Simulating the effect of formation of amyloid plaques on aggregation of tau protein

Numerical and Analytical Simulation of the Growth of Amyloid-β Plaques

Numerical and analytical solutions were employed to calculate the radius of an amyloid-β (Aβ) plaque over time. To the author's knowledge, this study presents the first model simulating the growth of Aβ plaques. Findings indicate that the plaque can attain a diameter of 50 μm after 20 years of growth, provided the Aβ monomer degradation machinery is malfunctioning. A mathematical model incorporates nucleation and autocatalytic growth processes using the Finke-Watzky model. The resulting system of ordinary differential equations was solved numerically, and for the simplified case of infinitely long Aβ monomer half-life, an analytical solution was found. Assuming that Aβ aggregates stick together and using the distance between the plaques as an input parameter of the model, it was possible to calculate the plaque radius from the concentration of Aβ aggregates. This led to the "cube root hypothesis," positing that Aβ plaque size increases proportionally to the cube root of time. This hypothesis helps explain why larger plaques grow more slowly. Furthermore, the obtained results suggest that the plaque size is independent of the kinetic constants governing Aβ plaque agglomeration, indicating that the kinetics of Aβ plaque agglomeration is not a limiting factor for plaque growth. Instead, the plaque growth rate is limited by the rates of Aβ monomer production and degradation.

Why slow axonal transport is bidirectional - can axonal transport of tau protein rely only on motor-driven anterograde transport?

Slow axonal transport (SAT) moves multiple proteins from the soma, where they are synthesized, to the axon terminal. Due to the great lengths of axons, SAT almost exclusively relies on active transport, which is driven by molecular motors. The puzzling feature of slow axonal transport is its bidirectionality. Although the net direction of SAT is anterograde, from the soma to the terminal, experiments show that it also contains a retrograde component. One of the proteins transported by SAT is the microtubule-associated protein tau. To better understand why the retrograde component in tau transport is needed, we used the perturbation technique to analyze how the full tau SAT model can be simplified for the specific case when retrograde motor-driven transport and diffusion-driven transport of tau are negligible and tau is driven only by anterograde (kinesin) motors. The solution of the simplified equations shows that without retrograde transport the tau concentration along the axon length stays almost uniform (decreases very slightly), which is inconsistent with the experimenal tau concentration at the outlet boundary (at the axon tip). Thus kinesin-driven transport alone is not enough to explain the empirically observed distribution of tau, and the retrograde motor-driven component in SAT is needed.

Personalized Computational Causal Modeling of the Alzheimer Disease Biomarker Cascade

BACKGROUND

Mathematical models of complex diseases, such as Alzheimer's disease, have the potential to play a significant role in personalized medicine. Specifically, models can be personalized by fitting parameters with individual data for the purpose of discovering primary underlying disease drivers, predicting natural history, and assessing the effects of theoretical interventions. Previous work in causal/mechanistic modeling of Alzheimer's Disease progression has modeled the disease at the cellular level and on a short time scale, such as minutes to hours. No previous studies have addressed mechanistic modeling on a personalized level using clinically validated biomarkers in individual subjects.

OBJECTIVES

This study aimed to investigate the feasibility of personalizing a causal model of Alzheimer's Disease progression using longitudinal biomarker data.

DESIGN/SETTING/PARTICIPANTS/MEASUREMENTS

We chose the Alzheimer Disease Biomarker Cascade model, a widely-referenced hypothetical model of Alzheimer's Disease based on the amyloid cascade hypothesis, which we had previously implemented mathematically as a mechanistic model. We used available longitudinal demographic and serial biomarker data in over 800 subjects across the cognitive spectrum from the Alzheimer's Disease Neuroimaging Initiative. The data included participants that were cognitively normal, had mild cognitive impairment, or were diagnosed with dementia (probable Alzheimer's Disease). The model consisted of a sparse system of differential equations involving four measurable biomarkers based on cerebrospinal fluid proteins, imaging, and cognitive testing data.

RESULTS

Personalization of the Alzheimer Disease Biomarker Cascade model with individual serial biomarker data yielded fourteen personalized parameters in each subject reflecting physiologically meaningful characteristics. These included growth rates, latency values, and carrying capacities of the various biomarkers, most of which demonstrated significant differences across clinical diagnostic groups. The model fits to training data across the entire cohort had a root mean squared error (RMSE) of 0.09 (SD 0.081) on a variable scale between zero and one, and were robust, with over 90% of subjects showing an RMSE of < 0.2. Similarly, in a subset of subjects with data on all four biomarkers in at least one test set, performance was high on the test sets, with a mean RMSE of 0.15 (SD 0.117), with 80% of subjects demonstrating an RMSE < 0.2 in the estimation of future biomarker points. Cluster analysis of parameters revealed two distinct endophenotypic groups, with distinct biomarker profiles and disease trajectories.

