  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.
