The study of salivary amino acid profiles has attracted the attention of researchers, since amino acids are actively involved in most metabolic processes, including breast cancer. In this study, we analyzed the amino acid profile of saliva in a sample including all molecular biological subtypes of breast cancer to obtain a more complete picture and evaluate the potential utility of individual amino acids or their combinations for diagnostic purposes. This study included 116 patients with breast cancer, 24 patients with benign breast disease, and 25 healthy controls. From all patients, strictly before the start of treatment, saliva samples were collected, and the quantitative content of 26 amino acids was determined. Statistically significant differences between the three groups are shown in the content of Asp, Gly, Leu + Ile, Orn, Phe, Pro, Thr, and Tyr. To differentiate the three groups from each other, a decision tree was built. To construct it, we selected those amino acids for which the change in concentrations in the subgroups was multidirectional (GABA, Hyl, Arg, His, Pro, and Car). For the first time, it is shown that the amino acid profile of saliva depends on the molecular biological subtype of breast cancer. The most significant differences are shown for the luminal B HER2-positive and TNBC subgroups. In our opinion, it is critically important to consider the molecular biological subtype of breast cancer when searching for potential diagnostic markers.

Background: Changes in salivary metabolites in patients after surgery can provide important information for fundamental research. Objectives: To analyze changes in the salivary metabolic profile before and after breast tumor surgery. Methods: The study involved 660 breast cancer patients (54.6 ± 1.9 years) and 127 healthy volunteers (49.3 ± 1.5 years). Saliva samples were collected from all patients before treatment, and levels of total protein, urea, α-amino acids, imidazole compounds, and nitric oxide, as well as gamma-glutamyl transferase (GGT), lactate dehydrogenase (LDH), α-amylase, and catalase activity, were determined. In 139 breast cancer patients, these parameters were re-measured in saliva 4 weeks after surgical removal of the tumor (radical mastectomy). Results: In breast cancer, the activity of GGT (+17.6%, p < 0.0001), catalase (+14.7%, p = 0.0485), urea content (+24.6%, p = 0.0006), total protein (+13.6%, p = 0.0432), α-amino acids (+3.1%, p < 0.0001) increased in saliva, but LDH activity (−16.9%, p < 0.0001) and the content of imidazole compounds (−25.2%, p < 0.0001) decreased. It was found that after surgical treatment of breast cancer, a number of the biochemical parameters of saliva are restored. It is shown that the greatest deviations of the metabolic profile of saliva from the norm before surgery correspond to the early stages and the most prognostically favorable phenotypes of breast cancer. After surgery, metabolic differences are most pronounced for triple-negative breast cancer. Conclusions: A time interval of 4 weeks is not sufficient for complete recovery, but for most biochemical parameters the values are intermediate between those for healthy controls and primary breast cancer.

IR spectrometry is successfully used for the taxonomic identification of various groups of invertebrates: butterflies, flies, grasshoppers, mosquitoes, weevils, and termites. Therefore, the purpose of this study was to assess the fundamental possibility of taxonomic identification of earthworms based on the study of the functional-group composition of their bodies by the method of IR-Fourier spectroscopy. Seven species of earthworms from five genera were the objects of the study: Aporrectodea caliginosa, Eisenia fetida, Eisenia nordenskioldi, Lumbricus rubellus, Rhiphaeodrilus diplotetrathecus, Octolasion lacteum, Eisenia ventripapillata. IR absorption spectra were registered on an FT-801 IR-Fourier spectrometer in the range of 500–4000 cm–1. The position, intensity, and area of the absorption bands were determined on all spectra. The maximum differences between individual species in intensity and position of protein and nucleic acid absorption bands were established. Using the principal component method, it was shown that the classification was based on the characteristics of the absorption bands of stretching and bending vibrations of methyl and methylene groups, as well as amide groups in the structure of worms. At the same time, the division of earthworms into subgroups with similar spectral characteristics could be explained by several reasons: their origin, kinship, and the influence of the environment. The data obtained during the study confirm the possibility of using IR spectrometry for the identification of earthworms.

Рангом планарности многообразия полугрупп называем наибольшее число образующих свободной полугруппы многообразия, относительного которых полугруппа допускает планарный граф Кэли. За прошедшее время, с момента когда Л. М. Мартыновым была сформулирована задача описания рангов планарности полугрупповых многообразий, было получено много конкретных результатов в этом направлении. Модулярным многообразием полугрупп называется многообразие полугрупп с модулярной решеткой подмногообразий. В настоящей статье вычислены точные значения рангов планарности бесконечного счетного множества всех возможных модулярных многообразий полугрупп. Оказалось, что эти значения не превосходят 3. При доказательстве в большинстве своём используются машинные вычисления. Средствами Prover9 и Mace4 проверяются равенства элементов свободных полугрупп многообразий, задаваемых тождествами в большом количестве. Для доказательства непланарности графов используется критерий Понтрягина – Куратовского, а при обосновании планарности косвенно задействован инвариант Колен де Вердьера.

Доказаны характеристические свойства внешнепланарности и обобщенной внешнепланарности графов Кэли прямых произведений циклических полугрупп в терминах копредставлений. Основная идея доказательства теорем, приведенных в статье, заключается в следующем: если обнаруженные в результате исследования условия выполнены, то полугруппа допускает обобщенную внешнеплоскую [соответственно, внешнеплоскую] укладку её графа Кэли (то есть такую укладку, при которой каждое ребро принадлежит одной грани хотя бы одной из своих вершин, и ребра не пересекаются в плоскости) [соответственно, такую укладку, при которой все вершины принадлежат одной грани, а ребра не пересекаются в плоскости]; обратно, по закону контрапозиции, если найденные условия не выполнены, то указывается подграф, гомеоморфный одной из запрещенных конфигураций. Рассуждения ведутся по аналогии с исследованиями полугрупп, допускающих планарные графы, при этом запрещенные конфигурации меняются на новые, в силу критерия Чартрэнда–Харари и Седлачека.