Tau Function Research Articles

Specific phosphorylation of microtubule-associated protein 2c by extracellular signal–regulated kinase reduces interactions at its Pro-rich regions

Microtubule-associated protein 2 (MAP2) is an important neuronal target of extracellular signal–regulated kinase 2 (ERK2) involved in Raf signaling pathways, but mechanistic details of MAP2 phosphorylation are unclear. Here, we used NMR spectroscopy to quantitatively describe the kinetics of phosphorylation of individual serines and threonines in the embryonic MAP2 variant MAP2c. We carried out real-time monitoring of phosphorylation to discover major phosphorylation sites that were not identified in previous studies relying on specific antibodies. Our comparison with the phosphorylation of MAP2c by a model cyclin-dependent kinase CDK2 and with phosphorylation of the MAP2c homolog Tau revealed differences in phosphorylation profiles that explain specificity of regulation of biological functions of MAP2c and Tau. To probe the molecular basis of the regulatory effect of ERK2, we investigated the interactions of phosphorylated and unphosphorylated MAP2c by NMR with single-residue resolution. As ERK2 phosphorylates mostly outside the regions binding microtubules, we studied the binding of proteins other than tubulin, namely regulatory subunit RIIα of cAMP-dependent PKA, adapter protein Grb2, Src homology domain 3 of tyrosine kinases Fyn and Abl, and ERK2 itself. We found ERK2 phosphorylation interfered mostly with binding to proline-rich regions of MAP2c. Furthermore, our NMR experiments in SH-SY5Y neuroblastoma cell lysates showed that the kinetics of dephosphorylation are compatible with in-cell NMR studies and that residues targeted by ERK2 and PKA are efficiently phosphorylated in the cell lysates. Taken together, our results provide a deeper characterization of MAP2c phosphorylation and its effects on interactions with other proteins.

Functional Cooperation of α-Synuclein and Tau Is Essential for Proper Corticogenesis.

Alpha-synuclein (αSyn) and tau are abundant multifunctional neuronal proteins, and their intracellular deposits have been linked to many neurodegenerative diseases, including Alzheimer's disease and Parkinson's disease. Despite the disease relevance, their physiological roles remain elusive, as mice with knock-out of either of these genes do not exhibit overt phenotypes. To reveal functional cooperation, we generated αSyn-/-tau-/- double-knock-out mice and characterized the functional cross talk between these proteins during brain development. Intriguingly, deletion of αSyn and tau reduced Notch signaling and accelerated interkinetic nuclear migration of G2 phase at early embryonic stage. This significantly altered the balance between the proliferative and neurogenic divisions of progenitor cells, resulting in an overproduction of early born neurons and enhanced neurogenesis, by which the brain size was enlarged during the embryonic stage in both sexes. On the other hand, a reduction in the number of neural progenitor cells in the middle stage of corticogenesis diminished subsequent gliogenesis in the αSyn-/-tau-/- cortex. Additionally, the expansion and maturation of macroglial cells (astrocytes and oligodendrocytes) were suppressed in the αSyn-/-tau-/- postnatal brain, which in turn reduced the male αSyn-/-tau-/- brain size and cortical thickness to less than the control values. Our study identifies important functional cooperation of αSyn and tau during corticogenesis.SIGNIFICANCE STATEMENT Correct understanding of the physiological functions of αSyn and tau in CNS is critical to elucidate pathogenesis involved in the etiology of neurodegenerative diseases including Alzheimer's disease and Parkinson's disease. We show here that αSyn and tau are cooperatively involved in brain development via maintenance of progenitor cells. αSyn and tau double-knock-out mice exhibited an overproduction of early born neurons and accelerated neurogenesis at early corticogenesis. Furthermore, loss of αSyn and tau also perturbed gliogenesis at later embryonic stage, as well as the subsequent glial expansion and maturation at postnatal brain. Our findings provide new mechanistic insights and extend therapeutic opportunities for neurodegenerative diseases caused by aberrant αSyn and tau.

