Tau Function Research Articles

KP solitons from tropical limits

We study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemann's theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the Hirota variety which parametrizes all tau functions arising from such a sum. We compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces and we present an algorithm that finds a soliton solution from a rational nodal curve.

Polynomial Tau Functions of Symplectic KP and Multi-component Symplectic KP Hierarchies

Polynomial Tau Functions of Symplectic KP and Multi-component Symplectic KP Hierarchies

Caspase-cleaved tau is senescence-associated and induces a toxic gain of function by putting a brake on axonal transport

The microtubule-associated protein tau plays a central role in tauopathies such as Alzheimer’s disease (AD). The exact molecular mechanisms underlying tau toxicity are unclear, but aging is irrefutably the biggest risk factor. This raises the question of how cellular senescence affects the function of tau as a microtubule regulator. Here we report that the proportion of tau that is proteolytically cleaved at the caspase-3 site (TauC3) doubles in the hippocampus of senescent mice. TauC3 is also elevated in AD patients. Through quantitative live-cell imaging, we show that TauC3 has a drastically reduced dynamics of its microtubule interaction. Single-molecule tracking of tau confirmed that TauC3 has a longer residence time on axonal microtubules. The reduced dynamics of the TauC3-microtubule interaction correlated with a decreased transport of mitochondria, a reduced processivity of APP-vesicle transport and an induction of region-specific dendritic atrophy in CA1 neurons of the hippocampus. The microtubule-targeting drug Epothilone D normalized the interaction of TauC3 with microtubules and modulated the transport of APP-vesicles dependent on the presence of overexpressed human tau. The results indicate a novel toxic gain of function, in which a post-translational modification of tau changes the dynamics of the tau-microtubule interaction and thus leads to axonal transport defects and neuronal degeneration. The data also introduce microtubule-targeting drugs as pharmacological modifiers of the tau-microtubule interaction with the potential to restore the physiological interaction of pathologically altered tau with microtubules.

In vitro characterization and molecular dynamics simulation reveal mechanism of 14-3-3ζ regulated phase separation of the tau protein

As a major microtubule-associated protein, tau is involved in the assembly of microtubules in the central nervous system. However, under pathological conditions tau assembles into amyloid filaments. Liquid droplets formed by liquid−liquid phase separation (LLPS) are a recently identified assembly state of tau and may have a major effect on the physiological function of tau and the formation of tau aggregates. 14-3-3 proteins are ubiquitously expressed in various tissues and regulate a wide variety of biological processes. In this work, we demonstrate that 14-3-3ζ is recruited into tau droplets and regulates tau LLPS by in vitro assays. While the mobility of tau molecules inside the droplets is not affected in the presence of 14-3-3ζ, the amount and size of droplets can vary significantly. Mechanistic studies reveal that 14-3-3ζ regulates tau LLPS by electrostatic interactions and hydrophobic interactions with the proline-rich domain and the microtubule-binding domain of tau. Surprisingly, the disordered C-terminal tail rather than the amphipathic binding groove of 14-3-3ζ plays a key role. Our findings not only provide a novel dimension to understand the interactions between 14-3-3 proteins and tau, but also suggest that 14-3-3 proteins may play an important role in regulating the LLPS of their binding partners.

Alzheimer's Disease and other Tauopathies: Exploring Efficacy of Medicinal Plant-derived Compounds in Alleviating Tau-mediated Neurodegeneration.

