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

TL;DR

This paper introduces a high-order Discontinuous Galerkin method for approximating the heterodimer model of protein interactions, demonstrating stability, convergence, and applicability to complex biological geometries relevant to neurodegenerative diseases.

Contribution

It develops a novel flexible discretization approach with proven stability and error estimates for the heterodimer model, applicable to realistic brain geometries.

Findings

The method achieves stable and accurate approximations.

Convergence tests confirm theoretical error bounds.

Successfully simulates protein spread in brain geometries.

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 $\theta$-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.