Побудова асимптотики розв’язку для системи сингулярно збурених рівнянь методом істотно особливих функцій

Singularly perturbed problems with turning points arise as mathematical models for various physical phenomena. The internal turning point problem is a one-dimensional version of the steady-state convection-diffusion problem with a dominant convective term and a velocity field that changes sign in the reservoir. Boundary turning point problems, on the other hand, arise in geophysics and in the modeling of thermal boundary layers in laminar flow. The paper analyzes the results from the asymptotic analysis of singularly perturbed problems with turning points. For a homogeneous system of singularly perturbed differential equations with a small parameter at the highest derivative and a turning point, the conditions for constructing a uniform asymptotic solution are obtained. We consider the case when the spectrum of the limit operator contains multiple and identically zero elements. The asymptotics are constructed by the method of essentially singular functions, which allows using the Airy model operator in the vicinity of the turning point. The construction of asymptotic solutions contains arbitrary constants, which are determined uniquely during the solution of the iterative equations. At the same time, the conditions for the existence of a solution of a system of differentials with a small parameter for the highest derivative and for the presence of a turning point are obtained, provided that the turning point is located on the interval [0; l]. An example of constructing the asymptotic of a homogeneous system of differential equations is given. Pages of the article in the issue: 184 - 192 Language of the article: UkrainianСингулярно збурені задачі з точками звороту виникають як математичні моделі для різних фізичних явищ. Задача з внутрішньою точкою звороту є одновимірною версією стаціонарної конвекційно-дифузійної задачі з домінуючим конвективним членом і полем швидкості, яке змінює свій знак у водозбірнику. Задачі з граничними точками звороту, з іншого боку, виникають у геофізиці та при моделюванні теплових прикордонних шарів у ламінарному потоці. У роботі проведено аналіз результатів з асимптотичного аналізу сингулярно збурених задач з точками звороту. Для однорідної системи сингулярно збурених диференціальних рівнянь з малим параметром при старшій похідній і точкою звороту, одержані умови для побудови рівномірної асимптотики розв’язку. Розглядається випадок, коли спектр граничного оператора містить кратні і тотожно рівні нулю елементи. Асимптотику побудовано методом істотно особливих функцій, який дозволяє в околі точки звороту використати модельний оператор Ейрі. Конструкція асимптотичних розв’язків містить довільні сталі, які визначаються однозначно під час розв’язання ітераційних рівнянь. Водночас отримано умови існування розв’язку системи диференціальних з малим параметром при старшій похідній та при наявності точки звороту, за умови, що точка звороту міститься на інтервалі [0; l]. Наведено приклад побудови асимптотики однорідної системи диференціальних рівнянь

DISEASE-ASSOCIATED HLA-DR POLYMORPHISM, CLINICAL AND IMMUNOLOGICAL CHARACTERISTICS OF MULTIPLE SCLEROSIS PATIENTS IN THE NORTHEASTERN REGION OF UKRAINE