Alzheimer's disease (AD), a major form of dementia, has been reported to affect more than 50 million people worldwide. It is characterized by the presence of amyloid-β (Aβ) plaques and hyperphosphorylated Tau-associated neurofibrillary tangles in the brain. Apart from AD, microtubule (MT)-associated protein Tau is also involved in other neurodegenerative diseases called tauopathies, including Pick's disease, frontotemporal lobar degeneration, progressive supranuclear palsy, and corticobasal degeneration. The recent unsuccessful phase III clinical trials related to Aβ- targeted therapeutic drugs have indicated that alternative targets, such as Tau, should be studied to discover more effective and safer drugs. Recent drug discovery approaches to reduce AD-related Tau pathologies are primarily based on blocking Tau aggregation, inhibiting Tau phosphorylation, compensating impaired Tau function with MT-stabilizing agents, and targeting the degradation pathways in neuronal cells to degrade Tau protein aggregates. Owing to several limitations of the currently available Tau-directed drugs, further studies are required to generate further effective and safer Tau-based disease-modifying drugs. Here, we review the studies focused on medicinal plant- derived compounds capable of modulating the Tau protein, which is significantly elevated and hyperphosphorylated in AD and other tauopathies. We have mainly considered the studies focused on Tau protein as a therapeutic target. We have reviewed several pertinent papers retrieved from PubMed and ScienceDirect using relevant keywords, with a primary focus on the Tau-targeting compounds from medicinal plants. These compounds include indolines, phenolics, flavonoids, coumarins, alkaloids, and iridoids, which have been scientifically proven to be Tau-targeting candidates for the treatment of AD.

Tau Toxicity in Neurodegeneration.

Tau is a microtubule-associated protein widely distributed in the central nervous system (CNS). The main function of tau is to promote the assembly of microtubules and stabilize their structure. After a long period of research on neurodegenerative diseases, the function and dysfunction of the microtubule-associated protein tau in neurodegenerative diseases and tau neurotoxicity have attracted increasing attention. Tauopathies are a series of progressive neurodegenerative diseases caused by pathological changes in tau, such as abnormal phosphorylation. The pathological features of tauopathies are the deposition of abnormally phosphorylated tau proteins and the aggregation of tau proteins in neurons. This article first describes the normal physiological function and dysfunction of tau proteins and then discusses the enzymes and proteins involved in tau phosphorylation and dephosphorylation, the role of tau in cell dysfunction, and the relationships between tau and several neurodegenerative diseases. The study of tau neurotoxicity provides new directions for the treatment of tauopathies.

Symplectic aspects of the tt*-Toda equations

We evaluate explicitly, in terms of the asymptotic data, the ratio of the constant pre-factors in the large and small x asymptotics of the tau functions for global solutions of the tt*-Toda equations. This constant problem for the sinh–Gordon equation, which is the case n = 1 of the tt*-Toda equations, was solved by Tracy (1991 Commun. Math. Phys. 142 297–311). We also introduce natural symplectic structures on the space of asymptotic data and on the space of monodromy data for a wider class of solutions, and show that these symplectic structures are preserved by the Riemann-Hilbert correspondence.

Inhibition of CK2 mitigates Alzheimer\u2019s tau pathology by preventing NR2B synaptic mislocalization

Alzheimer’s disease (AD) is a neurodegenerative disorder that exhibits pathological changes in both tau and synaptic function. AD patients display increases in hyperphosphorylated tau and synaptic activity. Previous studies have individually identified the role of NR2B subunit-containing NMDA receptors in AD related synaptic dysfunction and aggregated tau without reconciling the conflicting differences and implications of NR2B expression. Inhibition of extrasynaptically located NR2B mitigates tau pathology in AD models, whereas the inhibition of synaptic NR2B replicates tau-associated hyperactivity. This suggests that a simultaneous increase in extrasynaptic NR2B and decrease in synaptic NR2B may be responsible for tau pathology and synaptic dysfunction, respectively. The synaptic location of NR2B is regulated by casein kinase 2 (CK2), which is highly expressed in AD patients. Here, we used patient brains diagnosed with AD, corticobasal degeneration, progressive supranuclear palsy or Pick’s disease to characterize CK2 expression across these diverse tauopathies. Human derived material was also utilized in conjunction with cultured hippocampal neurons in order to investigate AD-induced changes in NR2B location. We further assessed the therapeutic effect of CK2 inhibition on NR2B synaptic distribution and tau pathology. We found that aberrant expression of CK2, and synaptically translocated NR2B, is unique to AD patients compared to other tauopathies. Increased CK2 was also observed in AD-tau treated neurons in addition to the mislocalization of NR2B receptors. Tau burden was alleviated in vitro by correcting synaptic:extrasynaptic NR2B function. Restoring NR2B physiological expression patterns with CK2 inhibition and inhibiting the function of excessive extrasynaptic NR2B with Memantine both mitigated tau accumulation in vitro. However, the combined pharmacological treatment promoted the aggregation of tau. Our data suggests that the synaptic:extrasynaptic balance of NR2B function regulates AD-tau pathogenesis, and that the inhibition of CK2, and concomitant prevention of NR2B mislocalization, may be a useful therapeutic tool for AD patients.

