This paper is a survey of contemporary results and applications of the theory of homotopes. The notion of a well-tempered element of an associative algebra is introduced, and it is proved that the category of representations of the homotope constructed by a well-tempered element is the heart of a suitably glued -structure. The Hochschild and global dimensions of homotopes are calculated in the case of well-tempered elements. The homotopes constructed from generalized Laplace operators in Poincaré groupoids of graphs are studied. It is shown that they are quotients of Temperley–Lieb algebras of general graphs. The perverse sheaves on a punctured disc and on a 2-dimensional sphere with a double point are identified with representations of suitable homotopes. Relations of the theory to orthogonal decompositions of the Lie algebras into a sum of Cartan subalgebras, to classifications of configurations of lines, to mutually unbiased bases, to quantum protocols, and to generalized Hadamard matrices are discussed. Bibliography: 56 titles.
{"title":"Theory of homotopes with applications to mutually unbiased bases, harmonic analysis on graphs, and perverse sheaves","authors":"A. Bondal, I. Zhdanovskiy","doi":"10.1070/RM9983","DOIUrl":"https://doi.org/10.1070/RM9983","url":null,"abstract":"This paper is a survey of contemporary results and applications of the theory of homotopes. The notion of a well-tempered element of an associative algebra is introduced, and it is proved that the category of representations of the homotope constructed by a well-tempered element is the heart of a suitably glued -structure. The Hochschild and global dimensions of homotopes are calculated in the case of well-tempered elements. The homotopes constructed from generalized Laplace operators in Poincaré groupoids of graphs are studied. It is shown that they are quotients of Temperley–Lieb algebras of general graphs. The perverse sheaves on a punctured disc and on a 2-dimensional sphere with a double point are identified with representations of suitable homotopes. Relations of the theory to orthogonal decompositions of the Lie algebras into a sum of Cartan subalgebras, to classifications of configurations of lines, to mutually unbiased bases, to quantum protocols, and to generalized Hadamard matrices are discussed. Bibliography: 56 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"76 1","pages":"195 - 259"},"PeriodicalIF":0.9,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45792580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the 1930s and 40s, one and the same delay differential equation appeared in papers by two mathematicians, Karl Dickman and Vasily Leonidovich Goncharov, who dealt with completely different problems. Dickman investigated the limit value of the number of natural numbers free of large prime factors, while Goncharov examined the asymptotics of the maximum cycle length in decompositions of random permutations. The equation obtained in these papers defines, under a certain initial condition, the density of a probability distribution now called the Dickman–Goncharov distribution (this term was first proposed by Vershik in 1986). Recently, a number of completely new applications of the Dickman–Goncharov distribution have appeared in mathematics (random walks on solvable groups, random graph theory, and so on) and also in biology (models of growth and evolution of unicellular populations), finance (theory of extreme phenomena in finance and insurance), physics (the model of random energy levels), and other fields. Despite the extensive scope of applications of this distribution and of more general but related models, all the mathematical aspects of this topic (for example, infinite divisibility and absolute continuity) are little known even to specialists in limit theorems. The present survey is intended to fill this gap. Both known and new results are given. Bibliography: 62 titles.
