Protein-protein interactions in neurodegenerative diseases: A conspiracy theory

Abstract

Neurodegenerative diseases such as Alzheimer's or Parkinson's are associated with the prion-like propagation and aggregation of toxic proteins. A long standing hypothesis that amyloid-beta drives Alzheimer's disease has proven the subject of contemporary controversy; leading to new research in both the role of tau protein and its interaction with amyloid-beta. Conversely, recent work in mathematical modeling has demonstrated the relevance of nonlinear reaction-diffusion type equations to capture essential features of the disease. Such approaches have been further simplified, to network-based models, and offer researchers a powerful set of computationally tractable tools with which to investigate neurodegenerative disease dynamics. Here, we propose a novel, coupled network-based model for a two-protein system that includes an enzymatic interaction term alongside a simple model of aggregate transneuronal damage. We apply this theoretical model to test the possible interactions between tau proteins and amyloid-beta and study the resulting coupled behavior between toxic protein clearance and proteopathic phenomenology. Our analysis reveals ways in which amyloid-beta and tau proteins may conspire with each other to enhance the nucleation and propagation of different diseases, thus shedding new light on the importance of protein clearance and protein interaction mechanisms in prion-like models of neurodegenerative disease.

Fig 1. Front propagation in primary tauopathy; synthetic rectangular domain.

Each subfigure consists of a toxic Aβ concentration distribution (top left), toxic τP concentration distribution (bottom left) and a plot (solid line: Aβ, dashed line: τP) of the concentration level on the x−axis. Dark blue indicates the minimum concentration of c = 0.0 while bright red indicates the maximum of c = 0.5. See the Methods section (Front propagation) for a comparison to theory. (See also: supplementary S1 Video).

Fig 2. Front propagation in primary tauopathy; brain connectome.

Each subfigure consists of a toxic Aβ concentration distribution (subfigure left) besides a toxic τP concentration distribution (subfigure right). Dark blue indicates the minimum concentration of c = 0.0 while bright red indicates the maximum of c = 0.5. (See also: supplementary S2 Video and supplementary fle S2 Data).

Fig 3. Front propagation in secondary tauopathy; rectangular domain.

Each subfigure consists of a toxic Aβ concentration distribution (top left), toxic τP concentration distribution (bottom left) and a plot (solid line: Aβ, dashed line: τP) of the concentration level on the x−axis. Dark blue indicates the minimum concentration of c = 0.0 while bright red indicates the maximum of c = 0.5 for toxic Aβ and $c=0.58\overline{3}=7/12$ for toxic τP. See the Methods section (Front propagation) for a comparison to theory. (See also: supplementary S3 Video).

Fig 4. The onset effect due to b₃ in secondary tauopathy.

Saturation % (y-axis) vs Simulation time (x-axis).

Fig 5. Front propagation in secondary tauopathy; brain connectome.

Each subfigure consists of a toxic Aβ concentration distribution (subfigure left) besides a toxic τP concentration distribution (subfigure right). Dark blue indicates the minimum concentration of c = 0.0 while bright red indicates the maximum of c = 0.5 for toxic Aβ and $c=0.58\overline{3}=7/12$ for toxic τP. (See also: supplementary S4 Video and supplementary S2 Data).

Fig 6. Simulated seeding sites for a model of Alzheimer’s disease.

Toxic Aβ (left) and toxic τP (right) seeding sites.

Fig 7. Characteristic progression of of Aβ and τP lesions.

3-stage Aβ (top) progression and τP NFT (bottom) progression.

Fig 8. Protein-protein interaction in primary tauopathy.

Fig 9. Toxic proteopathy progression dynamics in the primary tauopathy patient.

Toxic Aβ (top row) and opacity exaggerated toxic Aβ progression (second row); Toxic τP (third row) and opacity exaggerated toxic τP progression (last row). Color scale is identical to Fig 1. (See also: supplementary S5 Video, supplementary S6 Video and supplementary S2 Data).

Fig 10. Protein-protein interaction in secondary tauopathy.

Fig 11. Prodromal window variations with b₃, secondary tauopathy.

Invasion starting (left) and ending (right) time vs. b₃.