A GENERALISATION OF WITTEN’S CONJECTURE FOR THE PIXTON CLASS AND THE NONCOMMUTATIVE KDV HIERARCHY

AbstractIn this paper, we formulate and present ample evidence towards the conjecture that the partition function (i.e. the exponential of the generating series of intersection numbers with monomials in psi classes) of the Pixton class on the moduli space of stable curves is the topological tau function of the noncommutative Korteweg-de Vries hierarchy, which we introduced in a previous work. The specialisation of this conjecture to the top degree part of Pixton’s class states that the partition function of the double ramification cycle is the tau function of the dispersionless limit of this hierarchy. In fact, we prove that this conjecture follows from the double ramification/Dubrovin–Zhang equivalence conjecture. We also provide several independent computational checks in support of it.

Explicit closed algebraic formulas for Orlov–Scherbinn-point functions

We derive a new explicit formula in terms of sums over graphs for the $n$-point correlation functions of general formal weighted double Hurwitz numbers coming from the Kadomtsev-Petviashvili tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions). Notably, we use the change of variables suggested by the associated spectral curve, and our formula turns out to be a polynomial expression in a certain small set of formal functions defined on the spectral curve.

Effects of toxic apolipoprotein E fragments on Tau phosphorylation and cognitive impairment in neonatal mice under sevoflurane anesthesia.

BackgroundAnesthesia induces Tau phosphorylation and cognitive impairment in young, but not adult, mice. Apolipoprotein E (ApoE) may play a protective role in neuronal activity and injury repair, whereas its toxic fragments are reported to induce neurodegeneration and neurocognitive impairment in patients with Alzheimer's disease (AD). Therefore, we set out to test the hypothesis that the difference in ApoE fragments, but not the full‐length ApoE, contributes to the difference in Tau phosphorylation and neurocognitive functions following sevoflurane anesthesia in young mice.MethodsSevoflurane was administered to wild‐type (WT), ApoE‐knockout (ApoE‐KO), ApoE3‐targeted replacement (ApoE3 expresses both full‐length and fragmented ApoE), and ApoE2‐targeted replacement (ApoE2 only expresses full‐length ApoE) mice. The mRNA and protein levels of ApoE, phosphorylated Tau (pTau), and cognitive function were tested in the mice.ResultsSevoflurane anesthesia enhanced ApoE mRNA, total ApoE, full‐length ApoE, ApoE fragments, Tau phosphorylation (AT8 and PHF1), and cognitive impairment in young mice, but not in adult mice. ApoE2, but not ApoE3 or ApoE‐KO, mice showed reduced sevoflurane‐induced pTau elevation and cognitive impairment.ConclusionThese data suggest that elevated ApoE fragments rather than full‐length ApoE might be one of the underlying mechanisms of age‐dependent Tau phosphorylation and cognitive impairment in young mice following sevoflurane anesthesia.

Entanglement entropies of an interval in the free Schrödinger field theory at finite density

We study the entanglement entropies of an interval on the infinite line in the free fermionic spinless Schrödinger field theory at finite density and zero temperature, which is a non-relativistic model with Lifshitz exponent z = 2. We prove that the entanglement entropies are finite functions of one dimensionless parameter proportional to the area of a rectangular region in the phase space determined by the Fermi momentum and the length of the interval. The entanglement entropy is a monotonically increasing function. By employing the properties of the prolate spheroidal wave functions of order zero or the asymptotic expansions of the tau function of the sine kernel, we find analytic expressions for the expansions of the entanglement entropies in the asymptotic regimes of small and large area of the rectangular region in the phase space. These expansions lead to prove that the analogue of the relativistic entropic C function is not monotonous. Extending our analyses to a class of free fermionic Lifshitz models labelled by their integer dynamical exponent z, we find that the parity of this exponent determines the properties of the bipartite entanglement for an interval on the line.