Recent studies of atomic‐resolution structures of tau protein and structure‐based inhibitors

BackgroundAlzheimer’s disease (AD) is one of the most popular tauopathies. Neurofibrillary tangles and senile plaques are widely recognized as the pathological hallmarks of AD, which are mainly composed of tau and β‐amyloid (Aβ) respectively. Recent failures of drugs targeting Aβ have led scientists to scrutinize the crucial impact of tau in neurodegenerative diseases. Mutated or abnormal phosphorylated tau protein loses affinity with microtubules and assembles into pathological accumulations. The aggregation process closely correlates to two amyloidogenic core of PHF6 (306VQIVYK311) and PHF6* (275VQIINK280) fragments. Moreover, tau accumulations display diverse morphological characteristics in different diseases, which increases the difficulty of providing a unifying neuropathological criterion for early diagnosis.ResultsThis review mainly summarizes atomic‐resolution structures of tau protein in the monomeric, oligomeric and fibrillar states, as well as the promising inhibitors designed to prevent tau aggregation or disaggregate tau accumulations, recently revealed by experimental and computational studies. We also systematically sort tau functions, their relationship with tau structures and the potential pathological processes of tau protein.ConclusionThe current progress on tau structures at atomic level of detail expands our understanding of tau aggregation and related pathology. We discuss the difficulties in determining the source of neurotoxicity and screening effective inhibitors. We hope this review will inspire new clues for designing medicines against tau aggregation and shed light on AD diagnosis and therapies.

Gromov-Witten invariants of [formula omitted] coupled to a KdV tau function

We consider the pull-back of a natural sequence of cohomology classes Θg,n∈H2(2g−2+n)(M‾g,n,Q) to the moduli space of stable maps M‾g,n(P1,d). These classes are related to the Brézin-Gross-Witten tau function of the KdV hierarchy via ZBGW(ħ,t0,t1,...)=exp⁡∑ħ2g−2n!∫M‾g,nΘg,n⋅∏j=1nψjkj∏tkj. Insertions of the pull-backs of the classes Θg,n into the integrals defining Gromov-Witten invariants define new invariants which we show in the case of target P1 are given by a random matrix integral and satisfy the Toda equation.

Bilinear equations in Darboux transformations by Boson–Fermion correspondence

Bilinear equation is an important property for integrable nonlinear evolution equation. Many famous research objects in mathematical physics, such as Gromov–Witten invariants, can be described in terms of bilinear equations to show their connections with the integrable systems. Here in this paper, we mainly discuss the bilinear equations of the transformed tau functions under the successive applications of the Darboux transformations for the KP hierarchy, the modified KP hierarchy (Kupershmidt–Kiso version) and the BKP hierarchy, by the method of the Boson–Fermion correspondence. The Darboux transformations are considered in the Fermionic picture, by multiplying the different Fermionic fields on the tau functions. Here the Fermionic fields are corresponding to the (adjoint) eigenfunctions, whose changes under the Darboux transformations are showed to be the ones of the squared eigenfunction potentials in the Bosonic picture, used in the spectral representations of the (adjoint) eigenfunctions. Then the successive applications of the Darboux transformations are given in the Fermionic picture. Based upon this, some new bilinear equations in the Darboux chain are derived, besides the ones of (l−l′) -th modified KP hierarchy. The corresponding examples of these new bilinear equations are given.

