Spatially-extended nucleation-aggregation-fragmentation models for the dynamics of prion-like neurodegenerative protein-spreading in the brain and its connectome

Abstract

The prion-like hypothesis of neurodegenerative diseases states that the accumulation of misfolded proteins in the form of aggregates is responsible for tissue death and its associated neurodegenerative pathology and cognitive decline. Some disease-specific misfolded proteins can interact with healthy proteins to form long chains that are transported through the brain along axonal pathways. Since aggregates of different sizes have different transport properties and toxicity, it is important to follow independently their evolution in space and time. Here, we model the spreading and propagation of aggregates of misfolded proteins in the brain using the general Smoluchowski theory of nucleation, aggregation, and fragmentation. The transport processes considered here are either anisotropic diffusion along axonal bundles or discrete Laplacian transport along a network. In particular, we model the spreading and aggregation of both amyloid-β and τ molecules in the brain connectome. We show that these two models lead to different size distributions and different propagation along the network. A detailed analysis of these two models reveals the existence of four different stages with different dynamics and invasive properties.

Prion-like spreading of Alzheimer’s disease within the brain’s connectome

“…Here, we adopt a simplification of the Smoluchowski model (2.10), the nucleated polymerization model [26,27] with a nucleus size of two and spontaneous nucleation, to model the nucleation, aggregation, and fragmentation of tau proteins in Alzheimer’s disease [17] and make the following simplifying assumptions: we assume that diffusion is size-independent, D i = D ; production is only possible for healthy monomers, k 01 = k 0 for i = 1, but not for misfolded particles of any other size, k 0 i = 0 for i > 1; clearance occurs at k 1 for healthy monomers, is size-independent k 2 = k i for larger particles 1 < i < n ; and impossible k n = 0 for the largest particle size i = n ; nucleation of two monomers occurs at a nucleation rate a 11 = κ and is irreversible; aggregation of larger particles is size-independent a ij = a , but can only occur by adding single monomers for i = 1 or j = 1 and is impossible a ij = 0 otherwise; fragmentation into monomers is impossible f ij = 0 for i = 1 or j = 1, fragmentation into larger particles is size-independent f ij = f for 1 < i , j < n , and fragmentation is impossible f ij = 0 for the largest particle size i , j = n . This results in the following explicit set of equations for the concentrations of sizes 1, 2, i = 3, …, n − 1, and n [28],4pt1emleft leftdc1dt=leftdivfalse(D⋅normal∇c1false)+k0−k1c1−2κc12−ac1∑j=2n−1c0.056emjleft leftdc2dt=leftdivfalse(D⋅normal∇c2…”

“…There is a large uncertainty about the true values for these rate constants and they may differ highly between variants [26]. Naturally, the choice of these rates will affect the sequence of events and the interplay of nucleation, aggregation, fragmentation and spread [28]. Here, we chose the parameter values such that their order of magnitude closely followed reported values in the literature [43], where the largest rate constant is the monomer production k 0 followed by the monomer clearance k 1 , the polymer clearance k i , the fragmentation f and the nucleation κ .…”

“…There is a large uncertainty about the true values for these rate constants and they may differ highly between variants [26]. Naturally, the choice of these rates will affect the sequence of events and the interplay of nucleation, aggregation, fragmentation and spread [28].…”