{"title":"The Dickman–Goncharov distribution","authors":"S. Molchanov, V. Panov","doi":"10.1070/RM9976","DOIUrl":"https://doi.org/10.1070/RM9976","url":null,"abstract":"In the 1930s and 40s, one and the same delay differential equation appeared in papers by two mathematicians, Karl Dickman and Vasily Leonidovich Goncharov, who dealt with completely different problems. Dickman investigated the limit value of the number of natural numbers free of large prime factors, while Goncharov examined the asymptotics of the maximum cycle length in decompositions of random permutations. The equation obtained in these papers defines, under a certain initial condition, the density of a probability distribution now called the Dickman–Goncharov distribution (this term was first proposed by Vershik in 1986). Recently, a number of completely new applications of the Dickman–Goncharov distribution have appeared in mathematics (random walks on solvable groups, random graph theory, and so on) and also in biology (models of growth and evolution of unicellular populations), finance (theory of extreme phenomena in finance and insurance), physics (the model of random energy levels), and other fields. Despite the extensive scope of applications of this distribution and of more general but related models, all the mathematical aspects of this topic (for example, infinite divisibility and absolute continuity) are little known even to specialists in limit theorems. The present survey is intended to fill this gap. Both known and new results are given. Bibliography: 62 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"75 1","pages":"1089 - 1132"},"PeriodicalIF":0.9,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41688728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A number of directions were initiated by the author and his students in their papers of 1981–1982. However, one of them, concerning the properties of closed orbits on the sphere and in the groups and , has not been sufficiently developed. This paper revives the discussion of these questions, states unsolved problems, and explains what was regarded as fallacies in old papers. In general, magnetic orbits have been poorly discussed in the literature on dynamical systems and theoretical mechanics, but Grinevich has pointed out that in theoretical physics one encounters similar situations in the theory related to particle accelerators such as proton cyclotrons. It is interesting to look at Chap. III of Landau and Lifshitz’s Theoretical physics, vol. 2, Field theory (Translated into English as The classical theory of fields [12]. where mathematical relatives of our situations occur, but the physics is completely different and there are actual strong magnetic fields. Bibliography: 12 titles.
{"title":"Spinning tops and magnetic orbits","authors":"S. Novikov","doi":"10.1070/RM9977","DOIUrl":"https://doi.org/10.1070/RM9977","url":null,"abstract":"A number of directions were initiated by the author and his students in their papers of 1981–1982. However, one of them, concerning the properties of closed orbits on the sphere and in the groups and , has not been sufficiently developed. This paper revives the discussion of these questions, states unsolved problems, and explains what was regarded as fallacies in old papers. In general, magnetic orbits have been poorly discussed in the literature on dynamical systems and theoretical mechanics, but Grinevich has pointed out that in theoretical physics one encounters similar situations in the theory related to particle accelerators such as proton cyclotrons. It is interesting to look at Chap. III of Landau and Lifshitz’s Theoretical physics, vol. 2, Field theory (Translated into English as The classical theory of fields [12]. where mathematical relatives of our situations occur, but the physics is completely different and there are actual strong magnetic fields. Bibliography: 12 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"75 1","pages":"1133 - 1141"},"PeriodicalIF":0.9,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44194390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