Impaired Glucose Homeostasis in a Tau Knock-In Mouse Model.

Alzheimer’s disease (AD) is the leading cause of dementia. While impaired glucose homeostasis has been shown to increase AD risk and pathological loss of tau function, the latter has been suggested to contribute to the emergence of the glucose homeostasis alterations observed in AD patients. However, the links between tau impairments and glucose homeostasis, remain unclear. In this context, the present study aimed at investigating the metabolic phenotype of a new tau knock-in (KI) mouse model, expressing, at a physiological level, a human tau protein bearing the P301L mutation under the control of the endogenous mouse Mapt promoter. Metabolic investigations revealed that, while under chow diet tau KI mice do not exhibit significant metabolic impairments, male but not female tau KI animals under High-Fat Diet (HFD) exhibited higher insulinemia as well as glucose intolerance as compared to control littermates. Using immunofluorescence, tau protein was found colocalized with insulin in the β cells of pancreatic islets in both mouse (WT, KI) and human pancreas. Isolated islets from tau KI and tau knock-out mice exhibited impaired glucose-stimulated insulin secretion (GSIS), an effect recapitulated in the mouse pancreatic β-cell line (MIN6) following tau knock-down. Altogether, our data indicate that loss of tau function in tau KI mice and, particularly, dysfunction of pancreatic β cells might promote glucose homeostasis impairments and contribute to metabolic changes observed in AD.

AN APPLICATION OF BINARY QUADRATIC FORMS OF DISCRIMINANT TO MODULAR FORMS

AbstractIn this note, we use Dedekind’s eta function to prove a congruence relation between the number of representations by binary quadratic forms of discriminant $-31$ and Fourier coefficients of a weight $16$ cusp form. Our result is analogous to the classical result concerning Ramanujan’s tau function and binary quadratic forms of discriminant $-23$ .

Protein conformations in Tau condensates probed by smFRET and FLIM-FRET

Tau-rich neurofibrillary tangles are the defining neuropathologic feature of Alzheimer’s disease. Liquid-liquid phase separation play critical roles in Tau function and dysfunction by promoting microtubule assembly and acting as an intermediate step towards protein fibrillation and disease pathology. To track Tau conformational and morphological evolution during LLPS-mediated microtubule assembly and aggregation, we used FRET as a conformational reporter both at the single-molecule resolution (smFRET) and the ensemble regime (FLIM-FRET).

Tau mRNA Metabolism in Neurodegenerative Diseases: A Tangle Journey.

Tau proteins are known to be mainly involved in regulation of microtubule dynamics. Besides this function, which is critical for axonal transport and signal transduction, tau proteins also have other roles in neurons. Moreover, tau proteins are turned into aggregates and consequently trigger many neurodegenerative diseases termed tauopathies, of which Alzheimer’s disease (AD) is the figurehead. Such pathological aggregation processes are critical for the onset of these diseases. Among the various causes of tau protein pathogenicity, abnormal tau mRNA metabolism, expression and dysregulation of tau post-translational modifications are critical steps. Moreover, the relevance of tau function to general mRNA metabolism has been highlighted recently in tauopathies. In this review, we mainly focus on how mRNA metabolism impacts the onset and development of tauopathies. Thus, we intend to portray how mRNA metabolism of, or mediated by, tau is associated with neurodegenerative diseases.

Mechanism of psychoactive substance-induced cognitive disorders: does tau protein play a role?