The development of multiple sclerosis is the result of complex interactions between environmental factors, genetic factors that determine individual disease susceptibility, and immunological and physiological characteristics of the patient. multiple sclerosis treatment requires obtaining information about the main pathological processes, therefore, the evaluation of biochemical, immunological, and genetic markers of such processes, and not just clinical indicators, can become the basis of improved approaches for diagnosis and monitoring of the disease. Polymorphism of the disease-associated genes is considered as one of the key factors in multiple sclerosis pathogenesis, capable of influencing the risk of development, clinical manifestations and nature of the course of the disease, treatment response. The HLA genes polymorphism was found to be the most important and playing an essential role in the development of autoimmune diseases. There are data on the population characteristics of these types of polymorphisms, which require conducting relevant research in Ukraine and its regions to identify relevant genetic markers. The aim of the study was to provide a comparative clinical and immunological characteristic of multiple sclerosis patients in the northeastern region of Ukraine, depending on the presence of the disease-associated HLA-DR polymorphism. Materials and methods. 39 patients with multiple sclerosis, inhabitants of Kharkiv and Kharkiv region, were examined, of which 7 men and 32 women of medium age of 33.7 ± 7.7 and 42.1 ± 11.9 years, respectively; control group was consisted of 27 practically healthy persons of both sexes with an average age of 30.1 ± 8.2 years. Clinical characteristics of multiple sclerosis patients included the determining the form and the actual type of the disease, its duration, and disability assessment based on the EDSS scale. The rate of the disease progression was determined as the ratio of the EDSS score to the duration of the disease. Biological material was blood and buccal epithelium samples from multiple sclerosis patients and practically healthy people. In addition to assessing the clinical status of the patients, design of the study also included evaluation of the systemic immunity indicators, levels of nonspecific and antinuclear antibodies in the serum, and analysis of HLA polymorphism, in particular, detection of the HLA-DR15 haplotype for its specific marker SNP rs9271366. Isolation of high molecular DNA was performed on a magnetically sensitive sorbent using the NeoPrep100 DNA Magnet kit (NeoGene, Ukraine). SNP polymorphism rs9271366 A/G was typed by the allele-specific amplification method with subsequent electrophoretic detection of the results. Primer selection was performed using the method by Liu Jing et al. (2012) adding a mismatch in the third position at the 3’ends. The primers MS92АF 5`-CACGTAATATAA-ATGGTTGCAAAGGA-3`, MS92GF 5`-CACGTAATATAAATGGTTGCA-AAGGG-3` and MS92R5` AACCCTGATGTAACAGA(C/T)CTCTA-3` (Eurofins Genomics), as well as Taq-mut polymerase (Liteh, Russian Federation), were used in the study. The amplification was performed on the multichannel amplifier Tercyc (Russian Federation). Amplification mode: denaturation at 96°C for 3 minutes and 35 cycles that included denaturation at 95°C for 30 seconds, annealing at 58°C for 30 seconds, and synthesis at 72°C for 30 seconds. The length of the amplicon was 233 bp. Electrophoresis was performed in 2% agarose gel in TAE buffer. Electrophoregram analysis was carried out on transilluminator UVT 1 (Biocom, Russian Federation). Serum IgM, IgG, IgA levels were evaluated by ELISA using a test system by NPL Granum (Ukraine) and the immunoassay analyzer Stat-Fax 303 (USA). The determination of peripheral blood lymphocytes subpopulations was carried out using the test systems by NPL Granum, Ukraine in accordance with the manufacturer's instructions. Circulating immune complexes in serum were evaluated by selective precipitation of antigen complexes with 3.5% PEG-6000 solution (AppliChem GmbH, Germany). The content of lymphocytotoxic autoantibodies was determined by a lymphocytotoxic macrotest. The complementary activity of the blood serum was evaluated by means of 50% of sheep erythrocytes hemolysis in the presence of homologous antibodies. The statistical treatment was performed using STATISTICA 11.0 (StatSoft, Inc). To determine the reliability of the differences between the indexes in the studied samples, the non-parametric Mann-Whitney's criterion was used, while the Student's T-criterion was used for the normal distribution. Results. At the stage of clinical examination of multiple sclerosis patients, 25 patients (64.1%) were diagnosed with RRMS, 4 patients (10.3%) with PPMS, and another 10 persons (25,6%) with SPMS. The familial form of the disease was established in 11 cases (28.2%), in 28 cases (71.8%) the form of the disease was classified as sporadic. The average disease duration of the examined persons was 10.16 ± 9.57 years. The disability score assessed according to the EDSS scale was 3.54 ± 1.67 points; the rate of progression of the disease was 0.93 ± 1.26 and 0.83 ± 0.85 points, respectively. Analysis of the HLA polymorphism indicated that the disease-associated (minor) allele G SNP rs9271366 was present in 43.6% of the subjects examined (haplotype AG, G+ group), and another 56.4% of the patients were homozygous by the major allele A (haplotype AA, G- group). The form of the disease, the EDSS score, and the rate of the disease progression in the examined patients did not have association with the allele G SNP rs9271366. Patients with familial and sporadic forms of multiple sclerosis did not have significant differences in immune status and clinical indicators. The average EDSS score was 3.45 ± 1.95 points in G+ patients and 3.65 ± 1.26 points in G- patients and the rate of disease progression was 0.83 ± 0.85 points and 0.93 ± 1.26 points, respectively. The study of cellular immunity indicators revealed a decrease in relative number of CD4+ and CD8+ lymphocytes in multiple sclerosis patients compared to that in control group, and it was more pronounced in subjects with the presence of the SNP rs9271366 minor allele. The relative content of CD4 + cells was 28.00 ± 3.45% in heterozygous patients, 30.94 ± 4.42% in patients homozygous for allele A, in the control group 42.3 ± 1.8% (p<0.05), and the CD8+ cells content was 18,55 ± 4.08%, 20.65 ± 3.52% and 29.4 ± 2.2%, respectively (p<0.05). At the same time, the reduction in the relative number of CD4+ lymphocytes was observed in 90.9% of G+ patients, and in 64.7% cases in G- patients (p<0.05). Decreased CD8+ lymphocyte content was also more common in patients with G+ (77.3% of cases) than in G- (41.2% of cases), p<0.05. Acute inflammatory process was determined in 38.5% of patients with multiple sclerosis, as evidenced by an elevated IgM level in 20.5% of cases, as well as an increase in complement activity in 62.5% of subjects. The levels of lymphocytotoxic autoantibodies in patients with multiple sclerosis were significantly higher compared to the control group (18.33 ± 3.67% of subjects and 5.82 ± 3.12% of subjects, respectively, p<0.05), and the levels of circulating immune complexes were elevated in 10.3% of patients. The study of the levels of antinuclear antibodies in multiple sclerosis patients revealed a diagnostically significant levels of antibodies to native DNA found in 92.3% of the cases, including all G + patients and 83.3% of G- patients. In this case, 16.7% of patients homozygous for allele A and 14.3% heterozygous patients were also positive for the presence of antibodies to denatured DNA. Similar results were also obtained concerning the presence of antibodies to formalinized DNA - 16.7% positive results in G- patients and 28.6% in G+ patients (p<0.05). At the same time, elevated levels of antibodies to native, denatured, and formalinized DNA were determined predominantly in patients with SPMS and PPMS (15.4% of the number of examined patients). The elevated levels of antinuclear antibodies may indicate an unfavorable course of the disease or its transition to the active phase. Conclusion. In the surveyed multiple sclerosis patients from the northeastern region of Ukraine, the prevalence of SNP rs9271366 and its association with a decrease in the relative number of CD4+ and CD8+ cells against the background of elevated levels of antinuclear and lymphocytotoxic antibodies were detected