Table 1.
Glossary of symbols used throughout the text.
Values marked with an asterisk were estimated by Konzack et al. [34] in cultured rodent neurons. Ant. = anterograde, ret. = retrograde, conc. = concentration, vel. = velocity.
Table 2.
Table indicating which processes are modeled in each biological compartment.
See Methods: Model Description for more details.
Fig 1.
Schematized version of the one-dimensional system that we simulate. We model two distinct species of pathological tau, soluble (red) and insoluble (blue), across within a multi-compartment model mimicking the two-neuron 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).
Fig 2.
Model regimes are determined by tau transport feedback parameters δ and ϵ.
(a) Simulation results where δ = 1 and ϵ = 0.01, which leads to a strong anterograde bias that emerges within hours and persists even at longer time scales. (b) Simulation results where δ = 0.01 and ϵ = 1, which leads to a strong retrograde bias that only emerges at intermediate-to-late time scales. (c) Simulation results where δ = 1 and ϵ = 0.35, which leads to an initial anterograde bias that is counteracted at intermediate time scales, leading to a uniform distribution of tau deposition at steady state.
Fig 3.
Temporal profiles of somatodendritic tau deposition.
(a) Time course of mean soluble (red) and insoluble (blue) tau deposition in the presynaptic (solid lines) and postsynaptic (dashed lines) SD compartments for each of the previous simulation conditions. (b) Total presynaptic (purple solid line) and postsynaptic (purple dashed line) tau can be used to calculate a net bias at each time point (magenta dotted line; Eq 15). The balance between transport parameters δ and ϵ determines the compartment in which tau preferentially accumulates and if there is a net bias over time. (c) Schematized versions of the end configurations of the system for each parameter regime.
Fig 4.
Steady-state bias analysis as a function of aggregation rate.
(a) Steady-state bias (postsynaptic SD tau—presynaptic SD tau / total SD tau) across a range of δ and ϵ parameter values where all other parameter values are identical to those of the previous simulations. There is a zero-bias linear manifold that emerges, whose best-fit line has a slope of approximately 2.8. (b) Steady-state bias for the same range of δ and ϵ parameter values where aggregation rate (γ) is doubled. The linear zero-bias manifold has a slope of ∼5.8, roughly twice that of the original aggregation rate. (c) Steady-state bias for the same range of δ and ϵ parameter values where γ is halved. Here the slope of the linear manifold is ∼1.4, or approximately half that of the original aggregation rate.
Fig 5.
Perturbation of initial conditions does not affect steady state.
We plot the relative pairwise error between model instances with randomly generated initial conditions using (a) the anterograde, (b) the retrograde, or (c) the net unbiased parameterizations (gray lines are representative sample traces). There is universal convergence at long time scales, suggesting that for these parameter values the model has a single fixed point.
Fig 6.
Tau transport feedback recaptures the directionality of mouse tauopathy models.
(a) The AD-like mouse models explored by [20] exhibit a strong retrograde bias that becomes more pronounced over time, which can be replicated in the two-neuron system with weak anterograde-directed transport feedback (low δ relative to ϵ). (b) The non-AD-like mouse models similarly have a trend towards increasing retrograde bias, although to a lesser extent than the AD-like studies and there is evidence of early anterograde bias, which is captured by fixing both δ and ϵ at high values. For all studies, we first linearly transform the bias parameter, s, used by [20] onto the [−1, 1] scale of our SD bias estimates before plotting. Refer to S2 Table for a full parameterization of both simulations and Methods: Analysis. Studies included: [11, 13, 36, 37].