Tau Function Research Articles

Tau functions of modified CKP hierarchy

Tau functions of modified CKP hierarchy

Bilinear Expansions of KP Multipair Correlators in BKP Correlators

I present a generalization of our joint works with John Harnad (2021) that relates Schur functions, KP tau functions and KP correlation functions to Schur's $Q$-functions, BKP tau functions and BKP correlation functions, respectively.

Polynomial Analogs of Theta Functions and Rational Solutions of the KP Equation

Abstract In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta divisor in terms of the singularities of the curve. Furthermore, we show a precise relation between such algebraic theta functions and the corresponding tau functions for the KP hierarchy.

Development and characterization of novel anti-acetylated tau monoclonal antibodies to probe pathogenic tau species in Alzheimer’s disease

Tauopathies, including Alzheimer’s disease (AD), are a class of neurodegenerative diseases characterized by the presence of insoluble tau inclusions. Tau phosphorylation has traditionally been viewed as the dominant post-translational modification (PTM) controlling tau function and pathogenesis in tauopathies. However, we and others have identified tau acetylation as a primary PTM regulating both normal tau function as well as abnormal pathogenic features including aggregation. Prior work showed robust tau acetylation in aggregation hotspots located within the 2nd and 3rd repeat regions of tau (residues K280 and K311) in tauopathy brains, including AD, compared to non-tauopathy controls. By screening thousands of hybridoma clones, we generated site-specific and modification-specific monoclonal antibodies targeting acetylated tau at residues K280 or K311. To validate these antibodies in a bona fide neuronal system, we targeted the acetyltransferase CBP to the cytoplasm of neurons to promote tau acetylation. Several antibody clones specifically detected CBP-acetylated tau and co-localized with ac-tau in neurons. Additionally, our lead optimal anti-acetylated-tau monoclonal antibodies detected robust tau pathology in tangles and neuritic plaques of human AD brains. Given the now emerging interest in acetylated tau as critical regulator of tau functions, these sensitive and highly specific tools will allow us to further unravel the tau PTM code and, importantly, could be deployed as diagnostic or disease-modifying agents.

Free superfermions construction for the coupled super KP and super modified KP hierarchies

In this paper, the Kac–van de Leur version of the coupled super KP (SKP) and super modified KP (SmKP) hierarchies are constructed. We define the tau functions as the orbit of a new group G(g) acting on the weight vectors, where G(g) is generated by elements in the even part of the tensor product of Clifford superalgebra and Grassmann algebra. The super Hirota bilinear equations of the coupled SKP and SmKP hierarchies are obtained by super boson-fermion correspondence of type A, and the bilinear identities with respect to the super Baker functions and tau functions in superbosonic Fock space are also constructed. Finally, the Miura transformation in terms of superfermions is also discussed.

Structural Insights into the Role of the Proline Rich Region in Tau Function.

Tau is a microtubule–associated protein that plays an important role in modulating axonal microtubules in neurons. Intracellular tau aggregates are found in a broad class of disorders, including Alzheimer&#8242s disease, termed tauopathies. Tau is an intrinsically disordered protein, and its structural disorder appears to be critical to its microtubule-related functions. Tubulin binding sites are found in tau&#8242s proline-rich region (PRR), microtubule binding repeats (MTBR: R1–R4), and pseudo–repeat, R′. While many post-translational modifications have been identified on tau, phosphorylation sites, which both regulate tubulin dimer and microtubule interactions and are correlated with disease, cluster with high frequency within the PRR. Here, we use fluorescence correlation spectroscopy and structural mass spectrometry techniques to characterize the impact of phosphomimic mutations in the PRR on tubulin dimer binding and probe the structure of the PRR–tubulin dimer complex. We find that phosphomimics cumulatively diminish tubulin dimer binding and slow microtubule polymerization. Additionally, we map two ~15 residue regions of the PRR as primary tubulin dimer binding sites and propose a model in which PRR enhances lateral interactions between tubulin dimers, complementing the longitudinal interactions observed for MTBR. Together these measurements provide insight into the previously overlooked relevance of tau&#8242s PRR in functional interactions with tubulin.

