Discontinuous Galerkin approximations of the heterodimer model for protein-protein interaction

TL;DR

A flexible high-order discretization method based on a Discontinuous Galerkin method on polygonal/polyhedral grids, which provides flexibility in handling complex geometries and proves stability and a-priori error estimates for the first time.

Abstract

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 $θ$-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 $α$-synuclein in a realistic test case of Parkinson's disease in a two-dimensional sagittal brain section geometry reconstructed from medical images.

Figures (9)

• Figure 1: Schematic representation of monomeric and polymeric seeding models.
• Figure 2: Test case 1: computed errors and computed convergence rates versus $h$ for different choices of the polynomial degree $p=1,2,3,4$ and for the solutions $c_h$ (left) and $q_h$ (right).
• Figure 3: Test case 1: computed errors and convergence rates with respect to $p$ (left) and $\Delta t$ (right). The results on the left panel have been obtained with the Crank-Nicolson (CN) time integration scheme; the results on the right panel have been obtained with both Backward Euler (BE) and CN schemes.
• Figure 4: Test case 2: computed errors at final time $T=10^{-3}$ and computed convergence rates versus $h$ for different choices of the polynomial degree $p=1,2,3,4$ and for the solutions $c_h$ (left) and $q_h$ (right).
• Figure 5: Test case 2: wavefront propagation approximated with $p = 2$.

Theorems & Definitions (11)

• Remark 1
• Proposition 1
• Proposition 2
• Proposition 3
• Theorem 1
• proof
• Remark 2
• Remark 3
• Theorem 2
• proof