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

Abstract

The prion hypothesis states that misfolded proteins can act as infectious agents that template the misfolding and aggregation of healthy proteins to transmit a disease. Increasing evidence suggests that pathological proteins in neurodegenerative diseases adopt prion-like mechanisms and spread across the brain along anatomically connected networks. Local kinetic models of protein misfolding and global network models of protein spreading provide valuable insight into several aspects of prion-like diseases. Yet, to date, these models have not been combined to simulate how pathological proteins multiply and spread across the human brain. Here, we create an efficient and robust tool to simulate the spreading of misfolded protein using three classes of kinetic models, the Fisher-Kolmogorov model, the Heterodimer model and the Smoluchowski model. We discretize their governing equations using a human brain network model, which we represent as a weighted Laplacian graph generated from 418 brains from the Human Connectome Project. Its nodes represent the anatomic regions of interest and its edges are weighted by the mean fibre number divided by the mean fibre length between any two regions. We demonstrate that our brain network model can predict the histopathological patterns of Alzheimer's disease and capture the key characteristic features of finite-element brain models at a fraction of their computational cost: simulating the spatio-temporal evolution of aggregate size distributions across the human brain throughout a period of 40 years takes less than 7 s on a standard laptop computer. Our model has the potential to predict biomarker curves, aggregate size distributions, infection times, and the effects of therapeutic strategies including reduced production and increased clearance of misfolded protein.

Keywords: Alzheimer’s disease; connectome; graph Laplacian; network; neurodegeneration; prion.

Figure 1.

Typical pattern of tau protein misfolding in Alzheimer’s disease. (a) Clinical observation [15], (b) continuum model [16] and (c) network spreading model display characteristic pattern with misfolded tau occurring first in the locus coeruleus and transentorhinal layer from where they spread to the transentorhinal region and the proper entorhinal cortex and ultimately affect all interconnected neocortical brain regions. (Online version in colour.)

Figure 2.

Kinetics of the Fisher–Kolmogorov model. The Fisher–Kolmogorov model has a single unknown, the misfolded protein concentration c. The model converts healthy to misfolded protein at a rate α. For the smallest perturbation from the healthy state, c > 0, all proteins will convert from the healthy to the misfolded state, c = 1. (Online version in colour.)

Figure 3.

Kinetics of the Heterodimer model. The Heterodimer model has two unknowns, the healthy concentration p and the misfolded concentration $\tilde{p}$. The model produces healthy protein at a rate k₀, clears healthy and misfolded protein at rates k₁ and ${\tilde{k}}_1$, and converts healthy to misfolded protein at a rate k₁₂, which collectively represents the processes of recruitment k_11′, misfolding k_1′2′ and fragmentation k_2′2. (Online version in colour.)

Figure 4.

Kinetics of the Smoluchowski model. The Smoluchowski model has n unknowns c_i, one for the concentration of each size i = 1, …, n, where c₁ represents the concentration of healthy monomers. The model produces healthy protein at a rate k₀, clears healthy and misfolded protein at rates k₁ and k_i, nucleates two misfolded proteins from two healthy proteins at a rate κ, aggregates by adding single healthy proteins to misfolded filaments at a rate a, and fragments misfolded filaments at a rate f. (Online version in colour.)

Figure 5.

Brain network model. Misfolded tau proteins spread across the brain’s connectome represented as a weighted graph $\mathcal{G}$ with N = 83 nodes and E = 1130 edges. Edges are weighted by the mean fibre number n_IJ divided by the mean fibre length l_IJ averaged over 418 healthy brains from the Human Connectome Project. (Online version in colour.)

Figure 6.

Brain network model. The connectivity of the graph $\mathcal{G}$ is represented through the degree D_II, the number of edges per node, and the adjacency A_IJ = n_IJ/l_IJ, the ratio of fibre number and length. (a) Degree D_II of non-weighted and (b) connectivity-weighted graphs and (c) adjacency A_IJ of connectivity-weighted graph, averaged over 418 healthy brains from the Human Connectome Project. (Online version in colour.)

Figure 7.

Biomarker abnormality of Fisher–Kolmogorov model. Integrating the concentration of misfolded proteins c across individual lobes reveals the characteristic activation sequence in Alzheimer’s disease from the temporal lobe to the frontal, parietal and occipital lobes. The dashed grey and black lines highlight the biomarker abnormality $\mathcal{C}$ of the Fisher–Kolmogorov network and continuum model in figure 1 integrated across the entire brain. (Online version in colour.)

Figure 8.

Biomarker abnormality of Heterodimer model. Integrating the concentration of misfolded proteins $\tilde{p}$ across individual lobes reveals the characteristic activation sequence in Alzheimer’s disease from the temporal lobe to the frontal, parietal and occipital lobes. The dashed grey and black lines highlight the biomarker abnormality $\tilde{\mathcal{P}}$ of the Heterodimer network and continuum model in figure 1 integrated across the entire brain. (Online version in colour.)

