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IntroductionThis is the third of the series of the papers dealing with holonomic systems(* } . A holonomic system is, by definition, a left coherent (f-Module (or ^-Modules) ( * sS:) whose characteristic variety is Lagrangian. It shares the finiteness theorem with a linear ordinary differential equation, namely, all the cohomology groups associated with its solution sheaf are finite dimensional ([6], [12]). Hence the study of such a system will give us almost complete information concerning the functions which satisfy the system, as in the onedimensional case. Actually, analyzing special functions by the aid of the theory of ordinary differential equations is one of the most important subjects in the classical analysis. From this point of view, the study of holonomic systems with regular singularities is most important. However, even though the theory of linear ordinary differential equations with regular singularities has been developed quite successfully, the general theory of holonomic systems with regular singularities was not fully developed in the past, especially compared with the fruitful success attained in the one-dimensional case. Still it should be worth doing, and we hope we have established a solid basis for the theory in this paper. For example, we establish several basic results needed for the manipulation of holonomic systems with regular singularities, such as the integration and the restriction of such systems (Chapter V). We also give an analytic characterReceived November 25, 1980. * Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan.( * } The first one is [6] and the second one is [8]. (*#) g (resp., J^) denotes the sheaf of micro-differential (resp. linear differential) operators of finite order. See also the list of notations given at the end of this section. 814MASAKI KASHIWARA AND TAKAfflRO KAWAI ization of holonomic ^-Modules with regular singularities in terms of a comparison theorem, namely, we show that a holonomic ^-Module Jt is with regular singularities if and only if &<»t i x (^5 O X ) X = ^^SQ X (^, ®x,x) holds for any point xeX and for any j, where S XtX denotes the ring of formal power series at x. (Chapter VI.) In developing our theory, we make full use of the technique of micro-local analysis, i.e., the analysis on the cotangent bundle.We use the language of Sato-Kawai-Kashiwara [24], which shall be referred to as S-K-K [24] for brevity. Especially the use of micro-differential operators of infinite order is crucial in our study. Making use of such operators, we establish an important and interesting result to the effect that any holonomic system can be transformed into a holonomic system with regular singularities by micro-differential operators of infinite order (Chapters IV and V). The method of the proof of this result as well as the result itself is efficiently employed for establishing basic properties of a holonomic system with regular singularities mentioned earlier. In the course of our arguments, we also make essential use of the res...

Glucagon-like peptide-1 receptor (GLP-1R) agonists are efficacious antidiabetic medications that work by enhancing glucose-dependent insulin secretion and improving energy balance. Currently approved GLP-1R agonists are peptide based, and it has proven difficult to obtain small-molecule activators possessing optimal pharmaceutical properties. We report the discovery and mechanism of action of LY3502970 (OWL833), a nonpeptide GLP-1R agonist. LY3502970 is a partial agonist, biased toward G protein activation over β-arrestin recruitment at the GLP-1R. The molecule is highly potent and selective against other class B G protein–coupled receptors (GPCRs) with a pharmacokinetic profile favorable for oral administration. A high-resolution structure of LY3502970 in complex with active-state GLP-1R revealed a unique binding pocket in the upper helical bundle where the compound is bound by the extracellular domain (ECD), extracellular loop 2, and transmembrane helices 1, 2, 3, and 7. This mechanism creates a distinct receptor conformation that may explain the partial agonism and biased signaling of the compound. Further, interaction between LY3502970 and the primate-specific Trp33 of the ECD informs species selective activity for the molecule. In efficacy studies, oral administration of LY3502970 resulted in glucose lowering in humanized GLP-1R transgenic mice and insulinotropic and hypophagic effects in nonhuman primates, demonstrating an effect size in both models comparable to injectable exenatide. Together, this work determined the molecular basis for the activity of an oral agent being developed for the treatment of type 2 diabetes mellitus, offering insights into the activation of class B GPCRs by nonpeptide ligands.

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