u̇j = Auj + F (uj) + D[uj+1− 2uj + uj−1], j = 1, . . . , N ; u0 ≡ uN , uN+1 ≡ u1. (2) We associate the element uj(t) with the value of a function u(t, xj) of two variables, where xj = 2πjN−1 is the angular coordinate. The main assumption is that the number N of elements in (2) is sufficiently large, so that the parameter ε = 2πN−1 is sufficiently small: 0 < ε ≪ 1. This gives reason to switch from the discrete system (2) to the following system, which is continuous with respect to the spatial variable x:
{"title":"Bifurcations in spatially distributed chains of two-dimensional systems of equations","authors":"S. Kaschenko","doi":"10.1070/RM9986","DOIUrl":"https://doi.org/10.1070/RM9986","url":null,"abstract":"u̇j = Auj + F (uj) + D[uj+1− 2uj + uj−1], j = 1, . . . , N ; u0 ≡ uN , uN+1 ≡ u1. (2) We associate the element uj(t) with the value of a function u(t, xj) of two variables, where xj = 2πjN−1 is the angular coordinate. The main assumption is that the number N of elements in (2) is sufficiently large, so that the parameter ε = 2πN−1 is sufficiently small: 0 < ε ≪ 1. This gives reason to switch from the discrete system (2) to the following system, which is continuous with respect to the spatial variable x:","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"75 1","pages":"1153 - 1155"},"PeriodicalIF":0.9,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42177421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Braverman, V. Buchstaber, M. Gromov, V. Ivrii, Y. Kordyukov, P. Kuchment, V. Maz'ya, S. Novikov, T. Sunada, L. Friedlander, A. Khovanskii
The prominent mathematician Mikhail Aleksandrovich Shubin passed away after a long illness on 13 May 2020. He was born on 19 December 1944 in Kuibyshev (now Samara) and raised by his mother and grandmother. His mother, Maria Arkadievna, was an engineer at the State Bearing Factory, where she was hired in 1941 after graduating from the Faculty of Mechanics and Mathematics at the Moscow State University. At that time the factory was evacuated from Moscow to Kuibyshev. She worked at the factory for many years as the Head of the Physics of Metals Laboratory. Later, she defended her Ph.D. thesis and moved to Kuibyshev Polytechnical Institute, where she worked as an associate professor. In his school years Shubin was mainly interested in music. He had absolute pitch. After finishing music school, he seriously considered entering a conservatory. However, in high school he developed an interest to mathematics, was successful in olympiads, and eventually decided to apply to the Faculty of Mechanics and Mathematics at the Moscow State University. He was admitted there in 1961. When the time came to choose an adviser, he became a student of M. I. Vishik. After graduating, he began postgraduate work there, and in 1969 defended his Ph.D. thesis. In the thesis he derived formulae for the index of matrix-valued Wiener–Hopf operators. In particular, for the study of families of such operators, he had to generalize a theorem of Birkhoff stating that a continuous matrix-valued function M(z) defined on the unit circle |z| = 1 can be factored as M(z) = A+(z)D(z)A−(z), where A+(z) and A−(z) are continuous and have analytic continuations to the interior of the unit circle and its exterior (infinity included), respectively, and D(z) is a diagonal matrix with entries zj on the diagonal, with integer nj . Shubin considered the problem of what happens when the matrix M depends continuously on an additional parameter t. The Birkhoff factorization cannot be made continuous
著名数学家米哈伊尔·亚历山德罗维奇·舒宾在长期患病后于2020年5月13日去世。他于1944年12月19日出生于古比雪夫(现萨马拉)，由母亲和祖母抚养长大。他的母亲玛丽亚·阿尔卡季耶夫娜(Maria Arkadievna)是国家轴承厂的一名工程师，1941年从莫斯科国立大学力学和数学系毕业后被聘用。当时工厂从莫斯科撤到古比雪夫。她在这家工厂担任金属物理实验室主任多年。后来，她为自己的博士论文辩护，搬到了古比雪夫理工学院，在那里担任副教授。在他的学生时代，舒斌主要对音乐感兴趣。他有绝对的音高。从音乐学校毕业后，他认真考虑过进入音乐学院。然而，在高中时，他对数学产生了兴趣，在奥林匹克竞赛中取得了成功，并最终决定申请莫斯科国立大学力学和数学学院。他于1961年被录取。到了选择导师的时候，他成了m.i. Vishik的学生。毕业后，他开始在那里攻读研究生，并于1969年为自己的博士论文辩护。在论文中，他推导了矩阵值Wiener-Hopf算子的指标公式。研究特别是家庭这样的运营商,他不得不推广定理比尔科夫指出,一个连续的矩阵值函数的M (z)在单位圆定义z | | = 1可以被分解为M (z) = a + (z) D (z)−(z) + (z)和−(z)连续,分析延续单位圆的内部和外部(包括无穷大),分别和D (z)是一个对角矩阵的条目zj对角线上,与整数新泽西。Shubin考虑了当矩阵M连续依赖于一个附加参数t时会发生什么问题。Birkhoff分解不能连续
