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 -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 -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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蛋白质-蛋白质相互作用异二聚体模型的不连续伽辽金近似


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）方面发挥着至关重要的作用。该模型是两个半线性抛物型偏微分方程组，描述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，用于能够处理复杂几何形状的异二聚体模型的数值近似。特别是，所提出的数值方案将用于空间离散的多边形/多面体网格上的不连续伽辽金方法与用于时间积分的方法结合起来。这项工作在数学文献方面呈现出新颖性和进展，因为首次对异二聚体模型进行了稳定性和先验误差分析。进行了多次数值测试，证明了理论收敛速度，并显示了该方法在逼近行波解方面的良好性能以及处理复杂几何形状的灵活性。最后，所提出的方案在源于神经科学应用的实际测试案例中进行了测试，即在从医学图像重建的二维矢状脑切片几何结构中模拟帕金森病的真实测试案例中β-突触核蛋白的传播。