Cancer cell hyper-proliferation disrupts the topological invariance of epithelial monolayers

Although the polygonal shape of epithelial cells has drawn the attention of scientists for several centuries, only recently, it has been demonstrated that distributions of polygon types (DOPTs) are similar in proliferative epithelia of many different plant and animal species. In this study we show that hyper-proliferation of cancer cells disrupts this universality paradigm and results in random epithelial structures. Examining non-synchronized and synchronized HeLa cervix cells, we suppose that the cell size spread is the single parameter controlling the DOPT in these monolayers. We test this hypothesis by considering morphologically similar random polygonal packings. By analyzing the differences between tumoral and non-tumoral epithelial monolayers, we uncover that the latter have more ordered structures and argue that the relaxation of mechanical stresses associated with cell division induces more effective ordering in the epithelia with lower proliferation rates. The proposed theory also explains the specific highly ordered structures of some post-mitotic unconventional epithelia.

The Image Milnor Number and Excellent Unfoldings

We show three basic properties of the image Milnor number µI(f) of a germ $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one can deduce the upper semi-continuity and the topological invariance for families. Second, we prove the weak Mond’s conjecture, which states that µI(f) = 0 if and only if f is stable. Finally, we show a conjecture by Houston that any family $f_t\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with $\mu_I(\,f_t)$ constant is excellent in Gaffney’s sense. For technical reasons, in the last two properties, we consider only the corank 1 case.

There are no deviations for the ergodic averages of Giulietti–Liverani horocycle flows on the two-torus

We show that the ergodic integrals for the horocycle flow on the two-torus associated by Giulietti and Liverani with an Anosov diffeomorphism either grow linearly or are bounded; in other words, there are no deviations. For this, we use the topological invariance of the Artin–Mazur zeta function to exclude resonances outside the open unit disc. Transfer operators acting on suitable spaces of anisotropic distributions and their Ruelle determinants are the key tools used in the proof. As a bonus, we show that for any $C^\infty $ Anosov diffeomorphism F on the two-torus, the correlations for the measure of maximal entropy and $C^\infty $ observables decay with a rate strictly smaller than $e^{-h_{\mathrm {top}}(F)}$ . We compare our results with very recent related work of Forni.

Topological invariance of the homological index

R. W. Carey and J. Pincus in [