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Conceived and designed the experiments: AR KSK FJD. Performed the experiments: TMY MRF. Analyzed the data: TMY PN MRF. Wrote the paper: TMY. Experiments are in silico: TMY.

The authors have declared that no competing interests exist.

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

Neurodegenerative disorders, particularly the tauopathy Alzheimer's disease, affect millions of people and cost billions of dollars a year in healthcare costs. Although effective treatments to delay or reverse cognitive decline are still unavailable, several approaches to address this medical need are being pursued. One such strategy involves ameliorating aberrant tau processing, as the characteristic tau tangles associated with the tauopathies are well-correlated with cognitive dysfunction, genetic mutations in tau lead directly to neurodegeneration, and experiments in animal models have yielded promising results. Two avenues are currently being explored: inhibition of kinase activity to reduce the presence of aberrant, hyperphosphorylated tau and means to prevent and reduce tau aggregation. We have taken a systems biology approach to understanding tau pathophysiology, creating a mathematical model to quantitatively explore the vulnerabilities in the tau network and identify effective intervention points. Our analysis of the resulting

Despite the fidelity of protein folding and the operation of quality control mechanisms to eliminate misfolded and otherwise abnormal proteins, a number of diseases can be traced to defects in these processes

A convincing body of evidence implicates defective tau processing and the formation of intraneuronal tau aggregates in cognitive decline. Mutations in the gene encoding tau protein are directly responsible for a number of genetic conditions collectively called primary tauopathies, among which is frontotemporal dementia and Parkinsonism linked to chromosome 17 (FTDP-17)

Tau is a neuronal, microtubule-associated protein (MAP) whose physiological function is to regulate microtubule dynamics (

The multifactorial nature of disease motivates our systems biology approach to understanding tau pathophysiology. We have developed a computational model that represents the network of interactions in which tau is involved as a system of ordinary differential equations that describe the deterministic chemical kinetics. The model was tuned to capture observed behavior in a healthy neuron and an aggregation-prone neuron. Although the class of tauopathies contains several diseases, specific experimental data from Alzheimer's disease studies informed this model. Sensitivity analysis tools were used to interrogate the model and ascertain the relative contributions of each component in the tau pathway from its synthesis to its post-translational modifications, to its degradation. Within both populations of neurons, and particularly the aggregation-prone population, we found ultrasensitive cellular conditions that are likely to be resistant to rescue.

As one of the first attempts at

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.

Abnormal 3R and 4R tau are bound by the chaperone Hsc70

Mass action kinetics described all reactions in the network except the phosphorylation and dephosphorylation reactions, which were described by Michaelis-Menten kinetics. For each species represented by our model, an ordinary differential equation that describes the species time-evolution was constructed as illustrated in Eq. S1. In total, the network contains 84 reactions, 93 parameters, and 45 states (i.e., differential equations). A full listing of the states, reactions, parameters, and differential equations can be found in

Parameter space for the healthy and aggregation-prone identifiability and optimization steps is different, as the chaperone and degradation machinery was considered to be operating homeostatically. As a result, before initiating each stage of the optimization, an

The results of both stages of this analysis confirm that the proposed model is _{1}, k_{84}, k_{10}) from the first stage of the procedure as they were highly correlated (>0.95).

In the next step, we optimized parameters to achieve steady-state behavior that represents healthy neuron function. Parameters associated with phosphorylation and dephosphorylation, microtubule binding and release, synthesis, and ubiquitin-independent degradation were estimated. We also estimated ATP synthesis and depletion. Parameters were generally assumed isoform-independent, with the exception of the microtubule binding parameters and aggregation parameters. Because evidence suggests that 4R tau has a greater affinity for microtubules

The objective function that mathematically quantifies the behavior of a healthy neuron was constructed to reflect known quantitative experimental data. It is well-established that aberrant tau species are undetectable in normal neurons; thus we require that free and microtubule-bound aberrant tau is minimized. From measurement of total tau in human brain homogenates

Necessarily, the solution in this case is not unique. Therefore, a set of 2500 optimizations was performed in which the model was run to steady-state, then evaluated against these objectives to generate a set of equally valid parameter vectors with which to initialize the model (

The median sensitivity of the population to perturbations in the parameters was calculated at steady-state, to provide insight into the triggers that disturb the system's homeostasis (

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.

The identifiability of the sensitivity coefficients is defined by the span of the confidence interval; if the interval does not contain zero, the coefficient is considered identifiable. Although some small sensitivity coefficients are identifiable, most are not and the converse is true for larger coefficients, particularly those >0.5 (

In

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.

For each model of a healthy neuron, we established a corresponding aggregation-prone model. The two models are coupled through the microtubule binding and release parameters. Synthesis, degradation, and phosphorylation and dephosphorylation were re-estimated because these activities are known to be altered in neurons containing tau aggregates. In addition, parameters associated with the chaperone and degradation machinery were estimated.

The objective function that quantifies the behavior of an aggregation-prone neuron is based on the data from several experiments. Quantification of tau in adult human brains affected by Alzheimer's was compared to that in control and showed that normal tau concentration was unaltered, but total tau concentration was 4–8 times normal tau; the increase is in the form of aberrant tau

Each result from the tuning of a neuron to healthy behavior was used to seed an optimization run designed to generate aggregation-prone behavior. For each run, the model was initialized to the steady-state concentrations achieved by the corresponding model of a healthy neuron. The simulation was run until quasi-steady-state was achieved and evaluated against the objective function to find parameters that instantiate an aggregation-prone model (

In general, a single primary route to establish the aggregation-prone behavior was not obvious. Rather, the nature of the changes required to establish aggregation-prone neurons was multifactorial, although definite trends were observed in a small subset of the parameters (

For parameters allowed to vary, the log_{2} ratios of the values in the aggregation-prone

The median sensitivity of the aggregation-prone population was calculated and the 95% confidence interval of the coefficients was used to determine their identifiability (

To compare the aggregation-prone and healthy populations, the ratios between the sensitivity coefficients in each pair of matched individuals was calculated and the medians are shown in

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.

