Mathematical models of protein-protein dynamics, such as the heterodimer model, play a crucial role in understanding many physical phenomena. This model is a system of two semilinear parabolic partial differential equations describing the evolution and mutual interaction of biological species. An example is the neurodegenerative disease progression in some significant pathologies, such as Alzheimer's and Parkinson's diseases, characterized by the accumulation and propagation of toxic prionic proteins. This article presents and analyzes a flexible high-order discretization method for the numerical approximation of the heterodimer model. We propose a space discretization based on a Discontinuous Galerkin method on polygonal/polyhedral grids, which provides flexibility in handling complex geometries. Concerning the semi-discrete formulation, we prove stability and a-priori error estimates for the first time. Next, we adopt a $\theta$-method scheme as a time integration scheme. Convergence tests are carried out to demonstrate the theoretical bounds and the ability of the method to approximate traveling wave solutions, considering also complex geometries such as brain sections reconstructed from medical images. Finally, the proposed scheme is tested in a practical test case stemming from neuroscience applications, namely the simulation of the spread of $\alpha$-synuclein in a realistic test case of Parkinson's disease in a two-dimensional sagittal brain section geometry reconstructed from medical images.

翻译：蛋白质动态过程的数学建模（如异源二聚体模型）在理解众多物理现象中具有关键作用。该模型由两个描述生物物种演化与相互作用的半线性抛物型偏微分方程构成，其典型应用场景包括阿尔茨海默病和帕金森病等重大神经退行性疾病的发展进程，这些疾病以毒性朊蛋白的积累与传播为特征。本文提出并分析了一种用于异源二聚体模型数值逼近的灵活高阶离散方法。我们采用基于多边形/多面体网格的不连续伽辽金方法进行空间离散，该方法在处理复杂几何构型时具有高度灵活性。针对半离散格式，我们首次证明了其稳定性与先验误差估计。随后采用$\theta$-法格式作为时间积分方案。通过收敛性测试验证了理论界值及该方法逼近行波解的能力，并考虑了医学影像重建脑切片等复杂几何结构。最后，将该方案应用于神经科学实际案例——基于医学影像重建的二维矢状脑切片几何构型中，模拟帕金森病真实病例中$\alpha$-突触核蛋白的扩散过程。