Third-order integrable difference equations generated by a pair of second-order equations

We show that the third-order difference equations proposed by Hirota, Kimura and Yahagi are generated by a pair of second-order difference equations. In some cases, the pair of the second-order equations are equivalent to the Quispel-Robert-Thomson(QRT) system, but in the other cases, they are irrelevant to the QRT system. We also discuss an ultradiscretization of the equations.Comment: 15 pages, 3 figures; Accepted for Publication in J. Phys.

Algebraic entropy for semi-discrete equations

We extend the definition of algebraic entropy to semi-discrete (difference-differential) equations. Calculating the entropy for a number of integrable and non integrable systems, we show that its vanishing is a characteristic feature of integrability for this type of equations

On the complexity of some birational transformations

Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines, and relies mainly on univariate polynomial algebra, the second approach is a singularity analysis, and the third method is more numerical, using integer arithmetics. Each method has its own domain of application, but they give corroborating results, and lead us to a conjecture on the complexity of a class of maps constructed from matrix inversions

A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice

We introduce a tropical analogue of Fay's trisecant identity for a special family of hyperelliptic tropical curves. We apply it to obtain the general solution of the ultra-discrete Toda lattice with periodic boundary conditions in terms of the tropical Riemann's theta function.Comment: 25 pages, 3 figure

Arp2/3 activity is necessary for efficient formation of E-cadherin adhesive contacts

Classical cadherin adhesion molecules are fundamental determinants of cell-cell recognition that function in cooperation with the actin cytoskeleton. Productive cadherin-based cell recognition is characterized by a distinct morphological process of contact zone extension, where limited initial points of adhesion are progressively expanded into broad zones of contact. We recently demonstrated that E-cadherin ligation recruits the Arp2/3 actin nucleator complex to the plasma membrane in regions where cell contacts are undergoing protrusion and extension. This suggested that Arp2/3 might generate the protrusive forces necessary for cell surfaces to extend upon one another during contact assembly. We tested this hypothesis in mammalian cells by exogenously expressing the CA region of N-WASP. This fragment, which potently inhibits Arp2/3-mediated actin assembly in vitro, also effectively reduced actin assembly at cadherin adhesive contacts. Blocking Arp2/3 activity by this strategy profoundly reduced the ability of cells to extend cadherin adhesive contacts but did not affect cell adhesiveness. These findings demonstrate that Arp2/3 activity is necessary for cells to efficiently extend and assemble cadherin-based adhesive contacts

Discrete integrable systems and Poisson algebras from cluster maps

We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutation-periodic with period 1. Such quivers were completely classified by Fordy and Marsh, who characterised them in terms of the skew-symmetric matrix that defines the quiver. The associated nonlinear recurrences are equivalent to birational maps, and we explain how these maps can be endowed with an invariant Poisson bracket and/or presymplectic structure. Upon applying the algebraic entropy test, we are led to a series of conjectures which imply that the entropy of the cluster maps can be determined from their tropical analogues, which leads to a sharp classification result. Only four special families of these maps should have zero entropy. These families are examined in detail, with many explicit examples given, and we show how they lead to discrete dynamics that is integrable in the Liouville-Arnold sense.Comment: 49 pages, 3 figures. Reduced to satisfy journal page restrictions. Sections 2.4, 4.5, 6.3, 7 and 8 removed. All other results remain, with minor editin

Propagating Cell-Membrane Waves Driven by Curved Activators of Actin Polymerization

Cells exhibit propagating membrane waves which involve the actin cytoskeleton. One type of such membranal waves are Circular Dorsal Ruffles (CDR) which are related to endocytosis and receptor internalization. Experimentally, CDRs have been associated with membrane bound activators of actin polymerization of concave shape. We present experimental evidence for the localization of convex membrane proteins in these structures, and their insensitivity to inhibition of myosin II contractility in immortalized mouse embryo fibroblasts cell cultures. These observations lead us to propose a theoretical model which explains the formation of these waves due to the interplay between complexes that contain activators of actin polymerization and membrane-bound curved proteins of both types of curvature (concave and convex). Our model predicts that the activity of both types of curved proteins is essential for sustaining propagating waves, which are abolished when one type of curved activator is removed. Within this model waves are initiated when the level of actin polymerization induced by the curved activators is higher than some threshold value, which allows the cell to control CDR formation. We demonstrate that the model can explain many features of CDRs, and give several testable predictions. This work demonstrates the importance of curved membrane proteins in organizing the actin cytoskeleton and cell shape

