We study the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation under the assumption that the solution vanishes as x → +∞ and approaches an oscillatory plane wave as x → −∞. We first develop an inverse scattering transform formalism for solutions satisfying such step-like boundary conditions. Using this formalism, we prove that there exists a global solution of the corresponding Cauchy problem and establish a representation for this solution in terms of the solution of a Riemann-Hilbert problem. By performing a steepest descent analysis of this Riemann-Hilbert problem, we identify three asymptotic sectors in the half-plane t ≥ 0 of the xtplane and derive asymptotic formulas for the solution in each of these sectors. Finally, by restricting the constructed solutions to the half-line x ≥ 0, we find a class of solutions with asymptotically time-periodic boundary values previously sought for in the context of the NLS half-line problem.

G protein-coupled receptors (GPCRs) control critical cellular signaling pathways. Therapeutic agents, such as antibodies (Abs), are being developed to modulate GPCR signaling pathways. However, validating the selectivity of anti-GPCR Abs is challenging due to sequence similarities of individual receptors within GPCR subfamilies. To address this, we developed a multiplexed immunoassay to test >400 anti-GPCR Abs from the Human Protein Atlas targeting a customized library of 215 expressed and solubilized GPCRs representing all GPCR subfamilies. We found that ~61% of Abs were selective for their intended target, ~11% to bind off-target, and ~28% not to bind any GPCR. Antigens of on-target Abs were, on average, significantly longer, more disordered, and less likely to be buried in the interior of the GPCR protein than the other Abs. These results provide important insights into the immunogenicity of GPCR epitopes and form a basis for the design of therapeutic Abs and the detection of pathological auto-antibodies.TEASERA multiplexed library-to-library selectivity analysis of 400 anti-GPCR antibodies within subfamilies of 200 solubilized receptors.

We study the defocusing nonlinear Schrödinger equation in the quarter plane with asymptotically periodic boundary values. We use the unified transform method, also known as the Fokas method, and the Deift‐Zhou nonlinear steepest descent method to construct solutions in a sector close to the boundary whose leading behavior is described by a single exponential plane wave. Furthermore, we compute the subleading terms in the long‐time asymptotics of the constructed solutions.

G protein–coupled receptors (GPCRs) control critical cellular signaling pathways. Therapeutic agents including anti-GPCR antibodies (Abs) are being developed to modulate GPCR function. However, validating the selectivity of anti-GPCR Abs is challenging because of sequence similarities among individual receptors within GPCR subfamilies. To address this challenge, we developed a multiplexed immunoassay to test >400 anti-GPCR Abs from the Human Protein Atlas targeting a customized library of 215 expressed and solubilized GPCRs representing all GPCR subfamilies. We found that ~61% of Abs tested were selective for their intended target, ~11% bound off-target, and ~28% did not bind to any GPCR. Antigens of on-target Abs were, on average, significantly longer, more disordered, and less likely to be buried in the interior of the GPCR protein than the other Abs. These results provide important insights into the immunogenicity of GPCR epitopes and form a basis for designing therapeutic Abs and for detecting pathological auto-Abs against GPCRs.