Reassessment of Neuronal Tau Distribution in Adult Human Brain and Implications for Tau Pathobiology

Tau is a predominantly neuronal, soluble and natively unfolded protein that can bind and stabilize microtubules in the central nervous system. Tau has been extensively studied over several decades, especially in the context of neurodegenerative diseases where it can aberrantly aggregate to form a spectrum of pathological inclusions. The presence of tau inclusions in the form of neurofibrillary tangles, neuropil threads and dystrophic neurites within senile plaques are essential and defining features of Alzheimer’s disease. The current dogma favors the notion that tau is predominantly an axonal protein, and that in Alzheimer’s disease there is a redistribution of tau towards the neuronal soma that is associated with the formation of pathological inclusions such as neurofibrillary tangles and neuropil threads. Using novel as well as previously established highly specific tau antibodies, we demonstrate that contrary to this overwhelmingly accepted fact, as asserted in numerous articles and reviews, in adult human brain, tau is more abundant in cortical gray matter that is enriched in neuronal soma and dendrites compared to white matter that is predominantly rich in neuronal axons. Additionally, in Alzheimer’s disease tau pathology is significantly more abundant in the brain cortical gray matter of affected brain regions compared to the adjacent white matter regions. These findings have important implications for the biological function of tau as well as the mechanisms involved in the progressive spread of tau associated with the insidious nature of Alzheimer’s disease.

On the integrable hierarchy for the resolved conifold

We provide a direct proof of a conjecture of Brini relating the Gromov–Witten (GW) theory of the resolved conifold to the Ablowitz–Ladik (AL) integrable hierarchy at the level of primaries. In doing so, we use a functional representation of the AL hierarchy as well as a difference equation for the GW potential. In particular, we express certain distinguished solutions of the difference equation in terms of an analytic function which is a specialization of a Tau function put forward by Bridgeland in the study of wall-crossing phenomena of Donaldson–Thomas invariants.

General breather and rogue wave solutions to the complex short pulse equation

In the present paper, we attempt to construct both the breather and rogue wave solutions to the focusing complex short pulse (fCSP) equation via the KP–Toda reduction method. Following a procedure of reducing the bilinear equations satisfied by tau functions of Kadomtsev–Petviashvili (KP)–Toda hierarchy to the ones of the fCSP equation with nonzero boundary condition, we first deduce the general breather solution of the fCSP equation starting from a specially arranged tau-function of the KP–Toda hierarchy, then we construct and prove the Nth order rogue wave solutions of the fCSP equation and express them in two different but equivalent forms of determinants. The dynamical behaviors of both the breather and rogue wave solutions are illustrated and analyzed.

On the tau function of the hypergeometric equation

The monodromy map for a rank-two system of differential equations with three Fuchsian singularities is classically solved by the Kummer formulæ for Gauss’ hypergeometric functions. We define the tau-function of such a system as the generating function of the extended monodromy symplectomorphism, using an idea recently developed. This formulation allows us to determine the dependence of the tau-function on the monodromy data. Using the explicit solution of the monodromy problem, the tau-function is then explicitly written in terms of Barnes G-function. In particular, if the Fuchsian singularities are placed to 0, 1 and ∞, this gives the structure constants of the asymptotical formula of Iorgov–Gamayun–Lisovyy for solutions of Painlevé VI equation.

Quantum Spectral Problems and Isomonodromic Deformations

We develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of 2times 2 linear systems (Riemann–Hilbert correspondence). Our technique applies to a variety of problems, though in this paper we only analyse in detail two examples. First we review the case of the (modified) Mathieu operator, which corresponds to a certain linear system on the sphere and makes contact with the Painlevé mathrm {III}_3 equation. Then we extend the analysis to the 2-particle elliptic Calogero–Moser operator, which corresponds to a linear system on the torus. By using the Kyiv formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions (epsilon _1+epsilon _2=0). Through blowup relations, we also find Nekrasov–Shatashvili type of quantizations (epsilon _2=0). In the case of the torus with one regular singularity we obtain certain results which are interesting by themselves. Namely, we derive blowup equations (filling some gaps in the literature) and we relate them to the bilinear form of the isomonodromic deformation equations. In addition, we extract the epsilon _2rightarrow 0 limit of the blowup relations from the regularized action functional and CFT arguments.

