A network-level transport model of tau progression in the Alzheimer's brain

TL;DR

A macroscopic version of a mathematical model of the axonal transport of toxic tau proteins that takes into account the effects tau exerts on the molecular motors is described and implemented, which is called the Network Transport Model (NTM).

Abstract

One of the hallmarks of Alzheimer's disease (AD) is the accumulation and spread of toxic aggregates of tau protein. The progression of AD tau pathology is thought to be highly stereotyped, which is in part due to the fact that tau can spread between regions via the white matter tracts that connect them. Mathematically, this phenomenon has been described using models of "network diffusion", where the rate of spread of tau between brain regions is proportional to its concentration gradient and the amount of white matter between them. Although these models can robustly predict the progression of pathology in a wide variety of neurodegenerative diseases, including AD, an under explored aspect of tau spreading is that it is governed not simply by diffusion but also active transport along axonal microtubules. Spread can therefore take on a directional bias, resulting in distinct patterns of deposition, but current models struggle to capture this phenomenon. Recently, we have developed a mathematical model of the axonal transport of toxic tau proteins that takes into account the effects tau exerts on the molecular motors. Here we describe and implement a macroscopic version of this model, which we call the Network Transport Model (NTM). A key feature of this model is that, while it predicts tau dynamics at a regional level, it is parameterized in terms of only microscopic processes such as aggregation and transport rates; that is, differences in brain-wide tau progression can be explained by its microscopic properties. We provide numerical evidence that, as with the two-neuron model that the NTM extends, there are distinct and rich dynamics with respect to the overall rate of spread and the staging of pathology when we simulated the NTM on the hippocampal subnetwork. The theoretical insights provided by the NTM have broad implications for understanding AD pathophysiology more generally.

Figures (8)

• Figure 1: Illustration of the model system. At the whole network level, brain regions are represented by nodes and white matter fiber projections between them by edges (top panel). Tau pathology propagates on this network in an anterograde or retrograde direction, depending on the cell polarity and the properties of tau itself. Instead of passive graph diffusion previously used to model the transmission along an edge, here we use the active axonal transport model from Torok, et al., which is schematized version in the bottom panel. It models two distinct species of pathological tau, soluble (red) and insoluble (blue), within a multi-compartment, two-neuron system mimicking the single-edge system shown in the top panel. The main biological phenomena captured in this model are diffusion (blue box), active transport (green box), species interconversion through fragmentation and aggregation (purple box), and a diffusion-based barrier to inter-compartmental spread (brown dashed lines). The full model involves iteratively solving the PDE-based, single-edge model for the fluxes at the boundaries of the system and the resulting solving the mass exchange problem at the nodes (right panel). Sol. - soluble tau; Ins. - insoluble tau; Diff. - diffusion; Conv. - tau interconversion; Diff. Barrier - diffusion barrier; Presyn. SD - presynaptic somatodendritic compartment; AIS - axon initial segment; SC - synaptic cleft; Postsyn. SD - postsynaptic somatodendritic compartment. Figure adapted from the original manuscript Torok2021.
• Figure 2: Comparison of end-timepoint simulations versus the analytical steady-state for the single-edge model. A. Spatial distributions of soluble ($n$) and insoluble ($m$) tau across compartments after simulating the Torok et al. single-edge mode to $t > 12$ months. B. Spatial distributions of soluble ($n$) and insoluble ($m$) tau across compartments for the derived steady-state solution of the Torok et al. model.
• Figure 3: Effect of $\lambda$ on NTM simulations. A. Total concentration of tau plotting against time in each of the 30 regions for the low lambda condition ($\lambda_{1} = \lambda_{2} = 0.005$). Left panel includes the seed region (left lateral part of the entorhinal cortex), right panel excludes the seed region. B. Tau concentration over time for the high lambda condition ($\lambda_{1} = \lambda_{2} = 0.1$). C. Heatmap representation of the per-region simulations shown in A., with the seed region excluded. D. Heatmap representation of the per-region simulations shown in B., with the seed region excluded.
• Figure 4: Effect of $\delta$ and $\epsilon$ on NTM simulations. A. Total concentration of tau plotting against time in each of the 30 regions for the anterograde-biased condition ($\delta = 100, \epsilon = 10$). Left panel includes the seed region (left lateral part of the entorhinal cortex), right panel excludes the seed region. B. Tau concentration over time for the retrograde-biased condition ($\delta = 10, \epsilon = 100$). C. Heatmap representation of the per-region simulations shown in A., with the seed region excluded. D. Heatmap representation of the per-region simulations shown in B., with the seed region excluded.
• Figure 5: Three-dimensional visualization of anterograde and retrograde biases A. Total concentration of tau over time, visualized as spheres centered in each region for the anterograde simulation shown in Figure \ref{['fig:antretmodels']}A, where sphere size is proportional to the amount of pathology. Arrows represent the directions and strengths of the upper 10% of fluxes between regions at each time point. All regions are colored in blue, with the exception of the seed region (in green) and the three non-seed regions with the most tau pathology at each time point (in magenta). B. Total concentration of tau per region in the retrograde simulation (Figure \ref{['fig:antretmodels']}B), with the same visualization conventions as in A.