Construction and Comprehensive Analysis of a Stratification System Based on AGTRAP in Patients with Hepatocellular Carcinoma

Background. With the development of sequencing technology, several signatures have been reported for the prediction of prognosis in patients with hepatocellular carcinoma (HCC). However, the above signatures are characterized by cumbersome application. Therefore, the study is aimed at screening out a robust stratification system based on only one gene to guide treatment. Methods. Firstly, we used the limma package for performing differential expression analysis on 374 HCC samples, followed by Cox regression analysis on overall survival (OS) and disease-free interval (PFI). Subsequently, hub prognostic genes were found at the intersection of the above three groups. In addition, the topological degree inside the PPI network was used to screen for a unique hub gene. The rms package was used to construct two visual stratification systems for OS and PFI, and Kaplan-Meier analysis was utilized to investigate survival differences in clinical subgroups. The ssGSEA algorithm was then used to reveal the relationship between the hub gene and immune cells, immunological function, and checkpoints. In addition, we also used function annotation to explore into putative biological functions. Finally, for preliminary validation, the hub gene was knocked down in the HCC cell line. Results. We discovered 6 prognostic genes (SKA1, CDC20, AGTRAP, BIRC5, NEIL3, and CDC25C) for constructing a PPI network after investigating survival and differential expression genes. According to the topological degree, AGTRAP was chosen as the basis for the stratification system, and it was revealed to be a risk factor with an independent prognostic value in Kaplan-Meier analysis and Cox regression analysis ( P < 0.05 ). In addition, we constructed two visualized nomograms based on AGTRAP. The novel stratification system had a robust predictive value for PFI and OS in ROC analysis and calibration curve ( P < 0.05 ). Meanwhile, AGTRAP upregulation was associated with T staging, N staging, M staging, pathological stage, grade, and vascular invasion ( P < 0.05 ). Notably, AGTRAP was overexpressed in tumor tissues in all pancancers with paired samples ( P < 0.05 ). Furthermore, AGTRAP was associated with immune response and may change immune microenvironment in HCC ( P < 0.05 ). Next, gene enrichment analysis suggested that AGTRAP may be involved in the biological process, such as cotranslational protein targeting to the membrane. Finally, we identified the oncogenic effect of AGTRAP by qRT-PCR, colony formation, western blot, and CCK-8 assay ( P < 0.05 ). Conclusion. We provided robust evidences that a stratification system based on AGTRAP can guide survival prediction for HCC patients.

Multiple Positive Solutions for a Class of Boundary Value Problem of Fractional p , q -Difference Equations under p , q -Integral Boundary Conditions

This paper is mainly concerned with a class of fractional p , q -difference equations under p , q -integral boundary conditions. Multiple positive solutions are established by using the topological degree theory and Krein–Rutman theorem. Finally, two examples are worked out to illustrate the main results.

Study of a mixed problem for a nonlinear elasticity system by topological degree

"In this paper, we consider a mixed problem for a nonlinear elasticity system with laws of general behavior. The coefficients of elasticity depends on x meanwhile the density of the volumetric forces depends on the displacement. The main aim of this paper is to apply the Schauder's fixed point theorem and the techniques of topological degree to prove a theorem of the existence and the uniqueness of the solution of the corresponding variational problem."

Periodic Third-Order Problems with a Parameter

This work concerns with the solvability of third-order periodic fully problems with a weighted parameter, where the nonlinearity must verify only a local monotone condition and no periodic, coercivity or super or sublinearity restrictions are assumed, as usual in the literature. The arguments are based on a new type of lower and upper solutions method, not necessarily well ordered. A Nagumo growth condition and Leray–Schauder’s topological degree theory are the existence tools. Only the existence of solution is studied here and it will remain open the discussion on the non-existence and the multiplicity of solutions. Last section contains a nonlinear third-order differential model for periodic catatonic phenomena, depending on biological and/or chemical parameters.

Estimates of the topological degree of a class of piecewise linear maps with applications

We provide some new estimates for the topological degree of a class of continuous and piecewise linear maps based on a classical integral computation formula. We provide applications to some nonlinear problems that exhibit a local [Formula: see text] structure.

Nonlinear Differential Equations Modeling the Antarctic Circumpolar Current

The aim of this paper is to study one class of nonlinear differential equations, which model the Antarctic circumpolar current. We prove the existence results for such equations related to the geophysical relevant boundary conditions. First, based on the weighted eigenvalues and the theory of topological degree, we study the semilinear case. Secondly, the existence results for the sublinear and superlinear cases are proved by fixed point theorems.

TOPOLOGY OF 1-PARAMETER DEFORMATIONS OF NON-ISOLATED REAL SINGULARITIES

Let $f\,{:}\,(\mathbb R^n,0)\to (\mathbb R,0)$ be an analytic function germ with non-isolated singularities and let $F\,{:}\, (\mathbb{R}^{1+n},0) \to (\mathbb{R},0)$ be a 1-parameter deformation of f. Let $ f_t ^{-1}(0) \cap B_\epsilon^n$ , $0 < \vert t \vert \ll \epsilon$ , be the “generalized” Milnor fiber of the deformation F. Under some conditions on F, we give a topological degree formula for the Euler characteristic of this fiber. This generalizes a result of Fukui.