PICK1 inhibition of the Arp2/3 complex controls dendritic spine size and synaptic plasticity

Dendritic spine remodelling involves regulated actin dynamics and is essential for synaptic plasticity. The Arp2/3 inhibitor PICK1 is here shown to regulate spine size: inducing shrinkage during long-term depression

X-Linked thrombocytopenia causing mutations in WASP (L46P and A47D) impair T cell chemotaxis

BACKGROUND: Mutation in the Wiskott-Aldrich syndrome Protein (WASP) causes Wiskott-Aldrich syndrome (WAS), X-linked thrombocytopenia (XLT) and X-linked congenital neutropenia (XLN). The majority of missense mutations causing WAS and XLT are found in the WH1 (WASP Homology) domain of WASP, known to mediate interaction with WIP (WASP Interacting Protein) and CIB1 (Calcium and Integrin Binding). RESULTS: We analyzed two WASP missense mutants (L46P and A47D) causing XLT for their effects on T cell chemotaxis. Both mutants, WASP(R)(L46P) and WASP(R)(A47D) (S1-WASP shRNA resistant) expressed well in Jurkat(WASP-KD) T cells (WASP knockdown), however expression of these two mutants did not rescue the chemotaxis defect of Jurkat(WASP-KD) T cells towards SDF-1α. In addition Jurkat(WASP-KD) T cells expressing these two WASP mutants were found to be defective in T cell polarization when stimulated with SDF-1α. WASP exists in a closed conformation in the presence of WIP, however both the mutants (WASP(R)(L46P) and WASP(R)(A47D)) were found to be in an open conformation as determined in the bi-molecular complementation assay. WASP protein undergoes proteolysis upon phosphorylation and this turnover of WASP is critical for T cell migration. Both the WASP mutants were found to be stable and have reduced tyrosine phosphorylation after stimulation with SDF-1α. CONCLUSION: Thus our data suggest that missense mutations WASP(R)(L46P) or WASP(R)(A47D) affect the activity of WASP in T cell chemotaxis probably by affecting the turnover of the protein. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1186/s12929-014-0091-1) contains supplementary material, which is available to authorized users

Interleukin-6, tumour necrosis factor α and interleukin-1β in patients with renal cell carcinoma

As regulators of malignant cell behaviour and communication with stroma, cytokines have proved useful in understanding cancer biology and developing novel therapies. In renal cell carcinoma, patients with inflammatory reactions are known to have poor prognosis. In order to elucidate the relation between renal cell carcinoma and the host, serum levels of inflammatory cytokines, interleukin-6, tumour necrosis factor α, interleukin-1β, were measured. One hundred and twenty-two patients with renal cell carcinoma and 21 healthy control subjects were studied, and serum cytokine levels were measured using a highly sensitive ELISA kit. As a result, in the control group, interleukin-6, tumour necrosis factor α and interleukin-1β levels were 1.79±2.03, 2.74±0.94 and 0.16±0.17 pg ml−1, respectively. In the renal cell carcinoma patients, they were 8.91±13.12, 8.44±4.15 and 0.53±0.57 pg ml−1, respectively, and significantly higher. In the comparison of stage, interleukin-6 level was significantly higher in the stage IV group compared to the other stage groups including the control group, while tumour necrosis factor α level was significantly higher in each stage group compared to the control group. As for grade, interleukin-6 level was significantly higher in the grade 3 group compared to the control, grade 1 and grade 2 groups, while tumour necrosis factor α level was significantly higher in each grade group compared to the control group. All cytokines had a positive correlation with tumour size. In regard to the correlation with CRP, all cytokines had a positive correlation with CRP, while interleukin-6 had a particularly strong correlation. In conclusion, interleukin-6 may be one of the factors for the poor prognosis of patients with renal cell carcinoma. In addition, tumour necrosis factor α may be useful in the early diagnosis of renal cell carcinoma and post-operative follow-up