Fig 12. Toxic τP progression dynamics in the secondary tauopathy patient.

Toxic τP (first row) and opacity exaggerated toxic τP progression (second row). Color scale is identical to the τP case of Fig 3. (See also: supplementary S5 Video, supplementary S7 Video and supplementary S2 Data).

Fig 13. Skull-stripped, cross-sectional Alzheimer’s patient cohort SUVR intensity.

Top row: averaged SUVR data is shown. Bottom row: top 30% of SUVR intensities are visible. For both rows: (left side) 18F-AV45 florbetapir Aβ radiotracer SUVR and (right side) 18F-AV-1451 flortaucipir τP radiotracer SUVR. Darker colors correspond to higher SUVR values.

Fig 14. A connectome-graph view of the normalized patient SUVR data.

The (left side) 18F-AV45 florbetapir Aβ radiotracer SUVR and (right side) 18F-AV-1451 flortaucipir τP radiotracer SUVR. Highest 30% of connectome regional values are visible. Darker colors correspond to higher SUVR values.

Fig 15. Results of a mixed-modality simulation.

(left) Toxic Aβ population and (right) toxic τP population are shown at time t = 78. The top 30% of nodal values are visible; darker colors correspond to higher values.

Fig 16. Aggregate damage in primary and secondary tauopathy.

Aggregate damage (dashed; except k₄ = 1 × 10⁻³ solid, red) curves in the base primary (a) and secondary (b) tauopathy patients. Damage with increase toxic protein interaction, b₃, in primary (c) and secondary (d) tauopathy.

Fig 17. Damage progression in primary tauopathy.

Horizontal plane view (top row) with opacity exaggerated (second row) progression. sagittal view (third row) with opacity exaggerated (fourth row) progression. Dark blue indicates the minimal damage value of q = 0.0; bright red indicates the maximum of q = 1.0. Intermediate values are: purple (q = 0.14), sky blue (q = 0.29), green (q = 0.43), yellow (q = 0.57), orange (q = 0.71), and dark red (q = 0.86).

Fig 18. Damage progression in secondary tauopathy.

Horizontal plane view (top row) with opacity exaggerated (second row) progression. sagittal view (third row) with opacity exaggerated (fourth row) progression. The color scale is identical to that of Fig 17.

Fig 19. Toxic τP average regional concentration; six fixed time points.

83 (left) versus 1015 (right) vertex connectomes.

Fig 20. Kinetics of the heterodimer model.

Healthy protein (blue) and misfolded toxic protein (red) transition to two toxic proteins (long arrow) via, from left to right, the kinetics of: recruitment, induced misfolding, and fragmentation.

Fig 21. A high-resolution brain structural connectome graph.

(Bottom left) The average of 419 brain connectomes with V = 1, 015 vertices spanning (bottom right) 49 associated brain regions; the strongest 2,773 edge connections are shown. The weighted adjacency matrix (top) corresponding to the averaged connectome (bottom).

Fig 22. Patient pathology dynamics in primary tauopathy.

(Left) Phase plane $(\tilde{u},\tilde{v})$ with four equilibria. Homogeneous dynamics of the toxic states. Note that this is a two-dimensional slice of the four-dimensional phase space. (Right) When four different states co-exist, only the fully toxic state is stable as shown by the time-dynamics plot. (Parameters: a₀ = b₀ = a₁ = a₂ = b₁ = b₂ = 1, ${\tilde{a}}_1={\tilde{b}}_1=3/4$, b₃ = 1/2).

Fig 23. Patient pathology dynamics in secondary tauopathy.

(Left) Phase plane $(\tilde{u},\tilde{v})$ with three equilibria. (Right) When three different states co-exist, only the fully toxic state is stable as shown by the time-dynamics plot. (Parameters: a₀ = b₀ = a₁ = a₂ = b₁ = b₂ = 1, ${\tilde{a}}_1=3/4$, ${\tilde{b}}_1=4/3$, b₃ = 3). Note that trajectories are initialized by taking the initial condition ϵ = 0.005 away from an equilibrium point.

Fig 24. Front dynamics, primary tauopathy.

Fig 25. Front dynamics, secondary tauopathy.