CONCLUSION

Results support the feasibility of personalizing mechanistic models based on individual biomarker trajectories and suggest that this approach may be useful for reclassifying subjects on the Alzheimer's clinical spectrum. This computational modeling approach is not limited to the Alzheimer Disease Biomarker Cascade hypothesis, and can be applied to any mechanistic hypothesis of disease progression in the Alzheimer's field that can be monitored with biomarkers. Thus, it offers a computational platform to compare and validate various disease hypotheses, personalize individual biomarker trajectories and predict individual response to theoretical prevention and therapeutic intervention strategies.

Dynein Dysfunction Prevents Maintenance of High Concentrations of Slow Axonal Transport Cargos at the Axon Terminal: A Computational Study

Here, we report computational studies of bidirectional transport in an axon, specifically focusing on predictions when the retrograde motor becomes dysfunctional. We are motivated by reports that mutations in dynein-encoding genes can cause diseases associated with peripheral motor and sensory neurons, such as type 2O Charcot-Marie-Tooth disease. We use two different models to simulate bidirectional transport in an axon: an anterograde-retrograde model, which neglects passive transport by diffusion in the cytosol, and a full slow transport model, which includes passive transport by diffusion in the cytosol. As dynein is a retrograde motor, its dysfunction should not directly influence anterograde transport. However, our modeling results unexpectedly predict that slow axonal transport fails to transport cargos against their concentration gradient without dynein. The reason is the lack of a physical mechanism for the reverse information flow from the axon terminal, which is required so that the cargo concentration at the terminal could influence the cargo concentration distribution in the axon. Mathematically speaking, to achieve a prescribed concentration at the terminal, equations governing cargo transport must allow for the imposition of a boundary condition postulating the cargo concentration at the terminal. Perturbation analysis for the case when the retrograde motor velocity becomes close to zero predicts uniform cargo distributions along the axon. The obtained results explain why slow axonal transport must be bidirectional to allow for the maintenance of concentration gradients along the axon length. Our result is limited to small cargo diffusivity, which is a reasonable assumption for many slow axonal transport cargos (such as cytosolic and cytoskeletal proteins, neurofilaments, actin, and microtubules) which are transported as large multiprotein complexes or polymers.

Deep learning application for the classification of Alzheimer's disease using 18F-flortaucipir (AV-1451) tau positron emission tomography

The positron emission tomography (PET) with 18F-flortaucipir can distinguish individuals with mild cognitive impairment (MCI) and Alzheimer's disease (AD) from cognitively unimpaired (CU) individuals. This study aimed to evaluate the utility of 18F-flortaucipir-PET images and multimodal data integration in the differentiation of CU from MCI or AD through DL. We used cross-sectional data (18F-flortaucipir-PET images, demographic and neuropsychological score) from the ADNI. All data for subjects (138 CU, 75 MCI, 63 AD) were acquired at baseline. The 2D convolutional neural network (CNN)-long short-term memory (LSTM) and 3D CNN were conducted. Multimodal learning was conducted by adding the clinical data with imaging data. Transfer learning was performed for classification between CU and MCI. The AUC for AD classification from CU was 0.964 and 0.947 in 2D CNN-LSTM and multimodal learning. The AUC of 3D CNN showed 0.947, and 0.976 in multimodal learning. The AUC for MCI classification from CU had 0.840 and 0.923 in 2D CNN-LSTM and multimodal learning. The AUC of 3D CNN showed 0.845, and 0.850 in multimodal learning. The 18F-flortaucipir PET is effective for the classification of AD stage. Furthermore, the effect of combination images with clinical data increased the performance of AD classification.

