In [18] Jacobi introduced a third-order nonlinear ordinary differential equation which links two different moduli of an elliptic integral. In the present paper Lie group analysis is applied to that equation named Jacobi's modular differential equation. A six-dimensional Lie symmetry algebra is obtained and its symmetry generators are found to be given in terms of elliptic integrals. As a consequence the transformation between Jacobi's modular differential equation and the well-known Schwarzian differential equation is derived.

Obesity is a global health issue, in which modifications in gut microbiota composition have a key role. Different therapeutic strategies are being developed in combination with diet and exercise, including the use of plant extracts, such as those obtained from Morus alba L. leaves. Recent studies have revealed their anti-inflammatory and antioxidant properties. The aim of the present work was to evaluate whether the beneficial effects of M. alba L. leaf extract in high-fat diet-induced obesity in mice is correlated with its impact on gut microbiota. The extract reduced body weight gain and attenuated lipid accumulation, as well as increased glucose sensitivity. These effects were associated with an amelioration of the obesity-associated inflammatory status, most probably due to the described antioxidant properties of the extract. Moreover, M. alba L. leaf extract mitigated gut dysbiosis, which was evidenced by the restoration of the Firmicutes/Bacteroidota ratio and the decrease in plasma lipopolysaccharide (LPS) levels. Specifically, the extract administration reduced Alistipes and increased Faecalibaculum abundance, these effects being correlated with the beneficial effects exerted by the extract on the obesity-associated inflammation. In conclusion, anti-obesogenic effects of M. alba L. leaf extract may be mediated through the amelioration of gut dysbiosis.

Ulcerative colitis (UC) and Crohn’s Disease (CD) are chronic relapsing inflammatory diseases that are caused by genetic, environmental, and immune factors. Treatment strategies are currently based on symptomatic control by immunosuppression. The glucocorticoid-induced leucine zipper (GILZ), a mediator of several effects of glucocorticoids, was recently found to be secreted by goblet cells and play a role in inflammatory bowel disease (IBD). This study investigates which genes GILZ is associated with in its role in intestinal barrier functions. We examined datasets from the Gene Expression Omnibus (GEO) and ArrayExpress profiles of the gut of healthy subjects (HSs), as well as UC and CD patients. The human colonic epithelial HT29 cell line was used for in vitro validation experiments. GILZ was significantly correlated with MUC2, TLR2, and TLR4. In particular, an inverse correlation was found between the GILZ and MUC2 in HS and patients with IBD, mostly in those with an active disease. Further, direct pairwise correlations for GILZ/TLR2 and GILZ/TLR4 were found in HSs and UC patients, but not in CD patients. Overall, our results reveal the crosstalk at the transcription level between the GILZ, MUC2, and TLRs in the mucosal barrier through common pathways, and they open up new perspectives in terms of mucosal healing in IBD patients.