Figure 9.

Infection times. Biomarker curves for misfolded protein seeding in N = 83 seeding regions illustrate the regional vulnerability of the network model. Misfolded proteins spread fastest when seeded in the putamen or insula, red spheres and curves, and slowest when seeded in the frontal pole and entorhinal region, blue spheres and curves. The dashed grey line highlights the lower limit of the infection time associated with a homogeneous seeding across all regions. (Online version in colour.)

Figure 10.

Delaying conversion. The Fisher–Kolmogorov model predicts that lower conversion rates α delay the increase of misfolded protein c. Baseline Alzheimer’s disease (a) and Alzheimer’s disease with moderately (b) and markedly (c) reduced conversion α from the healthy to the misfolded state. (Online version in colour.)

Figure 11.

Reducing biomarker abnormality through delayed conversion. Decreasing the conversion α delays the accumulation of the misfolded protein concentration c and with it the biomarker abnormality $\mathcal{C}$. Irrespective of the conversion rate α, the misfolded protein concentration of the Fisher–Kolmogorov model always converges towards the fully misfolded state with a biomarker abnormality of $\mathcal{C}=100\text{\%}$. (Online version in colour.)

Figure 12.

Reducing misfolding. The Heterodimer model predicts that lower turnover rates k₁₂ delay and reduce the accumulation of misfolded protein $\tilde{p}$. Baseline Alzheimer’s disease (a) and Alzheimer’s disease with moderately (b) and markedly (c) reduced turnover k₁₂ from the healthy to the misfolded state. (Online version in colour.)

Figure 13.

Reducing biomarker abnormality through reduced misfolding. Decreasing the turnover k₁₂ delays and reduces the accumulation of misfolded tau protein $\tilde{p}$ and with it the biomarker abnormality $\tilde{\mathcal{P}}$. Depending on the turnover rate k₁₂, the misfolded protein concentration of the Heterodimer model can converge towards intermediate states with a reduced biomarker abnormality of $\tilde{\mathcal{P}}<100\text{\%}$. (Online version in colour.)

Figure 14.

Increasing clearance. The Heterodimer model predicts that higher clearance rates ${\tilde{k}}_1$ delay and reduce the accumulation of misfolded protein $\tilde{p}$. Baseline Alzheimer’s disease (a) and Alzheimer’s disease with moderately (b) and markedly (c) increased clearance ${\tilde{k}}_1$ of misfolded protein. (Online version in colour.)

Figure 15.

Reducing biomarker abnormality through increased clearance. Increasing the clearance ${\tilde{k}}_1$ delays and reduces the accumulation of misfolded tau protein $\tilde{p}$ and with it the biomarker abnormality $\tilde{\mathcal{P}}$. Depending on the clearance rate ${\tilde{k}}_1$, the misfolded protein concentration of the Heterodimer model can converge towards intermediate states with a reduced biomarker abnormality of $\tilde{\mathcal{P}}<100\text{\%}$. (Online version in colour.)

Figure 16.

Biomarker spectrum of Smoluchowski model. Integrating the mass of misfolded proteins i c_i larger than a critical toxic size i reveals the characteristic nucleation, aggregation, fragmentation dynamics in Alzheimer’s disease. The red line ${\mathcal{M}}_2$ indicates the total mass of all misfolded proteins c_i ≥ 2; the blue line ${\mathcal{M}}_{50}$ indicates the mass of all aggregates of the largest size c_i = 50. The dashed black line highlights the biomarker abnormality $\mathcal{C}$ of the Fisher–Kolmogorov model in figure 7 and $\tilde{\mathcal{P}}$ of the two-concentration model in figure 8. (Online version in colour.)

Figure 17.

Emerging aggregate size distribution. During the early stages of infection, the mean aggregate size increases; during the later stages, it decreases gradually towards a homeostatic mean. Colour-coded lines indicate the evolution of aggregate size and frequency; dots show the evolution of the mean aggregate size; dashed black line highlights the analytical solution [43]. (Online version in colour.)

Figure 18.

Typical pattern of tau aggregation in Alzheimer’s disease. The Smoluchowski model predicts that misfolded tau proteins occur first in the locus coeruleus and transentorhinal layer from where they spread to the transentorhinal region and the proper entorhinal cortex and ultimately affect all interconnected neocortical brain regions. Smaller aggregates emerge first (a) and grow progressively into moderate (b) and large (c) aggregates. (Online version in colour.)

Figure 19.

Reducing biomarker spectrum through size-targeted clearance. Increasing the clearance k_i of a single specific aggregate size delays and reduces the accumulation of misfolded tau protein c_i and with it the biomarker spectrum $\mathcal{M}$. Increasing the target size of clearance, here from c₂ to c₆ decelerates and reduces protein misfolding and narrows the overall size distribution. The red curves highlight the biomarker spectrum for the baseline model with a homogeneous clearance in figure 16. (Online version in colour.)