Periodic solutions of nonlinear autonomous systems in critical cases

We obtain coefficient conditions of existence and an iterational algorithm for constructing periodic solutions of weakly perturbed autonomous nonlinear differential systems in critical cases in the presence of multiple roots for generating amplitudes.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1180–1187, September, 1990.     

Weakly nonlinear boundary-value problems for operator equations with pulse influence

We consider the problem of finding conditions of solvability and algorithms for construction of solutions of weakly nonlinear boundary-value problems for operator equations (with the Noetherian linear part) with pulse influence at fixed times. The method of investigation is based on passing by methods of the Lyapunov—Schmidt type from a pulse boundary-value problem to an equivalent operator system that can be solved by iteration procedures based on the fixed-point principle. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 272–288, February, 1997.     

Weakly nonlinear boundary-value problems for differential equations in a Banach space in the critical case

We obtain necessary and sufficient conditions for the existence of solutions of weakly nonlinear boundary-value problems for differential equations in a Banach space. A convergent iterative procedure is proposed for the determination of solutions. We also establish a relationship between necessary and sufficient conditions.     

Computational-Based Discovery of the Anti-Cancer Activities of Pyrrole-Based Compounds Targeting the Colchicine-Binding Site of Tubulin

Despite the tubulin-binding agents (TBAs) that are widely used in the clinic for cancer therapy, tumor resistance to TBAs (both inherited and acquired) significantly impairs their effectiveness, thereby decreasing overall survival (OS) and progression-free survival (PFS) rates, especially for the patients with metastatic, recurrent, and unresectable forms of the disease. Therefore, the development of novel effective drugs interfering with the microtubules’ dynamic state remains a big challenge in current oncology. We report here about the novel ethyl 2-amino-1-(furan-2-carboxamido)-5-(2-aryl/tert-butyl-2-oxoethylidene)-4-oxo-4,5-dihydro-1H-pyrrole-3-carboxylates (EAPCs) exhibiting potent anti-cancer activities against the breast and lung cancer cell lines in vitro. This was due to their ability to inhibit tubulin polymerization and induce cell cycle arrest in M-phase. As an outcome, the EAPC-treated cancer cells exhibited a significant increase in apoptosis, which was evidenced by the expression of cleaved forms of PARP, caspase-3, and increased numbers of Annexin-V-positive cells. By using the in silico molecular modeling methods (e.g., induced-fit docking, binding metadynamics, and unbiased molecular dynamics), we found that EAPC-67 and -70 preferentially bind to the colchicine-binding site of tubulin. Lastly, we have shown that the EAPCs indicated above and colchicine utilizes a similar molecular mechanism to inhibit tubulin polymerization via targeting the T7 loop in the β-chain of tubulin, thereby preventing the conformational changes in the tubulin dimers required for their polymerization. Collectively, we identified the novel and potent TBAs that bind to the colchicine-binding site and disrupt the microtubule network. As a result of these events, the compounds induced a robust cell cycle arrest in M-phase and exhibited potent pro-apoptotic activities against the epithelial cancer cell lines in vitro.