{"title":"Mikhail Aleksandrovich Shubin","authors":"M. Braverman, V. Buchstaber, M. Gromov, V. Ivrii, Y. Kordyukov, P. Kuchment, V. Maz'ya, S. Novikov, T. Sunada, L. Friedlander, A. Khovanskii","doi":"10.1070/RM9968","DOIUrl":"https://doi.org/10.1070/RM9968","url":null,"abstract":"The prominent mathematician Mikhail Aleksandrovich Shubin passed away after a long illness on 13 May 2020. He was born on 19 December 1944 in Kuibyshev (now Samara) and raised by his mother and grandmother. His mother, Maria Arkadievna, was an engineer at the State Bearing Factory, where she was hired in 1941 after graduating from the Faculty of Mechanics and Mathematics at the Moscow State University. At that time the factory was evacuated from Moscow to Kuibyshev. She worked at the factory for many years as the Head of the Physics of Metals Laboratory. Later, she defended her Ph.D. thesis and moved to Kuibyshev Polytechnical Institute, where she worked as an associate professor. In his school years Shubin was mainly interested in music. He had absolute pitch. After finishing music school, he seriously considered entering a conservatory. However, in high school he developed an interest to mathematics, was successful in olympiads, and eventually decided to apply to the Faculty of Mechanics and Mathematics at the Moscow State University. He was admitted there in 1961. When the time came to choose an adviser, he became a student of M. I. Vishik. After graduating, he began postgraduate work there, and in 1969 defended his Ph.D. thesis. In the thesis he derived formulae for the index of matrix-valued Wiener–Hopf operators. In particular, for the study of families of such operators, he had to generalize a theorem of Birkhoff stating that a continuous matrix-valued function M(z) defined on the unit circle |z| = 1 can be factored as M(z) = A+(z)D(z)A−(z), where A+(z) and A−(z) are continuous and have analytic continuations to the interior of the unit circle and its exterior (infinity included), respectively, and D(z) is a diagonal matrix with entries zj on the diagonal, with integer nj . Shubin considered the problem of what happens when the matrix M depends continuously on an additional parameter t. The Birkhoff factorization cannot be made continuous","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"75 1","pages":"1143 - 1152"},"PeriodicalIF":0.9,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46819886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Bolotin, A. V. Borisov, A. Karapetyan, B. Kashin, E. I. Kugushev, Anatolii Iserovich Neishtadt, Dmitri Orlov, D. Treschev
On 1 January 2020 the prominent researcher and academician of the Russian Academy of Sciences Valerii Vasil’evich Kozlov observed his 70th birthday. Kozlov has made fundamental contributions to diverse areas of mathematics and mechanics: the theory of Hamiltonian systems, stability theory, the mechanics of non-holonomic systems, statistical mechanics. He has published about 300 papers on mathematics and mechanics and 8 monographs which are now classical. In this one article it is impossible to give even a brief account of all the directions of his research. Kozlov was born on 1 January 1950 in the village of Kostyli, in the Mikhailovskoe District of the Ryazan Oblast. His mother Ol’ga Arkhipovna was a teacher of mathematics, and his father Vasilii Nestorovich was a train-driver, and a veteran of World War II, from the first days when the Soviet Union was attacked until Victory Day. Valerii started his early school education in his native small village (where nobody lives now). There was only a primary school there, with one female teacher, who gave simultaneous lessons to grades I and III in the morning and to grades II and IV in the afternoon. As an 8-year boy, Kozlov moved with his parents to Lyublino-Dachnaya, close to Moscow. When the Moscow Ring Road was built (in 1961) this settlement, like many others, found itself inside the expanding Moscow. In this way Kozlov became a Moscow resident. During his last two years in secondary school he became deeply interested in mathematics and physics. Three times a week he travelled to lessons at a volunteer physics-mathematics evening school under the auspices of the Bauman Moscow State Technical School (now Technical University). This proved to be a remarkable school! (It was founded in 1962 and still exists.) Most teachers were students
