Can numerical modeling help understand the fate of tau protein in the axon terminal?

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.

Computational investigation of the effect of reduced dynein velocity and reduced cargo diffusivity on slow axonal transport

Contributions of three components of slow axonal transport (SAT) were studied using a computational model: the anterograde motor (kinesin)-driven component, the retrograde motor (dynein)-driven component and the diffusion-driven component. The contribution of these three components of SAT was investigated in three different segments of the axon: the proximal portion, the central portion, and the distal portion of the axon. MAP1B protein was used as a model system to study SAT because there are published experimental data reporting MAP1B distribution along the axon length and average velocity of MAP1B transport in the axon. This allows the optimization approach to be used to find values of model kinetic constants that give the best fit with published experimental data. The effects of decreasing the value of cargo diffusivity on the diffusion-driven component of SAT and decreasing the value of dynein velocity on the retrograde motor-driven component of SAT were investigated. We found that for the case when protein diffusivity and dynein velocity are very small, it is possible to obtain an analytical solution to model equations. We found that, in this case, the protein concentration in the axon is uniform. This shows that anterograde motor-driven transport alone cannot simulate a variation of cargo concentration in the axon. Most proteins are non-uniformly distributed in axons. They may exhibit, for example, an increased concentration closer to the synapse. The need to reproduce a non-uniform distribution of protein concentration may explain why SAT is bidirectional (in addition to an anterograde component, it also contains a retrograde component).

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.

Investigating sensitivity coefficients characterizing the response of a model of tau protein transport in an axon to model parameters

Evaluating the sensitivity of biological models to various model parameters is a critical step towards advancing our understanding of biological systems. In this paper, we investigated sensitivity coefficients for a model simulating transport of tau protein along the axon. This is an important problem due to the relevance of tau transport and agglomeration to Alzheimer's disease and other tauopathies, such as some forms of parkinsonism. The sensitivity coefficients that we obtained characterize how strongly three observables (the tau concentration, average tau velocity, and the percentage of tau bound to microtubules) depend on model parameters. The fact that the observables strongly depend on a parameter characterizing tau transition from the retrograde to the anterograde kinetic states suggests the importance of motor-driven transport of tau. The observables are sensitive to kinetic constants characterizing tau concentration in the free (cytosolic) state only at small distances from the soma. Cytosolic tau can only be transported by diffusion, suggesting that diffusion-driven transport of tau only plays a role in the proximal axon. Our analysis also shows the location in the axon in which an observable has the greatest sensitivity to a certain parameter. For most parameters, this location is in the proximal axon. This could be useful for designing an experiment aimed at determining the value of this parameter. We also analyzed sensitivity of the average tau velocity, the total tau concentration, and the percentage of microtubule-bound tau to cytosolic diffusivity of tau and diffusivity of bound tau along the MT lattice. The model predicts that at small distances from the soma the effect of these two diffusion processes is comparable.

A numerical study of sensitivity coefficients for a model of amyloid precursor protein and tau protein transport and agglomeration in neurons at the onset of Alzheimer's disease

Modeling of intracellular processes occurring during the development of Alzheimer's disease (AD) can be instrumental in understanding the disease and can potentially contribute to finding treatments for the disease. The model of intracellular processes in AD, which we previously developed, contains a large number of parameters. To distinguish between more important and less important parameters we performed a local sensitivity analysis of this model around the values of parameters that give the best fit with published experimental results. We show that the effect of model parameters on the total concentration of amyloid precursor protein (APP) and tau protein in the axon, respectively, is reciprocal to the effect of the same parameters on the average velocities of the same proteins during their transport in the axon. The results of our analysis also suggest that in the beginning of AD the aggregation of amyloid-ß and misfolded tau protein have little effect on transport of APP and tau in the axon, which suggests that early effects of AD may be reversible.

What mechanisms of tau protein transport could be responsible for the inverted tau concentration gradient in degenerating axons?

In tauopathies, such as Alzheimer's disease (AD), microtubule (MT)-associated protein tau detaches from MTs and aggregates, eventually forming insoluble neurofibrillary tangles. In a healthy axon, the tau concentration increases toward the axon terminal, but in a degenerating axon, the tau concentration gradient is inverted and the highest tau concentration is in the soma. In this article, we developed a mathematical model of tau transport in axons. We calibrated and tested the model by using published distributions of tau concentration and tau average velocity in a healthy axon. According to published research, the inverted tau concentration gradient may be one of the reasons leading to AD. We therefore used the model to investigate what modifications in tau transport can lead to the inverted tau concentration gradient. We investigated whether tau detachment from MTs due to tau hyperphosphorylation can cause the inverted tau concentration gradient. We found that the assumption that most tau molecules are detached from MTs does not consistently predict the inverted tau concentration gradient; the predicted tau distribution becomes more uniform if the axon length is increased. We then hypothesized that in degenerating axons some tau remains bound to MTs and participates in the component 'a' of slow axonal transport but that the rate of tau reversals from anterograde to retrograde motion increases. We demonstrated that this hypothesis results in a tau distribution where the tau concentration has its maximum value at the axon hillock and its minimum value at the axon terminal, in agreement with what is observed in AD. Our results thus suggest that defects in active transport of tau may be a contributing factor to the onset of neural degeneration.

