Vulnerabilities in the tau network and the role of ultrasensitive points in tau pathophysiology

Abstract

The multifactorial nature of disease motivates the use of systems-level analyses to understand their pathology. We used a systems biology approach to study tau aggregation, one of the hallmark features of Alzheimer's disease. A mathematical model was constructed to capture the current state of knowledge concerning tau's behavior and interactions in cells. The model was implemented in silico in the form of ordinary differential equations. The identifiability of the model was assessed and parameters were estimated to generate two cellular states: a population of solutions that corresponds to normal tau homeostasis and a population of solutions that displays aggregation-prone behavior. The model of normal tau homeostasis was robust to perturbations, and disturbances in multiple processes were required to achieve an aggregation-prone state. The aggregation-prone state was ultrasensitive to perturbations in diverse subsets of networks. Tau aggregation requires that multiple cellular parameters are set coordinately to a set of values that drive pathological assembly of tau. This model provides a foundation on which to build and increase our understanding of the series of events that lead to tau aggregation and may ultimately be used to identify critical intervention points that can direct the cell away from tau aggregation to aid in the treatment of tau-mediated (or related) aggregation diseases including Alzheimer's.

Figure 1. Model structure of tau pathophysiology for a single isoform.

The network captures tau phosphorylation and dephosphorylation, microtubule binding and release, uptake, rescue, and degradation by the chaperone machinery, and aggregation. Specifically, unphosphorylated tau (Tau0) can be degraded or phosphorylated, producing normally phosphorylated tau (TauN). TauN can also be degraded in a non-ubiquitin dependent fashion, dephosphorylated, or phosphorylated to create abnormal tau (TauH), which can likewise be degraded, dephosphorylated, or phosphorylated. Each of these free tau species undergoes a conformation change to produce a form with high affinity for microtubules; these species are denoted with a star as Tau0*, TauN*, and TauH*. Abnormal TauH is taken up by the chaperone Hsp70, which mediates the decision between rescue and degradation. Both isoforms participate in the same series of reactions, but at different rates of reaction, and their behavior is coupled through the chaperone and degradation machinery.

Figure 2. Relative, steady-state sensitivity for the healthy population of in silico neuron models.

Median sensitivity coefficient at steady-state is shown for pairs of states (proteins) and parameters (rate constants). States and parameters associated with the chaperone and degradation are not shown, as this network is not engaged when the model is behaving in a manner consistent with a healthy neuron.

Figure 3. Ultrasensitivity in the populations.

The cube root of the median sensitivity coefficients for each state across all parameters is shown with coefficients exceeding a 10-fold change in a state compared to the parameter perturbation highlighted. The ultrasensitive coefficients, which are denoted by the red and blue circled markers in the healthy and aggregation-prone populations, represent multiple individuals.

Figure 4. Parameter ratios.

For parameters allowed to vary, the log₂ ratios of the values in the aggregation-prone vs. the healthy population are given. The confidence intervals, maximum, and minimum for the population are also plotted.

Figure 5. Sensitivity ratios.

The ratio of the relative, median sensitivity coefficient for the aggregation prone population to the healthy population is shown for each state (protein concentration) and parameter (rate constant) pair.

Figure 6. Time-dependent response to perturbations.

A 5-fold increase in the parameter associated with the binding of normally phosphorylated, 3R tau to microtubules was applied and the response of unphosphorylated, microtubule bound 4R tau was monitored with respect to the basal concentration of this species. The perturbation results in a decrease in concentration with respect to basal in the healthy neuron, while in the aggregation-prone neuron, an increase in protein concentration is observed.