On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities

We study a non-local variant of a diffuse interface model proposed by Hawkins--Darrud et al. (2012) for tumour growth in the presence of a chemical species acting as nutrient. The system consists of a Cahn--Hilliard equation coupled to a reaction-diffusion equation. For non-degenerate mobilities and smooth potentials, we derive well-posedness results, which are the non-local analogue of those obtained in Frigeri et al. (European J. Appl. Math. 2015). Furthermore, we establish existence of weak solutions for the case of degenerate mobilities and singular potentials, which serves to confine the order parameter to its physically relevant interval. Due to the non-local nature of the equations, under additional assumptions continuous dependence on initial data can also be shown.Comment: 28 page

Longtime behavior of nonlocal Cahn-Hilliard equations

Here we consider the nonlocal Cahn-Hilliard equation with constant mobility in a bounded domain. We prove that the associated dynamical system has an exponential attractor, provided that the potential is regular. In order to do that a crucial step is showing the eventual boundedness of the order parameter uniformly with respect to the initial datum. This is obtained through an Alikakos-Moser type argument. We establish a similar result for the viscous nonlocal Cahn-Hilliard equation with singular (e.g., logarithmic) potential. In this case the validity of the so-called separation property is crucial. We also discuss the convergence of a solution to a single stationary state. The separation property in the nonviscous case is known to hold when the mobility degenerates at the pure phases in a proper way and the potential is of logarithmic type. Thus, the existence of an exponential attractor can be proven in this case as well

Global compactness for a class of quasi-linear elliptic problems

We prove a global compactness result for Palais-Smale sequences associated with a class of quasi-linear elliptic equations on exterior domains.Comment: 19 page

Global existence for a nonstandard viscous Cahn--Hilliard system with dynamic boundary condition

In this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter. This boundary condition models an additional nonconserving phase transition occurring on the surface of the domain. Different well-posedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies

Existence of continuous eigenvalues for a class of parametric problems involving the (p,2)-laplacian operator

We discuss a parametric eigenvalue problem, where the differential operator is of (p,2)-Laplacian type. We show that, when p≠2, the spectrum of the operator is a half line, with the end point formulated in terms of the parameter and the principal eigenvalue of the Laplacian with zero Dirichlet boundary conditions. Two cases are considered corresponding to p>2 and p2, and to infinity in the case of p<2

The domain architecture of large guanine nucleotide exchange factors for the small GTP-binding protein Arf

BACKGROUND: Small G proteins, which are essential regulators of multiple cellular functions, are activated by guanine nucleotide exchange factors (GEFs) that stimulate the exchange of the tightly bound GDP nucleotide by GTP. The catalytic domain responsible for nucleotide exchange is in general associated with non-catalytic domains that define the spatio-temporal conditions of activation. In the case of small G proteins of the Arf subfamily, which are major regulators of membrane trafficking, GEFs form a heterogeneous family whose only common characteristic is the well-characterized Sec7 catalytic domain. In contrast, the function of non-catalytic domains and how they regulate/cooperate with the catalytic domain is essentially unknown. RESULTS: Based on Sec7-containing sequences from fully-annotated eukaryotic genomes, including our annotation of these sequences from Paramecium, we have investigated the domain architecture of large ArfGEFs of the BIG and GBF subfamilies, which are involved in Golgi traffic. Multiple sequence alignments combined with the analysis of predicted secondary structures, non-structured regions and splicing patterns, identifies five novel non-catalytic structural domains which are common to both subfamilies, revealing that they share a conserved modular organization. We also report a novel ArfGEF subfamily with a domain organization so far unique to alveolates, which we name TBS (TBC-Sec7). CONCLUSION: Our analysis unifies the BIG and GBF subfamilies into a higher order subfamily, which, together with their being the only subfamilies common to all eukaryotes, suggests that they descend from a common ancestor from which species-specific ArfGEFs have subsequently evolved. Our identification of a conserved modular architecture provides a background for future functional investigation of non-catalytic domains

Modulation of Cell Adhesion and Migration by the Histone Methyltransferase Subunit mDpy-30 and Its Interacting Proteins