New cluster algebras from old: integrability beyond Zamolodchikov periodicity

Abstract We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of Zamolodchikov periodicity: they are affine birational maps for which every orbit is periodic with the same period. Following on from preliminary work by one of us with Kouloukas, here we present integrable maps obtained from deformations of cluster mutations related to the simple root systems 
$A_3$, $B_2$, $B_3$ and $D_4$. We further show how new cluster algebras arise, by considering Laurentification, that is, a lifting to a higher-dimensional map expressed in a set of new variables (tau functions), for which the dynamics exhibits the Laurent property. For the integrable map obtained by deformation of type $A_3$, which already appeared in our previous work, we show that there is a commuting map of Quispel-Roberts-Thompson (QRT) type which is built from a composition of mutations and a permutation applied to 
the same cluster algebra of rank 6, with an additional 2 frozen variables. Furthermore, both the deformed $A_3$ map and the QRT map correspond to translation by a generator in the Mordell-Weil group of a rational elliptic surface of rank two, and the underlying cluster algebra comes from a quiver that is mutation equivalent to the $q$-Painlev\'e III quiver found by Okubo. 
The deformed integrable maps of types $B_2$, $B_3$ and $D_4$ are also related to elliptic surfaces. 

From a dynamical systems viewpoint, the message of the paper is that special families of birational maps with completely periodic dynamics under iteration admit natural deformations that are aperiodic yet completely integrable.

On formal series solutions to 4th-order quadratic homogeneous differential equations and their convergence

Abstract It is known that all τ functions of the Painlevé equations satisfy the fourth-order quadratic differential equation. Among them, for the III, V, and VI equations, it is possible to express the formal series solutions combinatorially by using conformal blocks. In this paper, we show the convergence of the formal series, including the solutions of more general equations. The convergence of the conformal block function also follows in the case c = 1 by the absolute convergence of τ series, since it is a partial sum of the τ series. We also characterized the form of a homogeneous quadratic equation with a series solution similar to the tau functions of the Painlevé equations.

Exploring the Potential of Herbal Compounds as Autophagy Modulators in Alzheimer's Disease: A Comprehensive Review

Abstract: Alzheimer’s disease (AD) is a progressive neurodegenerative disorder that causes atrophy of brain cells, leading to their death, and has become a leading cause of death in aging populations worldwide. AD is characterized by β-amyloid (Aβ) deposition and tau phosphorylation in neural tissues, but the precise pathophysiology of the disease is still obscure. Autophagy is an evolutionarily targeted mechanism that is necessary for the elimination of neuronal and glial misfolded proteins as well as proteins. It also plays an essential role in synaptic plasticity. The aberrant autophagy primarily influences the process of aging and neurodegeneration. Autophagy significantly influences how Aβ and tau function physiologically, therefore, atypical autophagy is expected to perform an important role in Aβ deposition and tau phosphorylation characteristic in the development of AD. Bioactive phytoconstituents could majorly contribute as a natural yet effective alternative approach to slow down the progression of neurodegeneration and promote the active aging process in elderly patients. Over the recent years, it is well evidenced that different secondary metabolites including polyphenols, alkaloids, terpenes, and phenols exhibited neuroprotective effects, and attenuated brain damage, and cognitive impairment in vitro as well as in vivo. Additionally, the underlying mechanism of action shared by them is the regulation of competent autophagy via the removal of aggregated protein and mitochondrial dysfunction. The present article is structured as a reference for researchers keen to investigate and assess the new natural compound-mediated therapeutic approach for AD treatment through the modulation of autophagy.

Comprehensive review on Alzheimer's disease: From the posttranslational modifications of Tau to corresponding treatments

AbstractAlzheimer's disease (AD) is a neurodegenerative disease, which is mainly characterized by the abnormal deposition of β‐amyloid peptide (Aβ) and Tau. Since Tau aggregation is more closely associated with synaptic loss, neurodegeneration, and cognitive decline than Aβ, the correlation between Tau and cognitive function in AD has gradually gained attention. The posttranslational modifications (PTMs) of Tau are key factors contributing to its pathological changes, which include phosphorylation, acetylation, ubiquitination, glycosylation, glycation, small ubiquitin‐like modifier mediated modification (SUMOylation), methylation, succinylation, etc. These modifications change the structure of Tau, regulating Tau microtubule interactions, localization, degradation, and aggregation, thereby affecting its propensity to aggregate and leading to neuronal injury and cognitive impairments. Among numerous PTMs, drug development based on phosphorylation, acetylation, ubiquitination, and SUMOylation primarily involves enzymatic reactions, affecting either the phosphorylation or degradation processes of Tau. Meanwhile, methylation, glycosylation, and succinylation are associated with maintaining the structural stability of Tau. Current research is more extensive on phosphorylation, acetylation, ubiquitination, and methylation, with related drugs already developed, particularly focusing on phosphorylation and ubiquitination. In contrast, there is less research on SUMOylation, glycosylation, and succinylation, requiring further basic research, with the potential to become novel drug targets. In conclusion, this review summarized the latest research on PTMs of Tau and related drugs, highlighting the potential of targeting specific PTMs for developing novel therapeutic strategies in AD.