Psychoactive substances are a class of chemical substances which could cause public health threats. Cognitive disorders are a category of mental health disorders that primarily affect cognitive abilities. Tau protein could maintain neuronal cytoskeleton stabilization. Post-translational modification of tau, especially phosphorylation, is an important way to regulate the structure and function of tau and phosphorylated tau is closely related to cognitive function. Lots of studies have reported the phenomenon that psychoactive substances can cause cognitive function impairment. We reviewed recent related studies and discussed them by drug classification. We mainly focused on cognitive disorders caused by acute or chronic exposure of each drugs, animal experiments and the mechanisms associated with tau phosphorylation, then compared the similarities and differences among them, trying to find out the common rules. The results suggested that tau phosphorylation is involved in psychoactive substance-induced cognitive disorder and different psychoactive substances may act by affecting amount or activity of different kinases and phosphatases in the metabolic pathway of tau. We demonstrated that tau protein is a potential target for psychoactive substances induced cognitive disorder treatments.

Tau Function and Moduli of Meromorphic Quadratic Differential

The Bergman tau functions are applied to the study of the Picard group of moduli spaces of quadratic differentials with at most n simple poles on genus g complex algebraic curves. This generalizes our previous results on moduli spaces of holomorphic quadratic differentials.

Death-associated protein kinase 1 mediates Aβ42 aggregation-induced neuronal apoptosis and tau dysregulation in Alzheimer's disease.

The aggregation of amyloid-β (Aβ) peptides into oligomers and fibrils is a key pathological feature of Alzheimer's disease (AD). An increasing amount of evidence suggests that oligomeric Aβ might be the major culprit responsible for various neuropathological changes in AD. Death-associated protein kinase 1 (DAPK1) is abnormally elevated in brains of AD patients and plays an important role in modulating tau homeostasis by regulating prolyl isomerase Pin1 phosphorylation. However, it remains elusive whether and how Aβ species influence the function of DAPK1, and whether this may further affect the function and phosphorylation of tau in neurons. Herein, we demonstrated that Aβ aggregates (both oligomers and fibrils) prepared from synthetic Aβ42 peptides were able to upregulate DAPK1 protein levels and thereby its function through heat shock protein 90 (HSP90)-mediated protein stabilization. DAPK1 activation not only caused neuronal apoptosis, but also phosphorylated Pin1 at the Ser71 residue, leading to tau accumulation and phosphorylation at multiple AD-related sites in primary neurons. Both DAPK1 knockout (KO) and the application of a specific DAPK1 inhibitor could effectively protect primary neurons against Aβ aggregate-induced cell death and tau dysregulation, corroborating the critical role of DAPK1 in mediating Aβ aggregation-induced neuronal damage. Our study suggests a mechanistic link between Aβ oligomerization and tau hyperphosphorylation mediated by DAPK1, and supports the role of DAPK1 as a promising target for early intervention in AD.

Zeros of the isomonodromic tau functions in constructive conformal mapping of polycircular arc domains: the n-vertex case

The prevertices of the conformal map between a generic, n-vertex, simply connected, polycircular arc domain and the upper half plane are determined by finding the zeros of an isomonodromic tau function. The accessory parameters of the associated Fuchsian equation are then found in terms of logarithmic derivatives of this tau function. Using these theoretical results a constructive approach to the determination of the conformal map is given and the particular case of five vertices is considered in detail. A computer implementation of a construction of the isomonodromic tau function described by Gavrylenko and Lisovyy (2018 Commun. Math. Phys. 363 1–58) is used to calculate some illustrative examples. A procedural guide to constructing the conformal map to a given polycircular arc domain using the method presented here is also set out.

A note on the tau-additive measures

In this note, we point out that some results shown in Jónás et al. (2022) [2] are not valid. In that paper, a class of unary functions called tau function was introduced. By using the tau functions, the authors introduced a class of monotone measures called tau-additive measure and asserted that a tau function is either concave or convex and, as a consequence, a tau-additive measure is either submodular or supermodular. We present two examples to show that all these assertions are false.