Several Integrable Properties of the Discrete KP Hierarchy Obtained by Means of Gauge Transformations

The gauge transformation is a powerful method to solve integrable hierarchies. In the discrete KP hierarchy, there are two basic gauge transformations, T D ( q ) = Λ( q ) ∘ Δ ∘ q −1 and T I ( r ) = Λ −1 ( r −1 ) ∘ Δ −1 ∘ r. This paper is concerned with several integrable properties of the discrete KP hierarchy found by means of gauge transformations. The successive applications of T I to the constrained discrete KP hierarchy have been presented. Meanwhile, we derive two novel discrete equations as the orbits of gauge transformations T D and T I for the constrained KP hierarchy. Moreover, with the help of the spectral representations and the gauge transformations, Fay-similar equations of the tau functions for the constrained discrete KP have been investigated. In addition, we also discuss the relation between the gauge transformation and an additional symmetry for the discrete KP hierarchy.

Tau Acts in Concert With Kinase/Phosphatase Underlying Synaptic Dysfunction.

Alzheimer's disease (AD) is characterized by two pathological features: neurofibrillary tangles (NFTs), formed by microtubule-associated protein tau, and abnormal accumulation of amyloid-β (Aβ). Multiple evidence placed synaptic tau as the vital fact of AD pathology, especially at the very early stage of AD. In the present review, we discuss tau phosphorylation, which is critical for the dendritic localization of tau and synaptic plasticity. We review the related kinases and phosphatases implicated in the synaptic function of tau. We also review the synergistic effects of these kinases and phosphatases on tau-associated synaptic deficits. We aim to open a new perspective on the treatment of AD.

Hepatoprotective Effects of Taurine Against Cadmium-Induced Liver Injury in Female Mice.

Cadmium (Cd), a heavy metal contaminant, seriously threatens human and animal health. Taurine (Tau) has been used against hepatotoxicity caused by different environmental toxins. However, it has not been elucidated whether Tau exerts its protective function against Cd-induced hepatotoxicity. The aim of this study was thus to evaluate the ameliorative function of Tau (500mg/kg body weight intraperitoneally) on Cd-induced (2mg/kg body weight intraperitoneally) liver toxicity in mice for 14days. The histopathologic and ultrastructure changes as well as alterations in indexes related to liver function, antioxidant biomarkers, inflammatory, and apoptosis were evaluated. The results showed that Tau alleviated the vacuolar degeneration, nuclear condensation, mitochondria swelling, and cristae lysis of hepatocytes induced by Cd. In addition, Tau treatment significantly reduced the ALT, AST levels in serum, and inflammatory factor TNF-α and IL-1β in liver tissue. Furthermore, Tau treatment decreased the Bax/Bcl-2 ratio and cleaved caspase-3 protein expression levels. Taken together, these observations demonstrate that Tau has an important hepatic protective function against the inflammation and apoptosis induced by Cd.

Q-polynomial expansion for Brézin-Gross-Witten tau-function

In this paper, we prove a conjecture of Alexandrov that the generalized Brézin-Gross-Witten tau-functions are hypergeometric tau functions of BKP hierarchy after re-scaling. In particular, this shows that the original BGW tau-function, which has enumerative geometric interpretations, can be represented as a linear combination of Schur Q-polynomials with simple coefficients.

