A LFER Kinetic Study of The Reaction of 5‐Substituted Orotic Acids with Diazodiphenylmethane

Linear free energy relationships (LFER) were applied to the kinetic data for the reaction of 5‐substituted orotic acids, series 1 , with diazodiphenylmethane (DDM) in N,N–dimethylformamide and compared with results obtained for 2‐substituted benzoic acids, series 2 . The correlation analysis of the kinetic data with σ substituent parameters was carried out using SSP (single substituent parameter) methods. From the sign and value of proportinality constant ρ, lower sensitivity to the substituent effect was obtained in series 1 , 0.876, than in the series 2 , 1.877. Evaluation of substituent “ortho‐effect” was performed using the Charton model, which includes the steric substituent parameter, and Fujita and Nishioka's model, which describes the total ortho‐effect as contribution of ordinary polar effect, the ortho‐steric and ortho‐polar effects. Results of correlations, obtained by using the Charton model, showed highest contribution of the polar effect, 0.861 vs. 2.101, whereas the steric effect is of lowest significance, 0.117 vs. 0.055, for series 1 and 2 , respectively. Also, a low negative value of coefficient with the steric effect, –0.08, obtained from the Fujita–Nishioka model indicated low steric effect, influencing a decrease of the reaction rate in series 1 . The structural and substituent effects were also studied by using the density functional theory method, and together with kinetic data, it gave a better insight into the influence of the effect of both geometry and substituent on the π?electron density shift induced reactivity of investigated acids.     

The Voronoi Identity via the Laplace Transform

The classical Voronoi identity $$\Delta (x) = - \frac{2}{\pi }\sum\limits_{n = 1}^\infty {d(n)} \left( {\frac{x}{n}} \right)^{1/2} \left( {K_1 (4\pi \sqrt {xn} ) + \frac{\pi }{2}Y_1 (4\pi \sqrt {xn} )} \right)$$ is proved in a relatively simple way by the use of the Laplace transform. Here Δ(x) denotes the error term in the Dirichlet divisor problem, d(n) is the number of divisors of n and K_1, Y_1 are the Bessel functions. The method of proof may be used to yield other identities similar to Voronoi's.     

Size of supramolecular SNARE complex: membrane-directed self-assembly

Full length v-SNARE protein in lipid vesicles when exposed to t-SNARE-reconstituted lipid membrane results in the self-assembly of a t-/v-SNARE complex in a ring pattern, forming pores, and establishing continuity between the opposing bilayers. It is known that smaller vesicles fuse more efficiently than larger ones, and hence the curvature of secretory vesicles may dictate the potency and efficacy of their fusion at the cell plasma membrane. The diameter of t- and v-SNARE vesicles may, therefore, reflect the size of the t-/v-SNARE complex formed. In the present study, this hypothesis was tested, and results from the study demonstrate that the size of the t-/v-SNARE complex is directly proportional to the vesicle diameter (R2 = 0.9725).     

The Effect of Dia2 Protein Deficiency on the Cell Cycle,Cell Size,and Recruitment of Ctf4 Protein in Saccharomyces cerevisiae

Cells have evolved elaborate mechanisms to regulate DNA replication machinery and cell cycles in response to DNA damage and replication stress in order to prevent genomic instability and cancer. The E3 ubiquitin ligase SCF^Dia2 in S. cerevisiae is involved in the DNA replication and DNA damage stress response, but its effect on cell growth is still unclear. Here, we demonstrate that the absence of Dia2 prolongs the cell cycle by extending both S- and G2/M-phases while, at the same time, activating the S-phase checkpoint. In these conditions, Ctf4—an essential DNA replication protein and substrate of Dia2—prolongs its binding to the chromatin during the extended S- and G2/M-phases. Notably, the prolonged cell cycle when Dia2 is absent is accompanied by a marked increase in cell size. We found that while both DNA replication inhibition and an absence of Dia2 exerts effects on cell cycle duration and cell size, Dia2 deficiency leads to a much more profound increase in cell size and a substantially lesser effect on cell cycle duration compared to DNA replication inhibition. Our results suggest that the increased cell size in dia2∆ involves a complex mechanism in which the prolonged cell cycle is one of the driving forces.     

The story of majorizability as Karamata’s condition of convergence for Abel summable series

The greatest Serbian mathematician, Jovan Karamata (1902–1967), gained worldwide fame working on problems related to theorems of a Tauberian nature. His simple and elegant 1930 proof of the Hardy–Littlewood theorem found its place in the well-known monographs by Titchmarsh, Knopp, Doetsch, Widder, Hardy and Favard. It is less known that the method used in this proof was mentioned for the first time at a conference of the Academy of Natural Sciences of the Serbian Royal Academy of Sciences in Belgrade in 1929, where Karamata introduced the notion of majorizability as a new condition of convergence for Abel summable series. This fact holds the key to a historical insight into Karamata’s famous proof of the Hardy–Littlewood theorem.     

Triangulations of Seifert fibred manifolds

It is not completely unreasonable to expect that a computable function bounding the number of Pachner moves needed to change any triangulation of a given 3-manifold into any other triangulation of the same 3-manifold exists. In this paper we describe a procedure yielding an explicit formula for such a function if the 3-manifold in question is a Seifert fibred space.Revised version: 5 March 2004