Applications of symmetry to problem solving

Symmetry is an important mathematical concept which plays an extremely important role as a problem-solving technique. Nevertheless, symmetry is rarely used in secondary school in solving mathematical problems. Several investigations demonstrate that secondary school mathematics teachers are not aware enough of the importance of this elegant problem-solving tool. In this paper we present examples of problems from several branches of mathematics that can be solved using different types of symmetry. Teachers' attitudes and beliefs regarding the use of symmetry in the solutions of these problems are discussed.     

Advanced Sunscreens: UV Absorbers Encapsulated in Sol-Gel Glass Microcapsules

Undesirable phototoxic and photoallergic reactions accompanying a justified increased use of sunscreen active ingredients within cosmetic products have encouraged the development of new products safer for human use. The sol-gel microencapsulation technology developed utilizes an interfacial polymerization process, allowing for the achievement of transparent silica glass microcapsules with sizes ranging between 0.3–3 microns and a characteristic core-shell structure. Within the sol-gel microcapsule structure a UV absorber core, constituting roughly 80% of the final product weight, is enclosed within a silica shell. These advanced sunscreen actives are then incorporated into a suitable cosmetic vehicle to achieve high Sun Protection Factors (SPF), while affording an improved safety profile, as the penetration of the UV absorbers is markedly reduced.     

Single-molecule observation of the catalytic subunit of cAMP-dependent protein kinase binding to an inhibitor peptide

An engineered version of the staphylococcal alpha-hemolysin protein pore, bearing a peptide inhibitor near the entrance to the beta barrel, interacts with the catalytic (C) subunit of cAMP-dependent protein kinase. By monitoring the ionic current through the pore, binding events are detected at the single-molecule level. The kinetic and thermodynamic constants governing the binding interaction and the synergistic effect of MgATP are comparable but not identical to the values in bulk solution. Further, the values are strongly dependent on the applied membrane potential. Additional exploration of these findings may lead to a better understanding of the properties of enzymes at the lipid/water interface. Despite the complications, we suggest that the engineered pore might be used as a sensor element to screen inhibitors that act at either the substrate or ATP binding sites of the C subunit.     

Toward single molecule DNA sequencing: direct identification of ribonucleoside and deoxyribonucleoside 5'-monophosphates by using an engineered protein nanopore equipped with a molecular adapter

Individual nucleic acid molecules might be sequenced by the identification of nucleoside 5'-monophosphates as they are released by processive exonucleases. Here, we show that single molecule detection with a modified protein nanopore can be used to identify ribonucleoside and 2'-deoxyribonucleoside 5'-monophosphates, thereby taking a step along this path. Distinct levels of current block are observed for each of the four members of a set of nucleoside 5'-monophosphates when the molecules bind within a mutant alpha-hemolysin pore, (M113R)(7), equipped with the molecular adapter heptakis-(6-deoxy-6-amino)-beta-cyclodextrin. While our results compare favorably with alternative approaches, further work will be required to improve the accuracy of identification of the nucleic acid bases, to feed each released nucleotide into the pore, and to ensure that every nucleotide is captured by the adapter.     

On examples in mathematical thinking and learning

The importance of examples and exemplification in mathematical thinking, learning and teaching, is well recognized by mathematicians, epistemologists and mathematics educators. In the collection of papers on these topics presented in this issue, we aim to contribute to the debate on this theme, proposing original studies carried out from different approaches and perspectives, and linked to other relevant topics within mathematics education.     

Substitution of an Internal Disulfide Bridge with a Diselenide Enhances both Foldability and Stability of Human Insulin

Insulin analogues, mainstays in the modern treatment of diabetes mellitus, exemplify the utility of protein engineering in molecular pharmacology. Whereas chemical syntheses of the individual A and B chains were accomplished in the early 1960s, their combination to form native insulin remains inefficient because of competing disulfide pairing and aggregation. To overcome these limitations, we envisioned an alternative approach: pairwise substitution of cysteine residues with selenocysteine (Sec, U). To this end, Cys^A6 and Cys^A11 (which form the internal intrachain A6–A11 disulfide bridge) were each replaced with Sec. The A chain[C6U, C11U] variant was prepared by solid-phase peptide synthesis; while sulfitolysis of biosynthetic human insulin provided wild-type B chain-di-S-sulfonate. The presence of selenium atoms at these sites markedly enhanced the rate and fidelity of chain combination, thus solving a long-standing challenge in chemical insulin synthesis. The affinity of the Se-insulin analogue for the lectin-purified insulin receptor was indistinguishable from that of WT-insulin. Remarkably, the thermodynamic stability of the analogue at 25 °C, as inferred from guanidine denaturation studies, was augmented (ΔΔG_u ≈0.8 kcal mol⁻¹). In accordance with such enhanced stability, reductive unfolding of the Se-insulin analogue and resistance to enzymatic cleavage by Glu-C protease occurred four times more slowly than that of WT-insulin. 2D-NMR and X-ray crystallographic studies demonstrated a native-like three-dimensional structure in which the diselenide bridge was accommodated in the hydrophobic core without steric clash.     

A Catalase‐Like Metal‐Organic Framework Nanohybrid for O2‐Evolving Synergistic Chemoradiotherapy

Tumor hypoxia, the “Achilles’ heel” of current cancer therapies, is indispensable to drug resistance and poor therapeutic outcomes especially for radiotherapy. Here we propose an in situ catalytic oxygenation strategy in tumor using porphyrinic metal‐organic framework (MOF)‐gold nanoparticles (AuNPs) nanohybrid as a therapeutic platform to achieve O₂‐evolving chemoradiotherapy. The AuNPs decorated on the surface of MOF effectively stabilize the nanocomposite and serve as radiosensitizers, whereas the MOF scaffold acts as a container to encapsulate chemotherapeutic drug doxorubicin. In vitro and in vivo studies verify that the catalase‐like nanohybrid significantly enhances the radiotherapy effect, alleviating tumor hypoxia and achieving synergistic anticancer efficacy. This hybrid nanomaterial remarkably suppresses the tumor growth with minimized systemic toxicity, opening new horizons for the next generation of theranostic nanomedicines.