NP-Collabs: Investigating Counterion-Mediated Bridges in the Multiply Phosphorylated Tau-R2 Repeat.

Tau is an intrinsically disordered (IDP) microtubule-associated protein (MAP) that plays a key part in microtubule assembly and organization. The function of tau can be regulated by multiple phosphorylation sites. These post-translational modifications are known to decrease the binding affinity of tau for microtubules, and abnormal tau phosphorylation patterns are involved in Alzheimer's disease. Using all-atom molecular dynamics simulations, we compared the conformational landscapes explored by the tau R2 repeat domain (which comprises a strong tubulin binding site) in its native state and with multiple phosphorylations on the S285, S289, and S293 residues, with four different standard force field (FF)/water model combinations. We find that the different parameters used for the phosphate groups (which can be more or less flexible) in these FFs and the specific interactions between bulk cations and water lead to the formation of a specific type of counterion bridge, termed nP-collab (for nphosphate collaboration, with n being an integer), where counterions form stable structures binding with two or three phosphate groups simultaneously. The resulting effect of nP-collabs on the tau-R2 conformational space differs when using sodium or potassium cations and is likely to impact the peptide overall dynamics and how this MAP interacts with tubulins. We also investigated the effect of phosphoresidue spacing and ionic concentration by modeling polyalanine peptides containing two phosphoserines located one-six residues apart. Three new metrics specifically tailored for IDPs (proteic Menger curvature, local curvature, and local flexibility) were introduced, which allow us to fully characterize the impact of nP-collabs on the dynamics of disordered peptides at the residue level.

Expansions for semiclassical conformal blocks

We propose a relation the expansions of regular and irregular semiclassical conformal blocks at different branch points making use of the connection between the accessory parameters of the BPZ decoupling equations to the logarithm derivative of isomonodromic tau functions. We give support for these relations by considering two eigenvalue problems for the confluent Heun equations obtained from the linearized perturbation theory of black holes. We first derive the large frequency expansion of the spheroidal equations, and then compare numerically the excited quasi-normal mode spectrum for the Schwarzschild case obtained from the large frequency expansion to the one obtained from the low frequency expansion and with the literature, indicating that the relations hold generically in the complex modulus plane.

A family of integrable maps associated with the Volterra lattice

Recently Gubbiotti, Joshi, Tran and Viallet classified birational maps in four dimensions admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. The last three of the maps so obtained were shown to be Liouville integrable, in the sense of admitting a non-degenerate Poisson bracket with two first integrals in involution. Here we show how the first of these three Liouville integrable maps corresponds to genus 2 solutions of the infinite Volterra lattice, being the g = 2 case of a family of maps associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus g⩾1 . The continued fraction method provides explicit Hankel determinant formulae for tau functions of the solutions, together with an algebro-geometric description via a Lax representation for each member of the family, associating it with an algebraic completely integrable system. In particular, in the elliptic case (g = 1), as a byproduct we obtain Hankel determinant expressions for the solutions of the Somos-5 recurrence, but different to those previously derived by Chang, Hu and Xin. By applying contraction to the Stieltjes fraction, we recover integrable maps associated with Jacobi continued fractions on hyperelliptic curves, that one of us considered previously, as well as the Miura-type transformation between the Volterra and Toda lattices.

A KP-mKP hierarchy via pseudo-differential operators with two derivations

By using pseudo-differential operators containing two derivations, we extend the Kadomtsev–Petviashvili (KP) hierarchy to a certain KP-mKP hierarchy. For the KP-mKP hierarchy, we derive its Bäcklund transformations, bilinear equations of Baker–Akhiezer functions and Hirota equations of tau functions. Moreover, we show that this hierarchy is equivalent to a subhierarchy of the dispersive Whitham hierarchy associated to the Riemann sphere with its infinity point and one movable point marked.