Evaluation of the distribution of the coefficients revealed subset of individuals with very large magnitude sensitivities to changing parameters, or ultrasensitivity (

The

The population of healthy neurons is considered to be robust in several ways. The model generates healthy behavior in a relatively large domain of parameter space, a necessary property to maintain a phenotype given the inherent variation and noise in all biological systems. Likewise, the healthy population is robust and demands a vectorial assault to become pathological, as a multitude of perturbations to synthesis, degradation, and phosphorylation and dephosphorylation are required to generate a corresponding population of aggregation-prone neurons. In contrast, the aggregation-prone population is generally more sensitive to perturbations than the healthy population, as might be expected for a pathological phenotype (

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.

A subset of the aggregation-prone population displays extreme fragility (

The robustness of the tau network and the multifactorial nature of its vulnerability to pathological change presents a challenge to the selection of drug targets, and for a subset of patients the disease is likely to be nearly impossible to reverse after the network becomes ultrasensitive. The model analysis also suggests that stalling or reversing tau pathophysiology will be further complicated by the timing at which the intervention is begun; a treatment may have an opposite effect on the system than is expected due to the sign inversion observed for some sensitivity coefficients.

The systems biology approach we have taken here has highlighted the complex, nonlinear behavior that cellular networks can display and suggests the difficulties the pharmaceutical and biotechnology industries will face in attempting to treat diseases associated with their aberrant functioning. By modeling both the physiological and the pathological functioning of the network governing tau function, we have shown that the biological response to a perturbation is dependent on the condition of the network and that, therefore, the time at which a compensatory perturbation is made is potentially significant. This implication is particularly relevant in therapeutic treatment timing and approach. The population-based analyses we have completed also highlights the importance of variability in the study and treatment of disease and the need to characterize the variability of the network components, such as reaction rates, to more fully elucidate its nature. Such variations are distinct from stochastic variation and the extent of the variability is likely dependent on the biological network and the particular network component. From a modeling perspective,

Current knowledge about the molecular biology of tau protein was integrated into a deterministic, kinetic model that was realized as a set of 45 ordinary differential equations (ODE's) (

To validate the model construction effort, we used the method of Jacquez and Greif _{c} that establishes

Identifiable systems have correlations strictly < |1|. Here, the average correlation matrix is used to ascertain the identifiability of the system. Because the parameter sets were randomly generated, the resulting systems do not necessarily display biologically relevant behavior; therefore, the optimization objective function was calculated at each point in parameter space and used as weighting factors in calculating the average correlation. Given the model structure we established and the bounds on the parameter ranges, we can conclude that the model is

Using this framework, that reduces parametric dependence and assumes all states are experimentally measurable,

The model parameters were numerically fit using a hybrid stochastic-deterministic global optimization method

To assess the effect of parameter perturbations on the steady-state concentrations of protein in the healthy population and quasi-steady-state (due to the polymerization reaction) concentration in the aggregation-prone population, the local, relative sensitivity of this system, given by Eq. S4 (

The relative sensitivity coefficient gives the dependence of the protein concentration, “x_{i}”, on a parameter, “p_{j}” and is normalized with respect to the parameter and state values to facilitate. The non-normalized coefficients are calculated by applying the chain rule to Eq. S4 (_{x}, of size N_{x} (number of states) by N_{p} (number of parameters).

List of states, differential equations governing the time evolution of the states, and initial conditions for each state.

(0.11 MB PDF)

List of reactions and the rate equations for the model of tau pathophysiology.

(0.18 MB PDF)

Supporting equations for the construction of the ODE's and objective function and for the calculation of the correlation matrices and sensitivity coefficients.

(0.03 MB PDF)

Major events in the tau processing network. Phosphorylated (P) tau reversibly binds microtubules. In degenerating neurons, tau becomes abnormally and hyper-phosphorylated, misfolds, and is taken up by the chaperone system. Hsc70 mediates a decision between rescue and degradation.

(0.10 MB TIF)

Pseudo-global identifiability for the first stage of optimization to generate a population of healthy neuron models. The matrix shows the correlation between all pairs of parameters estimated during the optimization. A correlation of 1 or -1 indicates a non-identifiable parameter. No parameters were non-identifiable, but parameters that were highly correlated, i.e. >0.95 (circled), were nonetheless removed to improve the efficiency of the optimization.

(1.20 MB TIF)

Pseudo-global identifiability for the second stage of optimization to generate a population of aggregation-prone neuron models. The matrix shows the correlation between all pairs of parameters estimated during the optimization. A correlation of 1 or -1 indicates a non-identifiable parameter. No parameters were non-identifiable, nor did any parameter pairs have correlations greater than 0.95.

(1.60 MB TIF)

Identifiability of the median sensitivity coefficients for the healthy population, as computed from the 95% confidence intervals.

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Distribution of the median sensitivity coefficients, categorized by their identifiability.

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Relative, steady-state sensitivity for the aggregation-prone population of in silico neuron models. Median sensitivity coefficient at steady-state is shown for pairs of states (proteins) and parameters (rate constants). The parameters are grouped according to type.

(0.80 MB TIF)

Identifiability of the sensitivity coefficients, as computed from the 95% confidence intervals. If the confidence interval spanned 0, the coefficient was labeled unidentifiable.

(0.74 MB TIF)

Distribution of the median sensitivity coefficients of the aggregation-prone population according to their identifiability and magnitude.

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Optimization results.

(8.36 MB XLSX)