{"title":"Valerii Vasil’evich Kozlov","authors":"S. Bolotin, A. V. Borisov, A. Karapetyan, B. Kashin, E. I. Kugushev, Anatolii Iserovich Neishtadt, Dmitri Orlov, D. Treschev","doi":"10.1070/RM9949","DOIUrl":"https://doi.org/10.1070/RM9949","url":null,"abstract":"On 1 January 2020 the prominent researcher and academician of the Russian Academy of Sciences Valerii Vasil’evich Kozlov observed his 70th birthday. Kozlov has made fundamental contributions to diverse areas of mathematics and mechanics: the theory of Hamiltonian systems, stability theory, the mechanics of non-holonomic systems, statistical mechanics. He has published about 300 papers on mathematics and mechanics and 8 monographs which are now classical. In this one article it is impossible to give even a brief account of all the directions of his research. Kozlov was born on 1 January 1950 in the village of Kostyli, in the Mikhailovskoe District of the Ryazan Oblast. His mother Ol’ga Arkhipovna was a teacher of mathematics, and his father Vasilii Nestorovich was a train-driver, and a veteran of World War II, from the first days when the Soviet Union was attacked until Victory Day. Valerii started his early school education in his native small village (where nobody lives now). There was only a primary school there, with one female teacher, who gave simultaneous lessons to grades I and III in the morning and to grades II and IV in the afternoon. As an 8-year boy, Kozlov moved with his parents to Lyublino-Dachnaya, close to Moscow. When the Moscow Ring Road was built (in 1961) this settlement, like many others, found itself inside the expanding Moscow. In this way Kozlov became a Moscow resident. During his last two years in secondary school he became deeply interested in mathematics and physics. Three times a week he travelled to lessons at a volunteer physics-mathematics evening school under the auspices of the Bauman Moscow State Technical School (now Technical University). This proved to be a remarkable school! (It was founded in 1962 and still exists.) Most teachers were students","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"75 1","pages":"1165 - 1180"},"PeriodicalIF":0.9,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49615312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article contains a survey of a new algebro-geometric approach for working with iterated algebraic loop groups associated with iterated Laurent series over arbitrary commutative rings and its applications to the study of the higher-dimensional Contou-Carrère symbol. In addition to the survey, the article also contains new results related to this symbol. The higher-dimensional Contou-Carrère symbol arises naturally when one considers deformation of a flag of algebraic subvarieties of an algebraic variety. The non-triviality of the problem is due to the fact that, in the case 1$?> , for the group of invertible elements of the algebra of -iterated Laurent series over a ring, no representation is known in the form of an ind-flat scheme over this ring. Therefore, essentially new algebro-geometric constructions, notions, and methods are required. As an application of the new methods used, a description of continuous homomorphisms between algebras of iterated Laurent series over a ring is given, and an invertibility criterion for such endomorphisms is found. It is shown that the higher- dimensional Contou-Carrère symbol, restricted to algebras over the field of rational numbers, is given by a natural explicit formula, and this symbol extends uniquely to all rings. An explicit formula is also given for the higher-dimensional Contou-Carrère symbol in the case of all rings. The connection with higher-dimensional class field theory is described. As a new result, it is shown that the higher-dimensional Contou-Carrère symbol has a universal property. Namely, if one fixes a torsion-free ring and considers a flat group scheme over this ring such that any two points of the scheme are contained in an affine open subset, then after restricting to algebras over the fixed ring, all morphisms from the -iterated algebraic loop group of the Milnor -group of degree to the above group scheme factor through the higher-dimensional Contou-Carrère symbol. Bibliography: 67 titles.
