Mathematical models of protein–protein dynamics, such as the heterodimer model, play a crucial role in understanding many physical phenomena, e.g., the progression of some neurodegenerative diseases. This model is a system of two semilinear parabolic partial differential equations describing the evolution and mutual interaction of biological species. This article presents and analyzes a high-order discretization method for the numerical approximation of the heterodimer model capable of handling complex geometries. In particular, the proposed numerical scheme couples a Discontinuous Galerkin method on polygonal/polyhedral grids for space discretization, with a θ-method for time integration. This work presents novelties and progress with respect to the mathematical literature, as stability and a-priori error analysis for the heterodimer model are carried out for the first time. Several numerical tests are performed, which demonstrate the theoretical convergence rates, and show good performances of the method in approximating traveling wave solutions as well as its flexibility in handling complex geometries. Finally, the proposed scheme is tested in a practical test case stemming from neuroscience applications, namely the simulation of the spread of α-synuclein in a realistic test case of Parkinson’s disease in a two-dimensional sagittal brain section geometry reconstructed from medical images.

中文翻译：

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用于蛋白质-蛋白质相互作用的异二聚体模型的不连续 Galerkin 近似


蛋白质-蛋白质动力学的数学模型，例如异二聚体模型，在理解许多物理现象（例如某些神经退行性疾病的进展）中起着至关重要的作用。该模型是一个由两个半线性抛物线偏微分方程组成的系统，描述了生物物种的进化和相互相互作用。本文提出并分析了一种高阶离散化方法，用于处理复杂几何结构的异二聚体模型的数值近似。特别是，所提出的数值方案将多边形/多面体网格上的间断伽辽金方法与用于时间积分的 θ 方法耦合在一起。这项工作介绍了数学文献方面的新颖性和进展，因为首次对异二聚体模型进行了稳定性和先验误差分析。进行了几次数值测试，证明了理论收敛率，并表明该方法在近似行波解方面具有良好的性能，以及在处理复杂几何结构方面的灵活性。最后，在源于神经科学应用的实际测试用例中对所提出的方案进行了测试，即在帕金森病的真实测试用例中模拟 α-突触核蛋白的传播，该测试用例是从医学图像重建的二维矢状脑切片几何结构。