A Mathematical Model for the Proliferation, Accumulation and Spread of Pathogenic Proteins Along Neuronal Pathways with Locally Anomalous Trapping

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There is growing evidence that many neurodegenerative disease processes involve the proliferation, accumulation and spread of pathogenic proteins. The transport of proteins in the brain is typically hindered on small scales by micro-domain traps and binding sites but it may be enhanced on larger scales by neuronal pathways identified as white matter transport networks. We have introduced a mathematical network model to simulate a pathogenic protein neurodegenerative disease in the brain taking into account the anomalous transport. The proliferation and accumulation of pathogenic proteins is modelled using a set of reaction kinetics equations on the nodes of a network. Transport of the proteins on the network is modelled as a continuous time random walk with power law distributed waiting times on the nodes. This power law waiting time distribution is shown to be consistent with anomalously slowed diffusion on local scales but transport is enhanced on larger scales by the jumps between nodes. The model reveals that the disease spreads as a propagating front throughout the brain. The anomalous behaviour leads to a lessor variation in the concentration of pathogenic proteins. The enhanced transport on the network ensures that the approach to equilibrium is dominated by the short time behaviour of the waiting time density, hence the effects of subdiffusion are not as pronounced as in a spatial continuum.