A mathematical model of multisite phosphorylation of tau protein

Abstract

Abnormal tau metabolism followed by formation of tau deposits causes a number of neurodegenerative diseases called tauopathies including Alzheimer's disease. Hyperphosphorylation of tau protein precedes tau aggregation and is a topic of interest for the development of pharmacological interventions to prevent pathology progression at early stages. The development of a mathematical model of multisite phosphorylation of tau would be helpful for searching for the targets of pharmacological interventions and candidates for biomarkers of pathology progression. In the present study, we for the first time developed a model of multisite phosphorylation of tau protein and elucidated the relative contribution of kinases to phosphorylation of distinct sites. The model describes phosphorylation of tau or PKA-prephosphorylated tau by GSK3β and CDK5 and dephosphorylation by PP2A, accurately reproducing the data for short-term kinetics of tau (de)phosphorylation. Our results suggest that kinase inhibition may more specifically prevent tau hyperphosphorylation, e.g., on PHF sites, which are key biomarkers of pathological changes in Alzheimer's disease. The main features of our model are partial phosphorylation of tau residues and merging of random and sequential mechanisms of multisite phosphorylation within the framework of the probability-based approach assuming independent phosphorylation events.

Fig 1. Schematic representation of the tau residues and kinases incorporated into the mathematical model.

Fig 2. Mechanisms of multisite protein phosphorylation relevant to the independence of phosphorylation events.

GSK3β phosphorylates tau via a random mechanism (B) except for sites S404 and S396, which are phosphorylated by a sequential mechanism (A). CDK5 phosphorylates all the sites via a random mechanism (B). Phosphatase PP2A dephosphorylates tau protein by a random mechanism.

Fig 3. Phosphorylation stoichiometry (the ratio of phosphorylated tau to total tau, mol P/mol) for (A) GSK3β and (B) CDK5 with unphosphorylated tau (gray) or PKA-prephosphorylated tau (black) [20].

Experimental values are marked by points, and model predictions by lines. Two black and gray experimental points for CDK5 (B) coincide. Errors of experimental values were not provided by the authors [20].

Fig 4. Kinetics of tau phosphorylation at S404 by (A) GSK3β or (B) CDK5 with (black) or without (gray) PKA-prephosphorylation.

The transient peak in (A) is caused by sequential phosphorylation (first S404, then S396) by GSK3β in contrast to kinetics of tau phosphorylation at S404 with CDK5 when residues S396 and S404 are phosphorylated independently by a random mechanism. Errors of experimental values were not provided by the authors [20].

Fig 5. Phosphorylation kinetics of S396 (purple) and S404 (green) of tau (A) or PKA-prephosphorylated tau (B) by GSK3β.

Kinetics for pS404 with 95% confidence bands are represented. Errors of experimental values were not provided by the authors [20].

Fig 6. A bar chart of α_i parameters (proportion of opened states) for 10 sites with 95% confidence intervals (A), and a stacked bar chart for the same sites including a pseudo-residue (B).

Fig 7. Values of catalytic constants on a logarithmic scale with 95% confidence intervals.

Fig 8. Kinetics of tau dephosphorylation at individual phosphorylation sites by PP2A.

Experimental values are marked by points, and model predictions by lines. Errors of experimental values were not provided by the authors [10].

Fig 9. Heatmap representation of predicted sensitivity of tau residues and total tau phosphorylation (columns) to the reaction of (de)phosphorylation (row).