Quasi-Linear Differential-Deference Game of Approach

This is a post-peer-review, pre-copyedit version of a book chapter that is part of “V. A. Sadovnichiy, M. Z. Zgurovsky (eds.). Modern Mathematics and Mechanics. Understanding Complex Systems”. The final authenticated version is available online at: https://link.springer.com/chapter/10.1007/978-3-319-96755-4_26The paper is devoted to the games of approach. We consider a controlled object whose dynamics is described by the linear differential system with a pure time delay or the differential-difference system with commutative matrices in Euclidean space. The approaches to the solutions of these problems are proposed which based on the Method of Resolving Functions and the First Direct Method of L.S. Pontryagin. The guaranteed times of the game termination are found, and corresponding control laws are constructed. The results are illustrated by a model example

Gender-Specific Modulation of the Response to Arterial Injury by Soluble Guanylate Cyclase α1

Objective: Soluble guanylate cyclase (sGC), a heterodimer composed of α and β subunits, synthesizes cGMP in response to nitric oxide (NO). NO modulates vascular tone and structure but the relative contributions of cGMP-dependent versus cGMP-independent mechanisms remain uncertain. We studied the response to vascular injury in male (M) and female (F) mice with targeted deletion of exon 6 of the sGCα1 subunit (sGCα1-/-), resulting in a non-functional heterodimer. Methods: We measured aortic cGMP levels and mRNA transcripts encoding sGC α1, α2, and β1 subunits in wild type (WT) and sGCa1-/- mice. To study the response to vascular injury, BrdU-incorporation and neointima formation (maximum intima to media (I/M) ratio) were determined 5 and 28 days after carotid artery ligation, respectively. Results: Aortic cGMP levels were 4-fold higher in F than in M mice in both genotypes, and, within each gender, 4-fold higher in WT than in sGCa1-/-. In contrast, sGCα1, sGCα2, and sGCβ1 mRNA expression did not differ between groups. 3H-thymidine incorporation in cultured sGCa1-/- smooth muscle cells (SMC) was 27%±12% lower than in WT SMC and BrdU-incorporation in carotid arteries 5 days after ligation was significantly less in sGCa1-/- M than in WT M. Neointima area and I/M 28 days after ligation were 65% and 62% lower in sGCa1-/- M than in WT M mice (p<0,05 for both) but were not different in F mice. Conclusion: Functional deletion of sGCa1 resulted in reduced cGMP levels in male sGCa1-/- mice and a gender-specific effect on the adaptive response to vascular injury