We have previously shown that a subset of mDpy-30, an accessory subunit of the nuclear histone H3 lysine 4 methyltransferase (H3K4MT) complex, also localizes at the trans-Golgi network (TGN), where its recruitment is mediated by the TGN-localized ARF guanine nucleotide exchange factor (ArfGEF) BIG1. Depletion of mDpy-30 inhibits the endosome-to-TGN transport of internalized CIMPR receptors and concurrently promotes their accumulation at the cell protrusion. These observations suggest mDpy-30 may play a novel role at the crossroads of endosomal trafficking, nuclear transcription and adhesion/migration. Here we provide novel mechanistic and functional insight into this association. First, we demonstrate a direct interaction between mDpy-30 and BIG1 and locate the binding region in the N-terminus of BIG1. Second, we provide evidence that the depletion or overexpression of mDpy-30 enhances or inhibits cellular adhesion/migration of glioma cells in vitro, respectively. A similar increase in cell adhesion/migration is observed in cells with reduced levels of BIG1 or other H3K4MT subunits. Third, knockdown of mDpy-30, BIG1, or the RbBP5 H3K4MT subunit increases the targeting of β1 integrin to cell protrusions, and suppression of H3K4MT activity by depleting mDpy-30 or RbBP5 leads to increased protein and mRNA levels of β1 integrin. Moreover, stimulation of cell adhesion/migration via mDpy-30 knockdown is abolished after treating cells with a function-blocking antibody to β1 integrin. Taken together, these data indicate that mDpy-30 and its interacting proteins function as a novel class of cellular adhesion/migration modulators partially by affecting the subcellular distribution of endosomal compartments as well as the expression of key adhesion/migration proteins such as β1 integrin

Limiting problems for a nonstandard viscous Cahn--Hilliard system with dynamic boundary conditions

This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice and was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp.105--118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the Laplace-Beltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested to some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions to the corresponding solutions for the limit problem is proven. Then, we study the long-time behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omega-limit set in both cases

The Minimal Autoinhibited Unit of the Guanine Nucleotide Exchange Factor Intersectin

Intersectin-1L is a member of the Dbl homology (DH) domain guanine nucleotide exchange factors (GEF) which control Rho-family GTPase signaling. Intersectin-1L is a GEF that is specific for Cdc42. It plays an important role in endocytosis, and is regulated by several partners including the actin regulator N-WASP. Intact intersectin-1L shows low Cdc42 exchange activity, although the isolated catalytic DH domain shows high activity. This finding suggests that the molecule is autoinhibited. To investigate the mechanism of autoinhibition we have constructed a series of domain deletions. We find that the five SH3 domains of intersectin are important for autoinhibition, with the fifth domain (SH3(E)) being sufficient for the bulk of the autoinhibitory effect. This SH3 domain appears to primarily interact with the DH domain. We have determined the crystal structure of the SH3(E)-DH domain construct, which shows a domain swapped arrangement in which the SH3 from one monomer interacts with the DH domain of the other monomer. Analytical ultracentrifugation and gel filtration, however, show that under biochemical concentrations, the construct is fully monomeric. Thus we propose that the actual autoinhibited structure contains the related intramolecular SH3(E)-DH interaction. We propose a model in which this intramolecular interaction may block or distort the GTPase binding region of the DH domain

Investigating the Structural Impacts of I64T and P311S Mutations in APE1-DNA Complex: A Molecular Dynamics Approach

Elucidating the molecular dynamic behavior of Protein-DNA complex upon mutation is crucial in current genomics. Molecular dynamics approach reveals the changes on incorporation of variants that dictate the structure and function of Protein-DNA complexes. Deleterious mutations in APE1 protein modify the physicochemical property of amino acids that affect the protein stability and dynamic behavior. Further, these mutations disrupt the binding sites and prohibit the protein to form complexes with its interacting DNA.In this study, we developed a rapid and cost-effective method to analyze variants in APE1 gene that are associated with disease susceptibility and evaluated their impacts on APE1-DNA complex dynamic behavior. Initially, two different in silico approaches were used to identify deleterious variants in APE1 gene. Deleterious scores that overlap in these approaches were taken in concern and based on it, two nsSNPs with IDs rs61730854 (I64T) and rs1803120 (P311S) were taken further for structural analysis.Different parameters such as RMSD, RMSF, salt bridge, H-bonds and SASA applied in Molecular dynamic study reveals that predicted deleterious variants I64T and P311S alters the structure as well as affect the stability of APE1-DNA interacting functions. This study addresses such new methods for validating functional polymorphisms of human APE1 which is critically involved in causing deficit in repair capacity, which in turn leads to genetic instability and carcinogenesis