{"title":"Iterated Laurent series over rings and the Contou-Carrère symbol","authors":"S. Gorchinskiy, D. Osipov","doi":"10.1070/RM9975","DOIUrl":"https://doi.org/10.1070/RM9975","url":null,"abstract":"This article contains a survey of a new algebro-geometric approach for working with iterated algebraic loop groups associated with iterated Laurent series over arbitrary commutative rings and its applications to the study of the higher-dimensional Contou-Carrère symbol. In addition to the survey, the article also contains new results related to this symbol. The higher-dimensional Contou-Carrère symbol arises naturally when one considers deformation of a flag of algebraic subvarieties of an algebraic variety. The non-triviality of the problem is due to the fact that, in the case 1$?> , for the group of invertible elements of the algebra of -iterated Laurent series over a ring, no representation is known in the form of an ind-flat scheme over this ring. Therefore, essentially new algebro-geometric constructions, notions, and methods are required. As an application of the new methods used, a description of continuous homomorphisms between algebras of iterated Laurent series over a ring is given, and an invertibility criterion for such endomorphisms is found. It is shown that the higher- dimensional Contou-Carrère symbol, restricted to algebras over the field of rational numbers, is given by a natural explicit formula, and this symbol extends uniquely to all rings. An explicit formula is also given for the higher-dimensional Contou-Carrère symbol in the case of all rings. The connection with higher-dimensional class field theory is described. As a new result, it is shown that the higher-dimensional Contou-Carrère symbol has a universal property. Namely, if one fixes a torsion-free ring and considers a flat group scheme over this ring such that any two points of the scheme are contained in an affine open subset, then after restricting to algebras over the fixed ring, all morphisms from the -iterated algebraic loop group of the Milnor -group of degree to the above group scheme factor through the higher-dimensional Contou-Carrère symbol. Bibliography: 67 titles.","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"75 1","pages":"995 - 1066"},"PeriodicalIF":0.9,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48732400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-24eCollection Date: 2021-09-01DOI: 10.1093/nsr/nwaa284
Fan Yang, Shenglong Ling, Yingxin Zhou, Yanan Zhang, Pei Lv, Sanling Liu, Wei Fang, Wenjing Sun, Liaoyuan A Hu, Longhua Zhang, Pan Shi, Changlin Tian
G protein-coupled receptors (GPCRs) are responsible for most cytoplasmic signaling in response to extracellular ligands with different efficacy profiles. Various spectroscopic techniques have identified that agonists exhibiting varying efficacies can selectively stabilize a specific conformation of the receptor. However, the structural basis for activation of the GPCR-G protein complex by ligands with different efficacies is incompletely understood. To better understand the structural basis underlying the mechanisms by which ligands with varying efficacies differentially regulate the conformations of receptors and G proteins, we determined the structures of β2AR-Gαs[Formula: see text]γ bound with partial agonist salbutamol or bound with full agonist isoprenaline using single-particle cryo-electron microscopy at resolutions of 3.26 Å and 3.80 Å, respectively. Structural comparisons between the β2AR-Gs-salbutamol and β2AR-Gs-isoprenaline complexes demonstrated that the decreased binding affinity and efficacy of salbutamol compared with those of isoprenaline might be attributed to weakened hydrogen bonding interactions, attenuated hydrophobic interactions in the orthosteric binding pocket and different conformational changes in the rotamer toggle switch in TM6. Moreover, the observed stronger interactions between the intracellular loop 2 or 3 (ICL2 or ICL3) of β2AR and Gαs with binding of salbutamol versus isoprenaline might decrease phosphorylation in the salbutamol-activated β2AR-Gs complex. From the observed structural differences between these complexes of β2AR, a mechanism of β2AR activation by partial and full agonists is proposed to provide structural insights into β2AR desensitization.
G 蛋白偶联受体（GPCR）是对具有不同功效的细胞外配体做出反应的大部分细胞质信号。各种光谱技术发现，具有不同功效的激动剂可选择性地稳定受体的特定构象。然而，人们对不同功效的配体激活 GPCR-G 蛋白复合物的结构基础尚不完全清楚。为了更好地了解不同效力的配体对受体和 G 蛋白构象的不同调节机制的结构基础，我们使用单粒子冷冻电镜测定了与部分激动剂沙丁胺醇结合的 β2AR-Gαs[式中：见正文]γ 结构，或与完全激动剂异丙肾上腺素结合的 β2AR-Gαs[式中：见正文]γ 结构，分辨率分别为 3.26 Å 和 3.80 Å。β2AR-Gs-沙丁胺醇和β2AR-Gs-异丙肾上腺素复合物之间的结构比较表明，沙丁胺醇的结合亲和力和效力低于异丙肾上腺素，这可能是由于氢键相互作用减弱、正交结合口袋中的疏水相互作用减弱以及 TM6 中的转子拨动开关发生了不同的构象变化。此外，观察到沙丁胺醇与异丙肾上腺素结合时，β2AR 的胞内环 2 或 3（ICL2 或 ICL3）与 Gαs 之间的相互作用更强，这可能会降低沙丁胺醇激活的 β2AR-Gs 复合物的磷酸化。根据观察到的β2AR复合物之间的结构差异，提出了部分激动剂和完全激动剂激活β2AR的机制，为β2AR脱敏提供了结构上的启示。
