Universality of a mesenchymal transition signature in invasive solid cancers

In this brief communication, additional computational validation is provided consistent with the unifying hypothesis that a shared biological mechanism of mesenchymal transition, reflected by a precise gene expression signature, may be present in all types of solid cancers when they reach a particular stage of invasiveness

Nabla Discrete fractional Calculus and Nabla Inequalities

Here we define a Caputo like discrete nabla fractional difference and we produce discrete nabla fractional Taylor formulae for the first time. We estimate their remaiders. Then we derive related discrete nabla fractional Opial, Ostrowski, Poincare and Sobolev type inequalities

A subset of co-expressed genes in Slug-based cancer mesenchymal transition signature remains coexpressed in normal samples in a tissue-specific manner

A recently identified gene expression signature of EMT markers containing the transcription factor Slug was found present in samples from many publicly available cancer gene expression datasets of multiple cancer types except leukemia. We also found many of these genes co-expressed in human cancer xenografted cells, but not in mouse stroma cells, suggesting that the signature is largely produced by cancer cells undergoing some type of EMT. Here we report that a partial signature consisting of a subset of the co-expressed genes of the full signature, including at least Slug (SNAI2), collagens COL1A1, COL1A2, COL3A1, COL6A3 and genes DCN and LUM, is also present in leukemia, in which case it is also strongly associated with the chemokine CXCL12 (aka SDF1). The same subset of co-expressed genes is also strongly present even in normal samples in a tissue-specific manner, with lowest expression in brain tissues and highest expression in reproductive system tissues. The full signature, with prominent presence of COL11A1, THBS2 and INHBA appears to be triggered in solid cancers particularly when cancer cells encounter adipocytes

Estimates of the remainder in Taylor's theorem using the Henstock--Kurzweil integral

When a real-valued function of one variable is approximated by its $n^{th}$ degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue $p$-norms in cases where $f^{(n)}$ or $f^{(n+1)}$ are Henstock--Kurzweil integrable. When the only assumption is that $f^{(n)}$ is Henstock--Kurzweil integrable then a modified form of the $n^{th}$ degree Taylor polynomial is used. When the only assumption is that $f^{(n)}\in C^0$ then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1.Comment: To appear in Czechoslovak Mathematical Journa

Homoclinic points of 2-D and 4-D maps via the Parametrization Method

An interesting problem in solid state physics is to compute discrete breather solutions in $\mathcal{N}$ coupled 1--dimensional Hamiltonian particle chains and investigate the richness of their interactions. One way to do this is to compute the homoclinic intersections of invariant manifolds of a saddle point located at the origin of a class of $2\mathcal{N}$--dimensional invertible maps. In this paper we apply the parametrization method to express these manifolds analytically as series expansions and compute their intersections numerically to high precision. We first carry out this procedure for a 2--dimensional (2--D) family of generalized Henon maps ($\mathcal{N}$=1), prove the existence of a hyperbolic set in the non-dissipative case and show that it is directly connected to the existence of a homoclinic orbit at the origin. Introducing dissipation we demonstrate that a homoclinic tangency occurs beyond which the homoclinic intersection disappears. Proceeding to $\mathcal{N}=2$, we use the same approach to determine the homoclinic intersections of the invariant manifolds of a saddle point at the origin of a 4--D map consisting of two coupled 2--D cubic H\'enon maps. In dependence of the coupling the homoclinic intersection is determined, which ceases to exist once a certain amount of dissipation is present. We discuss an application of our results to the study of discrete breathers in two linearly coupled 1--dimensional particle chains with nearest--neighbor interactions and a Klein--Gordon on site potential.Comment: 24 pages, 10 figures, videos can be found at https://comp-phys.tu-dresden.de/supp