Connectome-based biophysics models of Alzheimer's disease diagnosis and prognosis

With the increasing prevalence of Alzheimer's disease (AD) among aging populations and the limited therapeutic options available to slow or reverse its progression, the need has never been greater for improved diagnostic tools for identifying patients in the preclinical and prodomal phases of AD. Biophysics models of the connectome-based spread of amyloid-beta (Aβ) and microtubule-associated protein tau (τ) have enjoyed recent success as tools for predicting the time course of AD-related pathological changes. However, given the complex etiology of AD, which involves not only connectome-based spread of protein pathology but also the interactions of many molecular and cellular players over multiple spatiotemporal scales, more robust, complete biophysics models are needed to better understand AD pathophysiology and ultimately provide accurate patient-specific diagnoses and prognoses. Here we discuss several areas of active research in AD whose insights can be used to enhance the mathematical modeling of AD pathology as well as recent attempts at developing improved connectome-based biophysics models. These efforts toward a comprehensive yet parsimonious mathematical description of AD hold great promise for improving both the diagnosis of patients at risk for AD and our mechanistic understanding of how AD progresses.

Simulation of a sudden drop-off in distal dense core vesicle concentration in Drosophila type II motoneuron terminals

Recent experimental observations have shown evidence of an unexpected sudden drop-off in the dense core vesicles (DCVs) content at the ends of certain types of axon endings. This article seeks to determine whether these observations may be explained without modifying the parameters characterizing the ability of distal en passant boutons to capture and accumulate DCVs. We developed a mathematical model that is based on the conservation of captured and transiting DCVs in boutons. The model consists of 77 ordinary differential equations and is solved using a standard Matlab solver. We hypothesize that the drop in DCV content in distal boutons is due to an insufficient supply of anterogradely moving DCVs coming from the soma. As anterogradely moving DCVs are captured (and eventually destroyed) in more proximal boutons on their way to the end of the terminal, the fluxes of anterogradely moving DCVs between the boutons become increasingly smaller, and the most distal boutons are left without DCVs. We tested this hypothesis by modifying the flux of DCVs entering the terminal and found that the number of most distal boutons left unfilled increases if the DCV flux entering the terminal is decreased. The number of anterogradely moving DCVs in the axon can be increased either by the release of a portion of captured DCVs into the anterograde component or by an increase of the anterograde DCV flux into the terminal. This increase could lead to having enough anterogradely moving DCVs such that they could reach the most distal bouton and then turn around by changing molecular motors that propel them. The model suggests that this could result in an increased concentration of resident DCVs in distal boutons beginning with bouton 2 (the most distal is bouton 1). This is because in distal boutons, DCVs have a larger chance to be captured from the transiting state as they pass the boutons moving anterogradely and then again as they pass the same boutons moving retrogradely.

Emergence of directional bias in tau deposition from axonal transport dynamics

Defects in axonal transport may partly underpin the differences between the observed pathophysiology of Alzheimer's disease (AD) and that of other non-amyloidogenic tauopathies. Particularly, pathological tau variants may have molecular properties that dysregulate motor proteins responsible for the anterograde-directed transport of tau in a disease-specific fashion. Here we develop the first computational model of tau-modified axonal transport that produces directional biases in the spread of tau pathology. We simulated the spatiotemporal profiles of soluble and insoluble tau species in a multicompartment, two-neuron system using biologically plausible parameters and time scales. Changes in the balance of tau transport feedback parameters can elicit anterograde and retrograde biases in the distributions of soluble and insoluble tau between compartments in the system. Aggregation and fragmentation parameters can also perturb this balance, suggesting a complex interplay between these distinct molecular processes. Critically, we show that the model faithfully recreates the characteristic network spread biases in both AD-like and non-AD-like mouse tauopathy models. Tau transport feedback may therefore help link microscopic differences in tau conformational states and the resulting variety in clinical presentations.

Modeling tau transport in the axon initial segment

By assuming that tau protein can be in seven kinetic states, we developed a model of tau protein transport in the axon and in the axon initial segment (AIS). Two separate sets of kinetic constants were determined, one in the axon and the other in the AIS. This was done by fitting the model predictions in the axon with experimental results and by fitting the model predictions in the AIS with the assumed linear increase of the total tau concentration in the AIS. The calibrated model was used to make predictions about tau transport in the axon and in the AIS. To the best of our knowledge, this is the first paper that presents a mathematical model of tau transport in the AIS. Our modeling results suggest that binding of free tau to microtubules creates a negative gradient of free tau in the AIS. This leads to diffusion-driven tau transport from the soma into the AIS. The model further suggests that slow axonal transport and diffusion-driven transport of tau work together in the AIS, moving tau anterogradely. Our numerical results predict an interplay between these two mechanisms: as the distance from the soma increases, the diffusion-driven transport decreases, while motor-driven transport becomes larger. Thus, the machinery in the AIS works as a pump, moving tau into the axon.