{"title":"Different conformational responses of the β<sub>2</sub>-adrenergic receptor-Gs complex upon binding of the partial agonist salbutamol or the full agonist isoprenaline.","authors":"Fan Yang, Shenglong Ling, Yingxin Zhou, Yanan Zhang, Pei Lv, Sanling Liu, Wei Fang, Wenjing Sun, Liaoyuan A Hu, Longhua Zhang, Pan Shi, Changlin Tian","doi":"10.1093/nsr/nwaa284","DOIUrl":"10.1093/nsr/nwaa284","url":null,"abstract":"<p><p>G protein-coupled receptors (GPCRs) are responsible for most cytoplasmic signaling in response to extracellular ligands with different efficacy profiles. Various spectroscopic techniques have identified that agonists exhibiting varying efficacies can selectively stabilize a specific conformation of the receptor. However, the structural basis for activation of the GPCR-G protein complex by ligands with different efficacies is incompletely understood. To better understand the structural basis underlying the mechanisms by which ligands with varying efficacies differentially regulate the conformations of receptors and G proteins, we determined the structures of β<sub>2</sub>AR-Gα<sub>s</sub>[Formula: see text]γ bound with partial agonist salbutamol or bound with full agonist isoprenaline using single-particle cryo-electron microscopy at resolutions of 3.26 Å and 3.80 Å, respectively. Structural comparisons between the β<sub>2</sub>AR-Gs-salbutamol and β<sub>2</sub>AR-Gs-isoprenaline complexes demonstrated that the decreased binding affinity and efficacy of salbutamol compared with those of isoprenaline might be attributed to weakened hydrogen bonding interactions, attenuated hydrophobic interactions in the orthosteric binding pocket and different conformational changes in the rotamer toggle switch in TM6. Moreover, the observed stronger interactions between the intracellular loop 2 or 3 (ICL2 or ICL3) of β<sub>2</sub>AR and Gα<sub>s</sub> with binding of salbutamol versus isoprenaline might decrease phosphorylation in the salbutamol-activated β<sub>2</sub>AR-Gs complex. From the observed structural differences between these complexes of β<sub>2</sub>AR, a mechanism of β<sub>2</sub>AR activation by partial and full agonists is proposed to provide structural insights into β<sub>2</sub>AR desensitization.</p>","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"52 1","pages":"nwaa284"},"PeriodicalIF":16.3,"publicationDate":"2020-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11261663/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80828902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
K e−ihξ dν(h) is the characteristic function of the measure ν. It was also shown that when (2) fails, there can be an infinite-dimensional kernel in this problem. The problem (1) is a natural generalization of boundary-value problems for elliptic differential-difference equations [5], [6] and functional-differential equations with contracted/extended independent variables [2], [3]. We note a connection between (possibly degenerate) elliptic functional-differential operators and Kato’s well-known problem of the square root of a regular accretive operator [6], [7].
K e−ihξ dν(h)是测度ν的特征函数。还表明，当(2)失效时，该问题可能存在一个无限维核。问题(1)是椭圆型微分-差分方程[5]，[6]和具有收缩/扩展自变量[2]，[3]的泛函-微分方程边值问题的自然推广。我们注意到(可能退化的)椭圆函数微分算子和加托著名的正则加积算子[6]，[7]的平方根问题之间的联系。
{"title":"The spectral radius of a certain parametric family of functional operators","authors":"N. B. Zhuravlev, L. Rossovskii","doi":"10.1070/RM9967","DOIUrl":"https://doi.org/10.1070/RM9967","url":null,"abstract":"K e−ihξ dν(h) is the characteristic function of the measure ν. It was also shown that when (2) fails, there can be an infinite-dimensional kernel in this problem. The problem (1) is a natural generalization of boundary-value problems for elliptic differential-difference equations [5], [6] and functional-differential equations with contracted/extended independent variables [2], [3]. We note a connection between (possibly degenerate) elliptic functional-differential operators and Kato’s well-known problem of the square root of a regular accretive operator [6], [7].","PeriodicalId":49582,"journal":{"name":"Russian Mathematical Surveys","volume":"62 1","pages":"971 - 973"},"PeriodicalIF":0.9,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59005123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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