A Time-Fractional Fischer-Kolmogorov Model for Protein Misfolding in Neuro-Degenerative Disorders: A Virtual Element Study

We investigate the anomalous propagation of misfolded proteins implicated in neurodegenerative conditions, with emphasis on tau-associated pathologies. To capture the sub-diffusive dynamics between misfolded and functional proteins, we formulate a two-dimensional time-fractional reaction-diffusion model governed by a Fisher-Kolmogorov type equation with a Caputo derivative of order α ∈ (0, 1]. The spatial domain is discretized using the Virtual Element Method, enabling flexible polygonal meshing under a Galerkin framework. For temporal integration, we employ a convolution-based backward Euler scheme aligned with the Grünwald-Letnikov approximation. The resulting fully discrete system is rigorously analyzed: we establish existence, uniqueness, and derive optimal convergence bounds. Leveraging discrete maximal regularity, we demonstrate unconditional stability and consistency. Numerical experiments confirm convergence in both <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> norms across varying mesh topologies, underscoring the robustness and applicability of the proposed scheme.