface network / anisotropic energy |
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| Outline of Annual Research Achievements |
Considering the gradient flow of anisotropic surface energy constrained by preservation of its enclosed volumes, we developed a numerical scheme for realizing anisotropic motion of an interface network based on a variational approach to vector-type Merriman-Bence-Osher (MBO) thresholding algorithm with volume penalization. Using this method, we generated a simulation of volume-preserving n-phase crystalline motion by introducing a smoothing technique. For n > 2, we found that an auxiliary scheme is necessary to satisfy the corresponding junction angle condition for a given anisotropy. Motivated by cellular pattern formations in sensory epithelium, we also considered anisotropic motions driven by cell-cell adhesion strengths. For 3-phase case, we introduced a correction algorithm to satisfy the required junction condition. However, due to its complexity in the multiphase case, we adopted an MBO-type convolution method with localized auction algorithm - corresponding to a topological constraint of preserving connectivity. In consultation with a cell biologist, we generated simulations of cell network dynamics resulting in either a checkerboard, football, or segregated pattern. Moreover, using experimental results on distribution patterns of beta-catenin in mouse olfactory epithelium, we simulated cellular rearrangements in its developmental stages. These algorithms were implemented using C programming language with Matlab as a plotting software. Results have been presented and well received in local and overseas conferences/meetings.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Overall, we were able to design an algorithm for simulating anisotropic motions of cell network not only for two-phase, but also for multiphase configurations. In fact, through collaboration with experts in cell and developmental biology, we were able to venture on a very interesting and important application of our work - an interface network model for realizing cellular rearrangements in forming experimentally observable mosaic patterns in sensory epithelial tissues by considering anisotropic motions of cell network driven by cell-cell adhesion strengths.
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| Strategy for Future Research Activity |
From here on, we shall revisit the theoretical framework of our currently designed algorithm - a variational vector-type Merriman-Bence-Osher (MBO) scheme with penalization technique - for realizing multiphase anisotropic motions of interface network constrained by the preservation of phase volumes, which will hopefully enable us to refine or redesign our algorithm so that for any multiphase configurations, the junction angle condition of a given anisotropy can be satisfied either implicitly or by a much simpler auxiliary correction algorithm. Moreover, because of its important and interesting application, we shall continue to work closely in collaboration with experts in cell and developmental biology (e.g. Prof. Hideru Togashi, Kobe University) to improve our interface network model for approximating anisotropic motions of cell network driven by cell-cell adhesion strengths - forming experimentally observable mosaic patterns in sensory epithelial tissues. The proposed and/or refined algorithms shall be implemented using the C programming language with Matlab or other plotting software. We shall then generate computational examples and also, conduct numerical experiments for error, convergence, and junction stability analysis. In addition, we shall rigorously investigate the stability of triple junctions under hyperbolic mean curvature flow. Research results shall be reported in domestic and/or overseas conferences/meetings and shall be published in reputable scientific journals.
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