Protein-protein interactions in neurodegenerative diseases: A conspiracy theory

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.


Numerical verification
In this appendix we test our computational platform by recovering the basic homogeneous dynamics of the full network model. To do this we use two hypothetical sets of illustrative, non-clinical parameters; one set of parameters for each regime. We will illustrate the four possible patient states (stationary points) discussed in the Methods section (An Analysis of the continuous model). In Section the primary and secondary tauopathy (Methods, Stability) patient state transitions are simulated and model patient dynamics are discussed in more detail. Front propagation in the brain connectome network is confirmed using synthetic left-right hemisphere initial seedings as discussed in the results section.

Patient states of the network system
We now briefly illustrate the four stationary states of the homogeneous system (Methods, An Analysis of the continuous model). To demonstrate that each of the predicted stationary points is indeed a stationary point of the homogeneous network system, we select illustrative parameters that satisfy the requisite characterizing inequalities. Every node in the brain network is then seeded with the initial value corresponding to the selected fixed point. We expect, and demonstrate, that the system remains stable at that fixed point. We will confirm the stationary points by selecting the effective diffusion constant, ρ of (7), as unity and solving (8)-(12) for t ∈ [0, 10] using one thousand time-steps. For the healthy Aβ-healthy τ P state, c.f. (14), we select a 0 = 0.75 and b 0 = 0.5; all other parameters are set to unity. All nodes were seeded with the corresponding initial value Figure 26a shows the plot of global mean tracer concentration with time and confirms that the healthy Aβ-healthy τ P state is stationary under the given conditions. For the healthy τ P-toxic Aβ fixed point, c.f. (15), we begin with the previous parameters and reduce the toxic Aβ clearance by 40%. We therefore haveã 1 = 0.6 and keep the previous parameters fixed. We then have The stationary behavior is again demonstrated; c.f. Figure 26b. For the third stationary state, given by (16), we begin once more with the parameters of the healthy Aβ-healthy τ P state and reduce the toxic tau clearance parameter by 60%. We then haveb 1 = 0.4 and keep all other parameters as in the healthy Aβ-healthy τ P state. All nodes are then set to the corresponding initial value Once more, Figure 26c, we see the stationary characteristic we expect. For the final stationary point, c.f. (19), we use the reduced toxic clearance parameters from the second and third stationary points above,ã 1 = 0.6 andb 1 = 0.4, in addition to the original production values, a 0 = 0.75 and b 0 = 0.5, of Aβ and τ P respectively. All other parameters not explicitly mentioned are again taken to be unity. Given these choices we can directly compute µ and v 4 , via (18)-(19), as Using the above, along with the expressions for v 1 , v 3 , u 1 , u 2 andũ 2 from (14)- (16), the value ofṽ 4 is given directly from the fourth entry of (19) as (u 4 ,ũ 4 , v 4 ,ṽ 4 ) = (u 2 ,ũ 2 , v 4 ,ṽ 4 ) = (0.6, 0.25, 0.32, 0.45).
The final plot, for the fourth stationary point, is shown in Figure 26d. Coronal and sagittal plane views of the stationary point verification computation at t = 10 are shown in Figure 27.

Patient pathology transitions of the network system
We briefly illustrate the homogeneous state dynamics of the network system; verifying the theoretical view of advanced in the Methods section (Methods, Stability and Disease Phenomenology) on the complex brain network geometry of Figure 21.

Primary tauopathy
We consider a hypothetical susceptible model patient characterized by the parameters previously chosen (S1 Appendix, Patient states of the network system). All four of the stationary points (Methods, An Analysis of the continuous model) coexist with this choice of parameters; hence, these parameters fall into the regime of primary tauopathy. In this section we verify the homogeneous state transitions, between the states of Figure 27, of (8)-(12) discretized on the brain network geometry of Figure 21. The selected illustrative primary tauopathy parameters are collected in Table 1 for posterity. The eigenvalues, (21) and (22), at the healthy Aβ-healthy τ P stationary point (u,ũ, v,ṽ) = (0.75, 0, 0.5, 0) can be calculated. We see that λ Aβ,1 , λ τ P,1 < 0, i.e. stable to healthy Aβ and τ P perturbations, while λ Aβ,2 , λ τ P,2 > 0 so that the otherwise healthy patient brain is susceptible to perturbations in both toxic Aβ and toxic τ P. Utilizing the given parameters to evaluate the stability properties at the second stationary point, (u,ũ, v,ṽ) = (0.6, 0.25, 0.5, 0) c.f. (15), we have λ Aβ,1 , λ Aβ,2 , λ τ P,1 < 0 and λ τ P,2 > 0; at this state the patient is susceptible only to a perturbation in toxic tau. Likewise at the third stationary point, (u,ũ, v,ṽ) = (0.75, 0, 0.4, 0.25) c.f. (16), we have λ Aβ,1 , λ τ P,1 , λ τ P,2 < 0 and λ Aβ,2 > 0 so that the patient in this state is only susceptible to an addition of toxic Aβ. Finally the fixed point (17) is fully stable, i.e. all eigenvalues are negative, and no further disease transition is possible from this state.
Verifications of the primary tauopathy homogeneous state transitions, first depicted in Figure 22, for the full connectome simulation are shown in Figure 28. For instance the healthy state, (u 1 ,ũ 1 , v 1 ,ṽ 1 ), perturbation with respect to both toxic Aβ and toxic τ P results in the fully toxic state, (u 4 ,ũ 4 , v 4 ,ṽ 4 ); this is shown in Figure 28c and appears in Figure 22 as the blue (diagonal) path.