Q-boson model and relations with integrable hierarchies

This work investigates the intricate relationship between the q-boson model, a quantum integrable system, and classical integrable systems such as the Toda and KP hierarchies. Initially, we analyze scalar products of off-shell Bethe states and explore their connections to tau functions of integrable hierarchies. Furthermore, we discuss correlation functions within this formalism, examining their representations in terms of tau functions, as well as their Schur polynomial expansions.

Cellular and pathological functions of tau.

Tau protein is involved in various cellular processes, including having a canonical role in binding and stabilization of microtubules in neurons. Tauopathies are neurodegenerative diseases marked by the abnormal accumulation of tau protein aggregates in neurons, as seen, for example, in conditions such as frontotemporal dementia and Alzheimer disease. Mutations in tau coding regions or that disrupt tau mRNA splicing, tau post-translational modifications and cellular stress factors (such as oxidative stress and inflammation) increase the tendency of tau to aggregate and interfere with its clearance. Pathological tau is strongly implicated in the progression of neurodegenerative diseases, and the propagation of tau aggregates is associated with disease severity. Recent technological advancements, including cryo-electron microscopy and disease models derived from human induced pluripotent stem cells, have increased our understanding of tau-related pathology in neurodegenerative conditions. Substantial progress has been made in deciphering tau aggregate structures and the molecular mechanisms that underlie protein aggregation and toxicity. In this Review, we discuss recent insights into the diverse cellular functions of tau and the pathology of tau inclusions and explore the potential for therapeutic interventions.

The disease-causing tau V337M mutation induces tau hypophosphorylation and perturbs axon morphology pathways.

Tau aggregation is a hallmark of several neurodegenerative diseases, including Alzheimer's disease and frontotemporal dementia. There are disease-causing variants of the tau-encoding gene, MAPT, and the presence of tau aggregates is highly correlated with disease progression. However, the molecular mechanisms linking pathological tau to neuronal dysfunction are not well understood due to our incomplete understanding of the normal functions of tau in development and aging and how these processes change in the context of causal disease variants of tau. To address these questions in an unbiased manner, we conducted multi-omic characterization of iPSC-derived neurons harboring the MAPT V337M mutation. RNA-seq and phosphoproteomics revealed that both V337M tau and tau knockdown consistently perturbed levels of transcripts and phosphorylation of proteins related to axonogenesis or axon morphology. Surprisingly, we found that neurons with V337M tau had much lower tau phosphorylation than neurons with WT tau. We conducted functional genomics screens to uncover regulators of tau phosphorylation in neurons and found that factors involved in axonogenesis modified tau phosphorylation in both MAPT WT and MAPT V337M neurons. Intriguingly, the p38 MAPK pathway specifically modified tau phosphorylation in MAPT V337M neurons. We propose that V337M tau might perturb axon morphology pathways and tau hypophosphorylation via a "loss of function" mechanism, which could contribute to previously reported cognitive changes in preclinical MAPT gene carriers.

Lambda-invariants of Mazur–Tate elements attached to Ramanujan’s tau function and congruences with Eisenstein series

Lambda-invariants of Mazur–Tate elements attached to Ramanujan’s tau function and congruences with Eisenstein series

Topological recursion for Kadomtsev–Petviashvili tau functions of hypergeometric type

AbstractWe study the ‐point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their ‐deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so‐called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of ‐deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)‐type formulas discussed in the literature.

Learning the weights using attribute order information for multi-criteria decision making tasks

In multi-criteria decision making, the importance of decision criteria (decision attributes) plays a crucial role. Ranking is a useful technique for expressing the importance of decision criteria in a decision-makers’ preference system. Since weights are commonly utilized for characterizing the importance of criteria, weight determination and assessment are important tasks in multi-criteria decision making and in voting systems as well. In this study, we concentrate on the connection between the preference order of decision criteria and the decision weights. Here, we present an easy-to-use procedure that can be used to produce a sequence of weights corresponding to a decision-makers’ preference order of decision criteria. The proposed method does not require pairwise comparisons, which is an advantageous property especially in cases where the number of criteria is large. This method is based on the application of a class of regular increasing monotone quantifiers, which we refer to as the class of weighting generator functions. We will show that the derivatives of these functions can be used for approximating the criteria weights. Also, we will demonstrate that using weighting generator functions, weights can be inverted in a consistent way. We will deduce the generators for arithmetic and geometric weight sequences, and we will present a one-parameter generator function known as the tau function in continuous-valued logic. We will show that using these weighting generator functions, the weight learning task can be turned into a simple, one-parameter optimization problem.