Towards the virtual human patient. Quantitative Systems Pharmacology in Alzheimer's disease

Development of successful therapeutic interventions in Central Nervous Systems (CNS) disorders is a daunting challenge with a low success rate. Probable reasons include the lack of translation from preclinical animal models, the individual variability of many pathological processes converging upon the same clinical phenotype, the pharmacodynamical interaction of various comedications and last but not least the complexity of the human brain. This paper argues for a re-engineering of the pharmaceutical CNS Research & Development strategy using ideas focused on advanced computer modeling and simulation from adjacent engineering-based industries. We provide examples that such a Quantitative Systems Pharmacology approach based on computer simulation of biological processes and that combines the best of preclinical research with actual clinical outcomes can enhance translation to the clinical situation. We will expand upon (1) the need to go from Big Data to Smart Data and develop predictive and quantitative algorithms that are actionable for the pharma industry, (2) using this platform as a "knowledge machine" that captures community-wide expertise in an active hypothesis-testing approach, (3) learning from failed clinical trials and (4) the need to go beyond simple linear hypotheses and embrace complex non-linear hypotheses. We will propose a strategy for applying these concepts to the substantial individual variability of AD patient subgroups and the treatment of neuropsychiatric problems in AD. Quantitative Systems Pharmacology is a new 'humanized' tool for supporting drug discovery and development in general and CNS disorders in particular.

Simulating tubulin-associated unit transport in an axon: using bootstrapping for estimating confidence intervals of best-fit parameter values obtained from indirect experimental data

In this paper, we first develop a model of axonal transport of tubulin-associated unit (tau) protein. We determine the minimum number of parameters necessary to reproduce published experimental results, reducing the number of parameters from 18 in the full model to eight in the simplified model. We then address the following questions: Is it possible to estimate parameter values for this model using the very limited amount of published experimental data? Furthermore, is it possible to estimate confidence intervals for the determined parameters? The idea that is explored in this paper is based on using bootstrapping. Model parameters were estimated by minimizing the objective function that simulates the discrepancy between the model predictions and experimental data. Residuals were then identified by calculating the differences between the experimental data and model predictions. New, surrogate 'experimental' data were generated by randomly resampling residuals. By finding sets of best-fit parameters for a large number of surrogate data the histograms for the model parameters were produced. These histograms were then used to estimate confidence intervals for the model parameters, by using the percentile bootstrap. Once the model was calibrated, we applied it to analysing some features of tau transport that are not accessible to current experimental techniques.

Utilization of the bootstrap method for determining confidence intervals of parameters for a model of MAP1B protein transport in axons

This paper develops a model of axonal transport of MAP1B protein. The problem of determining parameter values for the proposed model utilizing limited available experimental data is addressed. We used a global minimum search algorithm to find parameter values that minimize the discrepancy between model predictions and published experimental results. By analyzing the best fit parameter values it was established that some processes can be dropped from the model without losing accuracy, thus a simplified version of the model was formulated. In particular, our analysis suggests that reversals in MAP1B transport are infrequent and can be neglected. The detachment of anterogradely-biased MAP1B from microtubules (MTs) and the attachment of retrogradely-biased MAP1B to MTs can also be neglected. An analytical solution for the simplified model was obtained. Confidence intervals for the determined parameters were found by bootstrapping model residuals. The resultant analysis heavily constrained the values of some parameters while showing that some could vary without significantly impacting model error. For example, our analysis suggests that, above a certain threshold value, the kinetic constant determining the rate of MAP1B transition from the retrograde pausing state to the off-track state has little impact on model error. On the contrary, the kinetic constant describing MAP1B transition from a pausing to a running state has great impact on model error.

Mathematical models of α-synuclein transport in axons

To investigate possible effects of diffusion on α-synuclein (α-syn) transport in axons, we developed two models of α-syn transport, one that assumes that α-syn is transported only by active transport, as part of multiprotein complexes, and a second that assumes an interplay between motor-driven and diffusion-driven α-syn transport. By comparing predictions of the two models, we were able to investigate how diffusion could influence axonal transport of α-syn. The predictions obtained could be useful for future experimental work aimed at elucidating the mechanisms of axonal transport of α-syn. We also attempted to simulate possible defects in α-syn transport early in Parkinson's disease (PD). We assumed that in healthy axons α-syn localizes in the axon terminal while in diseased axons α-syn does not localize in the terminal (this was simulated by postulating a zero α-syn flux into the terminal). We found that our model of a diseased axon predicts the build-up of α-syn close to the axon terminal. This build-up could cause α-syn accumulation in Lewy bodies and the subsequent axonal death pattern observed in PD ('dying back' of axons).