Secondary tauopathy
The secondary tauopathy disease model arises when v 1 < v 3 , so that the stationary point (16) is in an unphysical state, while (14), (15) and (17) remain well defined. One   Fig 29. Hτ P-HAβ,ṽ stable way that this can be achieved is for b 3 , the coefficient mediating the effect of toxic Aβ protein on inducing healthy tau toxification, to be such that both v 4 < v 1 and v 4 < v 3 ; a decrease in b 2 can also accomplish this goal, c.f. (17).
(a) Hτ P-HAβ to Tτ P-TAβ (b) Hτ P-HAβ to Hτ P-TAβ (c) Hτ P-TAβ to Tτ P-TAβ  The condition v 1 < v 3 is equivalent to b 0 b 2 <b 1 b 1 . One can transform the primary tauopathy patient described by the parameters of Table 1 to a secondary tauopathy patient by decreasing b 2 by twenty-five percent; from 1.0 to 0.75. In this regime v 1 = 0.5 and v 3 = 0.53 and the stationary point (16)  We see that the first and second stationary points are identical to the case of primary tauopathy and the fourth is perturbed in the (v,ṽ) components. Strictly speaking, the healthy patient in this regime is susceptible only to toxic Aβ infection; that is λ Aβ,1 , λ τ P,1 , λ τ P,2 < 0 and λ Aβ,2 > 0 at (u 1 ,ũ 1 , v 1 ,ṽ 1 ). Verification of the healthy state robustness to perturbations in toxic tau,ṽ, is shown in Figure 29.
At the healthy state λ Aβ,2 > 0 holds. Thus, the susceptible, but otherwise healthy, secondary tauopathy patient is at risk of directly developing Aβ proteopathy. This is verified by perturbing the healthy state by a small concentration inũ; the pursuant transition from the Healthy τ P-Healthy Aβ state to the Healthy τ P-Toxic Aβ state is pictured in Figure 30b. Having arrived at (u 2 ,ũ 2 , v 2 ,ṽ 2 ) the patient is now susceptible to tauopathy as λ τ P,2 > 0 there; perturbingṽ then develops to the Toxic τ P-Toxic Aβ state as shown in Figure 30c.
In fact, as postulated, (Results, Stability and Disease Phenomenology Figure 23) the fully diseased state (u 4 ,ũ 4 , v 4 ,ṽ 4 ) is reachable from the healthy state provided that toxic Aβ is present alongside some toxic tau perturbation. This can be seen directly from λ τ P,2 in (22). Consider the Taylor expansion of (22), evaluated with b 2 = 0.75 and all other parameters as in Table 1, aboutṽ = 0. We first set θ =ũ + 0.6 and we let 0 ≤ � � 1 be denote a small perturbation inṽ. It is evident that the effect on λ τ P,2 due to a perturbation in toxic tau depends here on both toxic amyloid,ũ, and healthy tau, v, concentration levels. Then, using thatũ ≥ 0, and v ≥ 0, we approximate (22), to order � 2 , aroundṽ = 0 by If we presume, for instance, that the susceptible secondary tauopathy patient has healthy levels of tau protein, i.e. that v = v 1 = 0.5, we can directly visualize the effect of toxic Aβ on λ τ P,2 . Figure 31 shows the approximate value of λ τ P,2 (y-axis, c.f. (35)) versus the toxic Aβ value θ(ũ) =ũ + 0.75 (x-axis) for three given perturbations �.
Evidently, as � decreases the effect ofũ on increasing λ τ P,2 is not diminished. Thus an initial toxic τ P seed will develop into a full blown infection providedũ is present, or quickly develops, in sufficient quantity to evolve λ τ P,2 above zero. This is precisely the predicted behavior (Results, Figure 23). In accordance we see, c.f. Figure 30a, that perturbing bothũ andṽ simultaneously from the initial healthy state induces direct evolution to fully diseased state.