FTD-associated mutations in Tau result in a combination of dominant and recessive phenotypes

Although mutations in the microtubules-associated protein Tau have long been connected with several neurodegenerative diseases, the underlying molecular mechanisms causing these tauopathies are still not fully understood. Studies in various models suggested that dominant gain-of-function effects underlie the pathogenicity of these mutants; however, there is also evidence that the loss of normal physiological functions of Tau plays a role in tauopathies. Previous studies on Tau in Drosophila involved expressing the human Tau protein in the background of the endogenous Tau gene in addition to inducing high expression levels. To study Tau pathology in more physiological conditions, we recently created Drosophila knock-in models that express either wildtype human Tau (hTauWT) or disease-associated mutant hTau (hTauV337M and hTauK369I) in place of the endogenous Drosophila Tau (dTau). Analyzing these flies as homozygotes, we could therefore detect recessive effects of the mutations while identifying dominant effects in heterozygotes. Using memory, locomotion and sleep assays, we found that homozygous mutant hTau flies showed deficits already when quite young whereas in heterozygous flies, disease phenotypes developed with aging. Homozygotes also revealed an increase in microtubule diameter, suggesting that changes in the cytoskeleton underlie the axonal degeneration we observed in these flies. In contrast, heterozygous mutant hTau flies showed abnormal axonal targeting and no detectable changes in microtubules. However, we previously showed that heterozygosity for hTauV337M interfered with synaptic homeostasis in central pacemaker neurons and we now show that heterozygous hTauK369I flies have decreased levels of proteins involved in the release of synaptic vesicles. Taken together, our results demonstrate that both mutations induce a combination of dominant and recessive disease-related phenotypes that provide behavioral and molecular insights into the etiology of Tauopathies.

Alzheimer's disease associated AKAP9 I2558M mutation alters posttranslational modification and interactome of tau and cellular functions in CRISPR-edited human neuronal cells.

Alzheimer's disease (AD) is a pervasive neurodegeneration disease with high heritability. In this study, we employed CRISPR‐Cas9‐engineered technology to investigate the effects of a rare mutation (rs144662445) in the A kinase anchoring protein 9 (AKAP9) gene, which is associated with AD in African Americans (AA), on tau pathology and the tau interactome in SH‐SY5Y P301L neuron‐like cells. The mutation significantly increased the level of phosphorylated tau, specifically at the site Ser396/Ser404. Moreover, analyses of the tau interactome measured by affinity purification‐mass spectrometry revealed that differentially expressed tau‐interacting proteins in AKAP9 mutant cells were associated with RNA translation, RNA localization and oxidative activity, recapitulating the tau interactome signature previously reported with human AD brain samples. Importantly, these results were further validated by functional studies showing a significant reduction in protein synthesis activity and excessive oxidative stress in AKAP9 mutant compared with wild type cells in a tau‐dependent manner, which are mirrored with pathological phenotype frequently seen in AD. Our results demonstrated specific effects of rs14462445 on mis‐processing of tau and suggest a potential role of AKAP9 in AD pathogenesis.

Modified DKP hierarchy as modified BKP hierarchy

In this paper, we investigate some topics involving the modified D-type Kadomtsev-Petviashvili (DKP) hierarchy. Note that the modified DKP (mDKP) hierarchy is just the modified B-type Kadomtsev-Petviashvili (BKP) hierarchy in the bosonic forms. Based on this fact, we first proved the conjecture in You [Physica D 50, 429–462 (1991)], that is, the product of the two tau functions of the mDKP hierarchy is some tau function of the KP hierarchy. Then, we investigate the Darboux transformations of the DKP and mDKP hierarchy. Finally, the solutions of the constrained BKP hierarchy are given in the context of the fermionic representation of infinite Lie algebra d∞.

Ramanujan-type systems of nonlinear ODEs for [formula omitted] and [formula omitted

This paper aims to introduce two systems of nonlinear ordinary differential equations whose solution components generate the graded algebra of quasi-modular forms on Hecke congruence subgroups Γ0(2) and Γ0(3). Using these systems, we provide the generated graded algebras with an sl2(ℂ)-module structure. As applications, we introduce Ramanujan-type tau functions for Γ0(2) and Γ0(3), and obtain some interesting and non-trivial recurrence and congruence relations.

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.